Generalized Bekenstein–Hawking system: logarithmic correction

The European Physical Journal C, Jun 2014

The present work is a generalization of the recent work [arXiv.1206.1420] on the modified Hawking temperature on the event horizon. Here the Hawking temperature is generalized by multiplying the modified Hawking temperature by a variable parameter \(\alpha \) representing the ratio of the growth rate of the apparent horizon to that of event horizon. It is found that both the first and the generalized second law of thermodynamics are valid on the event horizon for any fluid distribution. Subsequently, the Bekenstein entropy is modified on the event horizon and the thermodynamical laws are examined. Finally, an interpretation of the parameters involved is presented.

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Generalized Bekenstein–Hawking system: logarithmic correction

Subenoy Chakraborty 0 0 Department of Mathematics, Jadavpur University , Kolkata, India The present work is a generalization of the recent work [arXiv.1206.1420] on the modified Hawking temperature on the event horizon. Here the Hawking temperature is generalized by multiplying the modified Hawking temperature by a variable parameter representing the ratio of the growth rate of the apparent horizon to that of event horizon. It is found that both the first and the generalized second law of thermodynamics are valid on the event horizon for any fluid distribution. Subsequently, the Bekenstein entropy is modified on the event horizon and the thermodynamical laws are examined. Finally, an interpretation of the parameters involved is presented. - In black hole physics a semi-classical description shows that a black hole behaves as a black body emitting thermal radiation with temperature (known as the Hawking temperature) and entropy (known as the Bekenstein entropy) proportional to the surface gravity at the horizon and area of the horizon [1,2], respectively. Further, this Hawking temperature and the Bekenstein entropy are related to the mass of the black hole through the first law of thermodynamics [3]. Due to this relationship between the physical parameters (namely, entropy and temperature) and the geometry of the horizon, there is natural speculation about the relationship between black hole thermodynamics and the Einstein field equations. A first step in this direction was put forward by Jacobson [4], who derived the Einstein field equations from the first law of thermodynamics: Q = T dS for all locally Rindler causal horizons with Q and T as the energy flux and Unruh temperature measured by an accelerated observer just inside the horizon. Subsequently, Padmanabhan [5,6] on the other side was able to derive the first law of thermodynamics on a e-mail: ; the horizon starting from the Einstein equations for a general static spherically symmetric space-time. This idea of equivalence between Einstein field equations and the thermodynamical laws has been extended in the context of cosmology. Usually, the universe bounded by the apparent horizon is assumed to be a thermodynamical system with Hawking temperature and the entropy as SA = where RA is the radius of the apparent horizon. It was shown that the first law of thermodynamics on the apparent horizon and the Friedmann equations are equivalent [7]. Subsequently, this equivalent idea was extended to higher dimensional space-time, namely gravity theory with a Gauss Bonnet term and Lovelock gravity theory [710]. It is presumed that such an inherent relationship between the thermodynamics at the apparent horizon and the Einstein field equations may lead to some clue on the properties of dark energy. Although the cosmological event horizon does not exist in the usual standard big bang cosmology, in the perspective of the recent observations [1116], the universe is in an accelerating phase dominated by dark energy (d < 1/3) and the event horizon distinct from the apparent horizon. By defining the entropy and temperature on the event horizon similar to those for the apparent horizon (given above) Wang et al. [17] showed that both the first and the second law of thermodynamics break down on the cosmological event horizon. They justified it arguing that the first law is applicable to nearby states of local thermodynamic equilibrium, while the event horizon reflects the global features of spacetime. As a result, the thermodynamical parameters on the non-equilibrium configuration of the event horizon may not be as simple as on the apparent horizon. Further, they speculated that the region bounded by the apparent horizon may be this normal space is (x ) = hi j (x )i R j R = 1 (H 2 + k/a2)R2 where k = 0, 1 stands for flat, closed or open model of the universe. The Friedmann equations are H 2 + k/a2 = H k/a2 = 4 G( + p) RA = TA = taken as the Bekenstein system, i.e., the Bekenstein entropy or mass bound, S < 2 RE , and the entropy or area bound, S < A/4, are satisfied in this region. Now due to universality of the Bekenstein bounds and as all gravitationally stable special regions with weak self-gravity should satisfy the above Bekenstein bounds, the corresponding thermodynamical system is termed a Bekenstein system. Further, due to the radius of the event horizon being larger than the apparent horizon, Wang et al. [17] termed the universe bounded by the event horizon a non-Bekenstein system. In the recent past there were published a series of works [1823] investigating the validity of the generalized second law of thermodynamics of the universe bounded by the event horizon for Einstein gravity [18,19] and in other gravity theories [1821] and for different fluid systems [18,19,22,23] (including dark energy [19,22,23]). In these works the validity of the first law of thermodynamics on the event horizon was assumed and it was possible to show the validity of the (...truncated)


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Subenoy Chakraborty. Generalized Bekenstein–Hawking system: logarithmic correction, The European Physical Journal C, 2014, pp. 2876, Volume 74, Issue 6, DOI: 10.1140/epjc/s10052-014-2876-5