Could you theoretically map the internal distribution of mass in a black hole using Hawking radiation?

Question

Assuming you could measure the qualities of the radiation emanating from all around a black hole, could this be used to determine the internal geometry or makeup of the mass inside?

Answer 1

In principle yes. But it is very, very hard: Computational Complexity and Black Hole Horizons

Quote from the paper:

Representing bulk physics by boundary data at a fixed time is difficult for well known reasons. It is especially difficult to represent the bulk degrees of freedom behind the horizon. And finally it is extraordinarily difficult to represent them at late time. The difficulty is connected with complexity and chaos. The purpose of this section is to explain these difficulties.

The conclusion of that section is, quote:

The bad news of this section is that chaos and complexity make it impractical to compute the gauge theory description of even a simple low energy observable behind Bob’s horizon at late time. The good news is that the difficulties are practical and do not present an in-principle obstacle to describing the interior

Answer 2

The answer is no, if our current understanding of Hawking radiation is correct. The problem is that Hawking radiation is thermal. This means that it comes in a statistical ensemble of quantum states. In particular it can't carry information about the evolution of quantum states you dropped into the black hole. This is the origin of the famous information paradox.

Of course nobody has solved the information paradox, although several competing explanations have been advocated. It's conceivable that perhaps there is hidden information in Hawking radiation within a more complete understanding.