Simplified Derivation of the Gravitational Wave Stress Tensor from the Linearized Einstein Field Equations

Simplified derivation of the gravitational wave stress tensor from the linearized Einstein field equations

Steven A. Balbusa,1

aAstrophysics, Department of Physics, University of Oxford, Oxford OX1 3RH, United Kingdom

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015.

Contributed by Steven A. Balbus, September 1, 2016 (sent for review August 9, 2016; reviewed by Thomas Baumgarte and Jean-Pierre Lasota) A conserved stress energy tensor for weak field gravitational waves gravitational constant by G, and the speed of light set to unity, the propagating in vacuum is derived directly from the linearized general Einstein field equation is the following: relativistic wave equation alone, for an arbitrary gauge. In any harmonic gauge, the form of the tensor leads directly to the classical Gμν = −8πGTμν, [3] expression for the outgoing wave energy. The method described here, however, is a much simpler, shorter, and more physically which upon expansion in hμν may be rewritten as follows: motivated approach than is the customary procedure, which involves a lengthy and cumbersome second-order (in wave-amplitude) calcu- ð1Þ G = −8πG Tμν + tμν . [4] lation starting with the Einstein tensor. Our method has the added μν

advantage of exhibiting the direct coupling between the outgoing ð1Þ wave energy flux and the work done by the gravitational field on Here, Gμν consists of the terms in Gμν that are linear in hμν, and the sources. For nonharmonic gauges, the directly derived wave stress tensor has an apparent index asymmetry. This coordinate 1 ð2Þ tμν = G + ... , [5] artifact may be straightforwardly removed, and the symmetrized 8πG μν (still gauge-invariant) tensor then takes on its widely used form. ð2Þ Angular momentum conservation follows immediately. For any where Gμν represents the Einstein tensor terms quadratic in hμν, harmonic gauge, however, the stress tensor found is manifestly and so forth. Following standard practice, we refer to tμν as a symmetric from the start, and its derivation depends, in its entirety, “pseudotensor,” because it is Lorentz covariant but not a true μν on the structure of the linearized wave equation. tensor, unlike T , under full coordinate transformations. To leading nonvanishing order, the pseudotensor tμν is then inter- gravitational radiation | general relativity | theoretical astrophysics preted as the stress energy of the gravitational radiation itself. The sum Tμν + tμν is often referred to as the energy-momentum pseudotensor; a yet more general version of the pseudotensor, he recent detection of gravitational radiation (1) has greatly using the full metric gμν, is presented in the textbook of Landau heightened interest in this subject. Deriving an expression for T and Lifschitz (3). There are by now many routes that lead to a the correct form of the energy flux carried off in the form of suitable definition of an appropriate stress tensor for gravita- gravitational waves is a famously difficult undertaking at both the tional radiation without the use of a pseudotensor formalism. conceptual and technical levels. The heart of the difficulty is that We make no pretense of doing anywhere near full justice to this the stress energy of the gravitational field is neither a unique nor elegant and sophisticated literature here; this is not the intent a localizable quantity, because local coordinates can be found for of this article. Our purpose, rather, is to show how to obtain a which the field can be made to vanish by the equivalence prin- ciple. It is not a source of spacetime curvature; it is part of the Significance curvature itself, which manifests globally. Indeed, for many years, debate abounded as to whether there was any true energy Gravitational radiation provides a probe of unprecedented propagated by gravitational radiation. We know now of course power with which to elucidate important astrophysical pro- that there is, but the hunt for a suitable stress energy is a bur- cesses that are otherwise completely dark (e.g., black hole densome demand for those approaching the subject either as mergers) or impenetrable (e.g., supernova and early universe nonspecialists or newcomers. The currently generally adopted dynamics). Historically, the gap between propagating fluctua- textbook approach (2) is to first write the metric tensor gμν as the tions in the spacetime metric and classical dynamical con- following sum*: cepts such as energy and angular momentum conservation has bedeviled this subject. By now, there is a vast literature on this topic, and there are many powerful methods available. Because gμν = ημν + hμν, [1] of their mathematical sophistication, however, they are not used η in introductory texts, which are forced instead to a follow a where μν is the usual Minkowski metric and hμν is the departure much more cumbersome path. We present here a derivation of therefrom, and then to treat the latter as a small quantity. We the most widely used form of the stress energy tensor of grav- work throughout in quasi-Cartesian coordinates that differ only itational radiation, using elementary methods only. infinitesimally in linear order from strictly Cartesian coordinates, η so that μν is a constant tensor.* The Einstein tensor, Author contributions: S.A.B. performed research and wrote the paper. Reviewers: T.B., Bowdoin University; and J.-P.L., Institut d’Astrophysique. gμνR Gμν ≡ Rμν − , [2] The author declares no conflict of interest. 2 1Email: [email protected]. μ *In this paper, Greek indices indicate spatiotemporal dimensions; Roman indices indicate where Rμν is the Ricci tensor and R is its R trace, is then expanded μ spatial dimensions. We use the sign convention ½− +++ for the diagonal Minkowski μν μν in powers of the amplitudes of hμν in its various forms. With metric ημν. Indices of hμν [h ] are raised [lowered] with η ½= ημν. For inline equations, we μ μ the material stress energy tensor denoted by Tμν, the Newtonian use the notation ∂ ≡∂=∂xμ, ∂μ ≡∂=∂x .

