### ¿Dónde se me habrá perdido la masa?

(Pedimos disculpas por escribir este post en inglés, pero en este caso nos ha  parecido más apropiado usar este idioma para comentar un artículo escrito en  él.)

After a period of silence that we have devoted to our real-life jobs, we are back, with  replenished energies, to comment the article Where is the mass of a black hole? Recently published by Francisco Villatoro (Francis) in the otherwise interesting blog Mapping Ignorance. We are afraid that in this case  the title of the blog describes the content of the article very well. Francis is a well-known science blogger who also give many popular science talks, participates in podcasts etc. We have commented some of his work in other
posts… to no avail. We will keep trying, because his articles start our alarms and red lights very often and many people read them...

This article contains so many and so deep misconceptions about the mass in General Relativity that it is not surprising that he arrives to such a great deal of wrong conclusions.  The main problem to describe the various problems present in Francis’ article (a task which takes a lot of unpaid (unlike his) time) is that the article does not start by giving a proper definition of the mass of a black hole that one can discuss in precise scientific terms (the
different masses he mentions are, in general, different ways of calculating the same mass at spatial infinity, for which there is a unique concept in GR, the exceptions being masses defined at null infinity, which are not meant to be used for black-hole spacetimes). As a result, the author will always be  able to bullshit his way out of any criticism. Nevertheless, we will try to do it.

First of all, in General relativity, the energy-momentum tensor that appears in the right-hand side of the Einstein equations is the energy-momentum tensor of matter. Its components can be identified as the matter energy, momentum and pressure densities in some reference frame. It is not possible to construct from this tensor a relativistic invariant that corresponds to the mass (or energy) of matter, but this is not a very relevant problem because of the
non-linearity of the gravitational field: matter sources the gravitational field, which also sources itself.

The total energy observed in some reference frame outside a star, for instance, is larger than the energy corresponding to the matter of the star alone. Part of this energy (mass) corresponds to the gravitational field itself. Since the gravitational field extends all the way to infinity, there is a contribution to the gravitational energy in the whole spacetime and, at
some distance from the source, spacetime is curved by all this energy and not just by that corresponding to the matter source.

This is why there can be vacuum solutions which are sourced by the self-interaction of the gravitational field alone.  This is similar to the vacuum solutions of non-Abelian Yang-Mills fields and the non-Abelian charge.

Thus, asking where the mass of a spacetime is, is just the wrong question: the mass, which is the energy of the spacetime, is everywhere. The mass of the Schwarzschild black hole is distributed somehow over the whole spacetime. Even if there was a distributional energy-momentum tensor at the singularity, it would not contain all the mass of the black hole. (By the way: in General Relativity “null” does not mean zero.... It means light-like)

Secondly, in General Relativity one can only define in a rigorous, unambiguous way, the total energy of a spacetime (which we will take to be asymptotically flat for simplicity), because it is not possible to define a local density of energy of the gravitational field.  The reason is that the Principle of Equivalence says that the gravitational field can be made to disappear at any given point using the appropriate coordinate system. If the gravitational field disappears, its local energy density would disappear as well. But this is not how an energy-density function behaves.

Over the years, all proposals of local energy density made have failed. This means that we can never say where the energy is concentrated or how it is distributed over the spacetime. But we can define the total energy of the spacetime (sources, if any, plus gravitational field self-interaction energy). This is the concept that the different “masses” mentioned by the
author try to represent in formulae: ADM, Komar or Abbott-Deser. In the black-hole spacetimes that the author discusses all these formulae give exactly the same value, as it should be.

The author says that

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.

This makes no sense at all. The total mass of the Kerr-Newman black-hole spacetime is M, whether you compute it use the Komar formula, the Abbott-Deser formula or the ADM formula. Period.

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.

We do not know about this consensus, really… There are lots of problems in the use of distributions in curved spacetime. Many results depend on the coordinates used. This is one of them.

Another problem with this statement is that the expression “spherical coordinates” does not define a coordinate system. Schwarzschild, Eddington-Finkelstein, isotropic, or Kruskal-Szekeres coordinates, all of them are spherically symmetric.  Since the maximal analytical extension of the Schwarzschild spacetime is given by the Kruskal-Szekeres coordinates, one
should probably use these. In these coordinates, the singularity at r=0 is not a point, as can be seen in the Penrose diagram, but, in any case, it is in the future (not in a given place, but in a given time) and it is hard to imagine how mass in the future can source gravity in the present… In addition r=0 corresponds to another singularity (the so-called “white-hole”
singularity). (Yes, there are two disconnected singularities in the Kruskal-Szekeres extension ode the Schwarzschild solution. One can safely ignore the last one we have mentioned if the black hole is the product of the gravitational collapse of a star, but the problems mentioned in relation to the other singularity are still there).

The author says:

In summary, we don’t know where the mass is located inside a black hole.

At this point, we are totally convinced that the author does not know where the mass of a spacetime is. Indeed. He can keep searching it inside the black hole. Not even a theory of Quantum Gravity will help him find it!