ON GLOBAL CONSERVATION LAWS AT NULL INFINITY

Joseph Katz^{*}^{*}E-mail: [email protected]
and Dorit Lerer

The Racah Institute of Physics, 91904 Jerusalem, Israel

9/12/1996

ABSTRACT

The “standard” expressions for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are also obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity to use or not to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to Nœther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, the conserved quantities are in principle explicitly related to the sources of gravity through Einstein’s equations, while standard definitions are not. The relations depend, however, on a rule for mapping spacetimes on backgrounds.

## 1. Introduction

In Science and Hypothesis, Poincaré (1904) imagines astronomers “whose vision would be bounded by the solar system” because of thick clouds that hide the fixed stars. There is thus no fixed frame of coordinates, only relative distances and relative angles are measurable. In those circumstances, says Poincaré, “we should be definitively led to conclude that the equations which define distances are of an order higher than the second. […] The values of the distances at any given moment depend upon their initial values, on that of their first derivatives, and something else. What is that something else? If we do not want it to be merely one of the second derivatives, we have only the choice of hypotheses. Suppose, as is usually done that this something else is the absolute orientation of the universe in space, or the rapidity with which this orientation varies; this may be, it certainly is, the most convenient solution for the geometer. But it is not the most satisfactory for the philosopher, because this orientation does not exist.”

The conservation of energy, linear and angular-momentum, so useful in classical mechanics and special relativity, are related to the homogeneity and isotropy properties attributed to an absolute “background” spacetime which does indeed not exist. In general relativity, the use of backgrounds is intrinsic to the definition of pseudo-tensor conservation laws. Rosen (1958) and Cornish (1964) used backgrounds explicitly to calculate the energy-momentum in more appropriate coordinates than orthogonal ones. While backgrounds have thus proven to be useful, they nevertheless got little support and attention from the community of general relativists. On the contrary, great efforts have been spend to get rid of backgrounds, to avoid those “additional structure completely counter spirit of general relativity” (Wald 1984). True, efforts to avoid introducing background geometries generated interesting works notably by Penrose (1965), Geroch (1976) and other mathematical physicists mentioned in Schmidt’s (1993) review on Asymptotic flatness — a Critical Appraisal. As a consequence, we now have a rather well understood, coordinate independant, picture of asymptotic flatness, and we have also asymptotic coordinate independant expressions for globally conserved quantities in particular at null infinity.

Conserved quantities at null infinity are given by integrals on spheres of
infinite radius at fixed null time . They represent total
energy, linear momentum, angular-momentum and the initial position of the
mass center (Synge 1964)
of spacetimes with isolated sources of curvature at a moment of “time”
. The expressions have become “standard” according to Dray and
Streubel (1984) [see also Dray (1985) and Shaw (1986)], and they are
invariant under coordinate transformations of the Bondi-Metzner-Sachs, or
BMS, group [Sachs (1962), Newman and Penrose (1966)]. Most standard
expressions draw their strength — beyond their esthetical appeal — from a
few physically interpretable quantities, which have all been obtained before
from the pseudo-tensors of Freud (1939) and of Landau and Lifshitz
(1951). The quantities are
(a) The Schwarzschild mass and more generally the
dominant
term in a multipole expansion of static solutions at spatial
infinity (Geroch 1970) and at null infinity (Schmidt 1993), which
was obtained from Einstein’s pseudo-tensor [see Tolman 1934].
(b) The Bondi mass where [see Bondi, Metzner and van
der Burg (1962) see already Bondi (1960)] and Sachs’s (1962a)
linear momentum at null infinity^{*}^{*}What is
actually defined in Sachs is . which have been calculated
by Møller (1972) using Freud’s superpotential;
(c) The Kerr angular-momentum and more generally the
dominant
“odd function” factor [Misner Thorne and Wheeler (1973)]
in a multipole expansion of stationary spacetimes at spatial or
null infinity, which is obtainable from Landau and Lifshitz
superpotential for angular momentum [Papapetrou 1965].

