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by HypothesisAugust 9th, 2024

**Authors:**

(1) Ankit Anand, Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India;

(2) Ruben Campos Delgado, Bethe Center for Theoretical Physics, Physikalisches Institut der Universit¨at Bonn, Nussallee 12, 53115 Bonn, Germany.

2 Modified Einstein equations with the Jacobson’s approach

3 Quantum gravitational corrections to the Schwarzschild metric

The Barrow hypothesis posits a fractal structure at the black hole horizon which modifies the form of the Bekenstein-Hawking entropy area law according to a parameter ∆. If ∆ is constant, then the Barrow hypothesis does not give any substantial modifications to general relativity. In this paper we showed that the assumption of ∆ as function of the radial distance leads to modified gravity theories beyond general relativity. All these theories have a common feature: they modify the classical Schwarzschild black hole metric with quantum gravitational corrections. Interestingly, we found corrections not only to the gtt and grr components, but also to the gθθ and gφφ components. This should not be a surprise, since the central point of the Barrow hypothesis is to replace spherical symmetry with fractal symmetry. The different theories that one can generate are encoded in the choice for I(r) (see (3.8)). However, the specific expression of I(r) remains indeterminate a priori, i.e. there is no best choice for I(r) and one can in principle obtain any form of quantum gravitational corrections. Furthermore, we assumed so far that the Barrow hypothesis is connected to already existing theories, but this does not necessarily have to be true. It may be that the Barrow hypothesis leads to completely new, still unknown quantum corrections. In this regard, an interesting future direction of research would be to compute some black hole thermodynamic quantities like temperature and pressure, and see whether the answer matches with the already available results. A similar calculation with constant ∆ has been recently carried out in [33]. Despite the aforementioned difficulties, we can rightly affirm that the Barrow hypothesis has become a rich and valid framework within the realm of quantum gravity.

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