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...

In this work, we investigate the thermodynamics of black p-branes (BB) in the context of Gravity's Rainbow. We investigate this using rainbow functions that have been motivated from loop quantum gravity and κ-Minkowski non-commutative spacetime. Then for the sake of comparison, we examine a couple of other rainbow functions that have also appeared in the literature. We show that, for consistency, Gravity's Rainbow imposes a constraint on the minimum mass of the BB, a constraint that we interpret here as implying the existence of a black p-brane remnant. This interpretation is supported by the computation of the black p-brane's heat capacity that shows that the latter vanishes when the Schwarzschild radius takes on a value that is bigger than its extremal limit. We found that the same conclusion is reached for the third version of rainbow functions treated here but not with the second one for which only standard black p-brane thermodynamics is recovered.

In this paper, we have constructed the Feynman path integral method for non-paraxial optics. This is done by using the mathematical analogy between a non-paraxial optical system and the generalized Schrödinger equation deformed by the existence a minimal measurable length. Using this analogy, we investigated the consequences of a minimal length in this optical system. This path integral has been used to obtain instanton solution for such a optical systems. Moreover, the Berry phase of this optical system has been investigated. These results may disclose a new way to use the path integral approach in optics. Furthermore, as such system with an intrinsic minimal length have been studied in quantum gravity, the ultra-focused optical pluses can be used as an optical analog of quantum gravity.

We investigate the derivation of Friedmann equations in Rainbow gravity following Jacobson thermodynamic approach. We do not restrict the rainbow functions to be constant as is customarily used, and show that the first law of thermodynamics with a corresponding 'classical' proportionality between entropy and surface area, supplemented eventually by a 'quantum' logarithmic correction, are not in general sufficient to obtain the equations in flat FRW metrics.

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