(p.402) APPENDIX A TOTAL MASS AND MOMENTUM IN GENERAL RELATIVITY
(p.402) APPENDIX A TOTAL MASS AND MOMENTUM IN GENERAL RELATIVITY
The concept of energy is of fundamental importance in most physical theories. Indeed, even in relativity the stress-energy tensor of matter T μν plays a key role both in the conservation laws, from which the dynamics can often by fully determined, and as a source of the gravitational field. In general relativity, the stress-energy tensor satisfies a conservation law of the form ∇ν T μν=0, but in contrast to special relativity this local conservation law does not lead to global conservation of energy integrated over a finite volume. The reason is clear, as T μν represents only the energy of matter and does not take into account the contribution from the gravitational field. However, it turns out that, in relativity, we can in general not define the energy density of the gravitational field itself.
On the other hand, in general relativity we can in fact define the total energy of an isolated system in a meaningful way. We will consider here two different approaches to defining both the total energy and the momentum of an asymptotically flat spacetime. The presentation here will be brief and not rigorous; for a more rigorous discussion see e.g. [31, 222, 306].
The first approach is motivated by considering weak gravitational fields for which g μν=ημν+h μν, with ∣h μν∣≪1. In such a case, it is clear that the total mass (energy) and momentum can be defined as the integral of the energy density ρ and momentum density j i of matter, respectively
Using now the Hamiltonian and momentum constraints of the 3+1 formalism, equations (2.4.10) and (2.4.11), plus the fact that, in the weak field limit, the extrinsic curvature K ij is itself a small quantity, we can rewrite this as (in all following expressions sum over repeated indices is understood)
where R is the Ricci scalar of the spatial metric, which in the linearized theory is given by
with h the trace of h ij. Notice that both the mass and momentum are now written as volume integrals of a divergence, so using Gauss’ theorem we can rewrite them as (p.403)
where the integrals are calculated over surfaces outside the matter sources, and where γij=δij+h ij is the spatial metric and dS i=s i dA, with s i the unit outward-pointing normal vector to the surface and dA the area element. The expressions above are called the ADM mass and ADM momentum of the spacetime (ADM stands for Arnowitt, Deser, and Misner ). Notice that the ADM mass depends only on the behavior of the spatial metric, while the ADM momentum depends on the extrinsic curvature instead. We should stress the fact that for these expressions to hold we must be working with quasi-Minkowski, i.e. Cartesian type, coordinates.
In order to derive the ADM integrals above, we started from the weak field theory and used Gauss’ theorem to transform the volume integrals into surface integrals. In the case of strong gravitational fields, the total mass and momentum can not be expected to be given directly by the volume integrals of the energy and momentum densities, precisely because these integrals fail to take into account the effect of the gravitational field. However, we in fact define the total mass and momentum of an isolated system through its gravitational effects on faraway test masses, and far away the weak field limit does hold, so the surface integrals above will still give the correct values we physically associate with the total mass and momentum, but they must be evaluated at infinity to guarantee that we are in the weak field regime. We then define the ADM mass and momentum in the general case as
Using the same idea we can also construct a measure of the total angular momentum J i of the system starting from the density of angular momentum given by εijk x j j k, with x i Cartesian coordinates (this is just in standard three-dimensional vector notation). The ADM angular momentum then becomes
Notice that, in fact, in the limit r→∞, the term proportional to K can be dropped out since at spatial infinity the Cartesian vector is collinear with the area element so that εijl x j dS l=0. The integral can then be rewritten as
In particular, for the projection along the z axis we find (p.404)
with the coordinate basis vector associated with the azimuthal angle ϕ.
There is a very important property of the ADM mass and momentum defined above. Since these quantities are defined at spatial infinity i 0, and since any gravitational waves taking energy and momentum away from the system will instead reach null infinity , the ADM mass and momentum will remain constant during the evolution of the system.
The expression for the ADM mass given in (A.5) has the disadvantage that it is not covariant and must be calculated in Cartesian-type coordinates. This can in fact be easily remedied. In an arbitrary curvilinear coordinate system define the tensor , with the metric of flat space expressed in the same coordinates. It is easy to convince oneself that h ij does indeed transform like a tensor. We can now rewrite the ADM mass as
where is the covariant derivative with respect to the flat metric in the corresponding curvilinear coordinates, and where indices are raised and lowered also with . Notice that the tensor h ij is not unique because we can still make infinitesimal coordinate transformations that will change h ij without changing (gauge transformations in the context of linearized theory).
From the last result we can also obtain a particularly useful expression for the ADM mass:
where dA is the area element of the surface S, k is the trace of the extrinsic curvature of S, and k 0 is the trace of the extrinsic curvature of S when embedded in flat space.108 It is important to mention that the surface S, when embedded in flat space, must have the same intrinsic curvature as it did when embedded in curved space. The derivation of this expression is not difficult and can be obtained by considering an adapted coordinate system, with one of the coordinates measuring proper distance along the normal direction and the other two coordinates transported orthogonally off S; see for example  (in 3+1 terms, we would have a “lapse” equal to 1 and zero “shift”, but in this case the orthogonal direction is spacelike and the surface is bi-dimensional).
where we have assumed that far away ψ becomes unity.
There is another common approach to defining global energy and momentum that works for spacetimes that have a Killing field ξμ. I will give here a rough derivation of the corresponding expressions; a more formal derivation can be found in e.g. . Notice first that for any antisymmetric tensor A μν we have
Integrating the second expression above and using Gauss’ theorem we find
where Ω is some four-dimensional region of spacetime, ∂Ω is its boundary and is the unit outward-pointing normal vector to the boundary, with the norm defined using a flat metric. Assume now that Ω is the region bounded by two spacelike hypersurfaces Σ1 and Σ2 (with Σ1 to the future of Σ2), and a timelike cylindrical world-tube σ. We then find
where in both the integrals over Σ1 and Σ2 the normal vector is taken to be future pointing.
