Black holes are vacuum solutions of the Einstein equation. Hence, the energy-momentum tensor for a black hole is null at every point of space. The only place where its mass can be located is where there is no space, i.e., at the singularity. Hence, for a Schwarzschild black hole, the mass is located at the origin in spherical coordinates, the centre of its circular event horizon. However, either for a charged, Newman black hole, or a rotating, Kerr black hole, such a definition results in some (apparent) paradoxes.
Mass is a parameter that characterizes a static, uncharged black hole. In the theory of general relativity there is no unique definition of this parameter (ADM mass, Bondi mass, Komar mass, etc.); in contrast to the theory of special relativity, where the mass is uniquely defined (based in either the equation E = m c², or in the non-relativistic limit) as the norm of the energy-momentum four-vector. For example, in a Kerr–Newman, charged, rotating black hole, the mass in E = m c² is equal to m² = M² + Q² + J², where J is the normalized angular momentum, Q is the normalized charge, and M is the irreducible mass; obviously, only for Q=0 and J=0 results in m=M.
The consensus is that in the case of Schwarzschild space-time, the energy-momentum tensor (that determines the density of mass) is a Dirac delta function at the singularity. Einstein equations in vacuum are satisfied at every space point except at the singularity at r=0 in spherical coordinates. In order for the solution to minimize Einstein action, a boundary condition is required at r=0, resulting in the appearance of an integration constant; the non-relativistic limit shows that its value coincides with that of the mass of the black hole. This calculation is analogous to that of the charge density of a point particle in classical electromagnetism.
For mathematicians, since electromagnetism is a linear theory, but general relativity is a strongly non-linear one, the use of generalized functions is a must. Balasin and Nachbagauer 1 use distributional techniques in order to calculate the energy-momentum tensor of the Schwarzschild geometry by using several regularizations; they confirm that it is a Dirac delta function, as expected. However, the coincidence in the result among several regularizations is not enough for a rigorous calculation. Fortunately, Balasin 2 uses the Colombeau’s calculus, that allows the multiplication of generalized functions, to obtain exactly the same result. Such a calculation will satisfy all mathematicians.
An electrically charged black hole is the vacuum solution of Einstein–Maxwell equations described by the Reissner–Nordström spacetime. For M² < Q² it is an unphysical, naked singularity; for M² ≥ Q² it has a point singularity where the mass (M) and the charge (Q) are located. However, the energy-momentum tensor is non-null in the space, due to the contribution of the electromagnetic field, being the sum of a Dirac delta plus a potential following an inverse quartic law 3.
The calculation of the energy-momentum tensor for a rotating black hole is more difficult. The Kerr–Newman spacetime is characterized by three parameters (M, J, Q) with a ring singularity for J ≠ 0. By means of using the Kerr–Schild decomposition of the metric and the theory of tensor-distributions, Balasin and Nachbagauer , have shown that, as expected, the energy-momentum tensor-distribution has its support on the singular region of the geometry, the ring. However, there is an additional contribution concentrated on the disk spanned by the ring; its origin is the topology of the maximal analytic extension of the Kerr solution, which leads to a branch singularity when the metric is written using the Kerr–Schild decomposition. This result has been confirmed by using the Colombeau’s theory ; however, there are some doubts since the embedding used is not unique and it was not shown that any reasonable embedding gave the same result 4.
In fact, the real world is quantum mechanical. In the vicinity of classical singularities, quantum effects are expected to become strong. Without a quantum theory of the gravitational field we can only resort to conjectures based on semiclassical analysis, like that of Belinski, Khalatnikov and Lifshitz (BKL) 5. The BKL singularity conjecture states that space-time shows a chaotic dynamics near singularities, where even the notion of space-time is not applicable. The BKL scenario is compatible with the results of numerical simulations 6. But there are no rigorous mathematical results on BKL singularities; it currently remains as an open problem 7.
In summary, we don’t know where the mass is located inside a black hole. We need a quantum theory of gravitation in order to solve this seemingly simple question. Unfortunately, current candidates to such a theory, like string theory and loop quantum gravity do not offer a clear answer to this question.
- Herbert Balasin, Herbert Nachbagauer, “The energy-momentum tensor of a black hole, or what curves the Schwarzschild geometry?” Classical and Quantum Gravity 10: 2271–2278 (1993), doi: 10.1088/0264-9381/10/11/010, arXiv:gr-qc/9305009. ↩
- Herbert Balasin, “Distributional energy – momentum tensor of the extended Kerr geometry,” Classical and Quantum Gravity 14: 3353–3362 (1997), doi: 10.1088/0264-9381/14/12/018, arXiv:gr-qc/9702060. ↩
- Herbert Balasin, Herbert Nachbagauer, “Distributional energy–momentum tensor of the Kerr–Newman spacetime family,” Classical and Quantum Gravity 11: 1453–1461 (1994), doi: 10.1088/0264-9381/11/6/010, arXiv:gr-qc/9312028. ↩
- Roland Steinbauer, James A. Vickers, “The use of generalized functions and distributions in general relativity,” Classical and Quantum Gravity 23: 91–R114 (2006), doi: 10.1088/0264-9381/23/10/R01, arXiv:gr-qc/0603078. ↩
- Andrzej Góźdź, Włodzimierz Piechocki, Grzegorz Plewa, “Quantum Belinski–Khalatnikov–Lifshitz scenario,” The European Physical Journal C 79: 45 (2019), doi:
10.1140/epjc/s10052-019-6571-4, arXiv:1807.07434 [gr-qc]. ↩
- David Garfinkle, “Numerical Simulations of Generic Singularities,” Physical Review Letters. 93: 161101 (2004), doi: 10.1103/PhysRevLett.93.161101, arXiv:gr-qc/0312117. ↩
- Alan A. Coley, “Mathematical General Relativity,” arXiv:1807.08628 [gr-qc]. ↩