The angular momentum at null infinity , as given for instance by Dray and Streubel (1984), has not been derived from previously known soperpotential. is not a well defined physical quantity because of supertranslation freedom. However, the weak field approximation fits in well with results deduced from Landau-Lifshitz’s superpotential (Creswell and Zimmerman 1986). The LL approximation is currently used in gravitational radiation calculations (Thorne 1980).

Here we show that , as well as — and thus also and — can be calculated using a single superpotential derived from a quadratic Lagrangian with an asymptotically flat background at null infinity. The derivation insures automatically Poincaré invariance, the absence of an anomalous factor of 2 in the ratio and zero value for all conserved quantities when the spacetime identifies with the background itself. As a result, and are the standard BMS invariant fluxes while , which is not BMS invariant, is nevertheless the same as the standard result. The position of the mass center, , associated with Lorentz rotations, and the corresponding flux given here are not the same as the standard formula, at least in the non-linear approximation.

Thus, the most important standard results, energy and angular momentum, can be deduced from Nœther conservation laws. One appealing aspect of Nœther conservation laws is that they are closer to classical intuition than the more abstract coordinate independant definitions at scri. Moreover, Nœther conserved quantities are directly related to the energy-momentum tensor of the matter through Einstein’s equations by differential conservation laws. “Standard” definitions are not.

Local differential conservation laws contain implicit definitions of local or quasi-local conserved quantities which depend, on the local mapping of the spacetime on the background. We give here no rule for local mappings. There are also several mapping-independent definitions of quasi-local energy in the literature, but they appear to be different from each other (Berqvist 1992).

## 2. Superpotential and Nœther Conservation Laws

Let us briefly review the elements that lead to our superpotential (Katz 1996). Full details are given in a work by Katz, Bic̆ák and Lynden-Bell (1996) — refered to below as KBL96 — which has its roots in earlier work by Katz (1985) on flat backgrounds. KBL96 is on curved backgrounds and the superpotential obtained there is new even in the limit where the backgrounds become flat.

## (i) Lagrangian density for gravitational fields on a curved background

Let be the metric of a spacetime with signature -2, and let be the metric of the background . Both are tensors with respect to arbitrary coordinate transformations. Once we have chosen a mapping so that points of map into points of , then we can use the convention that and shall always be given the same coordinates . This convention implies that a coordinate transformation on inevitably induces a coordinate transformation with the same functions on . With this convention, such expressions as , which can be looked at as “perturbations” of the background, become true tensors. However, if the particular mapping has been left unspecified we are still free to change it. The form of the equations for “perturbations” of the background must inevitably contain a gauge invariance corresponding to this freedom.

Let and be the curvature tensors of and . These are related as follows [see Rosen (1940, 1963), see also Choquet-Bruhat (1984) for mathematical aspects of background formalisms]:

Here are covariant derivatives with respect to , and is the difference between Christoffel symbols in and :

Our quadratic Lagrangian density for the gravitational field is then defined as

The mark means multiplication by , never by , and a bar over a symbol signifies that , etc. are replaced by , etc.. The vector density is given by

and its divergence cancels all second order derivatives of in . is the Lagrangian used by Rosen. is in which has been replaced by . When , is thus identically zero. The intention here is to obtain conservation laws in the background spacetime so that if , conserved vectors and superpotentials will be identically zero as in Minkowski space in special relativity.

The following formula, deduced from [2-3] and [2-1], shows explicitly how is quadratic in the first order derivatives of or, equivalently, quadratic in :

Notice that if and coordinates are such that , is nothing else than the familiar “” Lagrangian density [see for instance Landau and Lifshitz (1951)].

