Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Laszlo B. Szabados, published in LivingReviews in Relativity on 2004-03-16

This review is based on the talks given at the Erwin Schrodinger Institut, Vienna, in July 1997, at the Universitat Tubingen, in May 1998 and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.

1 Introduction

During the last 25 years one of the greatest achievements in classical general relativity is certainly the proof of the positivity of the total gravitational energy, both at spatial and null infinity. It is precisely its positivity that makes this notion not only important (because of its theoretical significance), but a useful tool as well in the everyday practice of working relativists. This success inspired the more ambitious claim to associate energy (or rather energy-momentum and, ultimately, angular momentum too) to extended but finite spacetime domains, i. e. at the quasi-local level.

Obviously, the quasi-local quantities could provide a more detailed characterization of the states of the gravitational ‘field’ than the global ones, so they (together with more general quasi-local observables) would be interesting in their own right.

Moreover, finding an appropriate notion of energy-momentum and angular momentum would be important from the point of view of applications as well. For example, they may play a central role in the proof of the full Penrose inequality (as they have already played in the proof of the Riemannian version of this inequality). The correct, ultimate formulation of black hole thermodynamics should probably be based on quasi-locally defined internal energy, entropy, angular momentum etc. In numerical calculations conserved quantities (or at least those for which balance equations can be derived) are used to control the errors. However, in such calculations all the domains are finite, i. e. quasi-local. Therefore, a solid theoretical foundation of the quasi-local conserved quantities is needed.

However, contrary to the high expectations of the eighties, finding an appropriate quasi-local notion of energy-momentum has proven to be surprisingly difficult. Nowadays, the state of the art is typically postmodern: Although there are several promising and useful suggestions, we have not only no ultimate, generally accepted expression for the energy-momentum and especially for the angular momentum, but there is no consensus in the relativity community even on general questions (for example, what should we mean e. g. by energy-momentum: Only a general expression containing arbitrary functions, or rather a definite one free of any ambiguities, even of additive constants), or on the list of the criteria of reasonableness of such expressions. The various suggestions are based on different philosophies, approaches and give different results in the same situation. Apparently, the ideas and successes of one construction have only very little influence on other constructions.

The aim of the present paper is therefore twofold. First, to collect and review the various specific suggestions, and, second, to stimulate the interaction between the different approaches by clarifying the general, potentially common points, issues, questions. Thus we wanted to write not only a ‘who-did-what’ review, but primarily we would like to concentrate on the understanding of the basic questions (such as why should the gravitational energy-momentum and angular momentum, or, more generally, any observable of the gravitational ‘field’, be necessarily quasi-local) and ideas behind the various specific constructions. Consequently, one-third of the present review is devoted to these general questions. We review the specific constructions and their properties only in the second part, and in the third part we discuss very briefly some (potential) applications of the quasi-local quantities. Although this paper is basically a review of known and published results, we believe that it contains several new elements, observations, suggestions etc.

Surprisingly enough, most of the ideas and concepts that appear in connection with the gravitational energy-momentum and angular momentum can be introduced in (and hence can be understood from) the theory of matter fields in Minkowski spacetime. Thus, in subsection 2.1 , we review the Belinfante–Rosenfeld procedure that we will apply to gravity in section 3 , introduce the notion of quasi-local energy-momentum and angular momentum of the matter fields and discuss their properties.The philosophy of quasi-locality in general relativity will be demonstrated in Minkowski spacetime where the energy-momentum and angular momentum of the matter fields are treated quasi-locally. Then we turn to the difficulties of gravitational energy-momentum and angular momentum, and we clarify why the gravitational observables should necessarily be quasi-local. The tools needed to construct and analyze the quasi-local quantities are reviewed in the fourth section. This closes the first, the general part of the review.

The second part is devoted to the discussion of the specific constructions (sections 5 – 12 ). Since most of the suggestions are constructions, they cannot be given as a short mathematical definition.

Moreover, there are important physical ideas behind them, without which the constructions may appear ad hoc. Thus we always try to explain these physical pictures, the motivations and interpretations. Although the present paper is intended to be a non-technical review, the explicit mathematical definitions of the various specific constructions will always be given. Then the properties and the applications are usually summarized only in a nutshell. Sometimes we give a review on technical aspects too, without which it would be difficult to understand even some of the conceptual issues. The list of references connected with this second part is intended to be complete. We apologize to all those whose results were accidentally left out.

The list of the (actual and potential) applications of the quasi-local quantities, discussed in section 13 , is far from being complete, and might be a little bit subjective. Here we consider the calculation of gravitational energy transfer, applications in black hole physics and a quasi-local characterization of the pp-wave metrics. We close this paper with a discussion of the successes and deficiencies of the general and (potentially) viable constructions. In contrast to the positivistic style of sections 5 – 12 , section 14 (as well as the choice for the matter of sections 2 – 4 ) reflects our own personal interest and view of the subject.

The theory of quasi-local observables in general relativity is far from being complete. The most important open problem is still the trivial one: ‘Find quasi-local energy-momentum and angular momentum expressions satisfying the points of the lists of subsection 4.3 ’. Several specific open questions in connection with the specific definitions are raised both in the corresponding sections and in section 14 , which could be worked out even by graduate students. On the other hand, any of their application to solve physical/geometrical problems (e. g. to some mentioned in section 13 ) would be a real success.

In the present paper we adopt the abstract index formalism. The signature of the spacetime metric

g_{a b}

- 2

; and the curvature and Ricci tensors and the curvature scalar of the covariant derivative

\nabla_{a}

are defined by

(\nabla_{c} \nabla_{d} - \nabla_{d} \nabla_{c}) X^{a} : = - R_{b c d}^{a} X^{b}

R_{b d} : = R_{b a d}^{a}

and

R : = R_{b d} g^{b d}

, respectively. Hence Einstein’s equations take the form

G_{a b} : = R_{a b} - \frac{1}{2} R g_{a b} = - 8 π G T_{a b}

, where

G

is Newton’s gravitational constant (and the speed of light is

c = 1

). However, in subsections 13.3 and 13.4 we use the traditional

c g s

system.

2 Energy-Momentum and Angular Momentum of Matter Fields

2.1 Energy-momentum and angular momentum density of matter fields

2.1.1 The symmetric energy-momentum tensor

It is a widely accepted view (appearing e. g. in excellent, standard textbooks on general relativity, too) that the canonical energy-momentum and spin tensors are well defined and have relevance only in flat spacetime, and hence usually are underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus first we introduce these quantities for the matter fields in a general curved spacetime.

To specify the state of the matter fields operationally two kinds of devices are needed: The first measures the value of the fields, while the other measures the spatio-temporal location of the first. Correspondingly, the fields on the manifold

M

of events can be grouped into two sharply distinguished classes: The first contains the matter field variables, e. g. finitely many

(r, s)

type tensor fields

{Φ_{N}}_{b_{1} . . . b_{s}}^{a_{1} . . . a_{r}}

; whilst the other contains the fields specifying the spacetime geometry, i. e.

the metric

g_{a b}

in Einstein’s theory. Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a Lagrangian

L_{m} = L_{m} (g^{a b}, Φ_{N}, \nabla_{e} Φ_{N}, . . ., \nabla_{e_{1}} . . . \nabla_{e_{k}} Φ_{N})

: If

I_{m} [g^{a b}, Φ_{N}]

is the volume integral of

L_{m}

on some open domain

D

with compact closure then the equations of motion are

{E^{N}}_{a . . .}^{b . . .} : = δ I_{m} / δ {Φ_{N}}_{b . . .}^{a . . .} = \sum_{n = 0}^{k} (-)^{n} \nabla_{e_{n}} . . . \nabla_{e_{1}} (\partial L_{m} / \partial (\nabla_{e_{1}} . . . \nabla_{e_{n}} {Φ_{N}}_{b . . .}^{a . . .})) = 0

, the Euler–Lagrange equations. The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by

\begin{matrix} T_{a b} : = \frac{2}{\sqrt{| g |}} \frac{δ I_{m}}{δ g^{a b}} = 2 \frac{\partial L_{m}}{\partial g^{a b}} - L_{m} g_{a b} + \frac{1}{2} \nabla^{e} (σ_{a b e} + σ_{b a e} - σ_{a e b} - σ_{b e a} - σ_{e a b} - σ_{e b a}), \end{matrix}

(1)

where we introduced the so-called canonical spin tensor

\begin{matrix} σ_{b}^{e a} : = \sum_{n = 1}^{k} \sum_{i = 1}^{n} (-)^{i} δ_{e_{i}}^{e} \nabla_{e_{i - 1}} . . . \nabla_{e_{1}} (\frac{\partial L_{m}}{\partial (\nabla_{e_{1}} . . . \nabla_{e_{n}} {Φ_{N}}_{d . . .}^{c . . .})}) \times {Δ_{b e_{i + 1} . . . e_{n} d . . .}^{a c . . .}}_{h . . .}^{f_{i + 1} . . . f_{n} g . . .} \nabla_{f_{i + 1}} . . . \nabla_{f_{n}} {Φ_{N}}_{g . . .}^{h . . .} . \end{matrix}

(2)

(The terminology will be justified in the next subsection.) Here

Δ_{d b_{1} . . . b_{q} e_{1} . . . e_{p}}^{c a_{1} . . . a_{p} f_{1} . . . f_{q}}

is the

(p + q + 1, p + q + 1)

type invariant tensor, built from the Kronecker deltas, appearing naturally in the expression of the Lie derivative of the

(p, q)

type tensor fields in terms of the torsion free covariant derivatives:

Ł_{K} Φ_{b . . .}^{a . . .} = \nabla_{K} Φ_{b . . .}^{a . . .} - \nabla_{c} K^{d} Δ_{d b . . . e . . .}^{c a . . . f . . .} Φ_{f . . .}^{e . . .}

. (For the general idea of deriving

T_{a b}

and ( 2 ), see e. g. section 3 of [170] .)

2.1.2 The canonical Noether current

Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that for any vector field

K^{a}

and the corresponding local 1-parameter family of diffeomorphisms

φ_{t}

one has

(φ_{t}^{*} L_{m}) (g^{a b}, Φ_{N}, \nabla_{e} Φ_{N}, . . .) - L_{m} (φ_{t}^{*} g^{a b}, φ_{t}^{*} Φ_{N}, φ_{t}^{*} \nabla_{e} Φ_{N}, . . .) = \nabla_{e} B_{t}^{e}

for some one parameter family of vector fields

B_{t}^{e} = B_{t}^{e} (g^{a b}, Φ_{N}, . . .)

. (

L_{m}

is called diffeomorphism invariant if

\nabla_{e} B_{t}^{e} = 0

, e. g. when

L_{m}

is a scalar.) Let

K^{a}

be any smooth vector field on

M

. Then, calculating the divergence

\nabla_{a} (L_{m} K^{a})

to determine the rate of change of the action functional

I_{m}

along the integral curves of

K^{a}

, by a tedious but straightforward computation one can derive the so-called Noether identity:

{E^{N}}_{a . . .}^{b . . .} Ł_{K} {Φ_{N}}_{b . . .}^{a . . .} + \frac{1}{2} T_{a b} Ł_{K} g^{a b} + \nabla_{e} C^{e} [K] = 0

, where

Ł_{K}

denotes the Lie derivative along

K^{a}

and

C^{a} [K]

, the so-called Noether current, is given explicitly by

\begin{matrix} C^{e} [K] = {\dot{B}}^{e} + θ^{e a} K_{a} + (σ^{e [a b]} + σ^{a [b e]} + σ^{b [a e]}) \nabla_{a} K_{b} . \end{matrix}

(3)

Here

{\dot{B}}^{e}

is the derivative of

B_{t}^{e}

with respect to

t

t = 0

, which may depend on

K_{a}

and its derivatives; and

θ_{b}^{a}

, the so-called canonical energy-momentum tensor, is defined by

\begin{matrix} θ_{b}^{a} : = - L_{m} δ_{b}^{a} - \sum_{n = 1}^{k} \sum_{i = 1}^{n} (-)^{i} δ_{e_{i}}^{e} \nabla_{e_{i - 1}} . . . \nabla_{e_{1}} (\frac{\partial L_{m}}{\partial (\nabla_{e_{1}} . . . \nabla_{e_{n}} {Φ_{N}}_{d . . .}^{c . . .})}) \nabla_{b} \nabla_{e_{i + 1}} . . . \nabla_{e_{n}} {Φ_{N}}_{d . . .}^{c . . .} . \end{matrix}

(4)

Note that, apart from the term

{\dot{B}}^{e}

, the current

C^{e} [K]

does not depend on higher than the first derivative of

K^{a}

, and the canonical energy-momentum and spin tensors could be introduced as the coefficients of

K_{a}

and its first derivative, respectively, in

C^{e} [K]

. (For the original introduction of these concepts, see [55, 56, 308] . If the torsion

Θ_{a b}^{c}

is not vanishing, then in the Noether identity there is a further term,

\frac{1}{2} S_{c}^{a b} Ł_{K} Θ_{a b}^{c}

, where the so-called dynamical spin tensor

S_{c}^{a b}

is defined by

\sqrt{| g |} S_{c}^{a b} : = 2 δ I_{m} / δ Θ_{a b}^{c}

, and the Noether current has a slightly different structure [188, 189] .) Obviously,

C^{e} [K]

is not uniquely determined by the Noether identity, because that contains only its divergence, and any identically conserved current may be added to it. In fact,

B_{t}^{e}

may be chosen to be an arbitrary non-zero (but divergence free) vector field even for diffeomorphism invariant Lagrangians. Thus, to be more precise, if

{\dot{B}}^{e} = 0

then we call the specific combination ( 3 ) the canonical Noether current. Other choices for the Noether current may contain higher derivatives of

K^{a}

too (see e. g. [222] ), but there is a specific one containing

K^{a}

algebraically (see the points 3 and 4 below). However,

C^{a} [K]

is sensitive to total divergences added to the Lagrangian, and, if the matter fields have gauge freedom, then in general it is not gauge invariant even if the Lagrangian is. On the other hand,

T^{a b}

is gauge invariant and is independent of total divergences added to

L_{m}

. Provided the field equations are satisfied, the Noether identity implies [55, 56, 308, 188, 189] that

1. $\nabla_{a} T^{a b} = 0$ ,
2. $T^{a b} = θ^{a b} + \nabla_{c} (σ^{c [a b]} + σ^{a [b c]} + σ^{b [a c]})$ ,
3. $C^{a} [K] = T^{a b} K_{b} + \nabla_{c} ((σ^{a [c b]} - σ^{c [a b]} - σ^{b [a c]}) K_{b})$ , where the second term on the right is an identically conserved current, and
4. $C^{a} [K]$ is conserved (i. e. divergence free) if $K^{a}$ is a Killing vector.

Hence

T^{a b} K_{b}

is also conserved and can equally be considered as a Noether current. (For a formally different, but essentially equivalent introduction of the Noether current and identity, see [371, 209, 137] .) The interpretation of the conserved currents

C^{a} [K]

and

T^{a b} K_{b}

depends on the nature of the Killing vector

K^{a}

. In Minkowski spacetime the ten dimensional Lie algebra

K

of the Killing vectors is well known to split to the semidirect sum of a four dimensional commutative ideal,

T

, and the quotient

K / T

, where the latter is isomorphic to

s o (1, 3)

. The ideal

T

is spanned by the constant Killing vectors, in which a constant orthonormal frame field

{E_{\underset{̲}{a}}^{a}}

M

\underset{̲}{a} = 0, . . ., 3

, forms a basis. (Thus the underlined Roman indices

\underset{̲}{a}

\underset{̲}{b}

, . . . are concrete, name indices.) By

g_{a b} E_{\underset{̲}{a}}^{a} E_{\underset{̲}{b}}^{b} = η_{\underset{̲}{a} \underset{̲}{b}} : = diag (1, - 1, - 1, - 1)

the ideal

T

inherits a natural Lorentzian vector space structure. Having chosen an origin

o \in M

, the quotient

K / T

can be identified as the Lie algebra

R_{o}

of the boost-rotation Killing vectors that vanish at

o

. Thus

K

has a ‘4+6’ decomposition into translations and boost-rotations, where the translations are canonically defined but the boost-rotations depend on the choice of the origin

o \in M

. In the coordinate system

{x^{\underset{̲}{a}}}

adapted to

{E_{\underset{̲}{a}}^{a}}

(i. e. for which the 1-form basis dual to

{E_{\underset{̲}{a}}^{a}}

has the form

ϑ_{a}^{\underset{̲}{a}} = \nabla_{a} x^{\underset{̲}{a}}

) the general form of the Killing vectors (or rather 1-forms) is

K_{a} = T_{\underset{̲}{a}} ϑ_{a}^{\underset{̲}{a}} + M_{\underset{̲}{a} \underset{̲}{b}} (x^{\underset{̲}{a}} ϑ_{a}^{\underset{̲}{b}} - x^{\underset{̲}{b}} ϑ_{a}^{\underset{̲}{a}})

for some constants

T_{\underset{̲}{a}}

and

M_{\underset{̲}{a} \underset{̲}{b}} = - M_{\underset{̲}{b} \underset{̲}{a}}

. Then the corresponding canonical Noether current is

C^{e} [K] = E_{\underset{̲}{e}}^{e} (θ^{\underset{̲}{e} \underset{̲}{a}} T_{\underset{̲}{a}} - (θ^{\underset{̲}{e} \underset{̲}{a}} x^{\underset{̲}{b}} - θ^{\underset{̲}{e} \underset{̲}{b}} x^{\underset{̲}{a}} - 2 σ^{\underset{̲}{e} [\underset{̲}{a} \underset{̲}{b}]}) M_{\underset{̲}{a} \underset{̲}{b}})

; and the coefficients of the translation and the boost-rotation parameters

T_{\underset{̲}{a}}

and

M_{\underset{̲}{a} \underset{̲}{b}}

are interpreted as the density of the energy-momentum and the sum of the orbital and spin angular momenta, respectively. Since, however, the difference

C^{a} [K] - T^{a b} K_{b}

is identically conserved and

T^{a b} K_{b}

has more advantageous properties, it is

T^{a b} K_{b}

that is used to represent the energy-momentum and angular momentum density of the matter fields.

Since in the de-Sitter and anti-de-Sitter spacetimes the (ten dimensional) Lie algebra of the Killing vector fields,

s o (1, 4)

and

s o (2, 3)

, respectively, are semisimple, there is no such natural notion of translations, and hence no natural ‘4+6’ decomposition of the ten conserved currents into energy-momentum and (relativistic) angular momentum density.

2.2 Quasi-local energy-momentum and angular momentum of the matter fields

In the next section we will see that well defined (i. e. gauge invariant) energy-momentum and angular momentum density cannot be associated with the gravitational ‘field’, and if we want to talk not only about global gravitational energy-momentum and angular momentum, then these quantities must be assigned to extended but finite spacetime domains.

In the light of modern quantum field theoretical investigations it has become clear that all physical observables should be associated with extended but finite spacetime domains [164, 163] .

Thus observables are always associated with open subsets of spacetime whose closure is compact, i. e. they are quasi-local. Quantities associated with spacetime points or with the whole spacetime are not observable in this sense. In particular, global quantities, such as the total energy or electric charge, should be considered as the limit of quasi-locally defined quantities. Thus the idea of quasi-locality is not new in physics. Although apparently in classical non-gravitational physics this is not obligatory, we adopt this view in talking about energy-momentum and angular momentum even of classical matter fields in Minkowski spacetime. Originally the introduction of these quasi-local quantities was motivated by the analogous gravitational quasi-local quantities [338, 342] . Since, however, many of the basic concepts and ideas behind the various gravitational quasi-local energy-momentum and angular momentum definitions can be understood from the analogous non-gravitational quantities in Minkowski spacetime, we devote the present subsection to the discussion of them and their properties.

2.2.1 The definition of the quasi-local quantities

To define the quasi-local conserved quantities in Minkowski spacetime, first observe that for any Killing vector

K^{a} \in K

the 3-form

ω_{a b c} : = K_{e} T^{e f} ɛ_{f a b c}

is closed, and hence, by the triviality of the third de Rham cohomology class,

H^{3} (R^{4}) = 0

, it is exact: For some 2-form

\cup [K]_{a b}

we have

K_{e} T^{e f} ɛ_{f a b c} = 3 \nabla_{[a}

\cup [K]_{b c]}

\lor^{c d} : = - \frac{1}{2} \cup [K]_{a b} ɛ^{a b c d}

may be called a superpotential for the conserved current 3-form

ω_{a b c}

. (The existence of globally defined superpotentials can be proven even without using the Poincare lemma [370] .) If

\tilde{\cup} [K]_{a b}

is (the dual of) another superpotential for the same current

ω_{a b c}

, then by

\nabla_{[a} (\cup [K]_{b c]} - \tilde{\cup} [K]_{b c]}) = 0

and

H^{2} (R^{4}) = 0

the dual superpotential is unique up to the addition of an exact 2-form. If therefore

S

is any closed orientable spacelike 2-surface in the Minkowski spacetime then the integral of

\cup [K]_{a b}

S

is free from this ambiguity.

Thus if

Σ

is any smooth compact spacelike hypersurface with smooth 2-boundary

S

, then

\begin{matrix} Q_{S} [K] : = \frac{1}{2} \oint_{S} \cup [K]_{a b} = \int_{Σ} K_{e} T^{e f} \frac{1}{3!} ɛ_{f a b c} \end{matrix}

(5)

depends only on

S

. Hence it is independent of the actual Cauchy surface

Σ

of the domain of dependence

D (Σ)

because all the spacelike Cauchy surfaces for

D (Σ)

have the same common boundary

S

. Thus

Q_{S} [K]

can equivalently be interpreted as being associated with the whole domain of dependence

D (Σ)

, and hence quasi-local in the sense of [164, 163] above. It defines the linear maps

P_{S} : T \to R

and

J_{S} : R_{o} \to R

Q_{S} [K] = : T_{\underset{̲}{a}} P_{S}^{\underset{̲}{a}} + M_{\underset{̲}{a} \underset{̲}{b}} J_{S}^{\underset{̲}{a} \underset{̲}{b}}

, i. e. they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates

P_{S}^{\underset{̲}{a}}

and

J_{S}^{\underset{̲}{a} \underset{̲}{b}}

transform as a Lorentz vector and anti-symmetric tensor, respectively, whilst under the translation

x^{\underset{̲}{a}} \mapsto x^{\underset{̲}{a}} + η^{\underset{̲}{a}}

of the origin

P_{S}^{\underset{̲}{a}}

is unchanged while

J_{S}^{\underset{̲}{a} \underset{̲}{b}} \mapsto J_{S}^{\underset{̲}{a} \underset{̲}{b}} + 2 η^{[\underset{̲}{a}} P_{S}^{\underset{̲}{b}]}

. Thus

P_{S}^{\underset{̲}{a}}

and

J_{S}^{\underset{̲}{a} \underset{̲}{b}}

may be interpreted as the quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike 2-surface

S

, or, equivalently, to

D (Σ)

. Then the quasi-local mass and Pauli–Lubanski spin are defined, respectively, by the usual formulae

m_{S}^{2} : = η_{\underset{̲}{a} \underset{̲}{b}} P_{S}^{\underset{̲}{a}} P_{S}^{\underset{̲}{b}}

and

S_{S}^{\underset{̲}{a}} : = \frac{1}{2} ɛ_{\underset{̲}{b} \underset{̲}{c} \underset{̲}{d}}^{\underset{̲}{a}} P_{S}^{\underset{̲}{b}} J_{S}^{\underset{̲}{c} \underset{̲}{d}}

. (If

m^{2} \neq 0

, then the dimensionally correct definition of the Pauli–Lubanski spin is

\frac{1}{m} S_{S}^{\underset{̲}{a}}

.) As a consequence of the definitions

η_{\underset{̲}{a} \underset{̲}{b}} P_{S}^{\underset{̲}{a}} S_{S}^{\underset{̲}{b}} = 0

holds, i. e. if

P_{S}^{\underset{̲}{a}}

is timelike then

S_{S}^{\underset{̲}{a}}

is spacelike or zero, but if

P_{S}^{\underset{̲}{a}}

is null (i. e.

m_{S}^{2} = 0

) then

S_{S}^{\underset{̲}{a}}

is spacelike or proportional to

P_{S}^{\underset{̲}{a}}

Obviously, we can form the flux integral of the current

T^{a b} ξ_{b}

on the hypersurface even if

ξ^{a}

is not a Killing vector, even in general curved spacetime:

\begin{matrix} E_{Σ} [ξ^{a}] : = \int_{Σ} ξ_{e} T^{e f} \frac{1}{3!} ɛ_{f a b c} . \end{matrix}

(6)

Then, however, the integral

E_{Σ} [ξ^{a}]

does depend on the hypersurface, because this is not connected with the spacetime symmetries. In particular, the vector field

ξ^{a}

can be chosen to be the unit timelike normal

t^{a}

Σ

. Since the component

μ : = T_{a b} t^{a} t^{b}

of the energy-momentum tensor is interpreted as the energy-density of the matter fields seen by the local observer

t^{a}

, it would be legitimate to interpret the corresponding integral

E_{Σ} [t^{a}]

as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to

Σ

’. Thus

E_{Σ} [t^{a}]

defines a different concept of the quasi-local energy: While that based on

Q_{S} [K]

is linked to some absolute element, namely to the translational Killing symmetries of the spacetime and the constant timelike vector fields can be interpreted as the observers ‘measuring’ this energy,

E_{Σ} [t^{a}]

is completely independent of any absolute element of the spacetime and is based exclusively on the arbitrarily chosen fleet of observers. Thus, while

P_{S}^{\underset{̲}{a}}

is independent of the actual normal

t^{a}

S

E_{Σ} [ξ^{a}]

(for non-Killing

ξ^{a}

) depends on

t^{a}

intrinsically and is a genuine 3-hypersurface rather than a 2-surface integral.

P_{b}^{a} : = δ_{b}^{a} - t^{a} t_{b}

, the orthogonal projection to

Σ

, then the part

j^{a} : = P_{b}^{a} T^{b c} t_{c}

of the energy-momentum tensor is interpreted as the momentum density seen by the observer

t^{a}

. Hence

(t_{a} T^{a b}) (t_{c} T^{c d}) g_{b d} = μ^{2} + h_{a b} j^{a} j^{b} = μ^{2} - {| j^{a} |}^{2}

is the square of the mass density of the matter fields, where

h_{a b}

is the spatial metric in the plane orthogonal to

t^{a}

. If

T^{a b}

satisfies the dominant energy condition (i. e.

T^{a b} V_{b}

is a future directed non-spacelike vector for any future directed non-spacelike vector

V^{a}

, see for example [170] ), then this is non-negative, and hence

\begin{matrix} M_{Σ} : = \int_{Σ} \sqrt{μ^{2} - {| j^{e} |}^{2}} \frac{1}{3!} t^{f} ɛ_{f a b c} \end{matrix}

(7)

can also be interpreted as the quasi-local mass of the matter fields seen by the fleet of observers being at rest with respect to

Σ

, even in general curved spacetime. However, although in Minkowski spacetime

E_{Σ} [K]

for the four translational Killing vectors gives the four components of the energy-momentum

P_{S}^{\underset{̲}{a}}

, the mass

M_{Σ}

is different from

m_{S}

. In fact, while

m_{S}

is defined as the Lorentzian norm of

P_{S}^{\underset{̲}{a}}

with respect to the metric on the space of the translations, in the definition of

M_{Σ}

first the norm of the current

T^{a b} t_{b}

is taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of the more advantageous properties (see 2.2.3 below), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form

Q_{S} [K]

instead of

E_{Σ} [t^{a}]

Thus even if there is a gauge invariant and unambiguously defined energy-momentum density of the matter fields, it is not a priori clear how the various quasi-local quantities should be introduced.

