In the Jacobson formalism general relativity is obtained from thermodynamics. This is done by using the Bekenstein-Hawking entropy-area relation. However, as a black hole gets smaller, its temperature will increase. This will cause the thermal fluctuations to also increase, and these will in turn correct the Bekenstein-Hawking entropy-area relation. Furthermore, with the reduction in the size of the black hole, quantum effects will also start to dominate. Just as the general relativity can be obtained from thermodynamics in the Jacobson formalism, we propose that the quantum fluctuations to the geometry can be obtained from thermal fluctuations.The entropy of a black hole is equal to the quarter of the area of its horizon in natural units [1,2]. This observation establishes a connection between the thermodynamics and the geometry of spacetime. This entropy associated with a black hole is also the maximum entropy that can be associated with any object of the same volume [3,4]. It is interesting to observe that this maximum entropy of a region of space scales with its area and not with its volume [5]. In fact, it is this observation that has motivated the holographic principle [6,7]. Even though the holographic principle is a very important principle in physics, it is expected that this holographic principle will get modified near the Planck scale due to quantum fluctuations [8,9]. This can also be observed from the fact that the relation between the entropy and area of a black hole is expected to get modified due to quantum fluctuations. The leading order correction to the relation between the area and entropy of a black hole is a logarithmic correction in almost all approaches to quantum gravity. In a e-mail: salwams@ksu.edu.sa particular, such logarithmic corrections have been obtained using non-perturbative quantum gravity [11], the Cardy formula [12], matter fields surrounding a black hole [13][14][15], string theory [16][17][18][19], dilatonic black holes [20] the partition function of a black hole [21], and the generalized uncertainty principle [10,22]. Even though the form of the corrections from various approaches to quantum gravity are logarithmic corrections, the coefficient of such a logarithmic correction is different for all these approaches to quantum gravity.It may be noted that such logarithmic corrections can also be obtained by considering the effects of thermal fluctuations on the entropy of a black hole [29][30][31]. Now it is well known that in the Jacobson formalism, spacetime emerges from thermodynamics [23], in that general relativity can be deduced from the Bekenstein-Hawking entropy-area relation combined with the first law of thermodynamics. Thus, the correction to the Bekenstein-Hawking entropy-area relation would generate corrections to the structure of spacetime. Furthermore, as the black hole becomes smaller due to hawking radiation, its temperature would increases, and this in turn would increase the contribution coming from the thermal corrections. However, as the black ho...
Motivated by a recent work of Scardigli, Lambiase and Vagenas (SLV), we derive the GUP parameter, i.e. α 0 , when the GUP has a linear and quadratic term in momentum. The value of the GUP parameter is obtained by conjecturing that the GUP-deformed black hole temperature of a Schwarzschild black hole and the modified Hawking temperature of a quantum-corrected Schwarzschild black hole are the same. The leading term in both cases is the standard Hawking temperature and since the corrections are considered as thermal, the modified and deformed expressions of temperature display a slight shift in the Hawking temperature. Finally, by equating the first correction terms, we obtain a value for the GUP parameter. In our analysis, the GUP parameter is not a pure number but depends on the ratio m p /M with m p to be the Planck mass and M the black hole mass. *
In this paper, we analyze the classical geometric flow as a dynamical system. We obtain an action for this system, such that its equation of motion is the Raychaudhuri equation. This action will be used to quantize this system. As the Raychaudhuri equation is the basis for deriving the singularity theorems, we will be able to understand the effects such a quantization will have on the classical singularity theorems. Thus, quantizing the geometric flow, we can demonstrate that a quantum space-time is complete (non-singular). This is because the existence of a conjugate point is a necessary condition for the occurrence of singularities, and we will be able to demonstrate that such conjugate points cannot occur due to such quantum effects.
In this paper, we compare the quantum corrections to the Schwarzschild black hole temperature due to quadratic and linear-quadratic generalized uncertainty principle, with the corrections from the quantum Raychaudhuri equation. The reason for this comparison is to connect the deformation parameters β0 and α0 with η which is the parameter that characterizes the quantum Raychaudhuri equation. The derived relation between the parameters appears to depend on the relative scale of the system (black hole), which could be read as a beta function equation for the quadratic deformation parameter β0. This study shows a correspondence between the two phenomenological approaches and indicates that quantum Raychaudhuri equation implies the existence of a crystal-like structure of spacetime.
In this paper, a four dimensional rotating Kaluza Klien (K-K) black hole was deformed using rainbow functions derived from loop quantum gravity and non-commutative geometry. We studied the thermodynamic properties and critical phenomena of this deformed black hole. The deformed temperature and entropy showed the existence of a Planckian remnant. The calculation of Gibbs free energy G for the ordinary and deformed black holes showed that both share a similar critical behaviour.
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