Black holes and singularities

Question

Is the mass of a black hole determined by the density of the singularity or is it determined by the mass of the singularity?

Answer 1

All solutions to the Einstein-Maxwell equations can be characterised by the mass $M$, charge $Q$ and angular momentum $J$ of a solution, due to the no hair theorem.

Thus, it is more appropriate to view these parameters as determining characteristics of the geometry of the black hole. We also cannot prescribe a mass to the singularity itself nor the density of the singularity; these notions are ill-defined as general relativity breaks down at that point when the curvature becomes singular. Nobody knows what happens at the singularity, yet.

That being said, there are a number of ways we may be able to infer these properties. For example, for a rotating, charged black hole, there is a 4-potential,

$$A_\mu = \frac{Qr^3}{r^4 + a^2 z^2}k_\mu$$

and obviously gives rise to an electric $\vec E = -\partial_t \vec A$ and magnetic $\vec B = \nabla\times \vec A$ field. Knowing the charge and angular momentum, one could find the mass since $a = J/Mc$.

Answer 2

You can not determine a black hole's mass by looking at the density of the singularity because most of the scientist believes that density of a singularity of a black hole is infinite. Even if it is not, how do you expect to measure the density or the mass of a singularity? There are other ways to determine the mass of a black hole. I recommend you to check this link How do we determine the mass of a black hole?

Answer 3

The mass of a black hole is determined by the mass-energy which has passed inside its event horizon. Indeed, the area of the event horizon is proportional to the mass-energy it has swallowed and is also proportional to the entropy of its dinner.

Whether this material has reached the singularity is immaterial; this is fortunate, because the timescales on which different observers might see this happening vary wildly.