We will see in the second part of the present review that there are specific suggestions for the gravitational quasi-local energy that are analogous to

P_{S}^{0}

, others to

E_{Σ} [t^{a}]

and some to

M_{Σ}

2.2.2 Hamiltonian introduction of the quasi-local quantities

In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [206, 377] and references therein) the configuration and momentum variables,

φ^{A}

and

π_{A}

, respectively, are fields on a connected 3-manifold

Σ

, which is interpreted as the typical leaf of a foliation

Σ_{t}

of the spacetime. The foliation can be characterized on

Σ

by a function

N

, called the lapse. The evolution of the states in the spacetime is described with respect to a vector field

K^{a} = N t^{a} + N^{a}

(‘evolution vector field’ or ’general time axis’), where

t^{a}

is the future directed unit normal to the leaves of the foliation and

N^{a}

is some vector field, called the shift, being tangent to the leaves. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Physical states can be only those that are on the constraint surface, specified by the vanishing of certain functions

C_{i} = C_{i} (φ^{A}, D_{e} φ^{A}, . . ., π_{A}, D_{e} π_{A}, . . .)

i = 1, . . ., n

, of the canonical variables and their derivatives up to some finite order, where

D_{e}

is the covariant derivative operator in

Σ

. Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form

\begin{matrix} H [K] = \int_{Σ} (μ N + j_{a} N^{a} + C_{i} N^{i} + D_{a} Z^{a}) d Σ . \end{matrix}

(8)

Here

d Σ

is the induced volume element, the coefficients

μ

and

j_{a}

are local functions of the canonical variables and their derivatives up to some finite order, the

N^{i}

’s are functions on

Σ

and

Z^{a}

is a local function of the canonical variables, the lapse, the shift, the functions

N^{i}

and their derivatives up to some finite order. The part

C_{i} N^{i}

of the Hamiltonian generates gauge motions in the phase space, and the functions

N^{i}

are interpreted as the freely specifiable ‘gauge generators’.

However, if we want to recover the field equations for

φ^{A}

(which are partial differential equations on the spacetime with smooth coefficients for the smooth field

φ^{A}

) on the phase space as the Hamilton equations and not some of their distributional generalizations, then the functional differentiability of

H [K]

must be required in the strong sense of [369] .^$1$ Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of

H [K]

requires some boundary conditions on the field variables, and may yield restrictions on the form of

Z^{a}

. It may happen that for a given

Z^{a}

only too restrictive boundary conditions would be able to ensure the functional differentiability of the Hamiltonian, and hence the ‘quasi-local phase space’ defined with these boundary conditions would contain only very few (or no) solutions of the field equations. In this case

Z^{a}

should be modified. In fact, the boundary conditions are connected to the nature of the physical situations considered.

For example, in electrodynamics different boundary conditions must be imposed if the boundary is to represent a conducting or an insulating surface. Unfortunately, no universal principle or ‘canonical’ way of finding the ‘correct’ boundary term and the boundary conditions is known.

In the asymptotically flat case the value of the Hamiltonian on the constraint surface defines the total energy-momentum and angular momentum, depending on the nature of

K^{a}

, in which the total divergence

D_{a} Z^{a}

corresponds to the ambiguity of the superpotential 2-form

\cup [K]_{a b}

: An identically conserved quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved). The energy density and the momentum density of the matter fields can be recovered as the functional derivative of

H [K]

with respect to the lapse

N

and the shift

N^{a}

, respectively. In principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising achievements of [6, 7, 312] for the Klein–Gordon, Maxwell and the Yang–Mills–Higgs fields, as far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still lacking.

^$1$ Sometimes in the literature this requirement is introduced as some new principle in the Hamiltonian formulation of the fields, but its real content is not more than to ensure that the Hamilton equations coincide with the field equations.

2.2.3 Properties of the quasi-local quantities

Suppose that the matter fields satisfy the dominant energy condition. Then

E_{Σ} [ξ^{a}]

is also non-negative for any non-spacelike

ξ^{a}

, and, obviously,

E_{Σ} [t^{a}]

is zero precisely when

T^{a b} = 0

Σ

, and hence, by the conservation laws (see for example page 94 of [170] ), on the whole domain of dependence

D (Σ)

. Obviously,

M_{Σ} = 0

if and only if

L^{a} : = T^{a b} t_{b}

is null on

Σ

. Then by the dominant energy condition it is a future pointing vector field on

Σ

, and

L_{a} T^{a b} = 0

holds. Therefore,

T^{a b}

Σ

has a null eigenvector with zero eigenvalue, i. e. its algebraic type on

Σ

is pure radiation.

The properties of the quasi-local quantities based on

Q_{S} [K]

in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [338, 342] that

1. $P_{S}^{\underset{̲}{a}}$ is a future directed nonspacelike vector, $m_{S}^{2} \geq 0$ ;
2. $P_{S}^{\underset{̲}{a}} = 0$ if and only if $T_{a b} = 0$ on $D (Σ)$ ;
3. $m_{S}^{2} = 0$ if and only if the algebraic type of the matter on $D (Σ)$ is pure radiation, i. e.
$T_{a b} L^{b} = 0$ holds for some constant null vector $L^{a}$ . Then $T_{a b} = τ L_{a} L_{b}$ for some non-negative function $τ$ , whenever $P_{S}^{\underset{̲}{a}} = e L^{\underset{̲}{a}}$ , where $L^{\underset{̲}{a}} : = L^{a} ϑ_{a}^{\underset{̲}{a}}$ and $e : = \int_{Σ} τ L^{a} \frac{1}{3!} ɛ_{a b c d}$ ;
4. For $m_{S}^{2} = 0$ the angular momentum has the form $J_{S}^{\underset{̲}{a} \underset{̲}{b}} = e^{\underset{̲}{a}} L^{\underset{̲}{b}} - e^{\underset{̲}{b}} L^{\underset{̲}{a}}$ , where $e^{\underset{̲}{a}} : = \int_{Σ} x^{\underset{̲}{a}} τ L^{a} \frac{1}{3!} ɛ_{a b c d}$ . Thus, in particular, the Pauli–Lubanski spin is zero.

Therefore, the vanishing of the quasi-local energy-momentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasi-local mass is equivalent to special configurations representing pure radiation.

Since

E_{Σ} [t^{a}]

and

M_{Σ}

are integrals of functions on a hypersurface, they are obviously additive, i. e. for example for any two hypersurfaces

Σ_{1}

and

Σ_{2}

(having common points at most on their boundaries

S_{1}

and

S_{2}

) one has

E_{Σ_{1} \cup Σ_{2}} [t^{a}] = E_{Σ_{1}} [t^{a}] + E_{Σ_{2}} [t^{a}]

. On the other hand, the additivity of

P_{S}^{\underset{̲}{a}}

is a slightly more delicate problem. Namely,

P_{S_{1}}^{\underset{̲}{a}}

and

P_{S_{2}}^{\underset{̲}{a}}

are elements of the dual space of the translations, and hence we can add them and, as in the previous case, we obtain additivity.

However, this additivity comes from the absolute parallelism of the Minkowski spacetime: The quasi-local energy-momenta of the different 2-surfaces belong to one and the same vector space.

If there were no natural connection between the Killing vectors on different 2-surfaces, then the energy-momenta would belong to different vector spaces, and they could not be added. We will see that the quasi-local quantities discussed in sections 7 , 8 and 9 belong to vector spaces dual to their own ‘quasi-Killing vectors’, and there is no natural way of adding the energy-momenta of different surfaces.

2.2.4 Global energy-momenta and angular momenta

Σ

extends either to spacelike or future null infinity, then, as is well known, the existence of the limit of the quasi-local energy-momentum can be ensured by slightly faster than

O (r^{- 3})

(for example by

O (r^{- 4})

) fall-off of the energy-momentum tensor, where

r

is any spatial radial distance. However, the finiteness of the angular momentum and centre-of-mass is not ensured by the

O (r^{- 4})

fall-off. Since the typical fall-off of

T_{a b}

– for example for the electromagnetic field – is

O (r^{- 4})

, we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the

O (r^{- 4})

fall-off, six global integral conditions for the leading terms of

T_{a b}

must be imposed. At the spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’

T_{; b}^{a b} = 0

(as evolution equations for the energy density and momentum density) preserve these fall-off and parity conditions [348] .

Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass the fields must be plane waves, furthermore by

T_{a b} = O (r^{- 4})

they must be asymptotically vanishing at the same time. However, a plane wave configuration can be asymptotically vanishing only if it is vanishing.

2.2.5 Quasi-local radiative modes of the matter fields

By the results of the previous subsection the vanishing of the quasi-local mass, associated with a closed spacelike 2-surface

S

, implies that the matter must be pure radiation on a four dimensional globally hyperbolic domain

D (Σ)

. Thus

m_{S} = 0

characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present subsection we raise the question if these states on

D (Σ)

can be characterized completely by data on the 2-surface

S

or not.

For the (real or complex) linear massless scalar field

φ

and the Maxwell field, represented by the symmetric spinor field

φ_{A B}

, the condition

T_{a b} L^{b} = 0

implies that they are completely determined on the whole

D (Σ)

by the value of these fields on

S

(and

φ_{A B}

is necessarily null:

φ_{A B} = φ O_{A} O_{B}

, where

φ

is a complex function and

O_{A}

is a constant spinor field such that

L_{a} = O_{A} {\bar{O}}_{A^{'}}

). Similarly, the null linear zero-rest-mass fields on

D (Σ)

with any spin,

φ_{A B . . . E} = φ O_{A} O_{B} . . . O_{E}

, are completely determined by their value on

S

(assuming, for the sake of simplicity, that

S

is future and past convex in the sense of subsection 4.1.3 below). Technically, these results are based on the unique complex analytic structure of the

u = const

2-surfaces foliating

Σ

, where

L_{a} = \nabla_{a} u

; and by the field equations the complex functions

φ

turn out to be holomorphic. Thus they are completely determined on

Σ

(and hence on

D (Σ)

too) by their value on

S = \partial Σ

[342] .

Since the independent radiative modes of the linear zero-rest-mass fields are just the

N

-type solutions of their field equations, characterized by the null wavevector

L^{a}

and the amplitude

φ (L^{a})

in their Fourier expansion, their radiative modes can equivalently be characterized by not only the familiar initial data on three dimensional spacelike hypersurfaces, but by data on closed 2-surfaces as well. Intuitively this result should be clear, because the two radiative (transversal) degrees of freedom of each Fourier mode may be expected to be characterized by two real functions on a two dimensional ‘screen’. This fact may be a specific manifestation in the classical non-gravitational physics of the holographic principle (see subsection 13.4.2 ). A more detailed discussion of these issues will be given elsewhere.

3 On the Energy-Momentum and Angular Momentum of Gravitating Systems

3.1 On the gravitational energy-momentum and angular momentum density: the difficulties

3.1.1 The root of the difficulties

The action

I_{m}

for the matter fields was a functional of both kinds of fields, thus one could take the variational derivatives both with respect to

{Φ_{N}}_{b . . .}^{a . . .}

and

g^{a b}

. The former gave the field equations, while the latter defined the symmetric energy-momentum tensor. Moreover,

g_{a b}

provided a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action

I_{g}

is, on the other hand, a functional of the metric alone, and its variational derivative with respect to

g^{a b}

yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of

g^{a b}

can be traced back to the principle of equivalence [21] , and introduces a huge gauge freedom in the dynamics of

g^{a b}

because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold

M

endowed with a Lorentzian metric

g_{a b}

, but the isomorphism class of such pairs, where

(M, g_{a b})

and

(M, φ^{*} g_{a b})

are considered to be equivalent for any diffeomorphism

φ

M

onto itself.^$2$ Thus we do not have, even in principle, any gravitational analog of the symmetric energy-momentum tensor of the matter fields. In fact, by its very definition,

T_{a b}

is the source-current for gravity, like the current

J_{A}^{a} : = δ I_{p} / δ A_{a}^{A}

in Yang–Mills theories (defined by the variational derivative of the action functional of the particles, e. g. of the fermions, interacting with a Yang–Mills field

A_{a}^{A}

), rather than energy-momentum.

The latter is represented by the Noether currents associated with special spacetime displacements.

Thus, in spite of the intimate relation between

T_{a b}

and the Noether currents, the proper interpretation of

T_{a b}

is only the source density for gravity, and hence it is not the symmetric energy-momentum tensor whose gravitational counterpart must be searched for. In particular, the Bel–Robinson tensor

T_{a b c d} : = ψ_{A B C D} {\bar{ψ}}_{A^{'} B^{'} C^{'} D^{'}}

, given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [318, 317] ), being a quadratic expression of the curvature (and its derivatives), is (are) expected to represent only ‘higher order’ gravitational energy-momentum. In fact, the physical dimension of the Bel–Robinson ‘energy-density’

T_{a b c d} t^{a} t^{b} t^{c} t^{d}

c m^{- 4}

, and hence (in the traditional units) there are no powers

A

and

B

such that

c^{A} G^{B} T_{a b c d} t^{a} t^{b} t^{c} t^{d}

would have energy-density dimension.

Here

c

is the speed of light and

G

is Newton’s gravitational constant. As we will see, the Bel–Robinson ‘energy-momentum density’

T_{a b c d} t^{b} t^{c} t^{d}

appears naturally in connection with the quasi-local energy-momentum and spin-angular momentum expressions for small spheres only in higher order terms. Therefore, if we want to associate energy-momentum and angular momentum with the gravity itself in a Lagrangian framework, then it is the gravitational counterpart of the canonical energy-momentum and spin tensors and the canonical Noether current built from them that should be introduced.

Hence it seems natural to apply the Lagrange–Belinfante–Rosenfeld procedure, sketched in the previous section, to gravity too [55, 56, 308, 188, 189, 336] .

^$2$ Since we do not have a third kind of device to specify the spatio-temporal location of the devices measuring the spacetime geometry, we do not have any further operationally defined, maybe non-dynamical background, just in accordance with the principle of equivalence.

If there were some non-dynamical background metric

g_{a b}^{0}

M

, then by requiring

g_{a b}^{0} = φ^{*} g_{a b}^{0}

we could reduce the almost arbitrary diffeomorphism

φ

(essentially four arbitrary functions of four variables) to a transformation depending on at most ten parameters.

3.1.2 Pseudotensors

The lack of any background geometric structure in the gravitational action yields, first, that any vector field

K^{a}

generates a symmetry of the matter plus gravity system. Its second consequence is the need for an auxiliary derivative operator, e. g. the Levi-Civita covariant derivative coming from an auxiliary, non-dynamical background metric (see for example [225, 302] ), or a background (usually torsion free, but not necessarily flat) connection (see for example [209] ), or the partial derivative coming from a local coordinate system (see for example [365] ). Though the natural expectation would be that the final results be independent of these background structures, as is well known, the results do depend on them.

In particular [336] , for Hilbert’s second order Lagrangian

L_{H} : = R / 16 π G

in a fixed local coordinate system

{x^{α}}

and derivative operator

\partial_{μ}

instead of

\nabla_{e}

( 4 ) gives precisely Møller’s energy-momentum pseudotensor

_{M} θ_{β}^{α}

, which was defined originally through the superpotential equation

\sqrt{| g |} (8 π G_{M} θ_{β}^{α} - G_{β}^{α}) = {\partial_{μ}}_{M} {\cup_{β}}^{α μ}

, where

_{M} {\cup_{β}}^{α μ} : = \sqrt{| g |} g^{α ρ} g^{μ ω} (\partial_{[ω} g_{ρ] β})

is the so-called Møller superpotential [260] . (For another simple and natural introduction of Møller’s energy-momentum pseudotensor see [101] .) For the spin pseudotensor ( 2 ) gives

8 π G {\sqrt{| g |}}_{M} σ_{β}^{μ α} = -_{M} {\cup_{β}}^{α μ} + \partial_{ν} (\sqrt{| g |} δ_{β}^{[μ} g^{ν] α}),

which is in fact only pseudotensorial. Similarly, the contravariant form of these pseudotensors and the corresponding canonical Noether current are also pseudotensorial. We saw in subsection 2.1.2 that a specific combination of the canonical energy-momentum and spin tensors gave the symmetric energy-momentum tensor, which is gauge invariant even if the matter fields have gauge freedom, and one might hope that the analogous combination of the energy-momentum and spin pseudotensors gives a reasonable tensorial energy-momentum density for the gravitational field. The analogous expression is, in fact, tensorial, but unfortunately it is just minus the Einstein tensor [336, 337] .^$3$ Therefore, to use the pseudotensors a ‘natural’ choice for a ‘preferred’ coordinate system would be needed. This could be interpreted as a gauge choice, or reference configuration.

A further difficulty is that the different pseudotensors may have different (potential) significance.

For example, for any fixed

k \in R

Goldberg’s

2 k

-th symmetric pseudotensor

t_{(2 k)}^{α β}

is defined by

2 {| g |}^{k + 1} (κ t_{(2 k)}^{α β} - G^{α β}) : = \partial_{μ} \partial_{ν} ({| g |}^{k + 1} \times (g^{α β} g^{μ ν} - g^{α ν} g^{β μ}))

(which, for

k = 0

, reduces to the Landau–Lifshitz pseudotensor, the only symmetric pseudotensor which is a quadratic expression of the first derivatives of the metric) [158] . However, by Einstein’s equations this definition implies that

\partial_{α} ({| g |}^{k + 1} (t_{(2 k)}^{α β} + T^{α β})) = 0

. Hence what is (coordinate-)divergence-free (i. e. ‘pseudo-conserved’) cannot be interpreted as the sum of the gravitational and matter energy-momentum densities.

Indeed, the latter is

{| g |}^{1 / 2} T^{α β}

, while the second term in the divergence equation has an extra weight

{| g |}^{k + 1 / 2}

. Thus there is only one pseudotensor in this series,

t_{(- 1)}^{α β}

, which satisfies the ‘conservation law’ with the correct weight. On the other hand, the pseudotensors coming from some action (the ‘canonical pseudotensors’) appear to be free of this kind of difficulties (see also [336, 337] ). Classical excellent reviews on these (and several other) pseudotensors are [365, 57, 8, 159] , and for some recent ones (using background geometric structures) see for example [133, 134, 77, 150, 151, 222, 302] .) We return to the discussion of pseudotensors in subsections 3.3.1 and 11.3.3 .

^$3$ Since Einstein’s Lagrangian is only weakly diffeomorphism invariant, the situation would even be worse if we used Einstein’s Lagrangian. The corresponding canonical quantities would still be coordinate dependent, though in certain ‘natural’ coordinate system they yield reasonable results (see for example [2] and references therein).

3.1.3 Strategies to avoid pseudotensors I: Background metrics/connections

One way of avoiding the use of the pseudotensorial quantities is to introduce an explicit background connection [209] or background metric [307, 223, 227, 225, 224, 301, 131] . The advantage of this approach would be that we could use the background not only to derive the canonical energy-momentum and spin tensors, but to define the vector fields

K^{a}

as the symmetry generators of the background.

Then the resulting Noether currents are without doubt tensorial. However, they depend explicitly on the choice of the background connection or metric not only through

K^{a}

: The canonical energy-momentum and spin tensors themselves are explicitly background-dependent. Thus, again, the resulting expressions would have to be supplemented by a ‘natural’ choice for the background, and the main question is how to find such a ‘natural’ reference configuration from the infinitely many possibilities.

3.1.4 Strategies to avoid pseudotensors II: The tetrad formalism

In the tetrad formulation of general relativity the

g_{a b}

-orthonormal frame fields

{E_{\underset{̲}{a}}^{a}}

\underset{̲}{a} = 0, . . ., 3

, are chosen to be the gravitational field variables [368, 230] . Re-expressing the Hilbert Lagrangian (i. e. the curvature scalar) in terms of the tetrad field and its partial derivatives in some local coordinate system, one can calculate the canonical energy-momentum and spin by ( 4 ) and ( 2 ), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [261] , the use of the tetrad fields as the field variables instead of the metric makes it possible to introduce a first order, scalar Lagrangian for Einstein’s field equations: If

γ_{\underset{̲}{e} \underset{̲}{b}}^{\underset{̲}{a}} : = E_{\underset{̲}{e}}^{e} γ_{e \underset{̲}{b}}^{\underset{̲}{a}} : = E_{\underset{̲}{e}}^{e} ϑ_{a}^{\underset{̲}{a}} \nabla_{e} E_{\underset{̲}{b}}^{a}

, the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is

\begin{matrix} L : = \frac{1}{16 π G} (R - 2 \nabla_{a} (E_{\underset{̲}{a}}^{a} η^{\underset{̲}{a} \underset{̲}{b}} γ_{\underset{̲}{c} \underset{̲}{b}}^{\underset{̲}{c}})) = \frac{1}{16 π G} (E_{\underset{̲}{a}}^{a} E_{\underset{̲}{b}}^{b} - E_{\underset{̲}{a}}^{b} E_{\underset{̲}{b}}^{a}) γ_{a \underset{̲}{c}}^{\underset{̲}{a}} γ_{b}^{\underset{̲}{c} \underset{̲}{b}} . \end{matrix}

(9)

Although it depends on the actual tetrad field

{E_{\underset{̲}{a}}^{a}}

, it is weakly

O (1, 3)

-invariant. Møller’s Lagrangian has a nice uniqueness property [285] : Any first order scalar Lagrangian built from the tetrad fields, whose Euler–Lagrange equations are the Einstein equations, is Møller’s Lagrangian. Using Dirac spinor variables Nester and Tung found a first order spinor Lagrangian [275] , which turned out to be equivalent to Møller’s Lagrangian [366] . The canonical energy-momentum

θ_{β}^{α}

derived from ( 9 ) using the components of the tetrad fields in some coordinate system as the field variables is still pseudotensorial, but, as Møller realized, it has a tensorial superpotential:

\begin{matrix} {\lor_{b}}^{a e} : = 2 (- γ_{\underset{̲}{b} \underset{̲}{c}}^{\underset{̲}{a}} η^{\underset{̲}{c} \underset{̲}{e}} + γ_{\underset{̲}{d} \underset{̲}{c}}^{\underset{̲}{d}} η^{\underset{̲}{c} \underset{̲}{s}} (δ_{\underset{̲}{b}}^{\underset{̲}{a}} δ_{\underset{̲}{s}}^{\underset{̲}{e}} - δ_{\underset{̲}{s}}^{\underset{̲}{a}} δ_{\underset{̲}{b}}^{\underset{̲}{e}})) ϑ_{b}^{\underset{̲}{b}} E_{\underset{̲}{a}}^{a} E_{\underset{̲}{e}}^{e} = {\lor_{b}}^{[a e]} . \end{matrix}

(10)

The canonical spin turns out to be essentially

{\lor_{b}}^{a e}

, i. e. a tensor. The tensorial nature of the superpotential makes it possible to introduce a canonical energy-momentum tensor for the gravitational ‘field’. Then the corresponding canonical Noether current

C^{a} [K]

will also be tensorial and satisfies

\begin{matrix} 8 π G C^{a} [K] = G^{a b} K_{b} + \frac{1}{2} \nabla_{c} (K^{b} {\lor_{b}}^{a c}) . \end{matrix}

(11)

Therefore, the canonical Noether current derived from Møller’s tetrad Lagrangian is independent of the background structure (i. e. the coordinate system) that we used to do the calculations (see also [336] ).

However,

C^{a} [K]

depends on the actual tetrad field, and hence a preferred class of frame fields, i. e. an

O (1, 3)

-gauge reduction, is needed. Thus the explicit background-dependence of the final result of other approaches has been transformed into an internal

O (1, 3)

-gauge dependence. It is important to realize that this difficulty always appears in connection with the gravitational energy-momentum and angular momentum, at least in disguise. In particular, the Hamiltonian approach in itself does not yield well defined energy-momentum density for the gravitational ‘field’ (see for example [271, 253] ). Thus in the tetrad approach the canonical Noether current should be supplemented by a gauge condition for the tetrad field. Such a gauge condition could be some spacetime version of Nester’s gauge conditions (in the form of certain partial differential equations) for the orthonormal frames of Riemannian manifolds [270, 273] . Furthermore, since

C^{a} [K] + T^{a b} K_{b}

is conserved for any vector field

K^{a}

, in the absence of the familiar Killing symmetries of the Minkowski spacetime it is not trivial to define the ‘translations’ and ‘rotations’, and hence the energy-momentum and angular momentum. To make them well defined additional ideas would be needed.

3.1.5 Strategies to avoid pseudotensors III: Higher derivative currents

Giving up the paradigm that the Noether current should depend only on the vector field

K^{a}

and its first derivative – i. e. if we allow a term

{\dot{B}}^{a}

to be present in the Noether current ( 3 ) even if the Lagrangian is diffeomorphism invariant – one naturally arrives at Komar’s tensorial superpotential

_{K} \lor [K]^{a b} : = \nabla^{[a} K^{b]}

and the corresponding Noether current ([235] , see also [57] ).

Although its independence of any background structure (viz. its tensorial nature) and uniqueness property (see Komar [235] quoting Sachs) is especially attractive, the vector field

K^{a}

is still to be determined.

3.2 On the global energy-momentum and angular momentum of gravitating systems: the successes

As is well known, in spite of the difficulties with the notion of the gravitational energy-momentum density discussed above, reasonable total energy-momentum and angular momentum can be associated with the whole spacetime provided it is asymptotically flat. In the present subsection we recall the various forms of them. As we will see, most of the quasi-local constructions are simply ‘quasi-localizations’ of the total quantities. Obviously, the technique used in the ‘quasi-localization’ does depend on the actual form of the total quantities, yielding mathematically inequivalent definitions for the quasi-local quantities. We return to the discussion of the tools needed in the quasi-localization procedures in subsections 4.2 and 4.3 . Classical, excellent reviews of global energy-momentum and angular momentum are [147, 159, 14, 276, 375, 299] , and a recent review of conformal infinity (with special emphasis on its applicability in numerical relativity) is [140] . Reviews of the positive energy proofs from the first third of the eighties are [196, 300] .

3.2.1 Spatial infinity: Energy-momentum

There are several mathematically inequivalent definitions of asymptotic flatness at spatial infinity [147, 328, 22, 47, 144] . The traditional definition is based on the existence of a certain asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable definition are asymptotically flat in the traditional sense too. A spacelike hypersurface

Σ

will be called

k

-asymptotically flat if for some compact set

K \subset Σ

the complement

Σ - K

is diffeomorphic to

R^{3}

minus a solid ball, and there exists a (negative definite) metric

_{0} h_{a b}

Σ

, which is flat on

Σ - K

, such that the components of the difference of the physical and the background metrics,

h_{i j} -_{0} h_{i j}

, and of the extrinsic curvature

χ_{i j}

in the

_{0} h_{i j}

-Cartesian coordinate system

{x^{k}}

fall off as

r^{- k}

and

r^{- k - 1}

, respectively, for some

k > 0

and

r^{2} : = δ_{i j} x^{i} x^{j}

[304, 46] . These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost–rotations.

Σ - K

together with the metric and extrinsic curvature is called the asymptotic end of

Σ

. In a more general definition of asymptotic flatness

Σ

is allowed to have finitely many such ends.

As is well known, finite and well defined ADM energy-momentum [10, 12, 11, 13] can be associated with any

k

-asymptotically flat spacelike hypersurface if

k > \frac{1}{2}

by taking the value on the constraint surface of the Hamiltonian

H [K^{a}]

, given for example in [304, 46] , with the asymptotic translations

K^{a}

(see [109, 36, 278, 110] ). In its standard form this is the

r \to \infty

limit of a 2-surface integral of the first derivatives of the induced 3-metric

h_{a b}

and of the extrinsic curvature

χ_{a b}

for spheres of large coordinate radius

r

. The ADM energy-momentum is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian 4-vector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian coordinates.