11662–11666 | PNAS | October 18, 2016 | vol. 113 | no. 42 www.pnas.org/cgi/doi/10.1073/pnas.1614681113 Downloaded by guest on September 25, 2021 widely used form of the stress energy tensor for gravitational with an uncertain overall normalization factor that must be

radiation, making use only of elementary methods and conserved determined by such considerations as the work done on the INAUGURAL ARTICLE fluxes emerging from linear wave theory. wave sources. Note in particular that the equation for the sec- The calculation of the energy from the pseudotensor is suffi- ond-order energy flux is entirely determined by a linear-in-f ciently cumbersome that it is rarely done explicitly in textbooks wave equation. (merely summarized), although the final answer is not unduly A linear scalar wave equation is yet more revealing, and only involved. In an arbitrary gauge and a background Minkowski slightly more complicated. With Φ the effective potential, and ρ spacetime (4), the source density, consider the wave equation of scalar gravity, "* + * + * + κλ λκ λκ 1 ∂hκλ ∂h ∂h ∂hκμ ∂h ∂hκν ∂2Φ tμν = − − □Φ ≡− + ∇2Φ = 4πGρ. [11] 32πG ∂xμ ∂xν ∂xλ ∂xν ∂xλ ∂xμ ∂t2 # [6] ∂ ∂ □ ∇2 ’ − 1 h h (Here, and are the usual d Alembertian and Laplacian μ ν , ð = π Þ∂ Φ 2 ∂x ∂x operators, respectively.) Then, if we multiply by 1 4 G t , 2 integrate ð∂tΦÞ∇ Φ by parts, and regroup, this leads to the following: where ημν μ ∂ ∂Φ 2 ∂Φ ∂Φ = − ≡ μ ≡ = − 1 2 1 hμν hμν h, h hμ, h hμ h. − + j∇Φj + ∇ · ∇Φ = ρ . [12] 2 8πG ∂t ∂t 4πG ∂t ∂t 6 As explained in standard textbooks, the use of Eq. as a stress However, tensor makes sense only if an average over many wavelengths is performed, so that oscillatory cross products do not contribute. ∂Φ ∂ðρΦÞ ∂ρ This averaging is indicated by the angle bracket hi notation. ρ = − Φ ∂ ∂ ∂ Moreover, although the expression [6] is gauge invariant, in t t t [13] solving explicitly for hμν, a choice of gauge must be made. The ∂ðρΦÞ “ ” = + Φ∇ · ðρvÞ PHYSICS harmonic gauge is a convenient choice for the study of gravita- ∂t tional waves, as it greatly simplifies the mathematics. If hμν de- μ pends on its coordinates as a plane wave of the form expðikμx Þ,a ∂ðρΦÞ μ = = + ∇ · ðρvΦÞ − ρv · ∇Φ [14] harmonic gauge is actually required if kμ is a null vector, kμk 0. ∂ , All physical, curvature-inducing radiation (as opposed to oscil- t lating coordinate transformations) has this property (4). The where v is the velocity and the usual mass conservation equation harmonic gauge is defined by the following condition: has been used in the second equality. A simple rearrangement then leads to the following: ∂hμν = 0 ðharmonic gauge conditionÞ. [7] ∂xμ ∂E + ∇ · F = ρv · ∇Φ, [15] ∂t That it is always possible to find such a gauge is well known (4); the proof is similar to that of being able to choose the Lorenz where gauge condition in electrodynamics. In the “transverse traceless” h = 2 (TT) gauge, there is the additional constraint 0, which leads 1 ∂Φ 1 ∂Φ to the following simple result: E = ρΦ + + j∇Φj2 F = ρvΦ − ∇Φ π ∂ , π ∂ . * + 8 G t 4 G t κλ [16] 1 ∂hκλ ∂h tμν = ðTT gaugeÞ. [8] 32πG ∂xμ ∂xν The right side of [15] is minus the volumetric rate at which work is being done on the sources. For the usual case of compact For linear gravitational plane waves propagating in vacuum sources, the left side may then be interpreted as a far-field 2 2 [although not more generally (4)], a transformation to the TT wave energy density of ½ð∂tΦÞ + j∇Φj =8πG and a wave energy gauge can always be found without departing from the harmonic flux of −ð∂tΦÞ∇Φ=4πG. The question we raise here is whether an constraint; there is also a precise electrodynamic counterpart. analogous formal “direct method” might be used to shed some By way of contrast, in classical wave problems, finding a con- light on the origin of Eq. 6, including, very importantly, a means served wave energy flux is much more straightforward. Consider of extracting the overall normalization factor. the simplest example of a wave equation for a quantity f, There does indeed seem to be such a formulation, which we now discuss. ∂2f ∂2f − = 0. [9] ∂t2 ∂x2 Analysis _ Conserved Densities and Fluxes. Begin with the standard, gauge- Start by looking for a conserved flux. If we multiply by ∂f=∂t ≡ f, _ invariant general weak field linearized wave equation (2, 4): integrate the second term −f∂2f=∂x2 by parts, and regroup, this leads to the following: λ λ λρ ∂2h ∂2 ∂2 " # □ − μ − hν + η h = −κ [17] hμν ν λ μ λ μν λ ρ Tμν, ∂ _2 ð ′Þ2 ∂ ∂x ∂x ∂x ∂x ∂x ∂x f + f − _ ′ = [10] ∂ ∂ ff 0, t 2 2 x where κ = 16πG. We restrict our attention throughout this work to the case of a small metric disturbance hμν on a background where f′ ≡∂f=∂x. This readily lends itself to the interpretation of Minkowski spacetime. The material stress tensor Tμν is treated as _ ′ _ an energy density ½ f 2 + f 2=2 and an energy flux −ff′, although completely Newtonian.