## (ii) Strong conservation laws and superpotential

If

represents an infinitesimal one parameter displacement generated by , the corresponding changes in tensors are given in terms of the Lie derivatives with respect to the vector field , $, etc.. The Lie derivatives may be written in terms of ordinary partial derivatives , covariant derivatives with respect to , or covariant derivatives with respect to . Thus,

Consider now the Lie differential of . With the variational principle in mind, we write $ in the form

where Einstein’s tensor density, , is the variational derivative of with respect to , is a vector density linear in whose detailed form will not concern us here. The Lie derivative of a scalar density like is just an ordinary divergence , Thus

Combining [2-9] with [2-8], we obtain

Bianchi identities imply so that with [2-7c], [2-10] can be written as the divergence of a vector density , say,

Hence, “generates” a vector density that is identically conserved. It has been obtained without using Einstein’s field equations; [2-11] is the kind of strong conservation law introduced by Bergmann (1949). We shall, of course, assume that Einstein’s equations are satisfied, and replace by the energy-momentum of matter

so that our strong conservation law [2-11] reads:

Equations [2-13] are, strictly speaking, not identities anymore. Given , [2-13] holds only for metrics that satisfy [2-12]. The vector density is linear in and its derivatives up to order 2.

Since as given by [2-11] is identically conserved whatever is , it must be the divergence of an antisymmetric tensor density that depends on the arbitrary ’s as well; thus

Indeed, is easy to find and is derived directly from
in Katz^{*}^{*}In the 1985 paper the background is assumed to be flat,
but the derivation of does not depend on that assumption.
(1985) [see also Chruściel (1986), Sorkin (1988) and Katz and Ori (1990)]:

The terms will be recognised as Komar’s(1959) superpotential. In terms of derivatives,

and, using expression [2-4] for , may be written in the form

Had we applied the identities [2-9] to instead of , we would have written everywhere instead of . We would have found strong, barred, conserved vector densities and barred superpotentials with the same ’s:

with

Strongly conserved vectors for are obtained by subtracting barred vectors and superpotentials from unbarred ones; in this way we define relative vectors and in particular relative superpotentials — relative to the background space. Setting

we have for the strongly conserved vector the following form:

which hold for any and any mapping of on ; in [2-21]

, and are given explicitly in appendix for the interested reader. The relative superpotential density is now given by

and can be also written in terms of , and :

in which

The tensors in [2-22] have a physical interpretation. On a flat background, in coordinates in which , [see appendix]

and reduces to Einstein’s pseudo-tensor density. appears therefore as the energy-momentum tensor of the gravitational field with respect to the background. The second tensor in [2-22], , is quadratic in the metric perturbations just like . It is also bilinear in the perturbed metric components () and their first order derivatives. resembles, in this respect, the helicity tensor density in electromagnetism. The factor of represents thus the helicity tensor density of the perturbations with respect to the background.

It should be noted again that all the components of and of the superpotential itself are identically zero if ; therefore strong conservation laws refer to “perturbations” only and not to the background.

## (iii) Nœther conservation laws

We now consider what happens when arbitrary ’s are replaced by Killing vectors of the background. defined in [2-22], which contains the physics of the conservation laws [see KLB96], is not, in general, a conserved vector density since the identically conserved vector density is and thus

However, when is a Killing vector of the background, , then [see appendix] and is conserved.

Our has been derived in the same way as “Nœther’s theorem” in classical field theory [see for instance Schweber, Bethe and Hoffmann (1956), or Bogoliubov and Shirkov (1959)]. Thus, by replacing in strongly conserved currents by Killing vectors, , of the background we obtain Nœther conserved vector densities in general relativity with mappings on curved backgrounds.

with given by

We can now integrate [2-28], on a part of a hypersurface , which spans a two-surface , and obtain integral conservation laws:

depends only on the gravitational field and its first derivatives on and on the mapping near the boundary. The relation with the matter tensor depends, however, on the mapping all the way down to the sources of gravity.

For weak fields on a flat background, the lowest order linear approximation of [2-30] is

If the spacetime is asymptotically flat, intergrals over the whole hypersurface extending to infinity define globally conserved quantities. It is then appropriate to map the spacetime near infinity on a flat Minkowski space with its ten Killing vectors associated with spacetime translations, spatial rotations and Lorentz rotations. The ten Killing vectors give ten different expressions , which can be interpreted respectively as the total energy , the linear momentum vector , the angular momentum vector and the initial mass center position on .