The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian analysis of general relativity is based on the 3+1 decomposition of the fields and the spacetime.

Thus it is not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular momentum and centre-of-mass, discussed below, form an anti-symmetric tensor).

One had to check a posteriori that the conserved quantities obtained in the 3+1 form are, in fact, Lorentz-covariant. To obtain manifestly Lorentz-covariant quantities one should not do the 3+1 decomposition. Such a manifestly Lorentz-covariant Hamiltonian analysis was suggested first by Nester [269] , and he was able to recover the ADM energy-momentum in a natural way (see also subsection 11.3 below).

Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [159] :

Taking the flux integral of the current

C^{a} [K] + T^{a b} K_{b}

on the spacelike hypersurface

Σ

, by ( 11 ) the flux can be rewritten as the

r \to \infty

limit of the 2-surface integral of Møller’s superpotential on spheres of large

r

with the asymptotic translations

K^{a}

. Choosing the tetrad field

E_{\underset{̲}{a}}^{a}

to be adapted to the spacelike hypersurface and assuming that the frame

E_{\underset{̲}{a}}^{a}

tends to a constant Cartesian one as

r^{- k}

, the integral reproduces the ADM energy-momentum. The same expression can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too: By the standard scenario one can construct the basic Hamiltonian [271] . This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of

C^{a} [K] + T^{a b} K_{b}

Σ

A particularly interesting and useful expression for the ADM energy-momentum is possible if the tetrad field is considered to be a frame field built from a normalized spinor dyad

{λ_{A}^{\underset{̲}{A}}}

\underset{̲}{A} = 0, 1

, on

Σ

which is asymptotically constant (see subsection 4.2.3 below). (Thus underlined capital Roman indices are concrete name spinor indices.) Then, for the components of the ADM energy-momentum in the constant spinor basis at infinity, Møller’s expression yields the limit of

\begin{matrix} P^{\underset{̲}{A} {\underset{̲}{B}}^{'}} = \frac{1}{4 π G} \oint_{S} \frac{i}{2} ({\bar{λ}}_{A^{'}}^{{\underset{̲}{B}}^{'}} \nabla_{B B^{'}} λ_{A}^{\underset{̲}{A}} - {\bar{λ}}_{B^{'}}^{{\underset{̲}{B}}^{'}} \nabla_{A A^{'}} λ_{B}^{\underset{̲}{A}}) \end{matrix}

(12)

as the 2-surface

S

is blown up to approach infinity. In fact, to recover the ADM energy-momentum in the form ( 12 ), the spinor fields

λ_{A}^{\underset{̲}{A}}

need not be required to form a normalized spinor dyad, it is enough that they form an asymptotically constant normalized dyad, and we have to use the fact that the generator vector field

K^{a}

has asymptotically constant components

K^{\underset{̲}{A} {\underset{̲}{A}}^{'}}

in the asymptotically constant frame field

λ_{\underset{̲}{A}}^{A} {\bar{λ}}_{{\underset{̲}{A}}^{'}}^{A^{'}}

. Thus

K^{a} = K^{\underset{̲}{A} {\underset{̲}{A}}^{'}} λ_{\underset{̲}{A}}^{A} {\bar{λ}}_{{\underset{̲}{A}}^{'}}^{A^{'}}

can be interpreted as an asymptotic translation. The complex valued 2-form in the integrand of ( 12 ) will be denoted by

u (λ^{\underset{̲}{A}}, {\bar{λ}}^{{\underset{̲}{B}}^{'}})_{a b}

, and is called the Nester–Witten 2-form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential: For any two spinor fields

α^{A}

and

β^{A}

one has

\begin{matrix} u {(α, \bar{β})}_{a b} - \bar{u {(β, \bar{α})}_{a b}} & = & - i \nabla_{[a} X_{b]}, \end{matrix}

(13)

\begin{matrix} u {(α, \bar{β})}_{a b} + \bar{u {(β, \bar{α})}_{a b}} & = & \nabla_{c} X_{d} \frac{1}{2} ɛ_{a b}^{c d} + i (ɛ_{A^{'} B^{'}} α_{(A} \nabla_{B) C^{'}} {\bar{β}}^{C^{'}} - ɛ_{A B} {\bar{β}}_{(A^{'}} \nabla_{B^{'}) C} α^{C}) . \end{matrix}

(14)

where

X_{a} : = α_{A} {\bar{β}}_{A^{'}}

and overline denotes complex conjugation. Thus, apart from the terms in ( 14 ) involving

\nabla_{A^{'} A} α^{A}

and

\nabla_{A A^{'}} {\bar{β}}^{A^{'}}

, the Nester–Witten 2-form

u (α, \bar{β})_{a b}

is just

- \frac{i}{2} (\nabla_{[a} X_{b]} + i \nabla_{[c} X_{d]} \frac{1}{2} ɛ_{a b}^{c d})

, i. e. the anti-self-dual part of the curl of

- \frac{i}{2} X_{a}

. (The original expressions by Witten and Nester were given using Dirac rather than two-component Weyl spinors [378, 268] .) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [313, 314, 378, 268, 196, 300, 218] , the simplest and most transparent ones are probably those based on the use of 2-component spinors: If the dominant energy condition is satisfied on the

k

-asymptotically flat spacelike hypersurface

Σ

, where

k > \frac{1}{2}

, then the ADM energy-momentum is future pointing and non-spacelike (i. e. the Lorentzian length of the energy-momentum vector, the ADM mass, is non-negative), and it is null if and only if the domain of dependence

D (Σ)

Σ

is flat [199, 305, 155, 306, 68] . Its proof may be based on the Sparling equation [329, 126, 299, 256] :

\nabla_{[a} u (λ, \bar{μ})_{b c]} = - \frac{1}{2} λ_{E} {\bar{μ}}_{E^{'}} G^{e f} \frac{1}{3!} ɛ_{f a b c} + Γ (λ, \bar{μ})_{a b c}

. The significance of this equation is that in the exterior derivative of the Nester–Witten 2-form the second derivatives of the metric appear only through the Einstein tensor, thus its structure is similar to that of the superpotential equations in the Lagrangian field theory, and

Γ (λ, \bar{μ})_{a b c}

, known as the Sparling 3-form, is a quadratic expression of the first derivatives of the spinor fields. If the spinor fields

λ_{A}

and

μ_{A}

solve the Witten equation on a spacelike hypersurface

Σ

, then the pull back of

Γ (λ, \bar{μ})_{a b c}

Σ

is positive definite.

This theorem has been extended and refined in various ways, in particular by allowing inner boundaries of

Σ

that represent future marginally trapped surfaces in black holes [155, 196, 300, 194] .

The ADM energy-momentum can also be written as the 2-sphere integral of certain parts of the conformally rescaled spacetime curvature [14, 15, 27] . This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [159] ): If

S

is any closed spacelike 2-surface with area element

d S

, then for any tensor fields

ω_{a b} = ω_{[a b]}

and

μ_{a b} = μ_{[a b]}

one can form the integral

\begin{matrix} I_{S} [ω, μ] : = \oint_{S} ω^{a b} R_{a b c d} μ^{c d} d S . \end{matrix}

(15)

Since the fall-off of the curvature tensor near spatial infinity is

r^{- k - 2}

, the integral

I_{S} [ω, μ]

at spatial infinity can give finite value precisely when

ω^{a b} μ^{c d}

blows up like

r^{k}

r \to \infty

. In particular, for the

1 / r

fall-off this condition can be satisfied by

ω^{a b} μ^{c d} = \sqrt{Area (S)} {\hat{ω}}^{a b} {\hat{μ}}^{c d}

, where

Area (S)

is the area of

S

and the hatted tensor fields are

O (1)

If the spacetime is stationary, then the ADM energy can be recovered as the

r \to \infty

limit of the 2-sphere integral of Komar’s superpotential with the Killing vector

K^{a}

of stationarity [159] too. On the other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time translation, the Komar expression does not reproduce the ADM energy.

However, by ( 13 )–( 14 ) such an additional restriction might be that

K^{a}

should be a constant combination of four future pointing null vector fields of the form

α^{A} {\bar{α}}^{A^{'}}

, where the spinor fields

α^{A}

are required to satisfy the Weyl neutrino equation

\nabla_{A^{'} A} α^{A} = 0

. This expression for the ADM energy-momentum was used to give an alternative, ‘four dimensional’ proof of the positivity of the ADM energy [199] .

3.2.2 Spatial infinity: Angular momentum

The value of the Hamiltonian of Beig and O Murchadha [46] together with the appropriately defined asymptotic rotation–boost Killing vectors [348] define the spatial angular momentum and centre-of-mass, provided

k \geq 1

and, in addition to the familiar fall-off conditions, certain global integral conditions are also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and Teitelboim [304] on the leading nontrivial parts of the metric

h_{a b}

and extrinsic curvature

χ_{a b}

: The components in the Cartesian coordinates

{x^{i}}

of the former must be even and the components of latter must be odd parity functions of

x^{i} / r

(see also [46] ). Thus in what follows we assume that

k = 1

. Then the value of the Beig–O Murchadha Hamiltonian parameterized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [304] , while that parameterized by the asymptotic boost Killing vectors deviate from the centre-of-mass of Beig and O Murchadha [46] by a term which is the spatial momentum times the coordinate time. (As Beig and O Murchadha pointed out [46] , the centre-of-mass of Regge and Teitelboim is not necessarily finite.) The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz 4-tensor, which transforms in the correct way under the 4-translation of the origin of the asymptotically Cartesian coordinate system, and it is conserved by the evolution equations [348] .

The centre-of-mass of Beig and O Murchadha was reexpressed recently [41] as the

r \to \infty

limit of 2-surface integrals of the curvature in the form ( 15 ) with

ω^{a b} μ^{c d}

proportional to the lapse

N

times

q^{a c} q^{b d} - q^{a d} q^{b c}

, where

q_{a b}

is the induced 2-metric on

S

(see subsection 4.1.1 below).

A geometric notion of centre-of-mass was introduced by Huisken and Yau [203] . They foliate the asymptotically flat hypersurface

Σ

by certain spheres with constant mean curvature. By showing the global uniqueness of this foliation asymptotically, the origin of the leaves of this foliation in some flat ambient Euclidean space

R^{3}

defines the centre-of-mass (or rather ‘centre-of-gravity’) of Huisken and Yau. However, no statement on its properties is proven. In particular, it would be interesting to see whether or not this notion of centre-of-mass coincides, for example, with that of Beig and O Murchadha. The Ashtekar–Hansen definition for the angular momentum is introduced in their specific conformal model of the spatial infinity as a certain 2-surface integral near infinity. However, their angular momentum expression is finite and unambiguously defined only if the magnetic part of the spacetime curvature tensor (with respect to the

Ω = const

timelike level hypersurfaces of the conformal factor) falls off faster than would follow from the

1 / r

fall-off of the metric (but they do not have to impose any global integral, e. g. a parity condition) [22, 14] .

If the spacetime admits a Killing vector of axis-symmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see for example [369] ). However, the value of the Komar integral is twice the expected angular momentum.

In particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr solution the integral is

m / G

, for the Killing vector of axis-symmetry it is

2 m a / G

instead of the expected

m a / G

(‘factor-of-two anomaly’) [223] .

3.2.3 Null infinity: Energy-momentum

The study of the gravitational radiation of isolated sources led Bondi to the observation that the 2-sphere integral of a certain expansion coefficient

m (u, θ, φ)

of the line element of a radiative spacetime in an asymptotically retarded spherical coordinate system

(u, r, θ, φ)

behaves as the energy of the system at the retarded time

u

: This notion of energy is not constant in time, but decreases with

u

, showing that gravitational radiation carries away positive energy (‘Bondi’s mass-loss’) [69, 70] . The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, nowadays known as the BMS group, having a structure very similar to that of the Poincare group [310] . The only difference is that while the Poincare group is a semidirect product of the Lorentz group and a four dimensional commutative group (of translations), the BMS group is the semidirect product of the Lorentz group and an infinite dimensional commutative group, called the group of the supertranslations. A four parameter subgroup in the latter can be identified in a natural way as the group of the translations. Just at the same time the study of asymptotic solutions of the field equations led Newman and Unti to another concept of energy at null infinity [277] . However, this energy (nowadays known as the Newman–Unti energy) does not seem to have the same significance as the Bondi (or Bondi–Sachs [299] or Trautman–Bondi [112, 113, 111] ) energy, because its monotonicity can be proven only between special, e. g. stationary, states. The Bondi energy, which is the time component of a Lorentz vector, the so-called Bondi–Sachs energy-momentum, has a remarkable uniqueness property [112, 113] .

Without additional conditions on

K^{a}

, Komar’s expression does not reproduce the Bondi–Sachs energy-momentum in non-stationary spacetimes either [376, 159] : For the ‘obvious’ choice for

K^{a}

Komar’s expression yields the Newman–Unti energy. This anomalous behaviour in the radiative regime could be corrected in, at least, two ways. The first is by modifying the Komar integral according to

\begin{matrix} L_{S} [K] : = \frac{1}{8 π G} \oint_{S} (\nabla^{[c} K^{d]} + α \nabla_{e} {K^{e}}^{⊥} ɛ^{c d}) \frac{1}{2} ɛ_{c d a b}, \end{matrix}

(16)

where

^{⊥} ɛ_{c d}

is the area 2-form on the Lorentzian 2-planes orthogonal to

S

(see subsection 4.1.1 ) and

α

is some real constant. For

α = 1

the integral

L_{S} [K]

is called the linkage [376] . In addition, to define physical quantities by linkages associated to a cut of the null infinity one should prescribe how the 2-surface

S

tends to the cut and how the vector field

K^{a}

should be propagated from the spacetime to null infinity into a BMS generator [376, 375] . The other way is considering the original Komar integral (i. e.

α = 0

) on the cut of infinity in the conformally rescaled spacetime and by requiring that

K^{a}

be divergence-free [149] . For such asymptotic BMS translations both prescriptions give the correct expression for the Bondi–Sachs energy-momentum.

The Bondi–Sachs energy-momentum can also be expressed by the integral of the Nester–Witten 2-form [208, 246, 247, 199] . However, in non-stationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [81] . Thus the spinor fields in the Nester–Witten 2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves be the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [81] , was to require the spinor fields to be the solutions of the so-called asymptotic twistor equation (see subsection 4.2.4 ). One can impose several such inequivalent conditions, and all these, based only on the linear first order differential operators coming from the two natural connections on the cuts (see subsection 4.1.2 ), are determined in [347] .

The Bondi–Sachs energy-momentum has a Hamiltonian interpretation as well. Although the fields on a spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable generalization of the standard Hamiltonian analysis could be developed [111] and used to recover the Bondi–Sachs energy-momentum.

Similarly to the ADM case, the simplest proofs of the positivity of the Bondi energy [315] are probably those that are based on the Nester–Witten 2-form [208] and, in particular, the use of two-components spinors [246, 247, 199, 197, 306] : the Bondi–Sachs mass (i. e. the Lorentzian length of the Bondi–Sachs energy-momentum) of a cut of future null infinity is non-negative if there is a spacelike hypersurface

Σ

intersecting null infinity in the given cut such that the dominant energy condition is satisfied on

Σ

, and the mass is zero iff the domain of dependence

D (Σ)

Σ

is flat.

3.2.4 Null infinity: Angular momentum

At null infinity there is no generally accepted definition for angular momentum, and there are various, mathematically inequivalent suggestions for it. Here we review only some of those total angular momentum definitions that can be considered as the null infinity limit of some quasi-local expression, and will be discussed in the main part of the review, namely in section 9 .

In their classic paper Bergmann and Thomson [58] raise the idea that while the gravitational energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic

O (1, 3)

symmetry. Thus, the angular momentum should be analogous with the spin. Based on the tetrad formalism of general relativity and following the prescription of constructing the Noether currents in Yang–Mills theories, Bramson suggested a superpotential for the six conserved currents corresponding to the internal Lorentz-symmetry [82, ?, 83] . (For another derivation of this superpotential from Møller’s Lagrangian ( 9 ) see [347] .) If

{λ_{A}^{\underset{̲}{A}}}

\underset{̲}{A} = 0, 1

, is a normalized spinor dyad corresponding to the orthonormal frame in ( 9 ), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable 2-surface

S

\begin{matrix} J_{S}^{\underset{̲}{A} \underset{̲}{B}} : = \frac{1}{8 π G} \oint_{S} - i λ_{(A}^{\underset{̲}{A}} λ_{B)}^{\underset{̲}{B}} ɛ_{A^{'} B^{'}}, \end{matrix}

(17)

where

ɛ_{A^{'} B^{'}}

is the symplectic metric on the bundle of primed spinors. We will denote its integrand by

w (λ^{\underset{̲}{A}}, λ^{\underset{̲}{B}})_{a b}

, and we call it the Bramson superpotential. To define angular momentum on a given cut of the null infinity by the formula ( 17 ) we should consider its limit when

S

tends to the cut in question and we should specify the spinor dyad, at least asymptotically. Bramson’s suggestion for the spinor fields was to take the solutions of the asymptotic twistor equation [81] . He showed that this definition yields a well defined expression, for stationary spacetimes which reduces to the generally accepted formula ( 31 ), and the corresponding Pauli–Lubanski spin, constructed from

ɛ^{{\underset{̲}{A}}^{'} {\underset{̲}{B}}^{'}} J^{\underset{̲}{A} \underset{̲}{B}} + ɛ^{\underset{̲}{A} \underset{̲}{B}} {\bar{J}}^{{\underset{̲}{A}}^{'} {\underset{̲}{B}}^{'}}

and the Bondi–Sachs energy-momentum

P^{\underset{̲}{A} {\underset{̲}{A}}^{'}}

(given for example in the Newman–Penrose formalism by ( 30 )), is invariant with respect to supertranslations of the cut (‘active supertranslations’). Note that since Bramson’s expression is based on the solutions of a system of partial differential equations on the cut in question, it is independent of the parameterization of the BMS vector fields. Hence, in particular, it is invariant with respect to the supertranslations of the origin cut (‘passive supertranslations’). Therefore, Bramson’s global angular momentum behaves like the spin part of the total angular momentum.

The construction based on the Winicour–Tamburino linkage ( 16 ) can be associated with any BMS vector field [376, 244, 29] . In the special case of translations it reproduces the Bondi–Sachs energy-momentum. The quantities that it defines for the proper supertranslations are called the super-momenta. For the boost-rotation vector fields they can be interpreted as angular momentum.

However, in addition to the factor-of-two anomaly, this notion of angular momentum contains a huge ambiguity (‘supertranslation ambiguity’): The actual form of both the boost-rotation Killing vector fields of Minkowski spacetime and the boost-rotation BMS vector fields at future null infinity depend on the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively. However, while the set of the origins of Minkowski spacetime is parameterized by four numbers, the set of the origins at null infinity requires a smooth function of the form

u : S^{2} \to R

. Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar origin-dependence (containing four parameters), the analogous transformation of the angular momentum defined by using the boost-rotation BMS vector fields depends on an arbitrary smooth real valued function on the 2-sphere. This makes the angular momentum defined at null infinity by the boost-rotation BMS vector fields ambiguous unless a natural selection rule for the origins, making them form a four parameter family of cuts, is found. Such a selection rule could be the suggestion by Dain and Moreschi [121] in the charge integral approach to angular momentum of Moreschi [262] .

Another promising approach might be that of Chrusciel, Jezierski and Kijowski [111] , which is based on a Hamiltonian analysis of general relativity on asymptotically hyperbolic spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian 4-space of origins, they appear to be the generators with respect to some fixed ‘centre-of-the-cut’; and the corresponding angular momentum appears to be the intrinsic angular momentum.

3.3 The necessity of quasi-locality for the observables in general relativity

3.3.1 Non-locality of the gravitational energy-momentum and angular momentum

One reaction to the non-tensorial nature of the gravitational energy-momentum density expressions was to consider the whole problem ill-defined and the gravitational energy-momentum meaningless.

However, the successes discussed in the previous subsection show that the global gravitational energy-momenta and angular momenta are useful notions, and hence it could also be useful to introduce them even if the spacetime is not asymptotically flat. Furthermore, the non-tensorial nature of an object does not imply that it is meaningless. For example, the Christoffel symbols are not tensorial, but they do have geometric, and hence physical content, namely the linear connection. Indeed, the connection is a non-local geometric object, connecting the fibres of the vector bundle over different points of the base manifold. Hence any expression of the connection coefficients, in particular the gravitational energy-momentum or angular momentum, must also be non-local. In fact, although the connection coefficients at a given point can be taken zero by an appropriate coordinate/gauge transformation, they cannot be transformed to zero on an open domain unless the connection is flat.

Furthermore, the superpotential of many of the classical pseudotensors (e. g. of the Einstein, Bergmann, Møller’s tetrad, Landau–Lifshitz pseudotensors), being linear in the connection coefficients, can be recovered as the pull back to the spacetime manifold of various forms of a single geometric object on the linear frame bundle, namely of the Nester–Witten 2-form, along various local sections [138, 256, 336, 337] , and the expression of the pseudotensors by their superpotentials are the pull backs of the Sparling equation [329, 126, 256] . In addition, Chang, Nester and Chen [101] found a natural quasi-local Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see subsection 11.3.3 ). Therefore, the pseudotensors appear to have been ‘rehabilitated’, and the gravitational energy-momentum and angular momentum are necessarily associated with extended subsets of the spacetime.

This fact is a particular consequence of a more general phenomenon [309, 207] : Since the physical spacetime is the isomorphism class of the pairs

(M, g_{a b})

instead of a single such pair, it is meaningless to speak about the ‘value of a scalar or vector field at a point

p \in M

’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), 2-surfaces (e. g. the area of a closed 2-surface) etc. Thus, if we want to associate energy-momentum and angular momentum not only to the whole (necessarily asymptotically flat) spacetime, then these quantities must be associated with extended but finite subsets of the spacetime, i. e. must be quasi-local.

3.3.2 Domains for quasi-local quantities

The quasi-local quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:

1. the globally hyperbolic domains $D \subset M$ ,
2. the compact spacelike surfaces $Σ$ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains $D$ ),
3. the closed, orientable spacelike 2-surfaces $S$ (interpreted as the boundary $\partial Σ$ of Cauchy surfaces for globally hyperbolic domains).

A typical example for 3 is any charge integral expression: The quasi-local quantity is the integral of some superpotential 2-form built from the data given on the 2-surface, as in ( 12 ), or the expression

Q_{S} [K]

for the matter fields given by ( 5 ). An example for 2 might be the integral of the Bel–Robinson ‘momentum’ on the hypersurface

Σ

\begin{matrix} E_{Σ} [ξ^{a}] : = \int_{Σ} ξ^{d} T_{d e f g} t^{e} t^{f} \frac{1}{3!} ɛ_{a b c}^{g} . \end{matrix}

(18)

This quantity is analogous to the integral

E_{Σ} [ξ^{a}]

for the matter fields given by ( 6 ) (though, by the remarks on the Bel–Robinson ‘energy’ in subsection 3.1.1 , its physical dimension cannot be of energy). If

ξ^{a}

is a future pointing nonspacelike vector then

E_{Σ} [ξ^{a}] \geq 0

. Obviously, if such a quantity were independent of the actual hypersurface

Σ

, then it could also be rewritten as a charge integral on the boundary

\partial Σ

. The gravitational Hamiltonian provides an interesting example for the mixture of type 2 and 3 expressions, because the form of the Hamiltonian is the 3-surface integral of the constraints on

Σ

and a charge integral on its boundary

\partial Σ

, thus if the constraints are satisfied then the Hamiltonian reduces to a charge integral. Finally, an example for 1 might be

\begin{matrix} E_{D} : = inf {E_{Σ} [t] | Σ is a Cauchy surface for D}, \end{matrix}

(19)

the infimum of the ‘quasi-local Bel–Robinson energies’, where the infimum is taken on the set of all the Cauchy surfaces

Σ

for

D

with given boundary

\partial Σ

. (The infimum always exists because the Bel–Robinson ‘energy density’

T_{a b c d} t^{a} t^{b} t^{c} t^{d}

is non-negative.) Quasi-locality in any of these three senses agrees with the quasi-locality of Haag and Kastler [163, 164] . The specific quasi-local energy-momentum constructions provide further examples both for charge integral type expressions and those based on spacelike hypersurfaces.

3.3.3 Strategies to construct quasi-local quantities

There are two natural ways of finding the quasi-local energy-momentum and angular momentum.

The first is to follow some systematic procedure, while the second is the ‘quasi-localization’ of the global energy-momentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasi-local quantities are integrals of some superpotential derived from the Lagrangian via a Noether-type analysis. The advantage of this approach could be its manifest Lorentz-covariance. On the other hand, since the Noether current is determined only through the Noether identity, which contains only the divergence of the current itself, the Noether current and its superpotential is not uniquely determined. In addition (as in any approach), a gauge reduction (for example in the form of a background metric or reference configuration) and a choice for the ‘translations’ and ‘boost-rotations’ should be made.

The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasi-local (covariant or not) Hamiltonian analysis we would have a Hamiltonian, and its value on the constraint surface in the phase space yields the expected quantities. Here the main idea is that of Regge and Teitelboim [304] that the Hamiltonian must reproduce the correct field equations as the flows of the Hamiltonian vector fields, and hence, in particular, the correct Hamiltonian must be functionally differentiable with respect to the canonical variables. This differentiability may restrict the possible ‘translations’ and ‘boost-rotations’ too. However, if we are not interested in the structure of the quasi-local phase space, then, as a short-cut, we can use the Hamilton–Jacobi method to define the quasi-local quantities. The resulting expression is a 2-surface integral. Nevertheless, just as in the Lagrangian approach, this general expression is not uniquely determined, because the action can be modified by adding an (almost freely chosen) boundary term to it. Furthermore, the ‘translations’ and ‘boost-rotations’ are still to be specified.

On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasi-local quantities would be some ‘quasi-localization’ of those expressions that gave the global energy-momentum and angular momentum of asymptotically flat spacetimes.

Therefore, respecting both strategies, it is also legitimate to consider the Winicour–Tamburino-type (linkage) integrals and the charge integrals of the curvature.

Since the global energy-momentum and angular momentum of asymptotically flat spacetimes can be written as 2-surface integrals at infinity (and, as we will see in subsection 7.1.1 , both the energy-momentum and angular momentum of the source in the linear approximation and the gravitational mass in the Newtonian theory of gravity can also be written as 2-surface integrals), the 2-surface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasi-local energy-momentum and angular momentum as 2-surface observables, then three things must be specified:

1. an appropriate general 2-surface integral (e. g. the integral of a superpotential 2-form in the Lagrangian approaches or a boundary term in the Hamiltonian approaches),
2. a gauge choice (in the form of a distinguished coordinate system in the pseudotensorial approaches, or a background metric/connection in the background field approaches or a distinguished tetrad field in the tetrad approach),
3. a definition for the ‘quasi-symmetries’ of the 2-surface (i. e. the ‘generator vector fields’ of the quasi-local quantities in the Lagrangian, and the lapse and the shift in the Hamiltonian approaches, respectively, which, in the case of timelike ‘generator vector fields’, can also be interpreted as a fleet of observers on the 2-surface).

In certain approaches the definition of the ‘quasi-symmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.

4 Tools to Construct and Analyze the Quasi-Local Quantities

Having accepted that the gravitational energy-momentum and angular momentum should be introduced at the quasi-local level, we next need to discuss the special tools and concepts that are needed in practice to construct (or even to understand the various special) quasi-local expressions.

Thus, first, we review the geometry of closed spacelike 2-surfaces, with special emphasis on the so-called 2-surface data. Then, in the remaining two subsections, we discuss the special situations where there is a more or less generally accepted ‘standard’ definition for the energy-momentum (or at least for the mass) and angular momentum. In these situations any reasonable quasi-local quantity should reduce to them.