Balbus PNAS | October 18, 2016 | vol. 113 | no. 42 | 11663 Downloaded by guest on September 25, 2021 Next, establish an identity by contracting Eq. 17 on μν: ! ! ∂ ∂μν ∂λμ ∂ ∂ ∂ S = − 1 hμν h + 1 h hμν + 1 h h [27] λρ ρ λ ρ , 2 4κ ∂x ∂xρ 2κ ∂x ∂xν 8κ ∂x ∂xρ ∂ h μ □ h + 2 = −κT ≡−κT. ∂xλ∂xρ μ and T ρσ is a flux tensor: Hence: ! ! ∂ ∂μν ∂λμ ∂ ∂ ∂ T = 1 hμν h − 1 h hμρ − 1 h h [28] λρ ρσ ρ σ λ σ ρ σ . ∂2h 1 κT 2κ ∂x ∂x κ ∂x ∂x 4κ ∂x ∂x = − □ h − , [18] ∂xλ∂xρ 2 2 Due to its second term, T ρσ is not symmetric in its indices. Index and we rewrite Eq. 17 as follows: asymmetry also arises in the development of the stress tensor of electromagnetic theory, and there are methods to correct this λ λ ∂2 2 η deficiency (5). Similar techniques may be brought to bear on hμ ∂ hν μν □ hμν − − − □ h = −κSμν, [19] the current problem, as we discuss below. For the moment, we ∂ ν∂ λ ∂ μ∂ λ μν x x x x 2 may note that, in a harmonic gauge ∂μh = 0 (not necessarily traceless), the asymmetry vanishes and the tensor becomes man- where the source function Sμν is the following: ifestly symmetric in ρσ. Notice that the wave stress tensor [6] is simply a symmetrized version of [28]. ημνT S T Sμν = Tμν − . [20] Rather than work with and ρσ each on its own, it is more 2 natural to form the composite tensor Uρσ,

