Abstract
We examine the entropy of self-gravitating radiation confined to a spherical box of radiusR in the context of general relativity. We expect that configurations (i.e., initial data) which extremize total entropy will be spherically symmetric, time symmetric distributions of radiation in local thermodynamic equilibrium. Assuming this is the case, we prove that extrema ofS coincide precisely with static equilibrium configurations of the radiation fluid. Furthermore, dynamically stable equilibrium configurations are shown to coincide with local maxima ofS. The equilibrium configurations and their entropies are calculated and their properties are discussed. However, it is shown that entropies higher than these local extrema can be achieved and, indeed, arbitrarily high entropies can be attained by configurations inside of or outside but arbitrarily near their own Schwarzschild radius. However, if we limit consideration to configurations which are outside their own Schwarzschild radius by at least one radiation wavelength, then the entropy is bounded and we find Smax ≲ MR, whereM is the total mass. This supports the validity for self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of material bodies.
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Sorkin, R.D., Wald, R.M. & Jiu, Z.Z. Entropy of self-gravitating radiation. Gen Relat Gravit 13, 1127–1146 (1981). https://doi.org/10.1007/BF00759862
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DOI: https://doi.org/10.1007/BF00759862