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

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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 À Á 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.
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