Abstract
In a previous paper by Pollock and Singh, it was proven that the total entropy of de Sitter space-time is equal to zero in the spatially flat case K=0. This result derives from the fundamental property of classical thermodynamics that temperature and volume are not necessarily independent variables in curved space-time, and can be shown to hold for all three spatial curvatures K=0,±1. Here, we extend this approach to Schwarzschild space-time, by constructing a non-vacuum interior space with line element ds 2=e2λ(r) dt 2−e−2λ(r) dr 2−r 2(dθ 2+sin2 θdϕ 2), where \(\mathrm{e}^{2{\lambda }(r)}=-\frac{1}{2}(1-\frac{r^{2}}{R_{0}^{2}})\), which matches onto the vacuum exterior Schwarzschild metric in such a way that e2λ and d(e2λ)/dr are both continuous at the Schwarzschild radius R 0=2M. Then we show that the volume entropy is equal to A/4, where \(A\equiv 4\pi R_{0}^{2}\) is the area of the apparent horizon, as found by Hawking.
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This paper was written at the University of Cambridge, Cambridge, England.
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Pollock, M.D. On the Entropy of Schwarzschild Space-Time. Found Phys 43, 615–630 (2013). https://doi.org/10.1007/s10701-013-9701-0
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DOI: https://doi.org/10.1007/s10701-013-9701-0