We seek an energy-like conservation equation from the wave U ≡ T + η S [29] equation in the form displayed in Eq. 19. Toward that end, mul- ρσ ρσ ρσ . ∂ μν μν σ tiply by σh , summing over as usual and leaving free. The 26 first term on the left side of [19] is then the following: The left side of Eq. may then be written more compactly as a ! 4-divergence: μν μν μν ∂2 ∂ ∂ ∂ ∂ ∂ ∂2 hμν h = hμν h − hμν h μν ρ σ ρ σ ρ σ, ∂Uρσ 1 ∂h ∂x ∂xρ ∂x ∂xρ ∂x ∂x ∂x ∂xρ∂x = − [30] ∂ Sμν ∂ σ . ! xρ 2 x ∂ ∂ ∂μν ∂ ∂ ∂μν = hμν h − hμν h [21] 30 ∂ ∂ ρ ∂ σ ∂ ρ ∂ σ ∂ , It should be noted that the content of Eq. is exactly the same xρ x x x x xρ as that of the wave equation [17]: no more, no less. At this stage, ! ! μν μν note that we have not done any spatial averaging. In the TT ∂ ∂ ∂ ∂ ∂ ∂ = hμν h − 1 hμν h gauge, Eq. 29 leads directly to the following: ∂ ∂ ρ ∂ σ ∂ σ ∂ ρ ∂ . xρ x x 2 x x xρ ! μν μν 1 ∂hμν ∂h ∂hμν ∂h The second term on the left is handled similarly. Juggling indices, U00 = + , [31] 4κ ∂xi ∂xi ∂t ∂t λ μν λμ 2 2 ∂ hμ ∂h ∂ h ∂hμν ! − = − , [22] μν ∂ ν∂ λ ∂ σ ∂ ∂ λ ∂ σ 1 ∂hμν ∂h x x x xν x x U = , [32] 0i 2κ ∂t ∂xi leads to the following: ! ! μν μν λ μν λμ λμ 2 2 1 ∂hμν ∂h δ ∂hμν ∂h ∂ hμ ∂h ∂ ∂h ∂hμν ∂h ∂ hμν U = − ij [33] − = − + [23] ij ρ . ν λ σ λ σ λ σ , 2κ ∂xi ∂xj 2 ∂x ∂xρ ∂x ∂x ∂x ∂xν ∂x ∂x ∂x ∂x ∂xν U or equivalently, By these canonical forms, the component 00 is readily interpreted as a wave energy density, U0i as a wave energy flux, and ! ! U λ μν λμ λμ ij as a wave momentum stress. However, in fact, the combina- ∂2h ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ μν ρ − μ h = − h hμρ + 1 h hμν [24] tion ð∂ hμνÞð∂ρh Þ (and ½∂ h½∂ρh inamoregeneralharmonic ν λ σ λ σ σ λ . ∂x ∂x ∂x ∂xρ ∂x ∂x 2 ∂x ∂x ∂xν gauge) must vanish when averaged over many wavelengths, because the Fourier wave vector components satisfy the null con- ρ = The third term is identical to the second upon summation over μ straint k kρ 0. In the end, there emerges the very simple and ν. The fourth and final term of the left side of Eq. 19 is as results: follows: * + ∂ ∂μν ∂ ∂ U = = 1 hμν h − 1 h h ð Þ ∂2 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρσ tρσ κ ∂ ρ ∂ σ ∂ ρ ∂ σ harmonic , − 1 h h = − 1 h h + 1 h h [25] 2 x x 2 x x ρ σ ρ σ σ ρ . 2 ∂x ∂xρ ∂x 2 ∂xρ ∂x ∂x 4 ∂x ∂x ∂xρ * + [34] μν 1 ∂hμν ∂h = ðTTÞ. Thus, after dividing by 2κ, Eq. 19 takes the form of 2κ ∂xρ ∂xσ ∂S ∂T ∂μν + ρσ = − 1 h [26] σ Sμν σ , [19] ∂x ∂xρ 2 ∂x Direct Energy Loss. In going from the wave equation to an energy equation [30], we divided by 2κ. How do we know that where S is a scalar density: this particular normalization is the proper one for producing a