Take now A μν=∇μξν, which will be antisymmetric if ξμ is a Killing vector. For the integrals over the timelike cylinder we then have
where R μν is the Ricci tensor and T μν the stress-energy tensor of matter. In the second line above we used the Ricci identity to relate the commutator of (p.406) covariant derivatives to the Riemann tensor (1.9.3) plus the fact that ξμ is a Killing field, and in the third line the Einstein field equations. If we now assume that we have an isolated source and the cylindrical world-tube is outside it then the above integral clearly vanishes. Equation (A.16) then implies that
This means that the spatial integral is in fact independent of the hypersurface Σ chosen, or in other words, the spatial integral is a conserved quantity.
To proceed further we choose coordinates such that . Using the fact that ∇μξν is antisymmetric, and applying again Gauss’ theorem, the spatial integral becomes
where ∂Σ is now the two-dimensional boundary of Σ and is the spatial unit outward-pointing normal vector to ∂Σ. In the last expression we can notice that , where now n μ and s μ are unit vectors with respect to the full curved metric g μν, and dA is the proper area element of ∂Σ. We can then write the integral as
This is known as the Komar integral  and as we have seen is a conserved quantity (the quantity A μν=∇μξν is also frequently called the Komar potential).109 The integral can in fact be calculated over any surface ∂Σ outside the matter sources. The normalization factor −1/4π has been chosen to ensure that, for the case of a timelike Killing field that has unit magnitude at infinity, the Komar integral will coincide with the ADM mass of the spacetime. If, on the other hand, ξμ is an axial vector associated with an angular coordinate φ (i.e. ), then the Komar integral turns out to be −2J, with J the total angular momentum (we can check that this is so by considering a Kerr black hole). The fact that, for a static spacetime, the Komar integral and the ADM mass coincide can be used to derive a general relativistic version of the virial theorem, but we will not consider this issue here (the interested reader can see e.g. ).
(p.407) Incidentally, we can use the same derivation that allowed us to show that the integral over the timelike cylinder σ vanishes to rewrite the Komar integral as a volume integral of the stress-energy of matter as
with dV the proper volume element of the spatial hypersurface.
The Komar integral can in fact also be used to define mass at null infinity in the case of non-stationary asymptotically flat spacetimes. This is known as the Bondi mass, and is defined as
where now ξμ is assumed to be the generator of an asymptotic time translation symmetry, and where the surface of integration is now taken to approach a cross section of (when ξμ is not an exact Killing field we must ask for an extra normalization condition, see e.g. ). While the ADM mass measures the total energy available in a spacetime, the Bondi mass represents the remaining energy at a retarded time. This means that, in particular, the Bondi mass can be used to calculate the change in the total energy of an isolated system that goes through a phase in which it radiates gravitational waves.
Before finishing this Appendix it is important to mention that the ADM expressions for mass and momentum given above, though correct, in practice converge very slowly as r goes to infinity, so that if we evaluate them at a finite radius in a numerical simulation the errors might be quite large. We can of course evaluate them at several radii and then extrapolate to the asymptotic value. For the mass, however, there are other approaches that work well in practice. One such approach is based on the use of the so-called Hawking mass, which is a quasi-local measure of energy defined for any given closed surface S as (see e.g. )
where A is the area of the surface, and H in and H out are the expansions of the ingoing and outgoing null geodesics which are given in terms of 3+1 quantities as (see equation (6.7.8))
with being the unit outward-pointing normal vector to S, and D i the standard three-dimensional covariant derivative. The Hawking mass is defined by thinking that the presence of a mass must cause light rays to converge, and it turns (p.408) out that the only gauge invariant measure of this on the surface is given by H in H out.110 We would then expect the mass to be given by an expression of the form M=A+B∫H in H out dS, with the constants A and B fixed by looking at special cases.
For spheres in Minkowski spacetime the Hawking mass vanishes identically since H in=−H out=2/r. For spheres in Schwarzschild (in standard coordinates) we have H in=−H out=2(1−2M/r)1/2/r and A=4πr 2, so that M H=M. That is, the Hawking mass gives us the correct mass at all radii, which makes it a more useful measure of energy than the ADM mass. The Hawking mass in general is not always positive definite, and is not always monotonic either, but for sufficiently “round” surfaces (and particularly in spherical symmetry) both these properties can be shown to be satisfied if the dominant energy condition holds.
Another less formal but very useful approach to obtaining a local expression for the mass of the spacetime is based on the fact that many astrophysically relevant spacetimes will not only be asymptotically flat, but they will also be asymptotically spherically symmetric. In such a case we know that the spacetime will approach Schwarzschild far away. For Schwarzschild we can easily prove the following exact relation between the mass M and the area A of spheres
where r is some arbitrary radial coordinate. In numerical simulations of spacetimes that are asymptotically spherically symmetric we can calculate the area A of coordinate spheres at finite radii, the rate of change of the area with radius dA/dr, and the average radial metric over the sphere , and then use the above expression to estimate the mass. In practice we often find that as r is increased this “Schwarzschild-like” mass converges very rapidly to the correct ADM mass, much more rapidly than the ADM integral itself. This works even for spacetimes with non-zero angular momentum like Kerr, as the angular momentum terms in the metric decay faster than the mass term.
(109) The Komar integral is usually written in differential form notation as
In fact, in the last expression, the area element dx α℘dx β is frequently not even written.
(110) More specifically, the product of the Newman–Penrose spin coefficients ρ and ρ′ is gauge invariant under a spin-boost transformation of the null tetrad, and in general it is given by ρρ′=H in H out/8.