4.1 The geometry of spacelike 2-surfaces

The first systematic study of the geometry of spacelike 2-surfaces from the point of view of quasi-local quantities is probably due to Tod [358, 363] . Essentially, his approach is based on the GHP formalism [148] . Although this is a very effective and flexible formalism [148, 298, 299, 200, 331] , its form is not spacetime covariant. Since in many cases the covariance of a formalism itself already gives some hint how to treat and solve the problem at hand, here we concentrate mainly on a spacetime–covariant description of the geometry of the spacelike 2-surfaces, developed gradually in [339, 341, 342, 343, 143] . The emphasis will be on the geometric structures rather than the technicalities. In the last paragraph, we comment on certain objects appearing in connection with families of spacelike 2-surfaces.

4.1.1 The Lorentzian vector bundle

The restriction

V^{a} (S)

to the closed, orientable spacelike 2-surface

S

of the tangent bundle

T M

of the spacetime has a unique decomposition to the

g_{a b}

-orthogonal sum of the tangent bundle

T S

S

and the bundle of the normals, denoted by

N S

. Then all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If

t^{a}

and

v^{a}

are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projection to

T S

and

N S

Π_{b}^{a} : = δ_{b}^{a} - t^{a} t_{b} + v^{a} v_{b}

and

O_{b}^{a} : = δ_{b}^{a} - Π_{b}^{a}

, respectively. The induced 2-metric and the corresponding area 2-form on

S

will be denoted by

q_{a b}

and

ɛ_{a b}

, respectively, while the area 2-form on the normal bundle will be

^{⊥} ɛ_{a b}

. The bundle

V^{a} (S)

together with the fibre metric

g_{a b}

and the projection

Π_{b}^{a}

will be called the Lorentzian vector bundle over

S

. For the discussion of the global topological properties of the closed orientable 2-manifolds, see for example [5] .

4.1.2 Connections

The spacetime covariant derivative operator

\nabla_{e}

defines two covariant derivatives on

V^{a} (S)

. The first, denoted by

δ_{e}

, is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces:

δ_{e} X^{a} : = Π_{b}^{a} Π_{e}^{f} \nabla_{f} (Π_{c}^{b} X^{c}) + O_{b}^{a} Π_{e}^{f} \nabla_{f} (O_{c}^{b} X^{c})

for any section

X^{a}

V^{a} (S)

. Obviously,

δ_{e}

annihilates both the fibre metric

g_{a b}

and the projection

Π_{b}^{a}

. However, since for 2-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’

t^{a} \mapsto t^{a} cosh u + v^{a} sinh u

v^{a} \mapsto t^{a} sinh u + v^{a} cosh u

. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection 1-form on

S

can be characterized, for example, by

A_{e} : = Π_{e}^{f} (\nabla_{f} t_{a}) v^{a}

. Therefore, the connection

δ_{e}

can be considered as a connection on

V^{a} (S)

coming from a connection on the

O (2) \otimes O (1, 1)

-principal bundle of the

g_{a b}

-orthonormal frames adapted to

S

The other connection,

Δ_{e}

, is analogous to the Sen connection [316] , and is defined simply by

Δ_{e} X^{a} : = Π_{e}^{f} \nabla_{f} X^{a}

. This annihilates only the fibre metric, but not the projection. The difference of the connections

Δ_{e}

and

δ_{e}

turns out to be just the extrinsic curvature tensor:

Δ_{e} X^{a} = δ_{e} X^{a} + Q_{e b}^{a} X^{b} - X^{b} {Q_{b e}}^{a}

. Here

Q_{e b}^{a} : = - Π_{c}^{a} Δ_{e} Π_{b}^{c} = τ_{e}^{a} t_{b} - ν_{e}^{a} v_{b}

, and

τ_{a b} : = Π_{a}^{c} Π_{b}^{d} \nabla_{c} t_{d}

and

ν_{a b} : = Π_{a}^{c} Π_{b}^{d} \nabla_{c} v_{d}

are the standard (symmetric) extrinsic curvatures corresponding to the individual normals

t_{a}

and

v_{a}

, respectively. The familiar expansion tensors of the future pointing outgoing and ingoing null normals,

l^{a} : = t^{a} + v^{a}

and

n^{a} : = \frac{1}{2} (t^{a} - v^{a})

, respectively, are

θ_{a b} = Q_{a b c} l^{c}

and

θ_{a b}^{'} = Q_{a b c} n^{c}

, and the corresponding shear tensors

σ_{a b}

and

σ_{a b}^{'}

are defined by their trace-free part. Obviously,

τ_{a b}

and

ν_{a b}

(and hence the expansion and shear tensors

θ_{a b}

θ_{a b}^{'}

σ_{a b}

and

σ_{a b}^{'}

) are boost-gauge dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination

Q_{e b}^{a}

is boost-gauge invariant. In particular, it defines a natural normal vector field to

S

Q_{b} : = Q_{a b}^{a} = τ t_{b} - ν v_{b} = θ^{'} l_{b} + θ n_{b}

, where

τ

ν

θ

and

θ^{'}

are the relevant traces.

Q_{a}

is called the main extrinsic curvature vector of

S

. If

{\tilde{Q}}_{b} : = ν t_{b} - τ v_{b} = - θ^{'} l_{a} + θ n_{a}

, then their norm is

Q_{a} Q_{b} g^{a b} = - {\tilde{Q}}_{a} {\tilde{Q}}_{b} g^{a b} = τ^{2} - ν^{2} = 2 θ θ^{'}

, and they are orthogonal to each other:

Q_{a} {\tilde{Q}}_{b} g^{a b} = 0

. It is easy to show that

Δ_{a} {\tilde{Q}}^{a} = 0

, i. e.

{\tilde{Q}}^{a}

is the uniquely pointwise determined direction orthogonal to the 2-surface in which the expansion of the surface is vanishing. If

Q_{a}

is not null, then

{Q_{a}, {\tilde{Q}}_{a}}

defines an orthonormal frame in the normal bundle (see for example [7] ). If

Q_{a}

is non-zero but (e. g. future pointing) null, then there is a uniquely determined null normal

S_{a}

S

such that

Q_{a} S^{a} = 1

, and hence

{Q_{a}, S_{a}}

is a uniquely determined null frame. Therefore, the 2-surface admits a natural gauge choice in the normal bundle unless

Q_{a}

is vanishing. Geometrically,

Δ_{e}

is a connection coming from a connection on the

O (1, 3)

-principal fibre bundle of the

g_{a b}

-orthonormal frames. The curvature of the connections

δ_{e}

and

Δ_{e}

, respectively, are

\begin{matrix} f_{b c d}^{a} & = & -^{⊥} ɛ_{b}^{a} (δ_{c} A_{d} - δ_{d} A_{c}) + {\frac{1}{2}}^{S} R (Π_{c}^{a} q_{b d} - Π_{d}^{a} q_{b c}), \end{matrix}

(20)

\begin{matrix} F_{b c d}^{a} & = & f_{b c d}^{a} - δ_{c} (Q_{d b}^{a} - {Q_{b d}}^{a}) + δ_{d} (Q_{c b}^{a} - {Q_{b c}}^{a}) + \end{matrix}

\begin{matrix} + Q_{c e}^{a} {Q_{b d}}^{e} + {Q_{e c}}^{a} Q_{d b}^{e} - Q_{d e}^{a} {Q_{b c}}^{e} - {Q_{e d}}^{a} Q_{c b}^{e}, \end{matrix}

(21)

where

^{S} R

is the curvature scalar of the familiar intrinsic Levi-Civita connection of

(S, q_{a b})

. The curvature of

Δ_{e}

is just the pull back to

S

of the spacetime curvature 2-form:

F_{b c d}^{a} = R_{b e f}^{a} Π_{c}^{e} Π_{d}^{f}

Therefore, the well known Gauss, Codazzi–Mainardi and Ricci equations for the embedding of

S

M

are just the various projections of ( 21 ).

4.1.3 Convexity conditions

To prove certain statements on quasi-local quantities various forms of the convexity of

S

must be assumed. The convexity of

S

in a 3-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If the embedding space is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of

S

. If

S

is in a Lorentzian spacetime then the weakest convexity conditions are conditions only on the mean null curvatures:

S

will be called weakly future convex if the outgoing null normals

l^{a}

are expanding on

S

, i. e.

θ : = q^{a b} θ_{a b} > 0

, and weakly past convex if

θ^{'} : = q^{a b} θ_{a b}^{'} < 0

[363] .

S

is called mean convex [177] if

θ θ^{'} < 0

S

, or, equivalently, if

{\tilde{Q}}_{a}

is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions

D : = det ∥ θ_{b}^{a} ∥ = \frac{1}{2} (θ_{a b} θ_{c d} - θ_{a c} θ_{b d}) q^{a b} q^{c d}

and

D^{'} : = det ‖ {θ^{'}}_{b}^{a} ‖ = \frac{1}{2} (θ_{a b}^{'} θ_{c d}^{'} - θ_{a c}^{'} θ_{b d}^{'}) q^{a b} q^{c d}

. Note that although the expansion tensors, and in particular the functions

θ

θ^{'}

D

and

D^{'}

are gauge dependent, their sign is gauge invariant. Then

S

will be called future convex if

θ > 0

and

D > 0

, and past convex if

θ^{'} < 0

and

D^{'} > 0

[363, 342] . A different kind of convexity condition, based on global concepts, will be used in subsection 6.1.3 .

4.1.4 The spinor bundle

The connections

δ_{e}

and

Δ_{e}

determine connections on the pull back

S^{A} (S)

S

of the bundle of unprimed spinors. The natural decomposition

V^{a} (S) = T S \oplus N S

defines a chirality on the spinor bundle

S^{A} (S)

in the form of the spinor

γ_{B}^{A} : = 2 t^{A A^{'}} v_{B A^{'}}

, which is analogous to the

γ_{5}

matrix in the theory of Dirac spinors. Then the extrinsic curvature tensor above is a simple expression of

Q_{e B}^{A} : = \frac{1}{2} (Δ_{e} γ_{C}^{A}) γ_{B}^{C}

and

γ_{B}^{A}

(and their complex conjugate), and the two covariant derivatives on

S^{A} (S)

are related to each other by

Δ_{e} λ^{A} = δ_{e} λ^{A} + Q_{e B}^{A} λ^{B}

. The curvature

F_{B c d}^{A}

Δ_{e}

can be expressed by the curvature

f_{B c d}^{A}

δ_{e}

, the spinor

Q_{e B}^{A}

and its

δ_{e}

-derivative. We can form the scalar invariants of the curvatures according to

\begin{matrix} f & : = f_{a b c d} \frac{1}{2} (ɛ^{a b} - i^{⊥} ɛ^{a b}) ɛ^{c d} = i γ_{B}^{A} f_{A c d}^{B} ɛ^{c d} & =^{S} R - 2 i δ_{c} (ɛ^{c d} A_{d}), \end{matrix}

(22)

\begin{matrix} F & : = F_{a b c d} \frac{1}{2} (ɛ^{a b} - i^{⊥} ɛ^{a b}) ɛ^{c d} = i γ_{B}^{A} F_{A c d}^{B} ɛ^{c d} & = f + θ θ^{'} - 2 σ_{e a}^{'} σ_{b}^{e} (q^{a b} + i ɛ^{a b}) . \end{matrix}

(23)

f

is four times the so-called complex Gauss curvature [298] of

S

, by means of which the whole curvature

f_{B c d}^{A}

can be characterized:

f_{B c d}^{A} = - \frac{i}{4} f γ_{B}^{A} ɛ_{c d}

. If the spacetime is space and time orientable, at least on an open neighbourhood of

S

, then the normals

t_{a}

and

v_{a}

can be chosen to be globally well defined, and hence

N S

is globally trivializable and the imaginary part of

f

is a total divergence of a globally well defined vector field.

An interesting decomposition of the

S O (1, 1)

connection 1-form

A_{e}

, i. e. the vertical part of the connection

δ_{e}

, was given by Liu and Yau [245] : There are real functions

α

and

γ

, unique up to additive constants, such that

A_{e} = {ɛ_{e}}^{f} δ_{f} α + δ_{e} γ

α

is globally defined on

S

, but in general

γ

is defined only on the local trivialization domains of

N S

that are homeomorphic to

R^{2}

. It is globally defined if

H^{1} (S) = 0

. In this decomposition

α

is the boost-gauge invariant part of

A_{e}

, while

γ

represents its gauge content. Since

δ_{e} A^{e} = δ_{e} δ^{e} γ

, the ‘Coulomb-gauge condition’

δ_{e} A^{e} = 0

uniquely fixes

A_{e}

. (See also subsection 10.4.1 .) By the Gauss–Bonnet theorem

\oint_{S} f d S = {\oint_{S}}^{S} R d S = 8 π (1 - g)

, where

g

is the genus of

S

. Thus geometrically the connection

δ_{e}

is rather poor, and can be considered as a part of the ‘universal structure of

S

’. On the other hand, the connection

Δ_{e}

is much richer, and, in particular, the invariant

F

carries information on the mass aspect of the gravitational ‘field’.

The 2-surface data for charge-type quasi-local quantities (i. e. for 2-surface observables) are the universal structure (i. e. the intrinsic metric

q_{a b}

, the projection

Π_{b}^{a}

and the connection

δ_{e}

) and the extrinsic curvature tensor

Q_{e b}^{a}

4.1.5 Curvature identities

The complete decomposition of

Δ_{A A^{'}} λ_{B}

into its irreducible parts gives

Δ_{A^{'} A} λ^{A}

, the Dirac–Witten operator, and

{T_{E^{'} E A}}^{B} λ_{B} : = Δ_{E^{'} (E} λ_{A)} + \frac{1}{2} γ_{E A} γ^{C D} Δ_{E^{'} C} λ_{D}

, the 2-surface twistor operator. A Sen–Witten type identity for these irreducible parts can be derived. Taking its integral one has

\begin{matrix} \oint_{S} {\bar{γ}}^{A^{'} B^{'}} ((Δ_{A^{'} A} λ^{A}) (Δ_{B^{'} B} μ^{B}) + ({T_{A^{'} C D}}^{E} λ_{E}) ({T_{B^{'}}}^{C D F} μ_{F})) d S = - \frac{i}{2} \oint_{S} λ^{A} μ^{B} F_{A B c d}, \end{matrix}

(24)

where

λ_{A}

and

μ_{A}

are two arbitrary spinor fields on

S

, and the right hand side is just the charge integral of the curvature

F_{B c d}^{A}

S

4.1.6 The GHP formalism

A GHP spin frame on the 2-surface

S

is a normalized spinor basis

ɛ_{A}^{A} : = {o^{A}, ι^{A}}

A = 0, 1

, such that the complex null vectors

m^{a} : = o^{A} {\bar{ι}}^{A^{'}}

and

{\bar{m}}^{a} : = ι^{A} {\bar{o}}^{A^{'}}

are tangent to

S

(or, equivalently, the future pointing null vectors

l^{a} : = o^{A} {\bar{o}}^{A^{'}}

and

n^{a} : = ι^{A} {\bar{ι}}^{A^{'}}

are orthogonal to

S

). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole

S

. This fact is connected with the non-triviality of the tangent bundle

T S

of the 2-surface. For example, on the 2-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors

m^{a}

and

{\bar{m}}^{a}

cannot form a globally defined basis on

S

. Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable 2-surface with globally trivial tangent bundle is the torus.

Fixing a GHP spin frame

{ɛ_{A}^{A}}

on some

U \subset S

, the components of the spinor and tensor fields on

U

will be local representatives of cross sections of appropriate complex line bundles

E (p, q)

of scalars of type

(p, q)

[148, 298] : A scalar

φ

is said to be of type

(p, q)

if under the rescaling

o^{A} \mapsto λ o^{A}

ι^{A} \mapsto λ^{- 1} ι^{A}

of the GHP spin frame with some nowhere vanishing complex function

λ : U \to C

the scalar transforms as

φ \mapsto λ^{p} {\bar{λ}}^{q} φ

. For example

ρ : = θ_{a b} m^{a} {\bar{m}}^{b} = - \frac{1}{2} θ

ρ^{'} : = θ_{a b}^{'} m^{a} {\bar{m}}^{b} = - \frac{1}{2} θ^{'}

σ : = θ_{a b} m^{a} m^{b} = σ_{a b} m^{a} m^{b}

and

σ^{'} : = θ_{a b}^{'} {\bar{m}}^{a} {\bar{m}}^{b} = σ_{a b}^{'} {\bar{m}}^{a} {\bar{m}}^{b}

are of type

(1, 1)

(- 1, - 1)

(3, - 1)

and

(- 3, 1)

, respectively. The components of the Weyl and Ricci spinors,

ψ_{0} : = ψ_{A B C D} o^{A} o^{B} o^{C} o^{D}

ψ_{1} : = ψ_{A B C D} o^{A} o^{B} o^{C} ι^{D}

ψ_{2} : = ψ_{A B C D} o^{A} o^{B} ι^{C} ι^{D}

, . . . ,

φ_{00} : = φ_{A B^{'}} o^{A} {\bar{o}}^{B^{'}}

φ_{01} : = φ_{A B^{'}} o^{A} {\bar{ι}}^{B^{'}}

, . . . , etc., also have definite

(p, q)

type. In particular,

Λ : = R / 24

has type

(0, 0)

. A global section of

E (p, q)

is a collection of local cross sections

{(U, φ), (U^{'}, φ^{'}), . . .}

such that

{U, U^{'}, . . .}

forms a covering of

S

and on the non-empty overlappings, e. g. on

U \cap U^{'}

the local sections are related to each other by

φ = ψ^{p} {\bar{ψ}}^{q} φ^{'}

, where

ψ : U \cap U^{'} \to C

is the transition function between the GHP spin frames:

o^{A} = ψ {o^{'}}^{A}

and

ι^{A} = ψ^{- 1} {ι^{'}}^{A}

The connection

δ_{e}

defines a connection

ð_{e}

on the line bundles

E (p, q)

[148, 298] . The usual edth operators,

ð

and

ð^{'}

, are just the directional derivatives

ð : = m^{a} ð_{a}

and

ð^{'} : = {\bar{m}}^{a} ð_{a}

on the domain

U \subset S

of the GHP spin frame

{ɛ_{A}^{A}}

. These locally defined operators yield globally defined differential operators, denoted also by

ð

and

ð^{'}

, on the global sections of

E (p, q)

. It might be worth emphasizing that the GHP spin coefficients

β

and

β^{'}

, which do not have definite

(p, q)

-type, play the role of the two components of the connection 1-form, and they are built both from the connection 1-form for the intrinsic Riemannian geometry of

(S, q_{a b})

and the connection 1-form

A_{e}

in the normal bundle.

ð

and

ð^{'}

are elliptic differential operators, thus their global properties, e. g. the dimension of their kernel, are connected with the global topology of the line bundle they act on, and, in particular, with the global topology of

S

. These properties are discussed in [143] for general, and in [128, 42, 340] for spherical topology.

4.1.7 Irreducible parts of the derivative operators

Using the projection operators

π_{B}^{\pm A} : = \frac{1}{2} (δ_{B}^{A} \pm γ_{B}^{A})

, the irreducible parts

Δ_{A^{'} A} λ^{A}

and

{T_{E^{'} E A}}^{B} λ_{B}

can be decomposed further into their right handed and left handed parts. In the GHP formalism these chiral irreducible parts are

\begin{matrix} \begin{matrix} - Δ^{-} λ & : = ð λ_{1} + ρ^{'} λ_{0}, Δ^{+} λ : = ð^{'} λ_{0} + ρ λ_{1}, & T^{-} λ & : = ð λ_{0} + σ λ_{1}, - T^{+} λ : = ð^{'} λ_{1} + σ^{'} λ_{0}, \end{matrix} \end{matrix}

(25)

where

λ : = (λ_{0}, λ_{1})

and the spinor components are defined by

λ_{A} = : λ_{1} o_{A} - λ_{0} ι_{A}

. The various first order linear differential operators acting on spinor fields, e. g. the 2-surface twistor operator, the holomorphy/anti-holomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [81] , are appropriate direct sums of these elementary operators.

Their global properties under various circumstances are studied in [42, 340, 347] .

4.1.8 $S O (1, 1)$ -connection 1-form versus anholonomicity

Obviously, all the structures we have considered can be introduced on the individual surfaces of oneor two-parameter families of surfaces, too. In particular [176] , let the 2-surface

S

be considered as the intersection

N^{+} \cap N^{-}

of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to

S

, and let the spacetime (or at least a neighbourhood of

S

) be foliated by two one-parameter families of smooth hypersurfaces

{ν_{+} = const}

and

{ν_{-} = const}

, where

ν_{\pm} : M \to R

, such that

N^{+} = {ν_{+} = 0}

and

N^{-} = {ν_{-} = 0}

. One can form the two normals,

n_{\pm a} : = \nabla_{a} ν_{\pm}

, which are null on

N^{+}

and

N^{-}

, respectively. Then we can define

β_{\pm e} : = (Δ_{e} n_{\pm a}) n_{\mp}^{a}

, for which

β_{+ e} + β_{- e} = Δ_{e} n^{2}

, where

n^{2} : = g_{a b} n_{+}^{a} n_{-}^{b}

. (If

n^{2}

is chosen to be 1 on

S

, then

β_{- e} = - β_{+ e}

is precisely the

S O (1, 1)

connection 1-form

A_{e}

above.) Then the so-called anholonomicity is defined by

ω_{e} : = \frac{1}{2 n^{2}} [n_{-}, n_{+}]^{f} q_{f e} = \frac{1}{2 n^{2}} (β_{+ e} - β_{- e})

. Since

ω_{e}

is invariant with respect to the rescalings

ν_{+} \mapsto exp (A) ν_{+}

and

ν_{-} \mapsto exp (B) ν_{-}

of the functions defining the foliations by those functions

A, B : M \to R

which preserve

\nabla_{[a} n_{\pm b]} = 0

, it was claimed in [176] that

ω_{e}

depends only on

S

. However, this implies only that

ω_{e}

is invariant with respect to a restricted class of the change of the foliations, and that

ω_{e}

is invariantly defined only by this class of the foliations rather than the 2-surface. In fact,

ω_{e}

does depend on the foliation: Starting with a different foliation defined by the functions

{\bar{ν}}_{+} : = exp (α) ν_{+}

and

{\bar{ν}}_{-} : = exp (β) ν_{-}

for some

α, β : M \to R

, the corresponding anholonomicity

{\bar{ω}}_{e}

would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities,

ω_{e}

and

{\bar{ω}}_{e}

, would be different:

{\bar{ω}}_{e} - ω_{e} = \frac{1}{2} Δ_{e} (α - β)

Therefore, the anholonomicity is still a gauge dependent quantity.

4.2 Standard situations to evaluate the quasi-local quantities

There are exact solutions to the Einstein equations and classes of special (e. g. asymptotically flat) spacetimes in which there is a commonly accepted definition of energy-momentum (or at least mass) and angular momentum. In this subsection we review these situations and recall the definition of these ‘standard’ expressions.

4.2.1 Round spheres

If the spacetime is spherically symmetric, then a 2-sphere which is a transitivity surface of the rotation group is called a round sphere. Then in a spherical coordinate system

(t, r, θ, φ)

the spacetime metric takes the form

g_{a b} = diag (exp (2 γ), - exp (2 α), - r^{2}, - r^{2} {sin}^{2} θ)

, where

γ

and

α

are functions of

t

and

r

. (Hence

r

is the so-called area-coordinate). Then with the notations of subsection 4.1 , one obtains

R_{a b c d} ɛ^{a b} ɛ^{c d} = \frac{4}{r^{2}} (1 - exp (- 2 α))

. Based on the investigations of Misner, Sharp and Hernandez [258, 193] , Cahill and McVitte [95] found

\begin{matrix} E (t, r) : = \frac{1}{8 G} r^{3} R_{a b c d} ɛ^{a b} ɛ^{c d} = \frac{r}{2 G} (1 - e^{- 2 α}) \end{matrix}

(26)

to be an appropriate (and hence suggested to be the general) notion of energy contained in the 2-sphere

S : = {t = const, r = const}

. In particular, for the Reissner–Nordstrom solution

G E (t, r) = m - e^{2} / 2 r

, while for the isentropic fluid solutions

E (t, r) = 4 π \int_{0}^{r} {r^{'}}^{2} μ (t, r^{'}) d r^{'}

, where

m

and

e

are the usual parameters of the Reissner–Nordstrom solutions and

μ

is the energy density of the fluid [258, 193] (for the static solution, see e. g. Appendix B of [170] ). Using Einstein’s equations nice and simple equations can be derived for the derivatives

\partial_{t} E (t, r)

and

\partial_{r} E (t, r)

, and if the energy-momentum tensor satisfies the dominant energy condition then

\partial_{r} E (t, r) > 0

. Thus

E (t, r)

is a monotonic function of

r

provided

r

is the area-coordinate. Since by the spherical symmetry all the quantities with non-zero spin weight, in particular the shears

σ

and

σ^{'}

, are vanishing and

ψ_{2}

is real, by the GHP form of ( 22 )–( 23 ) the energy function

E (t, r)

can also be written as

\begin{matrix} E (S) = \frac{1}{G} r^{3} ({\frac{1}{4}}^{S} R + ρ ρ^{'}) = \frac{1}{G} r^{3} (- ψ_{2} + φ_{11} + Λ) = \sqrt{\frac{Area (S)}{16 π G^{2}}} (1 + \frac{1}{2 π} \oint_{S} ρ ρ^{'} d S) . \end{matrix}

(27)

This expression is considered to be the ‘standard’ definition of the energy for round spheres.^$4$ Its expression in the second line does not depend on whether

r

is an area-coordinate or not.

E (S)

contains a contribution from the gravitational ‘field’ too. In fact, for example for fluids it is not simply the volume integral of the energy density

μ

of the fluid, because that would be

4 π \int_{0}^{r} {r^{'}}^{2} exp (α) μ d r^{'}

. This deviation can be interpreted as the contribution of the gravitational potential energy to the total energy.

Consequently,

E (S)

is not a globally monotonic function of

r

, even if

μ \geq 0

. For example, in the closed Friedmann–Robertson–Walker spacetime (where, to cover the whole space,

r

cannot be chosen to be the area–radius and

r \in [0, π]

)

E (S)

is increasing for

r \in [0, π / 2)

, taking its maximal value at

r = π / 2

, and decreasing for

r \in (π / 2, π]

This example suggests a slightly more exotic spherically symmetric spacetime. Its spacelike slice

Σ

will be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat metrics. The first is a ‘large’ spherically symmetric part of a

t = const

hypersurface of the closed Friedmann–Robertson–Walker spacetime with the line element

d l^{2} = Ω_{F R W}^{2} d l_{0}^{2}

, where

d l_{0}^{2}

is the line element for the flat 3-space and

Ω_{F R W}^{2} : = B (1 + \frac{r^{2}}{4 T^{2}})^{- 2}

with some positive constants

B

and

T^{2}

, and the range of the Euclidean radial coordinate

r

[0, r_{0}]

, where

r_{0} \in (2 T, \infty)

. It contains a maximal 2-surface at

r = 2 T

with round-sphere mass parameter

M : = G E (2 T) = \frac{1}{2} T \sqrt{B}

. The scalar curvature is

R = 6 / B T^{2}

, and hence, by the constraint parts of the Einstein equations and by the vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the metric of a piece of a

t = const

hypersurface in the Schwarzschild solution with mass parameter

m

(see [152] ):

d {\bar{l}}^{2} = Ω_{S}^{2} d {\bar{l}}_{0}^{2}

, where

Ω_{S}^{2} : = (1 + \frac{m}{2 \bar{r}})^{4}

and the Euclidean radial coordinate

\bar{r}

runs from

{\bar{r}}_{0}

and

\infty

, where

{\bar{r}}_{0} \in (0, m / 2)

. In this geometry there is a minimal surface at

\bar{r} = m / 2

, the scalar curvature is zero, and the round sphere energy is

E (\bar{r}) = m / G

. These two metrics can be matched to obtain a differentiable metric with Lipschitz-continuous derivative at the 2-surface of the matching (where the scalar curvature has a jump) with arbitrarily large ‘internal mass’

M / G

and arbitrarily small ADM mass

m / G

. (Obviously, the two metrics can be joined smoothly as well by an ‘intermediate’ domain between them.) Since this space looks like a big spherical bubble on a nearly flat 3-plane – like the capital Greek letter

Ω

– for later reference we call it an ‘

Ω_{M, m}

-spacetime’.

