Komar energy and Smarr formula for noncommutative Schwarzschild black hole

Abstract

We calculate the Komar energy $E$ for a noncommutative Schwarzschild black hole. A deformation from the conventional identity $E=2S{T}_H$ is found in the next to leading order computation in the noncommutative parameter $\theta$ (i.e. $\mathcal{O}(\sqrt{\theta }{e}^{-M^2/\theta })$) which is also consistent with the fact that the area law now breaks down. This deformation yields a nonvanishing Komar energy at the extremal point $T_H=0$ of these black holes. We then work out the Smarr formula, clearly elaborating the differences from the standard result $M=2S{T}_H$, where the mass ($M$) of the black hole is identified with the asymptotic limit of the Komar energy. Similar conclusions are also shown to hold for a deSitter--Schwarzschild geometry.

5 pages Latex

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