11664 | www.pnas.org/cgi/doi/10.1073/pnas.1614681113 Balbus Downloaded by guest on September 25, 2021 true energy flux? It is the right side of Eq. 26 that tells this story. □ hμν = −κTμνðharmonic gaugeÞ. [38] This is as follows: INAUGURAL ARTICLE μν μν μν This is certainly useful as a means to solve for hμν, but if we now 1 ∂h 1 ημν ∂h η ∂h μν − = − − − multiply by ∂σh , and regroup as before, we find the following: Sμν ∂ σ Tμν T ∂ σ ∂ σ 2 x 2 2 x 2 x [35] ! ! ∂ μν ∂ ∂ ∂μν ∂μν = − 1 h 1 hμν h = − 1 h ð Þ [39] Tμν σ . ρ σ σ Tμν harmonic gauge . 2 ∂x ∂xρ 2κ ∂x ∂x 2 ∂x We now set σ = 0, picking out the time component, and work in The difficulty now is that the right side has no obvious physical the Newtonian limit h00 ’ −2Φ, where Φ is the gravitational μν interpretation, and we are gauge-bound. It is only if we follow potential. We are then dominated by the 00 components of h the path of Eq. 37, retaining full gauge freedom, that we may and Tμν. Using the right arrow → to mean integrate by parts and simultaneously formulate a conserved flux on one side of the ignore the pure derivatives (as inconsequential for wave losses), “ ” 0μ equation with the readily interpretable work done combination and recalling the mass–energy conservation relation ∂μT = 0, μν −ðTμν=2Þ∂σh on the other. Within the same equation, the radi- we perform the following manipulations: ated waves and the effective Newtonian potential are best un- ∂ 00 ∂ ∂ 00 derstood each in their own gauge. Gauge selection (as an aid to 1 h 1 T00 00 1 T 00 − T00 → h = h interpretation) is the last, not the first, step of the analysis. Very 2 ∂x0 2 ∂x0 2 ∂x0 different gauges in very different regions are illustrative of the 1 ∂T0i nonlocal character of this problem. = − h00 2 ∂xi [36] Index Symmetry and Angular Momentum Conservation. The tensor 00 1 ∂h Uρσ lacks index symmetry for nonharmonic gauges. This is awk- → T0i 2 ∂xi ward for angular momentum conservation. To address this ’ −ρv · ∇Φ, problem, begin in Eq. 37 by setting σ = k, a spatial index. Next, multiply the equation by eijkxj, adhering to the usual summation PHYSICS which is the rate at which the effective Newtonian potential Φ convention for repeated spatial indices but not distinguishing does net work on the matter. (Here, ρ is the Newtonian mass between their covariant and contravariant placement. An in- density and v is the normal kinetic velocity. Averaging is under- tegration by parts then gives the following: stood; the hi notation has been suppressed in [36] for ease of ′ μν presentation.) This is negative if the force is oppositely directed ∂J ρ ∂ i − e U = − 1e h [40] to the velocity, so that the source is losing energy by generating ρ ijk jk ijkxjTμν , ∂x 2 ∂xk outgoing waves. Our TT gauge expression [8] for T 0i is also negative for an outward flowing wave of argument ðr − tÞ, r being where we have introduced a provisional angular momentum flux 0i spherical radius and t time. (By contrast, T would be positive.) of gravitational waves, A subtle but important point: can one be sure that such a potential actually exists? An ordinary Newtonian potential would ′ J iρ ≡ eijkxjUρk. [41] conserve mechanical energy over the course of the system’s evo-

