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3 The Stress-Energy-Momentum Tensor and the Einstein Equations

Any discussion of relativistic physics must include the stress-energy-momentum tensor Tμν. It is as important for general relativity as G μν in that it enters the Einstein equations in as direct a way as possible, i.e. 

G μν = 8πT μν. (65 )
Misner, Thorne, and Wheeler [80Jump To The Next Citation Point] refer to T μν as “…a machine that contains a knowledge of the energy density, momentum density, and stress as measured by any and all observers at that event”.

Without an a priori, physically-based specification for Tμν, solutions to the Einstein equations are devoid of physical content, a point which has been emphasized, for instance, by Geroch and Horowitz (in [52]). Unfortunately, the following algorithm for producing “solutions” has been much abused: (i) specify the form of the metric, typically by imposing some type of symmetry, or symmetries, (ii) work out the components of G μν based on this metric, (iii) define the energy density to be G00 and the pressure to be G11, say, and thereby “solve” those two equations, and (iv) based on the “solutions” for the energy density and pressure solve the remaining Einstein equations. The problem is that this algorithm is little more than a mathematical game. It is only by sheer luck that it will generate a physically viable solution for a non-vacuum spacetime. As such, the strategy is antithetical to the raison d’être of gravitational-wave astrophysics, which is to use gravitational-wave data as a probe of all the wonderful microphysics, say, in the cores of neutron stars. Much effort is currently going into taking given microphysics and combining it with the Einstein equations to model gravitational-wave emission from realistic neutron stars. To achieve this aim, we need an appreciation of the stress-energy tensor and how it is obtained from microphysics.

Those who are familiar with Newtonian fluids will be aware of the roles that total internal energy, particle flux, and the stress tensor play in the fluid equations. In special relativity we learn that in order to have spacetime covariant theories (e.g. well-behaved with respect to the Lorentz transformations) energy and momentum must be combined into a spacetime vector, whose zeroth component is the energy and the spatial components give the momentum. The fluid stress must also be incorporated into a spacetime object, hence the necessity for Tμν. Because the Einstein tensor’s covariant divergence vanishes identically, we must have also ∇ μT μν = 0 (which we will later see happens automatically once the fluid field equations are satisfied).

To understand what the various components of Tμν mean physically we will write them in terms of projections into the timelike and spacelike directions associated with a given observer. In order to project a tensor index along the observer’s timelike direction we contract that index with the observer’s unit four-velocity U μ. A projection of an index into spacelike directions perpendicular to the timelike direction defined by Uμ (see [105Jump To The Next Citation Point] for the idea from a “3 + 1” point of view, or [21Jump To The Next Citation Point] from the “brane” point of view) is realized via the operator ⊥ μ ν, defined as

μ μ μ μ ⊥ ν= δ ν + U U ν, U Uμ = − 1. (66 )
Any tensor index that has been “hit” with the projection operator will be perpendicular to the timelike direction associated with U μ in the sense that ⊥ μU ν = 0 ν. Figure 4View Image is a local view of both projections of a vector μ V for an observer with unit four-velocity μ U. More general tensors are projected by acting with μ U or μ ⊥ ν on each index separately (i.e. multi-linearly).
View Image

Figure 4: The projections at point P of a vector μ V onto the worldline defined by μ U and into the perpendicular hypersurface (obtained from the action of ⊥ μν).

The energy density ρ as perceived by the observer is (see Eckart [39Jump To The Next Citation Point] for one of the earliest discussions)

ρ = TμνU μU ν, (67 )
ρ ν 𝒫 μ = − ⊥ μ U Tρν (68 )
is the spatial momentum density, and the spatial stress 𝒮 μν is
σ ρ 𝒮μν = ⊥ μ⊥ν Tσρ. (69 )
The manifestly spatial component Sij is understood to be the ith-component of the force across a unit area that is perpendicular to the th j-direction. With respect to the observer, the stress-energy-momentum tensor can be written in full generality as the decomposition
Tμν = ρ UμU ν + 2U(μ𝒫ν) + 𝒮μν, (70 )
where 2U(μ𝒫 ν) ≡ U μ𝒫 ν + U ν𝒫μ. Because U μ𝒫μ = 0, we see that the trace T = T μμ is
T = 𝒮 − ρ, (71 )
where μ 𝒮 = 𝒮 μ. We should point out that use of ρ for the energy density is not universal. Many authors prefer to use the symbol ɛ and reserve ρ for the mass-density. We will later (in Section 6.2) use the above decomposition as motivation for the simplest perfect fluid model.

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