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 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 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 .

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 [3], 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 [2]; 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 .

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) . 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 . But there are no rigorous mathematical results on BKL singularities; it currently remains as an open problem .

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.