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On the thermodynamic origin of the Hawking entropy and a measurement of the Hawking temperature

Michael Petri
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Abstract

In the spherically symmetric case the Einstein field equations take on their simplest form for a matter-density rho = 1 / (8 pi r^2), from which a radial metric coefficient g_{rr} \propto r follows. The boundary of an object with such an interior matter-density is situated slightly outside of its gravitational radius. Its surface-redshift scales with z \propto \sqrt{r}, so that any such large object is practically indistinguishable from a black hole, as seen from exterior space-time. The interior matter has a well defined temperature, T \propto 1 / \sqrt{r}. Under the assumption, that the interior matter can be described as an ultra-relativistic gas, the object's total entropy and its temperature at infinity can be calculated by microscopic statistical thermodynamics. They are equal to the Hawking result up to a possibly different constant factor. The simplest solution of the field equations with rho = 1 / (8 pi r^2) is the so called holographic solution, short "holostar". It has an interior string equation of state. The strings are densely packed, explaining why the solution does not collapse to a singularity. The holographic solution has been shown to be a very accurate model for the universe as we see it today in Ref[7]. The factor relating the holostar's temperature at infinity to the Hawking temperature can be expressed in terms the holostar's interior (local) radiation temperature and its (local) matter-density, allowing an experimental verification of the Hawking temperature law. Using the recent experimental data for the CMBR-temperature and the total matter-density in the universe measured by WMAP, the Hawking formula is verified to an accuracy of 1%.

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Michael Petri. 2004-05-02. On the thermodynamic origin of the Hawking entropy and a measurement of the Hawking temperature. https://arxiv.org/abs/gr-qc/0405008

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