What causes the mass of a black hole?

Question

I heard in a lecture about the Higgs mechanism that the mass of a black hole has nothing to do with the Higgs mechanism. The point was made in relation to the proton mass being largely due to the energy content within the proton, not the mass of the constituent quarks.

I don't know what the claim 'the mass of a black hole has nothing to do with the Higgs mechanism' means. Once matter passes over the event horizon of a black hole, does the mass of the black hole arise through some other mechanism than simply 'adding' that mass to it?

Answer 1

The answer to your question is stress-energy, not mass. Mass is just a certain form, a manifestation of something deeper, being energy. A proton and a black hole, both have stress-energy, and both cause spacetime curvature.

It's a commonly made mistake that gravity, and therefore a black hole, is caused by matter. In fact the spacetime curvature is related to a quantity called the [stress-energy tensor][1]. This is usually represented by a matrix with ten independant values in it (it's a 4x4 matrix but it's symmetric so six of the elements in it are duplicated). Only one of the elements in the matrix, $T_{00}$, depends directly on the mass, and actually that element gives the energy density, where mass is counted as energy using Einstein's equation $e = mc^2$.

Why do photons add mass to a black hole?

In GR, we use the term stress-energy, and anything and everything, massive or massless that does have stress-energy (please note that in certain cases not all parts of the stress-energy momentum tensor contribute to curvature), will cause spacetime curvature.

The mass of a black hole can be determined in the same way as we determine the mass of any other astronomical object - by observing how it deflects the paths of other objects that pass close to it. These might be objects in closed elliptic orbits (like satellites around planets) or objects in hyperbolic trajectories that are just passing through.

Black hole mass

Please note that in certain cases, we use ADM (or Bondi) mass for black holes.

Note that mass is computed as the length of the energy–momentum four-vector, which can be thought of as the energy and momentum of the system "at infinity".

https://en.wikipedia.org/wiki/Mass_in_general_relativity

So basically, when we talk about the mass of a black hole, we really mean according to GR, its energy and momentum (length of the energy-momentum four vector), and its (gravitational) effects on paths of objects that are interacting with it (its gravitational field). No need to involve the Higgs mechanism. So the answer to your question is, what causes the mass of the black hole, is ultimately stress-energy.

Answer 2

General relativity does not distinguish between matter and energy. As far as gravitation goes they are the same and are related by Einstein's famous equation $E = mc^2$. The gravitational field of a proton is the same whether we treat it as a mass of $1.673 \times 10^{-27}$ kg or an energy of $938.2$ MeV.

So whether or not the Higgs field confers a rest mass to the proton is irrelevant once we've dropped that proton into the black hole. Its contribution to the gravitational field would be the same either way. This is probably what the statement you read refers to.