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I have a specific black hole solution (in AdS space) with a particular energy given. The energy was computed in the paper by assuming the validity of the first law ($dE = T dS + \Omega dJ + \Phi dQ$) and integrating it.

This energy is different from the one I get via Komar integral or ADM, when using the Killing vector $\xi^\mu = (\partial_t)^\mu$. I would like to find the timelike Killing vector that would give me the same energy calculated in the paper (when plugged into either the Komar or ADM formalism). So given a specific black hole energy, is there a systematic way to find the Killing vector field that would be responsible for reproducing it?

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  • $\begingroup$ Won't the specific black hole solution be determined by E, J of the solution? Moreover, integrating the first law gives the difference in energy between two infinitesimally close black hole solutions. And I don't know how you can find the exact energy. Could you give a link to the paper? $\endgroup$
    – physicscircus
    Commented Mar 2, 2020 at 16:55
  • $\begingroup$ @Abhikumbale $E$ and $J$ are calculated, they are not parameters in the solution. Complicated black hole solutions are usually found by looking for a solution to the convenient equations of motion (eg. Einstein and Maxwell if you are looking for a charged BH solution.) Conserved quantities like energy and angular momentum are then found using some formalism for calculating them. I recommend you check the paper arxiv.org/abs/hep-th/0408217. $\endgroup$
    – Y2H
    Commented Mar 2, 2020 at 17:25

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