. A statistical mechanical calculation of the configuration proposed yields α = 4π²/ ln(2) and a total entropy of the system S = kB N ln(2), in agreement with the Bekenstein entropy of a black hole. It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of λ = αRs is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass-energy distribution.]]>
Rs. Una posible configuración cuántica para el gas de fotones a R Rs se sugiere, la cual satisface todas las condiciones de agujero negro para la energía, entropía y temperatura. Finalmente, examinamos nuestros resultados desde el punto de vista de algunas ideas recientes de Loop Quantum Gravity.]]>
Investigación
Some statistical mechanical properties of photon black holes
X. Hernandez,ª C.S. Lopez-Monsalvoª, S. Mendozaª and R.A. Sussmanb
b Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, Distrito Federal 04510, México, e-mail: sussman@nuclecu.unam.mx
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Recibido el 3 de noviembre de 06 Aceptado el 17 de noviembre de 06
Abstract
We show that if the total internal energy of a blackhole is constructed as the sum of N photons all having a fixed wavelength chosento scale with the Schwarzschild radius as λ = αRs, then N will scale with. A statistical mechanical calculation of the configuration proposed yields α = 4π2/ ln(2) and a total entropy of the system S = kBN ln(2), in agreement with the Bekenstein entropy of a black hole. It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of λ = αRsis always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass-energy distribution.
Keywords: Physics of black holes; classical black holes; quantum aspects of black holes; evaporation; thermodynamics.
Resumen
En este artículo estudiamos la relación entre la energía y la entropía de un gas de fotones tipo cuerpo negro, contenido dentro de un recinto adiabatico de radio R , cuando es comprimido hacia un régimen auto-gravitacional. Mostramos que este régimen coincide aproximadamente con el régimen de un agujero negro para el sistema, i.e., R ˜Rs donde Rsrepresenta al radio de Schwarzschild del sistema. La entropía del sistema resulta estar siempre por debajo de la cota Holografíca, incluso cuando R Rs. Una posible configuración cuántica para el gas de fotones a R Rsse sugiere, la cual satisface todas las condiciones de agujero negro para la energía, entropía y temperatura. Finalmente, examinamos nuestros resultados desde el punto de vista de algunas ideas recientes de Loop Quantum Gravity.
Descriptores: Física de agujeros negros; agujeros negros classicos; aspectos cuánticos de agujeros negros; evaporación; termodinámica.
X. Hernandez acknowledges the support of grant UNAM DGAPA (IN117803-3), CONACyT (42809/A-1) and CONACyT (42748). C. Lopez-Monsalvo is grateful for economic support from UNAM DGAPA (IN119203). S. Mendoza gratefully acknowledges financial support from CONACyT (41443) and UNAM DGAPA (IN119203). We thank the anonymous referee for his comments and suggestions, which improved the final version of the paper.
References
1. S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences (Oxford University Press, 2000). [ Links ]
2. V.P Frolov & I.D. Novikov, Black Hole Physics (Kluwer Academic Publishers, AH Dordrecht The Netherlands, 1998). [ Links ]
3. G. 't Hooft, Dimensional reduction in quantum gravity, in Salam-festschrifft, A. Aly, J. Ellis, and S. Randjbar-Daemi, eds. (World Scientific, Singapore, 1993). [ Links ]
4. L. Susskind, J. Math. Phys. 36 (1995) 6377. [ Links ]
Rs se sugiere, la cual satisface todas las condiciones de agujero negro para la energía, entropía y temperatura. Finalmente, examinamos nuestros resultados desde el punto de vista de algunas ideas recientes de Loop Quantum Gravity.]]>
. A statistical mechanical calculation of the configuration proposed yields α = 4π2/ ln(2) and a total entropy of the system S = kBN ln(2), in agreement with the Bekenstein entropy of a black hole. It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of λ = αRs is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass-energy distribution.
Rs. Una posible configuración cuántica para el gas de fotones a R