12.3 Expressions in static spacetimes

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12.3.1 Tolman’s energy for static spacetimes

Let Ka be a hypersurface-orthogonal timelike Killing vector field, Σ a spacelike hypersurface to which Ka is orthogonal, and f2 := KaKa. Then − DaDaf = 4πG (μ + 3p − 4λπG)f, a field equation for f, follows from Einstein equations (see, e.g., pp. 71 – 74 of [240Jump To The Next Citation Point] or [199Jump To The Next Citation Point]). Here μ := Tabtatb and : 1 ab p = − 3Tabh, the energy density and the average spatial pressure of the matter fields, respectively, seen by the observer at rest with respect to Σ (or a K).

In the study of (‘quasi-static’) equilibrium configurations of self-gravitating systems Tolman [520, 521] found the integral

∫ ∮ ( --λ--) --1-- a ET (D ) := μ + 3p − 4πG fdΣ = 4πG v (Daf )d𝒮 (12.1 ) D ∂D
to be the energy of the system. Here D ⊂ Σ is a compact domain with smooth boundary ∂D, va is the outward pointing unit normal of ∂D in Σ; and the second expression follows from the field equation for f above. ET (D ) can in fact be interpreted as some form of a quasi-local energy [2Jump To The Next Citation Point, 3Jump To The Next Citation Point], called the Tolman energy. Clearly, for matter fields with non-negative energy density and average pressure and non-positive cosmological constant this is non-negative on the domain where a K is timelike. Using the defining equation of ET in terms of the two-surface integral, one can show that in asymptotically flat spacetimes it tends to the ADM energy as a non-decreasing set function. The second expression in Eq. (12.1View Equation) implies that, similarly to Komar’s expression, in vacuum ET depends only on the homology class of the 2-surface ∂D. Thus, in particular, it associates zero energy with vacuum domains. For spherically symmetric configurations on round spheres with the area radius r (see Section 4.2.1) it is r3f eα(8πGT vavb − λ + 1-[1 − e− 2α]) 2G ab r2, which in vacuum reduces to the Misner–Sharp energy.

The Tolman energy appeared to be a useful tool in practice: By means of ET Abreu and Visser gave remarkable entropy bounds for localized, but uncollapsed bodies [2Jump To The Next Citation Point, 3Jump To The Next Citation Point]. (We discuss this bound in Section 13.4.3.)

12.3.2 The Katz–Lynden-Bell–Israel energy for static spacetimes

Let 𝒮K := {f = K }, the set of those points of Σ where the length of the Killing field is the value K, i.e., 𝒮K are the equipotential surfaces in Σ. Let DK ⊂ Σ be the set of those points where the magnitude of Ka is not greater than K. Suppose that D K is compact and connected. Katz, Lynden-Bell, and Israel [309Jump To The Next Citation Point] associate a quasi-local energy to the two-surfaces 𝒮K as follows. Suppose that the matter fields can be removed from intDK and concentrated into a thin shell on 𝒮K in such a way that the space inside is flat but the geometry outside remains the same. Then, denoting the (necessarily distributional) energy-momentum tensor of the shell by Tab s and assuming that it satisfies the weak energy condition, the total energy of the shell, ∫ K Tabt dΣ DK a s b, is positive. Here ta is the future-directed unit normal to Σ. Then, using the Einstein equations, the energy of the shell can be rewritten in terms of geometric objects on the two-surface as

1 ∮ EKLI (𝒮K ) :=-----K [k]d𝒮K, (12.2 ) 8 πG 𝒮K
where [k ] is the jump across the two-surface of the trace of the extrinsic curvatures of the two-surface itself in Σ. Remarkably enough, the Katz–Lynden-Bell–Israel quasi-local energy EKLI in the form (12.2View Equation), associated with the equipotential surface 𝒮K, is independent of any distributional matter field, and can also be interpreted as follows. Let hab be the metric on Σ, kab the extrinsic curvature of 𝒮K in (Σ,hab) and k := habkab. Then, suppose that there is a flat metric h0ab on Σ such that the induced metric from h0 ab on 𝒮K coincides with that induced from hab, and 0 h ab matches continuously to hab on 𝒮K. (Thus, in particular, the induced area element d𝒮K determined on 𝒮K by hab, and h0 ab coincide.) Let the extrinsic curvature of 𝒮K in h0 ab be k 0 ab, and k0 := habk0 ab. Then E (𝒮 ) KLI K is the integral on 𝒮 K of K times the difference 0 k − k. Apart from the overall factor K, this is essentially the Brown–York energy.

In asymptotically flat spacetimes EKLI(𝒮K ) tends to the ADM energy [309]. However, it does not reduce to the round-sphere energy in spherically-symmetric spacetimes [374], and, in particular, gives zero for the event horizon of a Schwarzschild black hole.

12.3.3 Static spacetimes and post-Newtonian approximation

The Newtonian limit of general relativity is defined in [240Jump To The Next Citation Point], pp. 71 – 74, via static and (at spatial infinity) asymptotically flat spacetimes. The Newtonian scalar potential ϕ is identified with the logarithm of the length of the Killing vector, i.e., (in traditional units) it is 2 ϕ = c lnf. Decomposing the energy density μ as the sum of the rest-mass energy and the internal energy, 2 μ = c ρ + u, Einstein’s equations yield the field equation

ab 4πG-( c4λ-) -1 ab( )( ) − h DaDb ϕ = 4πG ρ + c2 u + 3p − 4πG + c2 h Daϕ Dbϕ . (12.3 )
Identifying the last term as 4πG- c2 (U + 3P), the sum of the energy density and three times of the average spatial stress of the field ϕ, (see the definitions (3.2View Equation) in the Newtonian case), the exact field equation (12.3View Equation) can be compared with the naïve, relativistically corrected Newtonian field equation (3.3View Equation); and one can read off the various relativistic corrections [199Jump To The Next Citation Point]. Then (3.4View Equation) motivates the definition of an ‘effective’ quasi-local energy as the integral of all the effective source terms on the right hand side of the exact field equation (12.3View Equation):
∫ ( c4λ 1 ) c2 ∮ ED := μ + 3p − -----− ----|Da ϕ|2 dΣ = ----- va (Da ϕ)d𝒮. (12.4 ) D 4πG 4πG 4πG 𝒮
If the spacetime is asymptotically flat at spatial infinity (in which case λ = 0) such that the hypersurface extends to spatial infinity, then, using e.g. the result of [60], one can show that E D tends to the ADM energy as D is enlarged to exhaust the whole Σ [199Jump To The Next Citation Point]. Since in the vacuum region the integrand of the 3-dimensional integral is negative definite, near infinity ED tends to the ADM energy as a monotonically decreasing set function. However, because of the extra relativistic correction term 4πG2-3P = 4π2GU c c in the source, the rate of change of this set function deviates from the one in the naïve relativistically corrected Newtonian theory of Section 3.1.1. In fact, for a two-sphere of radius r in the Schwarzschild spacetime with mass parameter m the quasi-local energy, for large r, is EDr = 2mG-(1 + mr) + 𝒪 (r−2), rather than EDr = 2mG(1 + 12mr) + 𝒪 (r−2) (see Section 3.1.1).

Though ED is negative in the vacuum regime, for spherically symmetric configurations, when the material source of the gravitational ‘field’ is contained in D, it is positive if an energy condition is satisfied; and it is zero if and only if the domain of dependence of D in the spacetime is flat. (For the details see [199].)

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