Abstract
In this work we derive general quantum phenomenological equations of gravitational dynamics and analyse its features. The derivation uses the formalism developed in thermodynamics of spacetime and introduces low energy quantum gravity modifications to it. Quantum gravity effects are considered via modification of Bekenstein entropy by an extra logarithmic term in the area. This modification is predicted by several approaches to quantum gravity, including loop quantum gravity, string theory, AdS/CFT correspondence and generalised uncertainty principle phenomenology, giving our result a general character. The derived equations generalise classical equations of motion of unimodular gravity, instead of the ones of general relativity, and they contain at most second derivatives of the metric. We provide two independent derivations of the equations based on thermodynamics of local causal diamonds. First one uses Jacobson's maximal vacuum entanglement hypothesis, the second one Clausius entropy flux. Furthermore, we consider questions of diffeomorphism and local Lorentz invariance of the resulting dynamics and discuss its application to a simple cosmological model, finding a resolution of the classical singularity.
Highlights
Has been attempted to extend this framework to general spacetimes in order to understand the relation between thermodynamics and geometry [8, 9]
The first one obtains the dynamics from thermodynamic equilibrium of geodesic local causal diamonds (GLCD), by performing a simultaneous variation of the entanglement entropy associated with the horizon of GLCD and the entanglement entropy of the matter present inside it [10]
The only assumption made about the effects of quantum gravity is the presence of a logarithmic correction term in the entanglement entropy associated with spherical local causal horizons
Summary
Considering ε ≈ lP , we obtain a result consistent with Bekenstein equation in the leading term, but with a negative correction logarithmic in the horizon area. In order to exploit the logarithmic term to obtain quantum corrections to Einstein equations, we need to consider a horizon with closed spatial cross-section, in contrast with the original Jacobson’s paper [10] Up to this point, we have only discussed the contribution coming from a single minimally coupled scalar field. For a spherical extremal black hole, the calculations leading to entropy of a sphere in flat spacetime can be reproduced with minimal modifications [39] These results include only the contribution of a single massless scalar field. Due to this sign ambiguity we will consider the most general form of the quantum modified Bekenstein entropy (2.3) in the following
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