Mass in general relativity (for black hole)

Based on my knowledge, we have several formalism to calculate the mass of black hole such as Brown-York formalism, ADM method, Hawking mass, Komar,...

What is the difference between them and how should we select one of them to calculate the mass of a black hole? For example, in order to calculate the mass of a Dilaton Maxwell black hole in AdS spacetime, why should we use the ADM method instead of B-Y formalism (http://arXiv.org/abs/0912.4199)?

• The fact that GR does not a unique way to compute the mass is not a bug, it is a feature. Computing the mass is overrated and customers don't need it. – t_d Jun 28 at 14:41

This won't be a complete answer, because I'm not familiar with Brown-York, but:

The Komar mass applies to stationary spacetimes.

The ADM mass is for asymptotically flat spacetimes, and includes gravitational radiation going to null infinity.

Bondi mass is similar to ADM, but doesn't include the radiation.

Komar and ADM were later shown to be consistent in the cases where both of them apply: https://arxiv.org/abs/1003.5015

The paper by Sheykhi that you link to is about black holes in anti-de Sitter space, which isn't asymptotically flat. In a spacetime that's not asymptotically flat, it's not at all obvious what would be meant by the mass. The way we normally get a description of the mass of an isolated object is by going far away, where the Newtonian limit applies. For example, you can have something in orbit around a black hole, at a large enough distance that the fact that it's a black hole doesn't matter. Then ordinary newtonian gravity tells you the mass of the object it's orbiting around. In a spacetime that's not asymptotically flat, you don't have this kind of natural definition.

• +1 but I would say that Bondi mass includes gravitational radiation that has not yet arrived at null infinity. That brings up the fact that ADM mass is a constant property of the spacetime (or more precisely, transforms as a four-vector, but does not depend on time coordinates in any way), whereas Bondi mass depends on the slice of null infinity on which it is measured (basically, it's a function of time). – Mike Jun 28 at 15:49