Spherically symmetric spacetimes admit a special vector field, the so-called Kodama vector field

K^{a}

, such that

K_{a} G^{a b}

is divergence free [234] . In asymptotically flat spacetimes

K^{a}

is timelike in the asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in fact, this is hypersurface-orthogonal), but in general it is not a Killing vector. However, by

\nabla_{a} (G^{a b} K_{b}) = 0

the vector field

S^{a} : = G^{a b} K_{b}

has a conserved flux on a spacelike hypersurface

Σ

. In particular, in the coordinate system

(t, r, θ, φ)

and line element above

K^{a} = exp (- (α + γ)) (\partial / \partial t)^{a}

. If

Σ

is the solid ball of radius

r

, then the flux of

S^{a}

is precisely the standard round sphere expression ( 26 ) for the 2-sphere

\partial Σ

[267] .

An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be given in terms of the energy function

E (t, r)

above (see for example [388, 252, 180] ). In particular, criteria for the existence and the formation of trapped surfaces and the presence and the nature of the central singularity can be given by

E (t, r)

^$4$ $E (S)$ can be thought of as the 0-component of some quasi-local energy-momentum 4-vector, but, just because of the spherical symmetry, its spatial parts are vanishing. Thus $E (S)$ can also be interpreted as the mass, the length of this energy-momentum 4-vector.

4.2.2 Small surfaces

In the literature there are two notions of small surfaces, the first is that of the small spheres (both in the light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [198] , and the other is the concept of the small ellipsoids in some spacelike hypersurface, considered first by Woodhouse in [229] . A small sphere in the light cone is a cut of the future null cone in the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given spacelike hypersurface, whose geodesic distance in the hypersurface from a given point

p

, the centre, is a small given value, and the geometry of this sphere is characterized by data at this centre. Small ellipsoids are 2-surfaces in a spacelike hypersurface with a more general shape.

To define the first, let

p \in M

be a point, and

t^{a}

a future directed unit timelike vector at

p

Let

N_{p} : = \partial I^{+} (p)

, the ‘future null cone of

p

M

’ (i. e. the boundary of the chronological future of

p

). Let

l^{a}

be the future pointing null tangent to the null geodesic generators of

N_{p}

such that, at the vertex

p

l^{a} t_{a} = 1

. With this condition we fix the scale of the affine parameter

r

on the different generators, and hence by requiring

r (p) = 0

we fix the parameterization completely.

Then, in an open neighbourhood of the vertex

p

N_{p} - {p}

is a smooth null hypersurface, and hence for sufficiently small

r

the set

S_{r} : = {q \in M | r (q) = r}

is a smooth spacelike 2-surface and homeomorphic to

S^{2}

S_{r}

is called a small sphere of radius

r

with vertex

p

. Note that the condition

l^{a} t_{a} = 1

fixes the boost gauge.

Completing

l^{a}

to a Newman–Penrose complex null tetrad

{l^{a}, n^{a}, m^{a}, {\bar{m}}^{a}}

such that the complex null vectors

m^{a}

and

{\bar{m}}^{a}

are tangent to the 2-surfaces

S_{r}

, the components of the metric and the spin coefficients with respect to this basis can be expanded as series in

r

.^$5$ Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the metric

q_{a b}

S_{r}

, the GHP spin coefficients

ρ

σ

τ

ρ^{'}

σ^{'}

and

β

and the higher order expansion coefficients of the curvature in terms of the lower order curvature components at

p

. Hence the expression of any quasi-local quantity

Q_{S_{r}}

for the small sphere

S_{r}

can be expressed as a series of

r

Q_{S_{r}} = \oint_{S} (Q^{(0)} + r Q^{(1)} + \frac{1}{2} r^{2} Q^{(2)} + \dots) d S,

where the expansion coefficients

Q^{(k)}

are still functions of the coordinates,

(ζ, \bar{ζ})

(θ, φ)

, on the unit sphere

S

. If the quasi-local quantity

Q

is spacetime–covariant, then the unit sphere integrals of the expansion coefficients

Q^{(k)}

must be spacetime covariant expressions of the metric and its derivatives up to some finite order at

p

and the ‘time axis’

t^{a}

. The necessary degree of the accuracy of the solution of the GHP equations depends on the nature of

Q_{S_{r}}

and on whether the spacetime is Ricci-flat in a neighbourhood of

p

or not.^$6$ These solutions of the GHP equations, with increasing accuracy, are given in [198, 229, 91, 344] .

Obviously, we can calculate the small sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime too. In particular [344] , the small sphere expressions for the quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields based on

Q_{S} [K]

, respectively, are

\begin{matrix} P_{S_{r}}^{\underset{̲}{A} {\underset{̲}{B}}^{'}} & = \frac{4 π}{3} r^{3} T^{A A^{'} B B^{'}} t_{A A^{'}} ℰ_{B}^{\underset{̲}{A}} {\bar{ℰ}}_{B^{'}}^{{\underset{̲}{B}}^{'}} + O (r^{4}), \end{matrix}

(28)

\begin{matrix} J_{S_{r}}^{\underset{̲}{A} \underset{̲}{B}} & = \frac{4 π}{3} r^{3} T_{A A^{'} B B^{'}} t^{A A^{'}} (r t^{B^{'} E} ɛ^{B F} ℰ_{(E}^{\underset{̲}{A}} ℰ_{F)}^{\underset{̲}{B}}) + O (r^{5}), \end{matrix}

(29)

where

{ℰ_{A}^{\underset{̲}{A}}}

\underset{̲}{A} = 0, 1

, is the ‘Cartesian spin frame’ at

p

. Here

K_{a}^{\underset{̲}{A} {\underset{̲}{B}}^{'}} = ℰ_{A}^{\underset{̲}{A}} {\bar{ℰ}}_{A^{'}}^{{\underset{̲}{B}}^{'}}

can be interpreted as the translation 1-forms, while

K_{a}^{\underset{̲}{A} \underset{̲}{B}} = r {t_{A^{'}}}^{E} ℰ_{(E}^{\underset{̲}{A}} ℰ_{A)}^{\underset{̲}{B}}

is an average on the unit sphere of the boost-rotation Killing 1-forms that vanish at the vertex

p

. Thus

P_{S_{r}}^{\underset{̲}{A} {\underset{̲}{B}}^{'}}

and

J_{S_{r}}^{\underset{̲}{A} \underset{̲}{B}}

are the 3-volume times the energy-momentum and angular momentum density with respect to

p

, respectively, that the observer with 4-velocity

t^{a}

sees at

p

Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in a large class of quasi-local spacetime covariant energy-momentum and angular momentum expressions. In fact, if

Q_{S}

is any coordinate-independent quasi-local quantity, built from the first derivatives of the metric, i. e.

Q_{S} = \oint_{S} (\partial_{μ} g_{α β}) d S

, then its expansion is

\begin{matrix} Q_{S_{r}} = & [\partial g] r^{2} + [\partial^{2} g, {(\partial g)}^{2}] r^{3} + [\partial^{3} g, (\partial^{2} g) (\partial g), {(\partial g)}^{3}] r^{4} + \end{matrix}

\begin{matrix} + [\partial^{4} g, (\partial^{3} g) (\partial g), {(\partial^{2} g)}^{2}, (\partial^{2} g) {(\partial g)}^{2}, {(\partial g)}^{4}] r^{5} + \dots, \end{matrix}

where

[A, B, . . .]

is a scalar. It depends linearly on the tensors constructed from

g_{α β}

and linearly from the coordinate dependent quantities

A

B

, . . . ; and it is a polynomial expression of

t^{a}

g_{a b}

and

ɛ_{a b c d}

at the vertex

p

. Since there is no non-trivial tensor built from the first derivative

\partial_{μ} g_{α β}

and

g_{α β}

, the leading term is of order

r^{3}

. Its coefficient

[\partial^{2} g, (\partial g)^{2}]

must be a linear expression of

R_{a b}

and

C_{a b c d}

, and polynomial in

t^{a}

g_{a b}

and

ɛ_{a b c d}

. In particular, if

Q_{S}

is to represent energy-momentum with generator

K^{c}

p

, then the leading term must be

Q_{S_{r}} [K] = r^{3} (a (G_{a b} t^{a} t^{b}) t_{c} + b R t_{c} + c (G_{a b} t^{a} P_{c}^{b})) K^{c} + O (r^{4})

for some unspecified constants

a

b

and

c

, where

P_{b}^{a} : = δ_{b}^{a} - t^{a} t_{b}

, the projection to the subspace orthogonal to

t^{a}

. If, in addition to the coordinate-independence of

Q_{S}

, it is Lorentz-covariant, i. e.

for example it does not depend on the choice for a normal to

S

(for example in the small sphere approximation on

t^{a}

) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer

t^{a}

in the same way. Therefore, either

a = c = 0

b = 0

, and in the second case the resulting expression will be Lorentz-covariant if in addition

c = a

also holds. However, in the first case we would have only energy but no linear momentum in any frame defined by the observer

t^{a}

, even in a general spacetime. This seems to be quite unreasonable, and we are lead to the second option. Thus, apart from the single parameter

a

, we arrived at ( 28 ): In the leading

r^{3}

order in non-vacuum any coordinate and Lorentz-covariant quasi-local energy-momentum expression must be proportional to the energy-momentum density of the matter fields seen by the observer

t^{a}

times the Euclidean volume of the 3-ball of radius

r

. Comparing this with ( 28 ) the factor of proportionality can be expected to be 1.

If a neighbourhood of

p

is vacuum, then the

r^{3}

order term is vanishing, and the fourth order term must be built from

\nabla_{e} C_{a b c d}

. However, the only scalar polynomial expression of

t^{a}

g_{a b}

ɛ_{a b c d}

\nabla_{e} C_{a b c d}

and the generator vector

K^{a}

, depending on the latter two linearly, is the zero. Thus the

r^{4}

order term in vacuum is also vanishing. In the fifth order the only non-zero terms are quadratic in the various parts of the Weyl tensor, yielding

Q_{S_{r}} [K] = r^{5} ((a E_{a b} E^{a b} + b H_{a b} H^{a b} + c E_{a b} H^{a b}) t_{c} + d E_{a e} H_{b}^{e} ɛ_{c}^{a b}) K^{c} + O (r^{6})

for some constants

a

b

c

and

d

, where

E_{a b} : = C_{a e b f} t^{e} t^{f}

is the electric and

H_{a b} : = * C_{a e b f} t^{e} t^{f} : = \frac{1}{2} {ɛ_{a e}}^{c d} C_{c d b f} t^{e} t^{f}

is the magnetic part of the Weyl curvature and

ɛ_{a b c} : = ɛ_{a b c d} t^{d}

is the induced volume 3-form. However, using the identities

C_{a b c d} C^{a b c d} = 8 (E_{a b} E^{a b} - H_{a b} H^{a b})

C_{a b c d} * C^{a b c d} = 16 E_{a b} H^{a b}

4 T_{a b c d} t^{a} t^{b} t^{c} t^{d} = E_{a b} E^{a b} + H_{a b} H^{a b}

and

T_{a b c d} t^{a} t^{b} t^{c} P_{e}^{d} = E_{a b} H_{c}^{a} ɛ_{e}^{b c}

, we can rewrite the above formula to

\begin{matrix} Q_{S_{r}} [K] = & r^{5} ((2 (a + b) T_{a b c d} t^{a} t^{b} t^{c} t^{d} + \frac{1}{16} (a - b) C_{a b c d} C^{a b c d} + \end{matrix}

\begin{matrix} + \frac{1}{16} c C_{a b c d} * C^{a b c d}) t_{e} + d T_{a b c d} t^{a} t^{b} t^{c} P_{e}^{d}) K^{e} + O (r^{6}) . \end{matrix}

Again, if

Q_{S}

does not depend on

t^{a}

intrinsically, then either

a = - b

and

d = 0

a = b

and

c = 0

, and in the latter case the remaining two terms form a Lorentz-covariant expression if

d = 2 (a + b)

also holds. Again, in the first case we would have only energy but no linear momentum in any frame defined by the observer

t^{a}

. Therefore, in vacuum in the leading

r^{5}

order any coordinate and Lorentz-covariant quasi-local energy-momentum expression must be proportional to the Bel–Robinson ‘momentum’

T_{a b c d} t^{a} t^{b} t^{c}

. Obviously, the same analysis can be repeated for any other quasi-local quantity. For quasi-local angular momentum

Q_{S}

has the structure

\oint_{S} (\partial_{μ} g_{α β}) r d S

, while the area of

S

\oint_{S} d S

. Then the leading term in the expansion of the angular momentum is

r^{4}

and

r^{6}

order in non-vacuum and vacuum, respectively; while the first non-trivial correction to the area

4 π r^{2}

is of order

r^{4}

and

r^{6}

in non-vacuum and vacuum, respectively.

On the small geodesic sphere

S_{r}

of radius

r

in the given spacelike hypersurface

Σ

one can introduce the complex null tangents

m^{a}

and

{\bar{m}}^{a}

above, and if

t^{a}

is the future pointing unit normal of

Σ

and

v^{a}

the outward directed unit normal of

S_{r}

Σ

, then we can define

l^{a} : = t^{a} + v^{a}

and

2 n^{a} : = t^{a} - v^{a}

. Then

{l^{a}, n^{a}, m^{a}, {\bar{m}}^{a}}

is a Newman–Penrose complex null tetrad, and the relevant GHP equations can be solved for the spin coefficients in terms of the curvature components at

p

The small ellipsoids are defined as follows [229] . If

f

is any smooth function on

Σ

with a non-degenerate minimum at

p \in Σ

with minimum value

f (p) = 0

, then, at least on an open neighbourhood

U

p

Σ

the level surfaces

S_{r} : = {q \in Σ | 2 f (q) = r^{2}}

are smooth compact 2-surfaces homeomorphic to

S^{2}

. Then, in the

r \to 0

limit, the surfaces

S_{r}

look like small nested ellipsoids centred in

p

. The function

f

is usually ‘normalized’ so that

h^{a b} D_{a} D_{b} f |_{p} = - 3

^$5$ If, in addition, the spinor constituent $o^{A}$ of $l^{a} = o^{A} {\bar{o}}^{A^{'}}$ is required to be parallel propagated along $l^{a}$ , then the tetrad becomes completely fixed, yielding the vanishing of several (combinations of the) spin coefficients.

^$6$ As we will see soon, the leading term of the small sphere expression of the energy-momenta in non-vacuum is of order $r^{3}$ , in vacuum it is $r^{5}$ , while that of the angular momentum is $r^{4}$ and $r^{6}$ , respectively.

4.2.3 Large spheres near the spatial infinity

Near spatial infinity we have the a priori

1 / r

and

1 / r^{2}

fall-off for the 3-metric

h_{a b}

and extrinsic curvature

χ_{a b}

, respectively, and both the evolution equations of general relativity and the conservation equation

T_{; b}^{a b} = 0

for the matter fields preserve these conditions. The spheres

S_{r}

of coordinate radius

r

Σ

are called large spheres if the values of

r

are large enough such that the asymptotic expansions of the metric and extrinsic curvature are legitimate.^$7$ Introducing some coordinate system, e. g. the complex stereographic coordinates, on one sphere and then extending that to the whole

Σ

along the normals

v^{a}

of the spheres, we obtain a coordinate system

(r, ζ, \bar{ζ})

Σ

. Let

ɛ_{A}^{A} = {o^{A}, ι^{A}}

A = 0, 1

, be a GHP spinor dyad on

Σ

adapted to the large spheres in such a way that

m^{a} : = o^{A} {\bar{ι}}^{A^{'}}

and

{\bar{m}}^{a} = ι^{A} {\bar{o}}^{A^{'}}

are tangent to the spheres and

t^{a} = \frac{1}{2} o^{A} {\bar{o}}^{A^{'}} + ι^{A} {\bar{ι}}^{A^{'}}

, the future directed unit normal of

Σ

. These conditions fix the spinor dyad completely, and, in particular,

v^{a} = \frac{1}{2} o^{A} {\bar{o}}^{A^{'}} - ι^{A} {\bar{ι}}^{A^{'}}

, the outward directed unit normal to the spheres tangent to

Σ

The fall-off conditions yield that the spin coefficients tend to their flat spacetime value like

1 / r^{2}

and the curvature components to zero like

1 / r^{3}

. Expanding the spin coefficients and curvature components as power series of

1 / r

, one can solve the field equations asymptotically (see [47, 43] for a different formalism). However, in most calculations of the large sphere limit of the quasi-local quantities only the leading terms of the spin coefficients and curvature components appear. Thus it is not necessary to solve the field equations for their second or higher order non-trivial expansion coefficients.

Using the flat background metric

_{0} h_{a b}

and the corresponding derivative operator

_{0} D_{e}

we can define a spinor field

_{0} λ_{A}

to be constant if

_{0} {D_{e}}_{0} λ_{A} = 0

. Obviously, the constant spinors form a two complex dimensional vector space. Then by the fall-off properties

{D_{e}}_{0} λ_{A} = O (r^{- 2})

. Hence we can define the asymptotically constant spinor fields to be those

λ_{A}

that satisfy

D_{e} λ_{A} = O (r^{- 2})

, where

D_{e}

is the intrinsic Levi-Civita derivative operator. Note that this implies that, with the notations of ( 25 ), all the chiral irreducible parts,

Δ^{+} λ

Δ^{-} λ

T^{+} λ

and

T^{-} λ

, of the derivative of the asymptotically constant spinor field

λ_{A}

are

O (r^{- 2})

^$7$ Because of the fall-off, no essential ambiguity in the definition of the large spheres arises from the use of the coordinate radius instead of the physical radial distance.

4.2.4 Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [286, 287, 288, 299] (see also [147] ), i. e. the physical spacetime can be conformally compactified by an appropriate boundary

ℑ^{+}

. Then future null infinity

ℑ^{+}

will be a null hypersurface in the conformally rescaled spacetime. Topologically it is

R \times S^{2}

, and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of

ℑ^{+}

is the metric of the unit sphere. Fixing such a slice

S_{0}

(called ‘the origin cut of

ℑ^{+}

’) the points of

ℑ^{+}

can be labelled by a null coordinate, namely the affine parameter

u \in R

along the null geodesic generators of

ℑ^{+}

measured from

S_{0}

and, for example, the familiar complex stereographic coordinates

(ζ, \bar{ζ}) \in S^{2}

, defined first on the unit sphere

S_{0}

and then extended in a natural way along the null generators to the whole

ℑ^{+}

. Then any other cut

S

ℑ^{+}

can be specified by a function

u = f (ζ, \bar{ζ})

. In particular, the cuts

S_{u} : = {u = const}

are obtained from

S_{0}

by a pure time translation.

The coordinates

(u, ζ, \bar{ζ})

can be extended to an open neighbourhood of

ℑ^{+}

in the spacetime in the following way. Let

N_{u}

be the family of smooth outgoing null hypersurfaces in a neighbourhood of

ℑ^{+}

such that they intersect the null infinity just in the cuts

S_{u}

, i. e.

N_{u} \cap ℑ^{+} = S_{u}

. Then let

r

be the affine parameter in the physical metric along the null geodesic generators of

N_{u}

Then

(u, r, ζ, \bar{ζ})

forms a coordinate system. The

u = const

r = const

2-surfaces

S_{u, r}

(or simply

S_{r}

if no confusion can arise) are spacelike topological 2-spheres, which are called large spheres of radius

r

near future null infinity. Obviously, the affine parameter

r

is not unique, its origin can be changed freely:

\bar{r} : = r + g (u, ζ, \bar{ζ})

is an equally good affine parameter for any smooth

g

. Imposing certain additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bondi type coordinate system’.^$8$ In many of the large sphere calculations of the quasi-local quantities the large spheres should be assumed to be large spheres not only in a general null, but in a Bondi type coordinate system. For the detailed discussion of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see for example [277, 276, 82] .

In addition to the coordinate system we need a Newman–Penrose null tetrad, or rather a GHP spinor dyad,

ɛ_{A}^{A} = {o^{A}, ι^{A}}

A = 0, 1

, on the hypersurfaces

N_{u}

. (Thus boldface indices are referring to the GHP spin frame.) It is natural to choose

o^{A}

such that

l^{a} : = o^{A} {\bar{o}}^{A^{'}}

be the tangent

(\partial / \partial r)^{a}

of the null geodesic generators of

N_{u}

, and

o^{A}

itself be constant along

l^{a}

. Newman and Unti [277] chose

ι^{A}

to be parallel propagated along

l^{a}

. This choice yields the vanishing of a number of spin coefficients (see for example the review [276] ). The asymptotic solution of the Einstein–Maxwell equations as a series of

1 / r

in this coordinate and tetrad system is given in [277, 130, 298] , where all the non-vanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the

u

-derivative

{\dot{σ}}^{0}

of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces

N_{u}

From the point of view of the large sphere calculations of the quasi-local quantities the choice of Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin frame to the family of the large spheres of constant ‘radius’

r

, i. e. to require

m^{a} : = o^{A} {\bar{ι}}^{A^{'}}

and

{\bar{m}}^{a} = ι^{A} {\bar{o}}^{A^{'}}

to be tangents of the spheres. This can be achieved by an appropriate null rotation of the Newman–Unti basis about the spinor

o^{A}

. This rotation yields a change of the spin coefficients and the metric and curvature components. As far as the present author is aware of, this rotation with the highest accuracy was done for the solutions of the Einstein–Maxwell system by Shaw [323] .

In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [81] . Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i. e. when

{\dot{σ}}^{0} = 0

. In the language of subsection 4.1.7 , this definition would be

{lim}_{r \to \infty} r Δ^{+} λ = 0

{lim}_{r \to \infty} r Δ^{-} λ = 0

{lim}_{r \to \infty} r T^{+} λ = 0

and

{lim}_{r \to \infty} r T^{-} λ = 0

. However, as Bramson showed [81] , half of these conditions can be imposed. Namely, at future null infinity

C^{+} λ : = (Δ^{+} \oplus T^{-}) λ = 0

(and at past null infinity

C^{-} λ : = (Δ^{-} \oplus T^{+}) λ = 0

) can always be imposed asymptotically, and it has two linearly independent solutions

λ_{A}^{\underset{̲}{A}}

\underset{̲}{A} = 0, 1

, on

ℑ^{+}

(or on

ℑ^{-}

, respectively). The space

S_{\infty}^{\underset{̲}{A}}

of its solutions turns out to have a natural symplectic metric

ɛ_{\underset{̲}{A} \underset{̲}{B}}

, and we refer to

(S_{\infty}^{\underset{̲}{A}}, ɛ_{\underset{̲}{A} \underset{̲}{B}})

as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations

{lim}_{r \to \infty} r C^{\pm} λ = 0

the future/past asymptotic twistor equations. At

ℑ^{+}

asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form

K^{\underset{̲}{A} {\underset{̲}{A}}^{'}} λ_{\underset{̲}{A}}^{A} {\bar{λ}}_{{\underset{̲}{A}}^{'}}^{A^{'}} = K^{\underset{̲}{A} {\underset{̲}{A}}^{'}} λ_{\underset{̲}{A}}^{1} {\bar{λ}}_{{\underset{̲}{A}}^{'}}^{1^{'}} ι^{A} {\bar{ι}}^{A^{'}}

for some constant Hermitian matrix

K^{\underset{̲}{A} {\underset{̲}{A}}^{'}}

. Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are

- σ_{i}^{\underset{̲}{A} \underset{̲}{B}} λ_{\underset{̲}{A}}^{1} λ_{\underset{̲}{B}}^{1}

, where

σ_{i}^{\underset{̲}{A} \underset{̲}{B}}

are the standard

S U (2)

Pauli matrices (divided by

\sqrt{2}

) [347] . Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from

Δ^{+}

Δ^{-}

T^{+}

and

T^{-}

too. In the present review we use only the fact that asymptotic spinors can be introduced as anti-holomorphic spinors (see also subsection 8.2.1 ), i. e. the solutions of

ℋ^{-} λ : = (Δ^{-} \oplus T^{-}) λ = 0

(and at past null infinity as holomorphic spinors), and as special solutions of the 2-surface twistor equation

T λ : = (T^{+} \oplus T^{-}) λ = 0

(see also subsection 7.2.1 ).

These operators, together with others reproducing the asymptotic spinors, are discussed in [347] .

The Bondi–Sachs energy-momentum given in the Newman–Penrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut

S

of a combination of the leading term

ψ_{2}^{0}

of the Weyl spinor component

ψ_{2}

, the asymptotic shear

σ^{0}

and its

u

-derivative, weighted by the first four spherical harmonics (see for example [276, 299] ):

\begin{matrix} P_{B S}^{\underset{̲}{A} {\underset{̲}{B}}^{'}} = - \frac{1}{4 π G} \oint (ψ_{2}^{0} + σ^{0} {\dot{\bar{σ}}}^{0}) λ_{0}^{\underset{̲}{A}} {\bar{λ}}_{0^{'}}^{{\underset{̲}{B}}^{'}} d S, \end{matrix}

(30)

where

λ_{0}^{\underset{̲}{A}} : = λ_{A}^{\underset{̲}{A}} o^{A}

\underset{̲}{A} = 0, 1

, are the

o^{A}

-component of the vectors of a spin frame in the space of the asymptotic spinors. (For the various realizations of these spinors see for example [347] .) Similarly, the various definitions for angular momentum at null infinity could be rewritten in this formalism. Although there is no generally accepted definition for angular momentum at null infinity in general spacetimes, in stationary spacetimes there is. It is the unit sphere integral on the cut

S

of the leading term of the Weyl spinor component

{\bar{ψ}}_{1^{'}}

, weighted by appropriate (spin weighted) spherical harmonics:

\begin{matrix} J^{\underset{̲}{A} \underset{̲}{B}} = \frac{1}{8 π G} \oint {\bar{ψ}}_{1^{'}}^{0} λ_{0}^{\underset{̲}{A}} λ_{0}^{\underset{̲}{B}} d S . \end{matrix}

(31)

In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [83] .

^$8$ In the so-called Bondi coordinate system the radial coordinate is the luminosity distance $r_{D} : = - 1 / ρ$ , which tends to the affine parameter $r$ asymptotically.

4.2.5 Other special situations

In the weak field approximation of general relativity [365, 21, 369, 299, 221] the gravitational field is described by a symmetric tensor field

h_{a b}

on Minkowski spacetime

(R^{4}, g_{a b}^{0})

, and the dynamics of the field

h_{a b}

is governed by the linearized Einstein equations, i. e. essentially the wave equation.

Therefore, the tools and techniques of the Poincare-invariant field theories, in particular the Noether–Belinfante–Rosenfeld procedure outlined in subsection 2.1 and the ten Killing vectors of the background Minkowski spacetime, can be used to construct the conserved quantities. It turns out that the symmetric energy-momentum tensor of the field

h_{a b}

is essentially the second order term in the Einstein tensor of the metric

g_{a b} : = g_{a b}^{0} + h_{a b}

. Thus in the linear approximation the field

h_{a b}

does not contribute to the global energy-momentum and angular momentum of the matter+gravity system, and hence these quantities have the form ( 5 ) with the linearized energy-momentum tensor of the matter fields. However, as we will see in subsection 7.1.1 , this energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized) curvature [333, 200, 299] .

pp-waves spacetimes are defined to be those that admit a constant null vector field

L^{a}

, and they are interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present then it is necessarily pure radiation with wavevector

L^{a}

, i. e.