lution. That a gauge does exist in which an appropriate effective Were Uρσ a symmetric tensor, eijkUjk would vanish identically, and potential function emerges is shown in standard texts (2, 4). This is an equation of strict angular momentum conservation would the Burke–Thorne potential (6, 7), which is proportional to the emerge. This suggests that the asymmetry is not truly fundamental. leading order “radiation reaction” term in an expansion of h00. Indeed, symmetry is easily restored. The method is to subtract The effective potential must emerge as part of the radiation re- “ ” U off an appropriate difference tensor from ρσ, which leaves the action terms in hμν if it is to deplete mechanical energy. This time- fundamental conservation equations intact. Begin by rewriting a dependent potential may be precisely defined in a suitable spatially averaged form of Uρσ as follows: “ ” Newtonian gauge. Although we make no explicit use of it here, * + it is given by the following (4): λμ 1 ∂h ∂hμσ ∂hμρ Uρσ = tρσ + ηρσS + − 2κ ∂xλ ∂xρ ∂xσ Φ = − 1 = G ðVÞ [42] h00 Ijk xjxk, 2 5 D ≡ tρσ + Uρσ, ðVÞ where Ijk is the traceless moment of inertia tensor, I refers to its fifth time derivative, and xj is a spatial Cartesian coordinate. where tρσ is the standard symmetric wave tensor given by Eq. 6 = κ D This justifies our overall normalization factor of 1 2 .Our and Uρσ, the (spatially averaged) difference tensor, is defined by D final energy equation in an arbitrary gauge thus takes the fol- this equation. It is easy to verify that Uρσ vanishes for the TT lowing form: gauge (we have already done so), but it is also a gauge-invariant quantity under the infinitesimal coordinate transformation ∂U ∂ μν λ λ λ ρσ = − 1 h [37] x → x + ξ , with the following: Tμν σ . ∂xρ 2 ∂x ∂ξ λ μ ∂ξν ∂ξ hμν → hμν − − + ημν . [43] The fact that the Newtonian gauge is not harmonic may have ∂xν ∂xμ ∂xλ contributed to this rather basic (work done) ↔ (wave flux) conservation equation, the analog of our scalar prototype in- Here, ξμ is a well-behaved but otherwise arbitrary vector func- troductory example, not being highlighted previously in the tion. In fact, it is a straightforward exercise to show that S and literature. If, for example, we follow custom and go directly the final ρσ-antisymmetric term in [42] are each gauge invariant to an harmonic gauge straight from Eq. 17, one obtains the on their own. A standard textbook problem is to show that tρσ is a following familiar result: gauge-invariant quantity (4); here, we have done so indirectly,

Balbus PNAS | October 18, 2016 | vol. 113 | no. 42 | 11665 Downloaded by guest on September 25, 2021 because Uρσ must be gauge invariant by virtue of its original The presented calculation also illuminates the physical con- D construction. Thus, if we evaluate Uρσ in the TT gauge, in which nection between the radiated gravitational waves and the effective it vanishes, and transform to any other gauge, the result must still Newtonian potential that serves to deplete mechanical energy vanish. We may therefore conclude that Uρσ = tρσ quite generally. from the matter source for these waves. The precise form of this Returning to the question of angular momentum conservation, “Burke–Thorne” potential does not itself play a role in our anal- U we replace ρσ with tρσ, and define the symmetrized angular ysis. Merely the fact that it exists, and that in common with any momentum flux tensor as follows: Newtonian potential function it is associated with −h00=2to leading order, is sufficient to determine the normalization con- J ρ ≡ e x tρ . [44] i ijk j k stant of the conserved flux tensor. Indeed, the entire calculation The precise statement of angular momentum conservation is then could be performed in a harmonic gauge, in which case −ð = Þ∂ μν as follows: Tμν 2 σh must be the generic expression for the work done on the sources, even if its manifestation is less transparent ∂J ∂ μν than in the Newtonian gauge. iρ = − 1e h [45] ρ ijkxjTμν . We have shown how to remove an apparent index asymmetry ∂x 2 ∂xk in Uρσ, which in its symmetrized and spatially averaged form The right side affords a direct method for computing angular reverts to tρσ. Angular momentum conservation readily follows. momentum loss via the explicit Burke–Thorne potential. The approach that is presented in this paper seems to be the simplest, the most concise, and ultimately the most physically Conclusion transparent route to understanding the form of the stress energy The linear wave equation that emerges from the Einstein field tensor of gravitational radiation, especially in its most natural equations, either in the form of [17] or [19], contains in itself all harmonic gauges, as embodied in Eq. 34. of the ingredients needed for determining a conserved gravitational wave energy flux tensor, propagating in a background ACKNOWLEDGMENTS. I am most grateful to J. Binney and P. Ferreira for Minkowski spacetime and produced by slowly moving sources. critical readings of an early draft of this work and for their many helpful The stress tensor that is calculated via a more lengthy and suggestions. It is likewise a pleasure to acknowledge stimulating conversa- complex second-order analysis of the Einstein tensor is, for any tions with R. Blandford, P. Dellar, C. Gammie, M. Hobson, D. Lynden-Bell, harmonic gauge, identical to that which emerges from our first- J. Magorrian, C. McKee, J. Papaloizou, and W. Potter. Finally, it is pleasure to U thank the referees T. Baumgarte and J.-P. Lasota for their excellent advice order calculation, that is, ρσ and tρσ are identical in this case. It and support. I acknowledge support from a gift from the Hintze Charitable is only for the construction of a symmetric wave stress tensor in Fund, from the Royal Society in the form of a Wolfson Research Merit nonharmonic gauges that an alteration of form is needed. Award, and from the Science and Technology Facilities Council.

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