12.1 The Komar integral for spacetimes with Killing vectors

UpdateJump To The Next Update Information Although the Komar integral (and, in general, the linkage (3.15View Equation) for some α) does not satisfy our general requirements discussed in Section 4.3.1, and does not always give the standard values in specific situations (see, for example, the ‘factor-of-two anomaly’ or the examples below), in the presence of a Killing vector, the Komar integral, built from the Killing field, could be a very useful tool in practice. (For Killing fields the linkage L 𝒮[K ] reduces to the Komar integral for any α.)

One of its most important properties is that in vacuum L [K ] 𝒮 depends only on the homology class of the two-surface (see, e.g., [534Jump To The Next Citation Point]). This follows directly from the explicit form of Komar’s canonical Noether current: 1 8πGCa [K ] = GabKb + ∇b ∇ [aKb ] = − 2RKa − ∇b (∇ (aKb ) − gab∇cKc ). In fact, if 𝒮 and 𝒮′ are any two two-surfaces such that 𝒮 − 𝒮 ′ = ∂Σ for some compact three-dimensional hypersurface Σ on which the energy-momentum tensor of the matter fields is vanishing and Ka is a Killing vector, then ′ L 𝒮[K ] = L 𝒮[K ]. (Note that, as we already stressed, the structure of the Noether current above dictates that the numerical coefficient in the definition (3.15View Equation) of the linkage would have to be -1-- 16πG rather than 8π1G-, i.e., the one that gives the correct value of angular momentum (rather than the mass) in Kerr spacetime.) In particular, the Komar integral for the static Killing field in the Schwarzschild spacetime is the mass parameter m of the solution for any two-surface 𝒮 surrounding the black hole, but it is zero if 𝒮 does not surround it. The explicit form of the current shows that, for timelike Killing field Ka, the small sphere expression of Komar’s quasi-local energy in the first non-trivial order is − 2πr3T gabtKc 3 ab c, i.e., it does not reproduce the expected result (4.9View Equation); moreover, in vacuum it always gives zero rather than, e.g., the Bel–Robinson ‘energy’ (see Section 4.2.2).

Furthermore [510], the analogous integral in the Reissner–Nordström spacetime on a metric two-sphere of radius r is m − e2∕r, which deviates from the generally accepted round-sphere value m − e2∕(2r). Similarly, in Einstein’s static universe for spheres of radius r on a t = const. hypersurface, L 𝒮[K ] is zero instead of the round sphere result 4π 3 3 r [μ + λ∕8πG ], where μ is the energy density of the matter and λ is the cosmological constant.

Accurate numerical calculations show that in stationary, axisymmetric asymptotically flat spacetimes describing a black hole or a rigidly-rotating dust disc surrounded by a perfect fluid ring the Komar energy of the black hole or the dust disc could be negative, even though the conditions of the positive energy theorem hold [21Jump To The Next Citation Point]. Moreover, the central black hole’s event horizon can be distorted by the ring so that the black hole’s Komar angular momentum is greater than the square of its Komar energy [20Jump To The Next Citation Point].

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