A lot of the text for this is from "How does one correctly interpret the behavior of the heat capacity of a charged black hole?" but this concerns a different question. The Reissner-Nordström black hole solution is: ds2=−(1−2Mr+Q2r2)dt2+(1−2Mr+Q2r2)−1dr2+r2dΩ22
Let us define f(r)≡(1−2Mr+Q2r2). Clearly, the solutions to f(r)=0 are r±=M±√M2−Q2, and these represent the two horizons of the charged black hole. If we are considering a point near r+, we can rewrite f(r) as follows: f(r+)∼(r+−r−)(r−r+)r2+
What I don't understand is how can derive the temperature of the black hole from this relation. The temperature is given by T=r+−r−4πr2+=12π√M2−Q2(M+√M2−Q2)2
I couldn't find a reasonable answer as to how we can obtain the temperature from f(r+). What are the steps and reasoning that are missing when making this jump?
Answer
The temperature of a black hole is related to its surface gravity. For stationary black holes, the surface gravity is given by κ2=−12DaξbDaξb|r=r+
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