T^{a b} L_{b} = 0

holds [236] . A remarkable feature of the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a two dimensional linear equation for a single function. In contrast to the approach adopted almost exclusively, Aichelburg [3] considered this field equation as an equation for a boundary value problem. As we will see, from the point of view of the quasi-local observables this is a particularly useful and natural standpoint. If a pp-wave spacetime admits an additional spacelike Killing vector

K^{a}

with closed

S^{1}

orbits, i. e. it is cyclically symmetric too, then

L^{a}

and

K^{a}

are necessarily commuting and are orthogonal to each other, because otherwise an additional timelike Killing vector would also be admitted [335] .

Since the final state of stellar evolution (the neutron star or the black hole state) is expected to be described by an asymptotically flat stationary, axis-symmetric spacetime, the significance of these spacetimes is obvious. It is conjectured that this final state is described by the Kerr–Newman (either outer or black hole) solution with some well defined mass, angular momentum and electric charge parameters [369] . Thus axis-symmetric 2-surfaces in these solutions may provide domains which are general enough but for which the quasi-local quantities are still computable. According to a conjecture by Penrose [291] , the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the Kerr–Newman black hole this area is

4 π (2 m^{2} - e^{2} + 2 m \sqrt{m^{2} - e^{2} - a^{2}})

. Thus, particularly interesting 2-surfaces in these spacetimes are the spacelike cross sections of the event horizon [60] .

There is a well defined notion of total energy-momentum not only in the asymptotically flat, but even in the asymptotically anti-de-Sitter spacetimes too. This is the Abbott–Deser energy [1] , whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [157] . The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically anti-de-Sitter spacetimes and to study their general, basic properties in [26] . A comparison and analysis of the various definitions of mass for asymptotically anti-de-Sitter metrics is given in [114] . Thus it is natural to ask whether a specific quasi-local energy-momentum expression is able to reproduce the Abbott–Deser energy-momentum in this limit or not.

4.3 On lists of criteria of reasonableness of the quasi-local quantities

In the literature there are various, more or less ad hoc, ‘lists of criteria of reasonableness’ of the quasi-local quantities (see for example [127, 108] ). However, before discussing them, it seems useful to formulate first some general principles that any quasi-local quantity should satisfy.

4.3.1 General expectations

In non-gravitational physics the notions of conserved quantities are connected with symmetries of the system, and they are introduced through some systematic procedure in the Lagrangian and/or Hamiltonian formalism. In general relativity the total energy-momentum and angular momentum are 2-surface observables, thus we concentrate on them even at the quasi-local level.

These facts motivate our three a priori expectations:

1. The quasi-local quantities that are 2-surface observables should depend only on the 2-surface data, but they cannot depend e. g. on the way that the various geometric structures on $S$ are extended off the 2-surface. There seems to be no a priori reason why the 2-surface would have to be restricted to have spherical topology. Thus, in the ideal case, the general construction of the quasi-local energy-momentum and angular momentum should work for any closed orientable spacelike 2-surface.
2. It is desirable to derive the quasi-local energy-momentum and angular momentum as the charge integral (Lagrangian interpretation) and/or as the value of the Hamiltonian on the constraint surface in the phase space (Hamiltonian interpretation). If they are introduced in some other way, they should have a Lagrangian and/or Hamiltonian interpretation.
3. These quantities should correspond to the ‘quasi-symmetries’ of the 2-surface. In particular, the quasi-local energy-momentum should be expected to be in the dual of the space of the ‘quasi-translations’, and the angular momentum in the dual of the space of the ‘quasi-rotations’.

To see that these conditions are non-trivial, let us consider the expressions based on the linkage integral ( 16 ).

L_{S} [K]

does not satisfy the first part of requirement 1 . In fact, it depends on the derivative of the normal components of

K^{a}

in the direction orthogonal to

S

for any value of the parameter

α

. Thus it depends not only on the geometry of

S

and the vector field

K^{a}

given on the 2-surface, but on the way in which

K^{a}

is extended off the 2-surface. Therefore,

L_{S} [K]

is ‘less quasi-local’ than

A_{S} [ω]

H_{S} [λ, \bar{μ}]

introduced in subsections 7.2.1 and 7.2.2 , respectively.

We will see that the Hawking energy satisfies 1 , but not 2 and 3 . The Komar integral (i. e.

the linkage for

α = 0

) has the form of the charge integral of a superpotential:

8 π G K_{S} [K] : = \oint_{S} \nabla^{[a} K^{b]} \frac{1}{2} ɛ_{a b c d}

, i. e. it has a Lagrangian interpretation. The corresponding conserved Komar-current is defined by

8 π G_{K} C^{a} [K]

: = G_{b}^{a} K^{b} + \nabla_{b} \nabla^{[a} K^{b]}

. However, its flux integral on some compact spacelike hypersurface with boundary

S : = \partial Σ

cannot be a Hamiltonian on the ADM phase space in general. In fact, it is

\begin{matrix} \begin{matrix} _{K} H [K] : & = {\int_{Σ}}_{K} C^{a} [K] t_{a} d Σ = & = \int_{Σ} (c N + c_{a} N^{a}) d Σ + \frac{1}{8 π G} \oint_{S} v_{a} (χ_{b}^{a} N^{b} - D^{a} N + \frac{1}{2 N} {\dot{N}}^{a}) d S . \end{matrix} \end{matrix}

(32)

Here

c

and

c_{a}

are, respectively, the Hamiltonian and momentum constraints of the vacuum theory,

t^{a}

is the future directed unit normal to

Σ

v^{a}

is the outward directed unit normal to

S

Σ

and

N

and

N^{a}

are the lapse and shift part of

K^{a}

, respectively, defined by

K^{a} = : N t^{a} + N^{a}

Thus

_{K} H [K]

is a well defined function of the configuration and velocity variables

(N, N^{a}, h_{a b})

and

(\dot{N}, {\dot{N}}^{a}, {\dot{h}}_{a b})

, respectively. However, since the velocity

{\dot{N}}^{a}

cannot be expressed by the canonical variables [377, 45] ,

_{K} H [K]

can be written as a function on the ADM phase space only if the boundary conditions at

\partial Σ

ensure the vanishing of the integral of

v_{a} {\dot{N}}^{a} / N

4.3.2 Pragmatic criteria

Since in certain special situations there are generally accepted definitions for the energy-momentum and angular momentum, it seems reasonable to expect that in these situations the quasi-local quantities reduce to them. One half of the pragmatic criteria is just this expectation, and the other is a list of some a priori requirements on the behaviour of the quasi-local quantities.

One such list for the energy-momentum and mass, based mostly on [127, 108] and the properties of the quasi-local energy-momentum of the matter fields of subsection 2.2 , might be the following:

1.1 The quasi-local energy-momentum $P_{S}^{\underset{̲}{a}}$ must be a future pointing nonspacelike vector (assuming that the matter fields satisfy the dominant energy condition on some $Σ$ for which $S = \partial Σ$ , and maybe some form of the convexity of $S$ should be required) (‘positivity’).
1.2 $P_{S}^{\underset{̲}{a}}$ must be zero iff $D (Σ)$ is flat, and null iff $D (Σ)$ has a pp-wave geometry with pure radiation (‘rigidity’).
1.3 $P_{S}^{\underset{̲}{a}}$ must give the correct weak field limit.
1.4 $P_{S}^{\underset{̲}{a}}$ must reproduce the ADM, Bondi–Sachs and Abbott–Deser energy-momenta in the appropriate limits (‘correct large sphere behaviour’).
1.5 For small spheres P S a ̲ must give the expected results (‘correct small sphere behaviour’):
- 1. $\frac{4}{3} π r^{3} T^{a b} t_{b}$ in non-vacuum and
- 2. $k r^{5} T^{a b c d} t_{b} t_{c} t_{d}$ in vacuum for some positive constant $k$ and the Bel–Robinson tensor $T^{a b c d}$ ;
1.6 For round spheres $P_{S}^{\underset{̲}{a}}$ must yield the ‘standard’ round sphere expression.
1.7 For marginally trapped surfaces the quasi-local mass $m_{S}$ must be the irreducible mass $\sqrt{Area (S) / 16 π G^{2}}$ .

1.7 is motivated by the expectation that the quasi-local mass associated with the apparent horizon of a black hole (i. e. the outermost marginally trapped surface in a spacelike slice) be just the irreducible mass [127, 108] . Usually,

m_{S}

is expected to be monotonic in some appropriate sense [108] .

For example, if

S_{1} = \partial Σ

for some achronal (and hence spacelike or null) hypersurface

Σ

in which

S_{2}

is a spacelike closed 2-surface and the dominant energy condition is satisfied on

Σ

, then

m_{S_{1}} \geq m_{S_{2}}

seems to be a reasonable expectation [127] . (But see also the next subsection.) On the other hand, in contrast to the energy-momentum and angular momentum of the matter fields on the Minkowski spacetime, the additivity of the energy-momentum (and angular momentum) is not expected. In fact, if

S_{1}

and

S_{2}

are two connected 2-surfaces, then, for example, the corresponding quasi-local energy-momenta would belong to different vector spaces, namely to the dual of the space of the quasi-translations of the first and of the second 2-surface, respectively. Thus, even if we consider the disjoint union

S_{1} \cup S_{2}

to surround a single physical system, then we can add the energy-momentum of the first to that of the second only if there is some physically/geometrically distinguished rule defining an isomorphism between the different vector spaces of the quasi-translations.

Such an isomorphism would be provided for example by some naturally chosen globally defined flat background. However, as we discussed in subsection 3.1.1 , general relativity itself does not provide any background: The use of such a background contradicts the complete diffeomorphism invariance of the theory. Nevertheless, the quasi-local mass and the length of the quasi-local Pauli–Lubanski spin of different surfaces can be compared, because they are scalar quantities.

Similarly, any reasonable quasi-local angular momentum expression

J_{S}^{\underset{̲}{a} \underset{̲}{b}}

may be expected to satisfy the following:

2.1 $J_{S}^{\underset{̲}{a} \underset{̲}{b}}$ must give zero for round spheres.
2.2 For 2-surfaces with zero quasi-local mass the Pauli–Lubanski spin should be proportional to the (null) energy-momentum 4-vector $P_{S}^{\underset{̲}{a}}$ .
2.3 $J_{S}^{\underset{̲}{a} \underset{̲}{b}}$ must give the correct weak field limit.
2.4 $J_{S}^{\underset{̲}{a} \underset{̲}{b}}$ must reproduce the generally accepted spatial angular momentum at the spatial infinity, and in stationary spacetimes it should reduce to the ‘standard’ expression at the null infinity as well (‘correct large sphere behaviour’).
2.5 For small spheres the anti-self-dual part of $J_{S}^{\underset{̲}{a} \underset{̲}{b}}$ is expected to give $\frac{4}{3} π r^{3} T_{c d} t^{c}$ $(r ɛ^{D (A} t^{B) D^{'}})$ in non-vacuum and $C r^{5} T_{c d e f} t^{c} t^{d} t^{e} (r ɛ^{F (A} t^{B) F^{'}})$ in vacuum for some constant $C$ (‘correct small sphere behaviour’).

Since there is no generally accepted definition for the angular momentum at null infinity, we cannot expect anything definite there in non-stationary spacetimes. Similarly, there are inequivalent suggestions for the centre-of-mass at the spatial infinity (see subsections 3.2.2 and 3.2.4 ).

4.3.3 Incompatibility of certain ‘natural’ expectations

As Eardley noted in [127] , probably no quasi-local energy definition exists which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasi-local energy definition which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed Friedmann–Robertson–Walker or the

Ω_{M, m}

spacetimes show explicitly. The points where the monotonicity breaks down are the extremal (maximal or minimal) surfaces, which represent event horizon in the spacetime. Thus one may argue that since the event horizon hides a portion of spacetime, we cannot know the details of the physical state of the matter+gravity system behind the horizon. Hence, in particular, the monotonicity of the quasi-local mass may be expected to break down at the event horizon. However, although for stationary systems (or at the moment of time symmetry of a time symmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general non-stationary systems the concepts of the event and apparent horizons deviate. Thus the causal argument above does not seem possible to be formulated in the hypersurface

Σ

of subsection 4.3.2 . Actually, the root of the non-monotonicity is the fact that the quasi-local energy is a 2-surface observable in the sense of 1 in subsection 4.3.1 above. This does not mean, of course, that in certain restricted situations the monotonicity (‘local monotonicity’) could not be proven. This local monotonicity may be based, for example, on Lie dragging of the 2-surface along some special spacetime vector field.

On the other hand, in the literature sometimes the positivity and the monotonicity requirements are confused, and there is an ‘argument’ that the quasi-local gravitational energy cannot be positive definite, because the total energy of the closed universes must be zero. However, this argument is based on the implicit assumption that the quasi-local energy is associated with a compact three dimensional domain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasi-local energy-momentum is associated with 2-surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression ( 26 ) in the closed Friedmann–Robertson–Walker spacetime, or, more generally, the Dougan–Mason energy-momentum (see subsection 8.2.3 ) are such examples.

5 The Bartnik Mass and its Modifications

5.1 The Bartnik mass

5.1.1 The main idea

One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [38, 37] . His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let

Σ

be a compact, connected 3-manifold with connected boundary

S

, and let

h_{a b}

be a (negative definite) metric and

χ_{a b}

a symmetric tensor field on

Σ

such that they, as an initial data set, satisfy the dominant energy condition: If

16 π G μ : = R + χ^{2} - χ_{a b} χ^{a b}

and

8 π G j^{a} : = D_{b} (χ^{a b} - χ h^{a b})

, then

μ \geq (- j_{a} j^{a})^{1 / 2}

. For the sake of simplicity we denote the triple

(Σ, h_{a b}, χ_{a b})

Σ

. Then let us consider all the possible asymptotically flat initial data sets

(\hat{Σ}, {\hat{h}}_{a b}, {\hat{χ}}_{a b})

with a single asymptotic end, denoted simply by

\hat{Σ}

, which satisfy the dominant energy condition, have finite ADM energy and are extensions of

Σ

above through its boundary

S

. The set of these extensions will be denoted by

ℰ (Σ)

. By the positive energy theorem

\hat{Σ}

has non-negative ADM energy

E_{A D M} (\hat{Σ})

, which is zero precisely when

\hat{Σ}

is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies,

inf {E_{A D M} (\hat{Σ}) | \hat{Σ} \in ℰ (Σ)}

, where the infimum is taken on

ℰ (Σ)

. Obviously, by the non-negativity of the ADM energies this infimum exists and is non-negative, and it is tempting to define the quasi-local mass of

Σ

by this infimum.^$9$ However, it is easy to see that, without further conditions on the extensions of

(Σ, h_{a b}, χ_{a b})

, this infimum is zero. In fact,

Σ

can be extended to an asymptotically flat initial data set

\hat{Σ}

with arbitrarily small ADM energy such that

\hat{Σ}

contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and

Σ

. In particular, in the ‘

Ω_{M, m}

-spacetime’, discussed in subsection 4.2.1 on the round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily large round-sphere mass

M / G

) has an asymptotically flat extension, the

Ω_{M, m}

-spacetime itself, with arbitrarily small ADM mass

m / G

Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding

Σ

from the outside. This led Bartnik [38, 37] to formulate his suggestion for the quasi-local mass of

Σ

. He concentrated on the time-symmetric data sets (i. e. those for which the extrinsic curvature

χ_{a b}

is vanishing), when the horizon appears to be a minimal surface of topology

S^{2}

\hat{Σ}

(see for example [152] ), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature:

R \geq 0

. Thus, if

ℰ_{0} (Σ)

denotes the set of asymptotically flat Riemannian geometries

\hat{Σ} = (\hat{Σ}, {\hat{h}}_{a b})

with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

\begin{matrix} m_{B} (Σ) : = inf {E_{A D M} (\hat{Σ}) | \hat{Σ} \in ℰ_{0} (Σ)} . \end{matrix}

(33)

The ‘no horizon’ condition on

\hat{Σ}

implies that topologically

Σ

is a 3-ball. Furthermore, the definition of

ℰ_{0} (Σ)

in its present form does not allow one to associate the Bartnik mass to those 3-geometries

(Σ, h_{a b})

that contain minimal surfaces inside

Σ

. Although formally the maximal 2-surfaces inside

Σ

are not excluded, any asymptotically flat extension of such a

Σ

would contain a minimal surface. In particular, the spherically symmetric 3-geometry with line element

d l^{2} = - d r^{2} - {sin}^{2} r (d θ^{2} + {sin}^{2} θ d φ^{2})

with

(θ, φ) \in S^{2}

and

r \in [0, r_{0}]

π / 2 < r_{0} < π

, has a maximal 2-surface at

r = π / 2

, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than

4 π {sin}^{2} r_{0}

. Thus the Bartnik mass (according to the original definition given in [38, 37] ) cannot be associated with every compact time symmetric data set

(Σ, h_{a b})

even if

Σ

is topologically trivial. Since for

0 < r_{0} < π / 2

this data set can be extended without any difficulty, this example shows that

m_{B}

is associated with the 3-dimensional data set

Σ

and not only to the 2-dimensional boundary

\partial Σ

Of course, to rule out this limitation, one can modify the original definition by considering the set

{\tilde{ℰ}}_{0} (S)

of asymptotically flat Riemannian geometries

\hat{Σ} = (\hat{Σ}, {\hat{h}}_{a b})

(with non-negative scalar curvature, finite ADM energy and with no stable minimal surface) which contain

(S, q_{a b})

as an isometrically embedded Riemannian submanifold, and define

{\tilde{m}}_{B} (S)

by ( 33 ) with

{\tilde{ℰ}}_{0} (S)

instead of

ℰ_{0} (Σ)

. Obviously, this

{\tilde{m}}_{B} (S)

could be associated with a larger class of 2-surfaces than the original

m_{B} (Σ)

to compact 3-manifolds, and

0 \leq {\tilde{m}}_{B} (\partial Σ) \leq m_{B} (Σ)

holds.

In [202, 40] the set

ℰ_{0} (Σ)

was allowed to include extensions

\hat{Σ}

Σ

having boundaries as compact outermost horizons, whenever the corresponding ADM energies are still non-negative [155] , and hence

m_{B} (Σ)

is still well defined and non-negative. (For another definition for

ℰ_{0} (Σ)

allowing horizons in the extensions but excluding them between

Σ

and the asymptotic end, see [84] and subsection 5.2 below.) Bartnik suggested a definition for the quasi-local mass of a spacelike 2-surface

S

(together with its induced metric and the two extrinsic curvatures), too [38] . He considered those globally hyperbolic spacetimes

\hat{M} : = (\hat{M}, {\hat{g}}_{a b})

that satisfy the dominant energy condition, admit an asymptotically flat (metrically complete) Cauchy surface

\hat{Σ}

with finite ADM energy, have no event horizon and in which

S

can be embedded with its first and second fundamental forms. Let

ℰ_{0} (S)

denote the set of these spacetimes. Since the ADM energy

E_{A D M} (\hat{M})

is non-negative for any

\hat{M} \in ℰ_{0} (S)

(and is zero precisely for flat

\hat{M}

), the infimum

\begin{matrix} m_{B} (S) : = inf {E_{A D M} (\hat{M}) | \hat{M} \in ℰ_{0} (S)} \end{matrix}

(34)

exists and is non-negative. Although it seems plausible that

m_{B} (\partial Σ)

is only the ‘spacetime version’ of

m_{B} (Σ)

, without the precise form of the no-horizon conditions in

ℰ_{0} (Σ)

and that in

ℰ_{0} (S)

they cannot be compared even if the extrinsic curvature were allowed in the extensions

\hat{Σ}

Σ

^$9$ Since we take the infimum, we could equally take the ADM masses, which are the minimum values of the zero-th component of the energy-momentum four-vectors in the different Lorentz frames, instead of the energies.

5.1.2 The main properties of $m_{B} (Σ)$

The first immediate consequence of ( 33 ) is the monotonicity of the Bartnik mass: If

Σ_{1} \subset Σ_{2}

, then

ℰ_{0} (Σ_{2}) \subset ℰ_{0} (Σ_{1})

, and hence

m_{B} (Σ_{1}) \leq m_{B} (Σ_{2})

. Obviously, by definition ( 33 ) one has

m_{B} (Σ) \leq m_{A D M} (\hat{Σ})

for any

\hat{Σ} \in ℰ_{0} (Σ)

. Thus if

m

is any quasi-local mass functional which is larger than

m_{B}

(i. e. which assigns a non-negative real to any

Σ

such that

m (Σ) \geq m_{B} (Σ)

for any allowed

Σ

), furthermore if

m (Σ) \leq m_{A D M} (\hat{Σ})

for any

\hat{Σ} \in ℰ_{0} (Σ)

, then by the definition of the infimum in ( 33 ) one has

m_{B} (Σ) \geq m (Σ) - ɛ \geq m_{B} (Σ) - ɛ

for any

ɛ > 0

. Therefore,

m_{B}

is the largest mass functional satisfying

m_{B} (Σ) \leq m_{A D M} (\hat{Σ})

for any

\hat{Σ} \in ℰ_{0} (Σ)

. Another interesting consequence of the definition of

m_{B}

, due to W. Simon, is that if

\hat{Σ}

is any asymptotically flat, time symmetric extension of

Σ

with non-negative scalar curvature satisfying

m_{A D M} (\hat{Σ}) < m_{B} (Σ)

, then there is a black hole in

\hat{Σ}

in the form of a minimal surface between

Σ

and the infinity of

\hat{Σ}

(see for example [40] ).

As we saw, the Bartnik mass is non-negative, and, obviously, if

Σ

is flat (and hence is a data set for the flat spacetime), then

m_{B} (Σ) = 0

. The converse of this statement is also true [202] : If

m_{B} (Σ) = 0

, then

Σ

is locally flat. The Bartnik mass tends to the ADM mass [202] : If

(\hat{Σ}, {\hat{h}}_{a b})

is an asymptotically flat Riemannian 3-geometry with non-negative scalar curvature and finite ADM mass

m_{A D M} (\hat{Σ})

, and if

{Σ_{n}}

n \in N

, is a sequence of solid balls of coordinate radius

n

\hat{Σ}

, then

{lim}_{n \to \infty} m_{B} (Σ_{n}) = m_{A D M} (\hat{Σ})

. The proof of these two results is based on the use of the Hawking energy (see subsection 6.1 ), by means of which a positive lower bound for

m_{B} (Σ)

can be given near the non-flat points of

Σ

. In the proof of the second statement one must use the fact that the Hawking energy tends to the ADM energy, which, in the time symmetric case, is just the ADM mass.

The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice application of the Riemannian Penrose inequality [202] : Let

Σ

be a spherically symmetric Riemannian 3-geometry with spherically symmetric boundary

S : = \partial Σ

. One can form its ‘standard’ round-sphere energy

E (S)

(see subsection 4.2.1 ), and take its spherically symmetric asymptotically flat vacuum extension

{\hat{Σ}}_{S S}

(see [38, 40] ). By the Birkhoff theorem the exterior part of

{\hat{Σ}}_{S S}

is a part of a

t = const

hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just

E (S)

. Then any asymptotically flat extension

\hat{Σ}

Σ

can also be considered as (a part of) an asymptotically flat time symmetric hypersurface with minimal surface, whose area is

16 π G^{2}

E_{A D M} ({\hat{Σ}}_{S S})

. Thus by the Riemannian Penrose inequality [202]

E_{A D M} (\hat{Σ})

\geq E_{A D M} ({\hat{Σ}}_{S S}) = E (S)

. Therefore, the Bartnik mass of

Σ

is just the ‘standard’ round sphere expression

E (S)

5.1.3 The computability of the Bartnik mass

Since for any given

Σ

the set

ℰ_{0} (Σ)

of its extensions is a huge set, it is almost hopeless to parameterize it. Thus, by the very definition, it seems very difficult to compute the Bartnik mass for a given, specific

(Σ, h_{a b})

. Without some computational method the potentially useful properties of

m_{B} (Σ)

would be lost from the working relativist’s arsenal.

Such a computational method might be based on a conjecture of Bartnik [38, 40] : The infimum in definition ( 33 ) of the mass

m_{B} (Σ)

is realized by an extension

(\hat{Σ}, {\hat{h}}_{a b})

(Σ, h_{a b})

such that the exterior region,

(\hat{Σ} - Σ, {\hat{h}}_{a b} |_{\hat{Σ} - Σ})

, is static, the metric is Lipschitz-continuous across the 2-surface

\partial Σ \subset \hat{Σ}

, and the mean curvatures of

\partial Σ

of the two sides are equal. Therefore, to compute

m_{B}

for a given

(Σ, h_{a b})

, one should find an asymptotically flat, static vacuum metric

{\hat{h}}_{a b}

satisfying the matching conditions on

\partial Σ

, and the Bartnik mass is the ADM mass of

{\hat{h}}_{a b}

. As Corvino showed [115] , if there is an allowed extension

\hat{Σ}

Σ

for which

m_{A D M} (\hat{Σ}) = m_{B} (Σ)

, then the extension

\hat{Σ} - \bar{Σ}

is static; furthermore, if

Σ_{1} \subset Σ_{2}

m_{B} (Σ_{1}) = m_{B} (Σ_{2})

and

Σ_{2}

has an allowed extension

\hat{Σ}

for which

m_{B} (Σ_{2}) = m_{A D M} (\hat{Σ})

, then

Σ_{2} - \bar{Σ_{1}}

is static. Thus the proof of Bartnik’s conjecture is equivalent to the proof of the existence of such an allowed extension. The existence of such an extension is proven in [257] for geometries

(Σ, h_{a b})

close enough to the Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still lacking. Bartnik’s conjecture is that

(Σ, h_{a b})

determines this exterior metric uniquely [40] . He conjectures [38, 40] that a similar computation method can be found for the mass

m_{B} (S)

, defined in ( 34 ), too, where the exterior metric should be stationary. This second conjecture is also supported by partial results [116] : If

(Σ, h_{a b}, χ_{a b})

is any compact vacuum data set, then it has an asymptotically flat vacuum extension which is a spacelike slice of a Kerr spacetime outside a large sphere near spatial infinity.

To estimate

m_{B} (Σ)

one can construct admissible extensions of

(Σ, h_{a b})

in the form of the metrics in quasi-spherical form [39] . If the boundary

\partial Σ

is a metric sphere of radius

r

with non-negative mean curvature

k

, then

m_{B} (Σ)

can be estimated from above in terms of

r

and

k

5.2 Bray’s modifications

Another, slightly modified definition for the quasi-local mass was suggested by Bray [84, 87] . Here we summarize his ideas.

