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Is it possible to have negative surface gravity $\kappa$? Also, since $T=\frac{\kappa}{2\pi}$ then if you have a negative surface gravity then that implies negative temperature too right?

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If you time reverse a black hole you get a white hole, and white holes do indeed have a negative surface gravity.

However the physical relevance of this is debatable since there is no process known that could form a white hole. It is an interesting question whether a white hole has a Hawking temperature. People have explored this, for example see Hawking Radiation in the Spacetime of White Holes, Kimet Jusufi, Gen. Relativ. Gravit. 50, 84 (2018). Also a question on this site Why are white holes the same thing as black holes in quantum gravity? However in the absence of a proper theory of quantum gravity I think we should regard this as rather speculative.

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