Does Stephen Hawking have prove that black holes are actually white?

Answer 1

Not exactly - What Stephen Hawking did was to promote a rationalization for an argument that black holes and white holes have similar natures. In quantum mechanics, the black hole emits Hawking radiation, and so can come to thermal equilibrium with a gas of radiation. Since a thermal equilibrium state is time reversal invariant, Stephen Hawking argued that the time reverse of a black hole in thermal equilibrium is again a black hole in thermal equilibrium. This implies that black holes and white holes are similar objects with the same nature.

However the classical consideration for white holes is that they are the reverse of black holes and theoretically support the wormhole hypothesis by pairing a black hole with a white hole.