Let

Σ = (Σ, h_{a b}, χ_{a b})

be any asymptotically flat initial data set with finitely many asymptotic ends and finite ADM masses, and suppose that the dominant energy condition is satisfied on

Σ

. Let

S

be any fixed 2-surface in

Σ

which encloses all the asymptotic ends except one, say the

i

-th (i. e. let

S

be homologous to a large sphere in the

i

-th asymptotic end). The outside region with respect to

S

, denoted by

O (S)

, will be the subset of

Σ

containing the

i

-th asymptotic end and bounded by

S

, while the inside region,

I (S)

, is the (closure of)

Σ - O (S)

. Next Bray defines the ‘extension’

{\hat{Σ}}_{e}

S

by replacing

O (S)

by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Similarly, the ‘fill-in’

{\hat{Σ}}_{f}

S

is obtained from

Σ

by replacing

I (S)

by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Finally, the surface

S

will be called outer-minimizing if for any closed 2-surface

\tilde{S}

enclosing

S

one has

Area (S) \leq Area (\tilde{S})

Let

S

be outer-minimizing, and let

ℰ (S)

denote the set of extensions of

S

in which

S

is still outer-minimizing, and

ℱ (S)

denote the set of fill-ins of

S

. If

{\hat{Σ}}_{f} \in ℱ (S)

and

A_{{\hat{Σ}}_{f}}

denotes the infimum of the area of the 2-surfaces enclosing all the ends of

{\hat{Σ}}_{f}

except the outer one, then Bray defines the outer and inner mass,

m_{out} (S)

and

m_{in} (S)

, respectively, by

\begin{matrix} m_{out} (S) & : = & {m_{A D M} ({\hat{Σ}}_{e}) | {\hat{Σ}}_{e} \in ℰ (S)}, \end{matrix}

\begin{matrix} m_{in} (S) & : = & {A_{{\hat{Σ}}_{f}} | {\hat{Σ}}_{f} \in ℱ (S)} . \end{matrix}

m_{out} (S)

deviates slightly from Bartnik’s mass ( 33 ) even if the latter would be defined for non-time-symmetric data sets, because Bartnik’s ‘no-horizon condition’ excludes apparent horizons from the extensions, while Bray’s condition is that

S

be outer-minimizing.

A simple consequence of the definitions is the monotonicity of these masses: If

S_{2}

and

S_{1}

are outer-minimizing 2-surfaces such that

S_{2}

encloses

S_{1}

, then

m_{in} (S_{2}) \geq m_{in} (S_{1})

and

m_{out} (S_{2}) \geq m_{out} (S_{1})

. Furthermore, if the Penrose inequality holds (for example in a time-symmetric data set, for which the inequality has been proved), then for outer-minimizing surfaces

m_{out} (S) \geq m_{in} (S)

[84, 87] . Furthermore, if

Σ_{i}

is a sequence such that the boundaries

\partial Σ_{i}

shrink to a minimal surface

S

, then the sequence

m_{out} (\partial Σ_{i})

tends to the irreducible mass

\sqrt{Area (S) / 16 π G^{2}}

[40] . Bray defines the quasi-local mass of a surface not simply to be a number, but the whole closed interval

[m_{in} (S), m_{out} (S)]

. If

S

encloses the horizon in the Schwarzschild data set, then the inner and outer masses coincide; and Bray expects that the converse is also true: If

m_{in} (S) = m_{out} (S)

then

S

can be embedded into the Schwarzschild spacetime with the given 2-surface data on

S

[87] .

6 The Hawking Energy and its Modifications

6.1 The Hawking energy

6.1.1 The definition

Studying the perturbation of the dust-filled

k = - 1

Friedmann–Robertson–Walker spacetimes, Hawking found that

\begin{matrix} \begin{matrix} E_{H} (S) : & = \sqrt{\frac{Area (S)}{16 π G^{2}}} (1 + \frac{1}{2 π} \oint_{S} ρ ρ^{'} d S) = & = \sqrt{\frac{Area (S)}{16 π G^{2}}} \frac{1}{4 π} \oint_{S} (σ σ^{'} + \bar{σ} {\bar{σ}}^{'} - ψ_{2} - {\bar{ψ}}_{2^{'}} + 2 φ_{11} + 2 Λ) d S \end{matrix} \end{matrix}

(35)

behaves as an appropriate notion of energy surrounded by the spacelike topological 2-sphere

S

[166] . Here we used the Gauss–Bonnet theorem and the GHP form of ( 22 )-( 23 ) for

F

to express

ρ ρ^{'}

by the curvature components and the shears. Thus the Hawking energy is genuinely quasi-local.

The Hawking energy has the following clear physical interpretation even in a general spacetime, and, in fact,

E_{H}

can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a spacelike 2-sphere

S

should be the measure of bending of the ingoing and outgoing light rays orthogonal to

S

, and recalling that under a boost gauge transformation

l^{a} \mapsto α l^{a}

n^{a} \mapsto α^{- 1} n^{a}

the convergences

ρ

and

ρ^{'}

transform as

ρ \mapsto α ρ

and

ρ^{'} \mapsto α^{- 1} ρ^{'}

, respectively, the energy must have the form

C + D \oint_{S} ρ ρ^{'} d S

, where the unspecified parameters

C

and

D

can be determined in some special situations. For metric 2-spheres of radius

r

in the Minkowski spacetime, for which

ρ = - 1 / r

and

ρ^{'} = 1 / 2 r

, we expect zero energy, thus

D = C / 2 π

. For the event horizon of a Schwarzschild black hole with mass parameter

m

, for which

ρ = 0 = ρ^{'}

, we expect

m / G

, which can be expressed by the area of

S

. Thus

C^{2} = Area (S) / 16 π G^{2}

, and hence we arrive at ( 35 ).

6.1.2 The Hawking energy for spheres

Obviously, for round spheres

E_{H}

reduces to the standard expression ( 26 ). This implies, in particular, that the Hawking energy is not monotonic in general. Since for a Killing horizon (e. g. for a stationary event horizon)

ρ = 0

, the Hawking energy of its spacelike spherical cross sections

S

\sqrt{Area (S) / 16 π G^{2}}

. In particular, for the event horizon of a Kerr–Newman black hole it is just the familiar irreducible mass

\sqrt{2 m^{2} - e^{2} + 2 m \sqrt{m^{2} - e^{2} - a^{2}}} / 2 G

For a small sphere of radius

r

with centre

p \in M

in non-vacuum spacetimes it is

\frac{4 π}{3} r^{3} T_{a b} t^{a} t^{b}

, while in vacuum it is

\frac{2}{45 G} r^{5} T_{a b c d} t^{a} t^{b} t^{c} t^{d}

, where

T_{a b}

is the energy-momentum tensor and

T_{a b c d}

is the Bel–Robinson tensor at

p

[198] . The first result shows that in the lowest order the gravitational ‘field’ does not have a contribution to the Hawking energy, that is due exclusively to the matter fields. Thus in vacuum the leading order of

E_{H}

must be higher than

r^{3}

. Then even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the

r^{k}

order term in the power series expansion of

E_{H}

(k - 1)

. However, there are no tensorial quantities built from the metric and its derivatives such that the total number of the derivatives involved would be three. Therefore, in vacuum, the leading term is necessarily of order

r^{5}

, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres

E_{H}

is positive definite both in non-vacuum (provided the matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in particular, that

E_{H}

should be interpreted as energy rather than as mass: For small spheres in a pp-wave spacetime

E_{H}

is positive, while, as we saw this for the matter fields in subsection 2.2.3 , a mass expression could be expected to be zero. (We will see in subsections 8.2.3 and 13.5 that, for the Dougan–Mason energy-momentum, the vanishing of the mass characterizes the pp-wave metrics completely.) Using the second expression in ( 35 ) it is easy to see that at future null infinity

E_{H}

tends to the Bondi–Sachs energy. A detailed discussion of the asymptotic properties of

E_{H}

near null infinity, both for radiative and stationary spacetimes is given in [323, 325] . Similarly, calculating

E_{H}

for large spheres near spatial infinity in an asymptotically flat spacelike hypersurface, one can show that it tends to the ADM energy.

6.1.3 Positivity and monotonicity properties

In general the Hawking energy may be negative, even in the Minkowski spacetime. Geometrically this should be clear, since for an appropriately general (e. g. concave) 2-surface

S

the integral

\oint_{S} ρ ρ^{'} d S

could be less than

- 2 π

. Indeed, in flat spacetime

E_{H}

is proportional to

\oint_{S} (σ σ^{'} + \bar{σ} {\bar{σ}}^{'}) d S

by the Gauss equation. For topologically spherical 2-surfaces in the

t = const

spacelike hyperplane of Minkowski spacetime

σ σ^{'}

is real and non-positive, and it is zero precisely for metric spheres; while for 2-surfaces in the

r = const

timelike cylinder

σ σ^{'}

is real and non-negative, and it is zero precisely for metric spheres.^$10$ If, however,

S

is ‘round enough’ (not to be confused with the round spheres in subsection 4.2.1 , which is some form of a convexity condition, then

E_{H}

behaves nicely [108] :

S

will be called round enough if it is a submanifold of a spacelike hypersurface

Σ

, and if among the two dimensional surfaces in

Σ

which enclose the same volume as

S

does,

S

has the smallest area. Then it is proven by Christodoulou and Yau [108] that if

S

is round enough in a maximal spacelike slice

Σ

on which the energy density of the matter fields is non-negative (for example if the dominant energy condition is satisfied), then the Hawking energy is non-negative.

Although the Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of 2-surfaces. Hawking considered one-parameter families of spacelike 2-surfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of

E_{H}

[166] . These calculations were refined by Eardley [127] . Starting with a weakly future convex 2-surface

S

and using the boost gauge freedom, he introduced a special family

S_{r}

of spacelike 2-surfaces in the outgoing null hypersurface

N

, where

r

will be the luminosity distance along the outgoing null generators. He showed that

E_{H} (S_{r})

is non-decreasing with

r

, provided the dominant energy condition holds on

N

. Similarly, for weakly past convex

S

and the analogous family of surfaces in the ingoing null hypersurface

E_{H} (S_{r})

is non-increasing. Eardley also considered a special spacelike hypersurface, filled by a family of 2-surfaces, for which

E_{H} (S_{r})

is non-decreasing.

By relaxing the normalization condition

l_{a} n^{a} = 1

for the two null normals to

l_{a} n^{a} = exp (f)

for some

f : S \to R

, Hayward obtained a flexible enough formalism to introduce a double-null foliation (see subsection 11.2 below) of a whole neighbourhood of a mean convex 2-surface by special mean convex 2-surfaces [177] . (For the more general GHP formalism in which

l_{a} n^{a}

is not fixed, see [298] .) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these 2-surfaces is non-decreasing in the outgoing, and non-increasing in the ingoing direction.

In contrast to the special foliations of the null hypersurfaces above, Frauendiener defined a special spacelike vector field, the inverse mean curvature vector in the spacetime [141] . If

S

is a weakly future and past convex 2-surface, then

q^{a} : = 2 Q^{a} / (Q_{b} Q^{b}) = - (1 / 2 ρ) l^{a} - (1 / 2 ρ^{'}) n^{a}

is an outward directed spacelike normal to

S

. Here

Q_{b}

is the trace of the extrinsic curvature tensor:

Q_{b} : = Q_{a b}^{a}

(see subsection 4.1.2 ). Starting with a single weakly future and past convex 2-surface, Frauendiener gives an argument for the construction of a one-parameter family

S_{t}

of 2-surfaces being Lie-dragged along its own inverse mean curvature vector

q^{a}

. Hence this family of surfaces would be analogous to the solution of the geodesic equation, where the initial point and direction in that point specify the whole solution, at least locally. Assuming that such a family of surfaces (and hence the vector field

q^{a}

on the 3-submanifold swept by

S_{t}

) exists, Frauendiener showed that the Hawking energy is non-decreasing along the vector field

q^{a}

if the dominant energy condition is satisfied. However, no investigation has been made to prove the existence of such a family of surfaces. Motivated by this result, Malec, Mars and Simon [251] considered spacelike hypersurfaces with an inverse mean curvature flow of Geroch thereon (see subsection 6.2.2 ). They showed that if the dominant energy condition and certain additional (essentially technical) assumptions hold, then the Hawking energy is monotonic. These two results are the natural adaptations for the Hawking energy of the corresponding results known for some time for the Geroch energy, aiming to prove the Penrose inequality. We return to this latter issue in subsection 13.2 only for a very brief summary.

^$10$ I thank Paul Tod for pointing out this to me.

6.1.4 Two generalizations

Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of the Bondi–Sachs energy-momentum are related to the Bondi energy:

\begin{matrix} P_{H}^{\underset{̲}{a}} (S) = \sqrt{\frac{Area (S)}{16 π G^{2}}} \frac{1}{4 π} \oint_{S} (σ σ^{'} + \bar{σ} {\bar{σ}}^{'} - ψ_{2} - {\bar{ψ}}_{2^{'}} + 2 φ_{11} + 2 Λ) W^{\underset{̲}{a}} d S, \end{matrix}

(36)

where

W^{\underset{̲}{a}}

\underset{̲}{a} = 0, . . ., 3

, are essentially the first four spherical harmonics:

\begin{matrix} W^{0} = 1, W^{1} = \frac{ζ + \bar{ζ}}{1 + ζ \bar{ζ}}, W^{2} = \frac{1}{i} \frac{ζ - \bar{ζ}}{1 + ζ \bar{ζ}}, W^{3} = \frac{1 - ζ \bar{ζ}}{1 + ζ \bar{ζ}}, \end{matrix}

(37)

where

ζ

and

\bar{ζ}

are the standard complex stereographic coordinates on

S \approx S^{2}

Hawking considered the extension of the definition of

E_{H} (S)

to higher genus 2-surfaces also by the second expression in ( 35 ). Then in the expression analogous to the first one in ( 35 ) the genus of

S

appears.

6.2 The Geroch energy

6.2.1 The definition

Suppose that the 2-surface

S

for which

E_{H}

is defined is embedded in the spacelike hypersurface

Σ

. Let

χ_{a b}

be the extrinsic curvature of

Σ

M

and

k_{a b}

the extrinsic curvature of

S

Σ

. (In subsection 4.1.2 we denoted the latter by

ν_{a b}

.) Then

8 ρ ρ^{'} = (χ_{a b} q^{a b})^{2} - (k_{a b} q^{a b})^{2}

, by means of which

\begin{matrix} \begin{matrix} E_{H} (S) & = \sqrt{\frac{Area (S)}{16 π G^{2}}} (1 - \frac{1}{16 π} \oint_{S} {(k_{a b} q^{a b})}^{2} d S + \frac{1}{16 π} \oint_{S} {(χ_{a b} q^{a b})}^{2} d S) \geq & \geq \sqrt{\frac{Area (S)}{16 π G^{2}}} (1 - \frac{1}{16 π} \oint_{S} {(k_{a b} q^{a b})}^{2} d S) = & = \frac{1}{16 π} \sqrt{\frac{Area (S)}{16 π G^{2}}} \oint_{S} (2^{S} R - {(k_{a b} q^{a b})}^{2}) d S = : E_{G} (S) . \end{matrix} \end{matrix}

(38)

In the last step we used the Gauss–Bonnet theorem for

S \approx S^{2}

E_{G} (S)

is known as the Geroch energy [146] . Thus it is not greater than the Hawking energy, and, in contrast to

E_{H}

, it depends not only on the 2-surface

S

, but the hypersurface

Σ

as well.

The calculation of the small sphere limit of the Geroch energy was saved by observing [198] that, by ( 38 ), the difference of the Hawking and the Geroch energies is

\sqrt{Area (S)} \oint_{S} (χ_{a b} q^{a b})^{2} d S

. Since, however,

χ_{a b} q^{a b}

– for the family of small spheres

S_{r}

– does not tend to zero in the

r \to 0

limit, in general this difference is

O (r^{3})

. It is zero if

Σ

is spanned by spacelike geodesics orthogonal to

t^{a}

p

. Thus, for general

Σ

, the Geroch energy does not give the expected

\frac{4 π}{3} r^{3} T_{a b} t^{a} t^{b}

result.

Similarly, in vacuum the Geroch energy deviates from the Bel–Robinson energy in

r^{5}

order even if

Σ

is geodesic at

p

Since

E_{H} (S) \geq E_{G} (S)

and since the Hawking energy tends to the ADM energy, the large sphere limit of

E_{G} (S)

in an asymptotically flat

Σ

cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [146] .

6.2.2 Monotonicity properties

The Geroch energy has interesting positivity and monotonicity properties along a special flow in

Σ

[146, 213] . This flow is the so-called inverse mean curvature flow defined as follows. Let

t : Σ \to R

be a smooth function such that

1. its level surfaces, $S_{t} : = {q \in Σ | t (q) = t}$ , are homeomorphic to $S^{2}$ ,
2. there is a point $p \in Σ$ such that the surfaces $S_{t}$ are shrinking to $p$ in the limit $t \to - \infty$ , and
3. they form a foliation of $Σ - {p}$ .

Let

n

be the lapse function of this foliation, i. e. if

v^{a}

is the outward directed unit normal to

S_{t}

Σ

, then

n v^{a} D_{a} t = 1

. Denoting the integral on the right hand side in ( 38 ) by

W_{t}

, we can calculate its derivative with respect to

t

. In general this derivative does not seem to have any remarkable property. If, however, the foliation is chosen in a special way, namely if the lapse is just the inverse mean curvature of the foliation,

n = 1 / k

where

k : = k_{a b} q^{a b}

, furthermore

Σ

is maximal (i. e.

χ = 0

) and the energy density of the matter is non-negative, then, as shown by Geroch [146] ,

W_{t} \geq 0

holds.

Jang and Wald [213] modified the foliation slightly such that

t \in [0, \infty)

, and the surface

S_{0}

was assumed to be future marginally trapped (i. e.

ρ = 0

and

ρ^{'} \geq 0

). Then they showed that, under the conditions above,

\sqrt{Area (S_{0})} W_{0} \leq \sqrt{Area (S_{t})} W_{t}

. Since

E_{G} (S_{t})

tends to the ADM energy as

t \to \infty

, these considerations were intended to argue that the ADM energy should be non-negative (at least for maximal

Σ

) and not less than

\sqrt{Area (S_{0}) / 16 π G^{2}}

(at least for time symmetric

Σ

), respectively. Later Jang [211] showed that if a certain quasi-linear elliptic differential equation for a function

w

on a hypersurface

Σ

admits a solution (with given asymptotic behaviour), then

w

defines a mapping between the data set

(Σ, h_{a b}, χ_{a b})

Σ

and a maximal data set

(Σ, {\bar{h}}_{a b}, {\bar{χ}}_{a b})

(i. e.

for which

{\bar{χ}}_{a b} {\bar{h}}^{a b} = 0

) such that the corresponding ADM energies coincide. Then Jang shows that a slightly modified version of the Geroch energy is monotonic (and tends to the ADM energy) with respect to a new, modified version of the inverse mean curvature foliation of

(Σ, {\bar{h}}_{a b})

The existence and the properties of the original inverse mean curvature foliation of

(Σ, h_{a b})

above were proven and clarified by Huisken and Ilmanen [201, 202] , giving the first complete proof of the Riemannian Penrose inequality; and, as proved by Schoen and Yau [313] , Jang’s quasi-linear elliptic equation admits a global solution.

6.3 The Hayward energy

We saw that

E_{H}

can be non-zero even in the Minkowski spacetime. This may motivate considering the following expression

\begin{matrix} I (S) : & = & \sqrt{\frac{Area (S)}{16 π G^{2}}} (1 + \frac{1}{4 π} \oint_{S} (2 ρ ρ^{'} - σ σ^{'} - \bar{σ} {\bar{σ}}^{'}) d S) = \end{matrix}

\begin{matrix} = & \sqrt{\frac{Area (S)}{16 π G^{2}}} \frac{1}{4 π} \oint_{S} (- ψ_{2} - {\bar{ψ}}_{2^{'}} + 2 φ_{11} + 2 Λ) d S . \end{matrix}

\begin{matrix}  \end{matrix}

(Thus the integrand is

\frac{1}{4} (F + \bar{F})

, where

F

is given by ( 23 ).) By the Gauss equation this is zero in flat spacetime, furthermore, it is not difficult to see that its limit at the spatial infinity is still the ADM energy. However, using the second expression of

I (S)

, one can see that its limit at the future null infinity is the Newman–Unti rather than the Bondi–Sachs energy.

In the literature there is another modification of the Hawking energy, due to Hayward [178] .

His suggestion is essentially

I (S)

with the only difference that the integrands above contain an additional term, namely the square of the anholonomicity

- ω_{a} ω^{a}

(see subsections 4.1.8 and 11.2.1 ). However, we saw that

ω_{a}

is a boost gauge dependent quantity, thus the physical significance of this suggestion is questionable unless a natural boost gauge choice, e. g. in the form of a preferred foliation, is made. (Such a boost gauge might be that given by the main extrinsic curvature vector

Q_{a}

and

{\tilde{Q}}_{a}

discussed in subsection 4.1.2 .) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [61, 63] seems to be gauge invariant, this is due only to an implicit gauge choice. The correct, general GHP form of the extra term is

- ω_{a} ω^{a} = 2 (β - {\bar{β}}^{'}) (\bar{β} - β^{'})

. If, however, the GHP spinor dyad is fixed as in the large sphere or in the small sphere calculations, then

β - {\bar{β}}^{'} = τ = - {\bar{τ}}^{'}

, and hence the extra term is, in fact,

2 τ \bar{τ}

Taking into account that

τ = O (r^{- 2})

near the future null infinity (see for example [323] ), it is immediate from the remark on the asymptotic behaviour of

I (S)

above that the Hayward energy tends to the Newman–Unti instead of the Bondi–Sachs energy at the future null infinity. The Hayward energy has been calculated for small spheres both in non-vacuum and vacuum [61] . In non-vacuum it gives the expected value

\frac{4 π}{3} r^{3} T_{a b} t^{a} t^{b}

. However, in vacuum it is

- \frac{8}{45 G} r^{5} T_{a b c d} t^{a} t^{b} t^{c} t^{d}

, which is negative.

7 Penrose’s Quasi-Local Energy-Momentum and Angular Momentum

The construction of Penrose is based on twistor-theoretical ideas, and motivated by the linearized gravity integrals for energy-momentum and angular momentum. Since, however, twistor-theoretical ideas and basic notions are still considered to be some ‘special knowledge’, the review of the basic idea behind the Penrose construction is slightly more detailed than that of the others. The basic references of the field are the volumes [298, 299] by Penrose and Rindler on ‘Spinors and Spacetime’, especially volume 2, the very well readable book by Hugget and Tod [200] and the comprehensive review article [360] by Tod.

7.1 Motivations

7.1.1 How do the twistors emerge?

In the Newtonian theory of gravity the mass contained in some finite 3-volume

Σ

can be expressed as the flux integral of the gravitational field strength on the boundary

S : = \partial Σ

\begin{matrix} m_{Σ} = \frac{1}{4 π G} \oint_{S} v^{a} (D_{a} φ) d S, \end{matrix}

(39)

where

φ

is the gravitational potential and

v^{a}

is the outward directed unit normal to

S

. If

S

is deformed in

Σ

through a source-free region, then the mass does not change. Thus the mass

m_{Σ}

is analogous to charge in electrostatics.

In the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the gravitational field, i. e. the linearized energy-momentum tensor, is still analogous to charge. In fact, the total energy-momentum and angular momentum of the source can be expressed as appropriate 2-surface integrals of the curvature at infinity [333] . Thus it is natural to expect that the energy-momentum and angular momentum of the source in a finite 3-volume

Σ

, given by ( 5 ), can also be expressed as the charge integral of the curvature on the 2-surface

S

However, the curvature tensor can be integrated on

S

only if at least one pair of its indices is annihilated by some tensor via contraction, i. e. according to ( 15 ) if some

ω^{a b} = ω^{[a b]}

is chosen and

μ^{a b} = ɛ^{a b}

. To simplify the subsequent analysis

ω^{a b}

will be chosen to be anti-self-dual:

ω^{a b} = ɛ^{A^{'} B^{'}} ω^{A B}

with

ω^{A B} = ω^{(A B)}

.^$11$ Thus our claim is to find an appropriate spinor field

ω^{A B}

S

such that

\begin{matrix} Q_{S} [K] : = \int_{Σ} K_{a} T^{a b} \frac{1}{3!} ɛ_{b c d e} = \frac{1}{8 π G} \oint_{S} ω^{A B} R_{A B c d} = : A_{S} [ω] . \end{matrix}

(40)

Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual of the

8 π G

times the integrand on the left, respectively, is

\begin{matrix} ɛ^{e c d f} \nabla_{e} (ω^{A B} R_{A B c d}) & = & - 2 i ψ_{A B C}^{F} \nabla^{F^{'} (A} ω^{B C)} + 2 {φ_{A B E^{'}}}^{F^{'}} i \nabla^{E^{'} F} ω^{A B} + 4 Λ i \nabla_{A}^{F^{'}} ω^{F A}, \end{matrix}

(41)

\begin{matrix} - 8 π G K_{a} T^{a f} & = & 2 φ^{F A F^{'} A^{'}} K_{A A^{'}} + 6 Λ K^{F F^{'}} . \end{matrix}

(42)

Equations ( 41 ) and ( 42 ) are equal if

ω^{A B}

satisfies

\begin{matrix} \nabla^{A^{'} A} ω^{B C} = - i ɛ^{A (B} K^{C) A^{'}} . \end{matrix}

(43)

This equation in its symmetrized form,

\nabla^{A^{'} (A} ω^{B C)} = 0

, is the valence 2 twistor equation, a specific example for the general twistor equation

\nabla^{A^{'} (A} ω^{B C . . . E)} = 0

for

ω^{B C . . . E} = ω^{(B C . . . E)}

. Thus, as could be expected,

ω^{A B}

depends on the Killing vector

K^{a}

, and, in fact,

K^{a}

can be recovered from

ω^{A B}

K^{A^{'} A} = \frac{2}{3} i \nabla_{B}^{A^{'}} ω^{A B}

. Thus

ω^{A B}

plays the role of a potential for the Killing vector

K^{A^{'} A}

. However, as a consequence of equation ( 43 ),

K_{a}

is a self-dual Killing 1-form in the sense that its derivative is a self-dual (or s.d.) 2-form: In fact, the general solution of equation ( 43 ) and the corresponding Killing vector are

\begin{matrix} \begin{matrix} ω^{A B} & = - i x^{A A^{'}} x^{B B^{'}} {\bar{M}}_{A^{'} B^{'}} + i x_{A^{'}}^{(A} T^{B) A^{'}} + Ω^{A B}, & K^{A A^{'}} & = T^{A A^{'}} + 2 x^{A B^{'}} {\bar{M}}_{B^{'}}^{A^{'}}, \end{matrix} \end{matrix}

(44)

where

{\bar{M}}_{A^{'} B^{'}}

T^{A A^{'}}

and

Ω^{A B}

are constant spinors, using the notation

x^{A A^{'}} : = x^{\underset{̲}{a}} σ_{\underset{̲}{a}}^{\underset{̲}{A} {\underset{̲}{A}}^{'}} ℰ_{\underset{̲}{A}}^{A} {\bar{ℰ}}_{{\underset{̲}{A}}^{'}}^{A^{'}}

, where

{ℰ_{\underset{̲}{A}}^{A}}

is a constant spin frame (the ‘Cartesian spin frame’) and

σ_{\underset{̲}{a}}^{\underset{̲}{A} {\underset{̲}{A}}^{'}}

are the standard

S L (2, C)

Pauli matrices (divided by

\sqrt{2}

). These yield that

K_{a}

is, in fact, self-dual,

\nabla_{A A^{'}} K_{B B^{'}} = ɛ_{A B} {\bar{M}}_{A^{'} B^{'}}

, and

T^{A A^{'}}

is a translation and

{\bar{M}}_{A^{'} B^{'}}

generates self-dual rotations. Then

Q_{S} [K] = T_{A A^{'}} P^{A A^{'}} + 2 {\bar{M}}_{A^{'} B^{'}} {\bar{J}}^{A^{'} B^{'}}

, implying that the charges corresponding to

Ω^{A B}

are vanishing; the four components of the quasi-local energy-momentum correspond to the real

T^{A A^{'}}

s; and the spatial angular momentum and centre-of-mass are combined into the three complex components of the self-dual angular momentum

{\bar{J}}^{A^{'} B^{'}}

, generated by

{\bar{M}}_{A^{'} B^{'}}

^$11$ The analogous calculations using tensor methods and the real $ω^{a b}$ instead of spinors and the anti-self-dual (or, shortly, a.s.d.) part of $ω^{a b}$ would be technically more complicated [293, 294, 299, 160] .

