One of its most important properties is that in vacuum depends only on the homology class of the two-surface (see, e.g., ). This follows directly from the explicit form of Komar’s canonical Noether current: . In fact, if and are any two two-surfaces such that for some compact three-dimensional hypersurface on which the energy-momentum tensor of the matter fields is vanishing and is a Killing vector, then . (Note that, as we already stressed, the structure of the Noether current above dictates that the numerical coefficient in the definition (3.15) of the linkage would have to be rather than , 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 of the solution for any two-surface surrounding the black hole, but it is zero if does not surround it. The explicit form of the current shows that, for timelike Killing field , the small sphere expression of Komar’s quasi-local energy in the first non-trivial order is , i.e., it does not reproduce the expected result (4.9); moreover, in vacuum it always gives zero rather than, e.g., the Bel–Robinson ‘energy’ (see Section 4.2.2).
Furthermore , the analogous integral in the Reissner–Nordström spacetime on a metric two-sphere of radius is , which deviates from the generally accepted round-sphere value . Similarly, in Einstein’s static universe for spheres of radius on a hypersurface, is zero instead of the round sphere result , where is the energy density of the matter and 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 . 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 .
Living Rev. Relativity 12, (2009), 4
This work is licensed under a Creative Commons License.