We shall now calculate the conserved quantities for Bondi’s asymptotic solution on a null hypersurface at infinity, using the right hand side of [2-30]. In what follows we intend to define all the quantities introduced. We shall not use cross-referencing for definitions which are often given with different symbols, factors of 2 or 1/ and other signs; most readers will appreciate this affort.

## 3. Elements of the Asymptotic BMS Metric in Newman-Unti Coordinates

## (i) The Newman-Unti asymptotic solution

In the coordinates used by Newman and Unti (1962) , the metric of Bondi, Metzner and Sachs (1962) has the following form:

The metric components are given by Newman and Unti in their formula (41). We shall make a few changes of notations because several indices which make sense in Newman and Unti are not useful here. Thus * The indices of the metric where shifted from to ( or any other lower-case greek letter) so that , , and * We have denoted real parts with a prime like , instead of Re in NU, and imaginary parts with a “second” like , instead of Im. Thus etc… * Complex conjugation is denoted here by an asterisk , because a bar over the symbol is reserved for the background. Thus . * The and of NU are here written and . Notice that and exist also in NU with a different meaning. * of NU is here written .

With these changes of notations, the metric components given by Newman and Unti become as follows:

is a function of , while , and are complex scalar functions of , , , defined in terms of the null tetrad components of the Weyl tensor. is defined in terms of and

and

Note that we have expanded to the fifth order of .

From [3-2] we have calculated the components:

A useful quantity in our calculation is the density

Further useful informations taken from Newman and Unti are (a) The -derivative equations derived from the Bianchi identities, in the form given in their equations (40k,l) or (42b,c). We shall most of the time use a dot on a symbol to denote a derivative of this function with respect to ; thus . With this change of notation the formulas are

where

(b) The metric keeps the same form under coordinate transformations of the Bondi-Metzner-Sachs, or BMS, group (Sachs 1962b, Newman and Penrose 1966). The leading terms in powers of of the transformation are given by NU in their eq. (46):

and are arbitrary functions and the induce conformal transformations in space, i.e.

From this follows that can be fixed with an appropriate conformal transformation [3-12], and can be fixed by choosing a spherical boundary for the surface . But is only defined up to a supertranslation . For a fixed and a fixed there are five independent scalar functions in the metric: and , and and . But they are defined up to a supertranslation and therefore there are actually four independent initial quantities among the ten ’s on . The imaginary part of , , is not independent; It is defined in terms of and [NU (40g)]:

The physical interpretation of these functions has been analyzed in details in a series of papers by Bondi, Metzner and Sachs [see especially Bondi van der Burg and Metzner (VII) 1962]

## 4. Elements of the Asymptotic Background

The asymptotic background is flat. In Minkowski coordinates , the metric element

An arrow designates spatial 3-vectors in Minkowski space. In coordinates, the Killing vector components of the ten translations are given by

and of rotations by

In NU-coordinates

where

and the metric of a sphere ( in conformal coordinates has the well known Riemann form for spaces of constant curvature [Eisenhart 1922]:

With given by [4-6] we find that

Eq. [4-7] somewhat simplifies given in [3-2]. Moreover, since satisfies the following equation

defined in [3-8] becomes also simpler:

has been defined in [3-3].

and are not uniquely defined nor are they uniquely related to the spherical coordinates in . By definition , and . We shall define as follows:

This choice makes the connection with the formalism of Newman and Penrose (1966) simple, as we shall see below. In terms of spherical coordinates that are sometimes useful in the calculations, becomes

Let us introduce a unit vector in the radial direction in Minkowski space, denoted by ,

and the complex 2-vector on the sphere

In terms of and , the ten Killing vector 4-components or the 2 real + 1 complex components in coordinates are as follows: (i) Time Translations :

(ii) 3-Space Translations :

(iii) 3-Space rotations :

(iv) 3-Spacetime (“Lorentz”) rotations :

These Killing vectors satisfy the Killing equations

which are very useful in further calculations; using the background metric [4-4], the Killing equations [4-20] can be written

From equation [4-25] applied to defined in [4-18], and from equation [4-24] applied to defined in [4-19], we deduce, respectively, the following important identities