7.1.2 Twistor space and the kinematical twistor

Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the pairs

Z^{α} : = (λ^{A}, π_{A^{'}})

of spinor fields, which solve the so-called valence 1 twistor equation

\nabla^{A^{'} A} λ^{B} = - i ɛ^{A B} π^{A^{'}}

. If

Z^{α}

is a solution of this equation, then

{\hat{Z}}^{α} : = (λ^{A}, π_{A^{'}} - i ϒ_{A^{'} A} λ^{A})

is a solution of the corresponding equation in the conformally rescaled spacetime, where

ϒ_{a} : = Ω^{- 1} \nabla_{a} Ω

and

Ω

is the conformal factor. In general the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complex parameter family of solutions. Its general solution in the Minkowski spacetime is

λ^{A} = Λ^{A} - i x^{A A^{'}} π_{A^{'}}

, where

Λ^{A}

and

π_{A^{'}}

are constant spinors. Thus the space

T^{α}

of valence 1 twistors, the so-called twistor-space, is four complex dimensional, and hence has the structure

T^{α} = S^{A} \oplus {\bar{S}}_{A^{'}}

T^{α}

admits a natural Hermitian scalar product: If

W^{β} = (ω^{B}, σ_{B^{'}})

is another twistor, then

H_{α β^{'}} Z^{α} {\bar{W}}^{β^{'}} : = λ^{A} {\bar{σ}}_{A} + π_{A^{'}} {\bar{ω}}^{A^{'}}

. Its signature is

(+, +, -, -)

, it is conformally invariant:

H_{α β^{'}} {\hat{Z}}^{α} {\bar{\hat{W}}}^{β^{'}} = H_{α β^{'}} Z^{α} {\bar{W}}^{β^{'}}

, and it is constant on Minkowski spacetime. The metric

H_{α β^{'}}

defines a natural isomorphism between the complex conjugate twistor space,

{\bar{T}}^{α^{'}}

, and the dual twistor space,

T_{β} : = S_{B} \oplus {\bar{S}}^{B^{'}}

, by

({\bar{λ}}^{A^{'}}, {\bar{π}}_{A}) \mapsto ({\bar{π}}_{A}, {\bar{λ}}^{A^{'}})

. This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate

{\bar{A}}_{α^{'} β^{'}}

of the covariant valence 2 twistor

A_{α β}

can be represented by the so-called conjugate twistor

{\bar{A}}^{α β} : = {\bar{A}}_{α^{'} β^{'}} H^{α^{'} α} H^{β^{'} β}

. We should mention two special, higher valence twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by

\begin{matrix} I^{α β} : = (\begin{matrix} ɛ^{A B} & 0 \\ 0 & 0 \end{matrix}), I_{α β} : = {\bar{I}}^{α^{'} β^{'}} H_{α^{'} α} H_{β^{'} β} = (\begin{matrix} 0 & 0 \\ 0 & ɛ^{A^{'} B^{'}} \end{matrix}) . \end{matrix}

(45)

The other is the completely anti-symmetric twistor

ɛ_{α β γ δ}

, whose component

ɛ_{0123}

in an

H_{α β^{'}}

-orthonormal basis is required to be one. The only non-vanishing spinor parts of

ɛ_{α β γ δ}

are those with two primed and two unprimed spinor indices:

ɛ_{C D}^{A^{'} B^{'}} = ɛ^{A^{'} B^{'}} ɛ_{C D}

{ɛ_{B}^{A^{'}}}_{D^{'}}^{C^{'}} = - ɛ^{A^{'} C^{'}} ɛ_{B D}

{ɛ_{A B}}^{C^{'} D^{'}} = ɛ_{A B} ɛ_{C^{'} D^{'}}

, . . . , etc. Thus for any four twistors

Z_{i}^{α} = (λ_{i}^{A}, π_{A^{'}}^{i})

i = 1, . . ., 4

, the determinant of the

4 \times 4

matrix whose

i

-th column is

(λ_{i}^{0}, λ_{i}^{1}, π_{0^{'}}^{i}, π_{1^{'}}^{i})

, where the

λ_{i}^{0}

, . . . ,

π_{1^{'}}^{i}

are the components of the spinors

λ_{i}^{A}

and

π_{A^{'}}^{i}

in some spin frame, is

\begin{matrix} ν : = det (\begin{matrix} λ_{1}^{0} & λ_{2}^{0} & λ_{3}^{0} & λ_{4}^{0} \\ λ_{1}^{1} & λ_{2}^{1} & λ_{3}^{1} & λ_{4}^{1} \\ π_{0^{'}}^{1} & π_{0^{'}}^{2} & π_{0^{'}}^{3} & π_{0^{'}}^{4} \\ π_{1^{'}}^{1} & π_{1^{'}}^{2} & π_{1^{'}}^{3} & π_{1^{'}}^{4} \end{matrix}) = \frac{1}{4} ε_{k l}^{i j} λ_{i}^{A} λ_{j}^{B} π_{A^{'}}^{k} π_{B^{'}}^{l} ɛ_{A B} ɛ^{A^{'} B^{'}} = \frac{1}{4} ɛ_{α β γ δ} Z_{1}^{α} Z_{2}^{β} Z_{3}^{γ} Z_{4}^{δ}, \end{matrix}

(46)

where

ε_{k l}^{i j}

is the totally antisymmetric Levi-Civita symbol. Then

I^{α β}

and

I_{α β}

are dual to each other in the sense that

I^{α β} = \frac{1}{2} ɛ^{α β γ δ} I_{γ δ}

, and by the simplicity of

I^{α β}

one has

ɛ_{α β γ δ} I^{α β} I^{γ δ} = 0

The solution

ω^{A B}

of the valence 2 twistor equation, given by ( 44 ), can always be written as a linear combination of the symmetrized product

λ^{(A} ω^{B)}

of the solutions

λ^{A}

and

ω^{A}

of the valence 1 twistor equation.

ω^{A B}

defines uniquely a symmetric twistor

ω^{α β}

(see for example [299] ). Its spinor parts are

ω^{α β} = (\begin{matrix} ω^{A B} & - \frac{1}{2} K_{B^{'}}^{A} \\ - \frac{1}{2} {K_{A^{'}}}^{B} & - i {\bar{M}}_{A^{'} B^{'}} \end{matrix}) .

However, ( 40 ) can be interpreted as a

C

-linear mapping of

ω^{α β}

into

C

, i. e. ( 40 ) defines a dual twistor, the (symmetric) kinematical twistor

A_{α β}

, which therefore has the structure

\begin{matrix} A_{α β} = (\begin{matrix} 0 & {P_{A}}^{B^{'}} \\ P_{B}^{A^{'}} & 2 i {\bar{J}}^{A^{'} B^{'}} \end{matrix}) . \end{matrix}

(47)

Thus the quasi-local energy-momentum and self-dual angular momentum of the source are certain spinor parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor, it has only ten real components as a consequence of its structure (its spinor part

A_{A B}

is identically zero) and the reality of

P^{A A^{'}}

. These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on

A_{α β}

: The vanishing of the spinor part

A_{A B}

is equivalent to

A_{α β} I^{α γ} I^{β δ} = 0

and the energy momentum is the

A_{α β} Z^{α} I^{β γ} H_{γ γ^{'}} {\bar{Z}}^{γ^{'}}

part of the kinematical twistor, while the whole reality condition (ensuring both

A_{A B} = 0

and the reality of the energy-momentum) is equivalent to

\begin{matrix} A_{α β} I^{β γ} H_{γ δ^{'}} = {\bar{A}}_{δ^{'} β^{'}} {\bar{I}}^{β^{'} γ^{'}} H_{γ^{'} α} . \end{matrix}

(48)

Using the conjugate twistors this can be rewritten (and, in fact, usually it is written) as

A_{α β} I^{β γ} = (H^{γ α^{'}} {\bar{A}}_{α^{'} β^{'}} H^{β^{'} δ}) (H_{δ δ^{'}} {\bar{I}}^{δ^{'} γ^{'}} H_{γ^{'} α}) = {\bar{A}}^{γ δ} I_{δ α}

. Finally, the quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [293] or as its determinant [354] :

\begin{matrix} m^{2} & = - {P_{A}}^{A^{'}} P_{A^{'}}^{A} = - \frac{1}{2} A_{α β} {\bar{A}}_{α^{'} β^{'}} H^{α α^{'}} H^{β β^{'}} = - \frac{1}{2} A_{α β} {\bar{A}}^{α β}, \end{matrix}

(49)

\begin{matrix} m^{4} & = 4 det A_{α β} = \frac{1}{3!} ɛ^{α β γ δ} ɛ^{μ ν ρ σ} A_{α μ} A_{β ν} A_{γ ρ} A_{δ σ} . \end{matrix}

(50)

Thus, to summarize, the various spinor parts of the kinematical twistor

A_{α β}

are the energy-momentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, were needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and, in particular, to define the mass. Furthermore, the Hermiticity condition ensuring

A_{α β}

to have the correct number of components (ten reals) were also formulated in terms of these additional structures.

7.2 The original construction for curved spacetimes

7.2.1 2-surface twistors and the kinematical twistor

In general spacetimes the twistor equations have only the trivial solution. Thus to be able to associate a kinematical twistor to a closed orientable spacelike 2-surface

S

in general, the conditions on the spinor field

ω^{A B}

had to be relaxed. Penrose’s suggestion [293, 294] is to consider

ω^{A B}

in ( 40 ) to be the symmetrized product

λ^{(A} ω^{B)}

of spinor fields that are solutions of the ‘tangential projection to

S

’ of the valence 1 twistor equation, the so-called 2-surface twistor equation. (The equation obtained as the ‘tangential projection to

S

’ of the valence 2 twistor equation ( 43 ) would be under-determined [294] .) Thus the quasi-local quantities are searched for in the form of a charge integral of the curvature:

\begin{matrix} A_{S} [λ, ω] : & = & \frac{i}{8 π G} \oint_{S} λ^{A} ω^{B} R_{A B c d} = \end{matrix}

(51)

\begin{matrix} = & \frac{1}{4 π G} \oint_{S} (λ^{0} ω^{0} (φ_{01} - ψ_{1}) + (λ^{0} ω^{1} + λ^{1} ω^{0}) (φ_{11} + Λ - ψ_{2}) + λ^{1} ω^{1} (φ_{21} - ψ_{3})) d S, \end{matrix}

where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the 2-surface

S

. Since the indices

c

and

d

on the right of the first expression are tangential to

S

, this is just the charge integral of

F_{A B c d}

in the spinor identity ( 24 ) of subsection 4.1.5 .

The 2-surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation

{T_{E^{'} E A}}^{B} λ_{B} = 0

. By ( 25 ) its GHP form is

T λ : = (T^{+} \oplus T^{-}) λ = 0

, which is a first order elliptic system, and its index is

4 (1 - g)

, where

g

is the genus of

S

[42] . Thus there are at least four (and in the generic case precisely four) linearly independent solutions to

T λ = 0

on topological 2-spheres. However, there are ‘exceptional’ 2-spheres for which there exist at least five linearly independent solutions [215] . For such ‘exceptional’ 2-spheres (and for higher genus 2-surfaces for which the twistor equation has only the trivial solution in general) the subsequent construction does not work. (The concept of quasi-local charges in Yang–Mills theory can also be introduced in an analogous way [353] ). The space

T_{S}^{α}

of the solutions to

{T_{E^{'} E A}}^{B} λ_{B} = 0

is called the 2-surface twistor space. In fact, in the generic case this space is four complex dimensional, and under conformal rescaling the pair

Z^{α} = (λ^{A}, i Δ_{A^{'} A} λ^{A})

transforms like a valence 1 contravariant twistor.

Z^{α}

is called a 2-surface twistor determined by

λ^{A}

. If

S^{'}

is another generic 2-surface with the corresponding 2-surface twistor space

T_{S^{'}}^{α}

, then although

T_{S}^{α}

and

T_{S^{'}}^{α}

are isomorphic as vector spaces, there is no canonical isomorphism between them. The kinematical twistor

A_{α β}

is defined to be the symmetric twistor determined by

A_{α β} Z^{α} W^{β} : = A_{S} [λ, ω]

for any

Z^{α} = (λ^{A}, i Δ_{A^{'} A} λ^{A})

and

W^{α} = (ω^{A}, i Δ_{A^{'} A} ω^{A})

from

T_{S}^{α}

. Note that

A_{S} [λ, ω]

is constructed only from the 2-surface data on

S

7.2.2 The Hamiltonian interpretation of the kinematical twistor

For the solutions

λ^{A}

and

ω^{A}

of the 2-surface twistor equation the spinor identity ( 24 ) reduces to Tod’s expression [293, 299, 360] for the kinematical twistor, making it possible to re-express

A_{S} [λ, ω]

by the integral of the Nester–Witten 2-form [340] . Indeed, if

\begin{matrix} H_{S} [λ, \bar{μ}] : = \frac{1}{4 π G} \oint_{S} u (λ, \bar{μ})_{a b} = - \frac{1}{4 π G} \oint_{S} {\bar{γ}}^{A^{'} B^{'}} {\bar{μ}}_{A^{'}} Δ_{B^{'} B} λ^{B} d S, \end{matrix}

(52)

then with the choice

{\bar{μ}}_{A^{'}} : = Δ_{A^{'} A} ω^{A}

this gives Penrose’s charge integral by ( 24 ):

A_{S} [λ, ω] = H_{S} [λ, \bar{μ}]

. Then, extending the spinor fields

λ^{A}

and

ω^{A}

from

S

to a spacelike hypersurface

Σ

with boundary

S

in an arbitrary way, by the Sparling equation it is straightforward to rewrite

A_{S} [λ, ω]

in the form of the integral of the energy-momentum tensor of the matter fields and the Sparling form on

Σ

. Since such an integral of the Sparling form can be interpreted as the Hamiltonian of general relativity, this is a quick re-derivation of Mason’s [255, 256] Hamiltonian interpretation of Penrose’s kinematical twistor:

A_{S} [λ, ω]

is just the boundary term in the total Hamiltonian of the matter+gravity system, and the spinor fields

λ^{A}

and

ω^{A}

(together with their ‘projection parts’

i Δ_{A^{'} A} λ^{A}

and

i Δ_{A^{'} A} ω^{A}

) on

S

are interpreted as the spinor constituents of the special lapse and shift, the so-called ‘quasi-translations’ and ‘quasi-rotations’ of the 2-surface, on the 2-surface itself.

7.2.3 The Hermitian scalar product and the infinity twistor

In general the natural pointwise Hermitian scalar product, defined by

〈 Z, \bar{W} 〉

: = - i (λ^{A} Δ_{A A^{'}} {\bar{ω}}^{A^{'}} - {\bar{ω}}^{A^{'}} Δ_{A A^{'}} λ^{A})

, is not constant on

S

, thus it does not define a Hermitian scalar product on the 2-surface twistor space. As is shown in [214, 217, 358] ,

〈 Z, \bar{W} 〉

is constant on

S

for any two 2-surface twistors if and only if

S

can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric and extrinsic curvatures. Such 2-surfaces are called non-contorted, while those that cannot be embedded are called contorted. One natural candidate for the Hermitian metric could be the average of

〈 Z, \bar{W} 〉

S

[293] :

H_{α β^{'}} Z^{α} {\bar{W}}^{β^{'}} : = (Area (S))^{- \frac{1}{2}} \oint_{S} 〈 Z, \bar{W} 〉 d S

, which reduces to

〈 Z, \bar{W} 〉

on non-contorted 2-surfaces. Interestingly enough,

\oint_{S} 〈 Z, \bar{W} 〉 d S

can also be re-expressed by the integral ( 52 ) of the Nester–Witten 2-form [340] . Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted 2-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. ( 49 )) is ambiguous unless a natural

H_{α β^{'}}

is found.

S

is non-contorted, then the scalar product

〈 Z, \bar{W} 〉

defines the totally anti-symmetric twistor

ɛ_{α β γ δ}

, and for the four independent 2-surface twistors

Z_{1}^{α}

,. . . ,

Z_{4}^{α}

the contraction

ɛ_{α β γ δ} Z_{1}^{α} Z_{2}^{β} Z_{3}^{γ} Z_{4}^{δ}

, and hence by ( 46 ) the determinant

ν

, is constant on

S

. Nevertheless,

ν

can be constant even for contorted 2-surfaces for which

〈 Z, \bar{W} 〉

is not. Thus, the totally anti-symmetric twistor

ɛ_{α β γ δ}

can exist even for certain contorted 2-surfaces. Therefore, an alternative definition of the quasi-local mass might be based on ( 50 ) [354] . However, although the two mass definitions are equivalent in the linearized theory, they are different invariants of the kinematical twistor even in de Sitter or anti-de-Sitter spacetimes. Thus, if needed, the former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by

m_{D}

If we want to have not only the notion of the mass but its reality is also expected, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition ( 48 ), one also needs the infinity twistor. However,

- ɛ^{A^{'} B^{'}} Δ_{A^{'} A} λ^{A} Δ_{B^{'} B} ω^{B}

is not constant on

S

even if it is non-contorted, thus in general it does not define any twistor on

T_{S}^{α}

. One might take its average on

S

(which can also be re-expressed by the integral of the Nester–Witten 2-form [340] ), but the resulting twistor would not be simple. In fact, even on 2-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant

λ

the natural definition for

I_{α β}

I_{α β} : = diag (λ ɛ_{A B}, ɛ^{A^{'} B^{'}})

[299, 297, 354] , while on round spheres in spherically symmetric spacetimes it is

I_{α β} Z^{α} W^{β} : = \frac{1}{2 r^{2}} (1 + 2 r^{2} ρ ρ^{'}) ɛ_{A B} λ^{A} ω^{B} - ɛ^{A^{'} B^{'}} Δ_{A^{'} A} λ^{A} Δ_{B^{'} B} ω^{B}

[347] . Thus no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [192] : Even if the spacetime is conformally flat (whenever the Hermitian metric exists) the Hermiticity condition would be fifteen algebraic equations for the (at most) twelve real components of the ‘would be’ infinity twistor. Then, since the possible kinematical twistors form an open set in the space of symmetric twistors, the Hermiticity condition cannot be satisfied even for non-simple

I^{α β}

s. However, in contrast to the linearized gravity case, the infinity twistor should not be given once and for all on some ‘universal’ twistor space, that may depend on the actual gravitational field. In fact, the 2-surface twistor space itself depends on the geometry of

S

, and hence all the structures thereon also.

Since in the Hermiticity condition ( 48 ) only the special combination

H_{β^{'}}^{α} : = I^{α β} H_{β β^{'}}

of the infinity and metric twistors (the so-called ‘bar-hook’ combination) appears, it might still be hoped that an appropriate

H_{β^{'}}^{α}

could be found for a class of 2-surfaces in a natural way [360] . However, as far as the present author is aware of, no real progress has been achieved in this way.

7.2.4 The various limits

Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea came from the linearized gravity, the construction gives the correct results in the weak field approximation.

The nonrelativistic weak field approximation, i. e. the Newtonian limit, was clarified by Jeffryes [216] . He considers a one parameter family of spacetimes with perfect fluid source such that in the

λ \to 0

limit of the parameter

λ

one gets a Newtonian spacetime, and, in the same limit, the 2-surface

S

lies in a

t = const

hypersurface of the Newtonian time

t

. In this limit the pointwise Hermitian scalar product is constant, and the norm-mass can be calculated. As could be expected, for the leading

λ^{2}

order term in the expansion of

m

as a series of

λ

he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a

λ^{4}

order correction.

The Penrose definition for the energy-momentum and angular momentum can be applied to the cuts

S

of the future null infinity

ℑ^{+}

of an asymptotically flat spacetime [293, 299] . Then every element of the construction is built from conformally rescaled quantities of the non-physical spacetime. Since

ℑ^{+}

is shear-free, the 2-surface twistor equations on

S

decouple, and hence the solution space admits a natural infinity twistor

I_{α β}

. It singles out precisely those solutions whose primary spinor parts span the asymptotic spin space of Bramson (see subsection 4.2.4 ), and they will be the generators of the energy-momentum. Although

S

is contorted, and hence there is no natural Hermitian scalar product, there is a twistor

H_{β^{'}}^{α}

with respect to which

A_{α β}

is Hermitian. Furthermore, the determinant

ν

is constant on

S

, and hence it defines a volume 4-form on the 2-surface twistor space [360] . The energy-momentum coming from

A_{α β}

is just that of Bondi and Sachs. The angular momentum defined by

A_{α β}

is, however, new. It has a number of attractive properties. First, in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two anomaly’ between the angular momentum and the energy-momentum. Since its definition is based on the solutions of the 2-surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the 2-surface twistor space on

S

reduces the BMS Lie algebra to one of its Poincare subalgebras. Thus the concept of the ‘translation of the origin’ is moved from null infinity to the twistor space (appearing in the form of a four parameter family of ambiguities in the potential for the shear

σ

), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. As was shown in [125] , Penrose’s angular momentum can be considered as a supertranslation of previous definitions. The corresponding angular momentum flux through a portion of the null infinity between two cuts was calculated in [125, 191] and it was shown that this is precisely that given by Ashtekar and Streubel [28] (see also [321, 322, 124] ).

The other way of determining the null infinity limit of the energy-momentum and angular momentum is to calculate them for the large spheres from the physical data, instead of the spheres at null infinity from the conformally rescaled data. These calculations were done by Shaw [323, 325] .

At this point it should be noted that the

r \to \infty

limit of

A_{α β}

vanishes, and it is

\sqrt{Area (S_{r})} A_{α β}

that yields the energy-momentum and angular momentum at infinity (see the remarks following eq. ( 15 )). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [354] . The 2-surfaces for which the mass was calculated are the

r = const

cuts of the geometrically distinguished outgoing null hypersurfaces

u = const

. Tod found that, for given

u

, the mass

m

is independent of

r

, as could be expected because of the lack of the incoming radiation.

The large sphere limit of the 2-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [328] , the Ashtekar–Hansen [22] and the Beig–Schmidt [47] models of spatial infinity in [319, 320, 322] . Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) non-contorted, and both the Hermitian scalar product and the infinity twistor are well defined. Thus the energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar–Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [324] .

The Penrose mass in asymptotically anti-de-Sitter spacetimes was studied by Kelly [228] . He calculated the kinematical twistor for spacelike cuts

S

of the infinity

ℑ

, which is now a timelike 3-manifold in the non-physical spacetime. Since

ℑ

admits global 3-surface twistors (see the next subsection),

S

is non-contorted. In addition to the Hermitian scalar product there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding Hermiticity condition.

The energy-momentum 4-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [26] . Therefore, the energy-momentum 4-vector is future pointing and timelike if there is a spacelike hypersurface extending to

ℑ

on which the dominant energy condition is satisfied. Consequently,

m^{2} \geq 0

. Kelly showed that

m_{D}^{2}

is also non-negative and in vacuum it coincides with

m^{2}

. In fact [360] ,

m \geq m_{D} \geq 0

holds.

7.2.5 The quasi-local mass of specific 2-surfaces

The Penrose mass has been calculated in a large number of specific situations. Round spheres are always non-contorted [358] , thus the norm-mass can be calculated. (In fact, axis-symmetric 2-surfaces in spacetimes with twist-free rotational Killing vector are non-contorted [217] .) The Penrose mass for round spheres reduces to the standard energy expression discussed in subsection 4.2.1 [354] . Thus every statement given in subsection 4.2.1 for round spheres is valid for the Penrose mass, and we do not repeat them. In particular, for round spheres in a

t = const

slice of the Kantowski–Sachs spacetime this mass is independent of the 2-surfaces [351] . Interestingly enough, although these spheres cannot be shrunk to a point (thus the mass cannot be interpreted as ‘the 3-volume integral of some mass density’), the time derivative of the Penrose mass looks like the mass conservation equation: It is minus the pressure times the rate of change of the 3-volume of a sphere in flat space with the same area as

S

[359] . In conformally flat spacetimes [354] the 2-surface twistors are just the global twistors restricted to

S

, and the Hermitian scalar product is constant on

S

. Thus the norm-mass is well defined.

The construction works nicely even if global twistors exist only on a (say) spacelike hypersurface

Σ

containing

S

. These twistors are the so-called 3-surface twistors [354, 356] , which are solutions of certain (overdetermined) elliptic partial differential equations, the so-called 3-surface twistor equations, on

Σ

. These equations are completely integrable (i. e. they admit the maximal number of linearly independent solutions, namely four) if and only if

Σ

with its intrinsic metric and extrinsic curvature can be embedded, at least locally, into some conformally flat spacetime [356] .

Such hypersurfaces are called non-contorted. It might be interesting to note that the non-contorted hypersurfaces can also be characterized as the critical points of the Chern–Simons functional built from the real Sen connection on the Lorentzian vector bundle or from the 3-surface twistor connection on the twistor bundle over

Σ

[48, 345] . Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a 2-surface

S

in a non-contorted

Σ

depends only on the homology class of

S

. In particular, if

S

can be shrunk to a point then the corresponding kinematical twistor is vanishing. Since

Σ

is non-contorted,

S

is also non-contorted, and hence the norm–mass is well defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any non-contorted 2-surface that can be deformed into a round sphere, and it is zero for those that do not link the black hole [358] . Thus, in particular, the Penrose mass can be zero even in curved spacetimes.

A particularly interesting class of non-contorted hypersurfaces is that of the conformally flat time-symmetric initial data sets. Tod considered Wheeler’s solution of the time symmetric vacuum constraints describing

n

‘points at infinity’ (or, in other words,

n - 1

black holes) and 2-surfaces in such a hypersurface [354] . He found that the mass is zero if

S

does not link any black hole, it is the mass

M_{i}

of the

i

-th black hole if

S

links precisely the

i

-th hole, it is

M_{i} + M_{j} - M_{i} M_{j} / d_{i j} + O (1 / d_{i j}^{2})

S

links precisely the

i

-th and the

j

-th holes, where

d_{i j}

is some appropriate measure of the distance of the holes, . . . , etc. Thus, the mass of the

i

-th and

j

-th holes as a single object is less than the sum of the individual masses, in complete agreement with our physical intuition that the potential energy of the composite system should contribute to the total energy with negative sign.

Beig studied the general conformally flat time symmetric initial data sets describing

n

‘points at infinity’ [44] . He found a symmetric trace-free and divergence-free tensor field

T^{a b}

and, for any conformal Killing vector

ξ^{a}

of the data set, defined the 2-surface flux integral

P (ξ)

T^{a b} ξ_{b}

S

. He showed that

P (ξ)

is conformally invariant, depends only on the homology class of

S

, and, apart from numerical coefficients, for the ten (locally existing) conformal Killing vectors these are just the components of the kinematical twistor derived by Tod in [354] (and discussed in the previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the length of the

P

’s with respect to the Cartan-Killing metric of the conformal group of the hypersurface.

Tod calculated the quasi-local mass for a large class of axis-symmetric 2-surfaces (cylinders) in various LRS Bianchi and Kantowski–Sachs cosmological models [359] and more general cylindrically symmetric spacetimes [361] . In all these cases the 2-surfaces are non-contorted, and the construction works. A technically interesting feature of these calculations is that the 2-surfaces have edges, i. e.

they are not smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately, and the resulting spinor fields are required to be continuous at the edges.

This matching reduces the number of linearly independent solutions to four. The projection parts of the resulting twistors, the

i Δ_{A^{'} A} λ^{A}

\times

f (L)

L

sinh ω L / ω L

sin ω L / ω L

ω