How Barrow Entropy Modifies Gravity: With Comments on Tsallis Entropy

Abstract

Barrow proposed that the area law of the horizon entropy might receive a "fractal correction" $S\propto A^{1+\Delta /2}$ due to quantum gravitational effects, with $0⩽\Delta ⩽1$ measures the deviation from the standard area law. While such a modification has been widely studied in the literature, its corresponding theory of gravity has not been discussed. We follow Jacobson's approach to derive the modified gravity theory (interpreted as an effective theory), and find that in the stationary case the resulting theory only differs from general relativity by a re-scaled cosmological constant. Consequently in asymptotically flat stationary spacetimes the theory is identical to general relativity. The theory is not applicable when there is no horizon; the multi-horizon case is complicated. We emphasize on the importance of identifying the correct thermodynamic mass in a theory with modified thermodynamics to avoid inconsistencies. We also comment on the Hawking evaporation rate beyond the effective theory. In addition, we show that the Bekenstein bound is satisfied if the thermodynamic mass is used as the energy, up to a constant prefactor. We briefly comment on the Tsallis entropy case as well. Interestingly, for the latter, the requirement that Bekenstein bound holds imposes a lower bound on the non-extensive parameter: $\delta >1/2$, which unfortunately rules out the previously suggested possibility that the expansion of the universe can accelerate with normal matter field alone.

6 pages: added comparisons with the Tsallis entropy case and discussions on the Bekenstein bound; references updated; version accepted by EPJC

References in corpus (6)

• Modified cosmology from extended entropy with varying exponent
• Corrected form of the first law of thermodynamics for regular black holes
• Area-law versus Rényi and Tsallis black hole entropies
• Thermodynamics of photons on fractals
• Prospecting Black Hole Thermodynamics with Fractional Quantum Mechanics
• Surfaces away from horizons are not thermodynamic

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• Generalized Entropy Implies Varying-G: Horizon Area Dependent Field Equations and Black Hole-Cosmology Coupling
• Look beyond additivity and extensivity of entropy for black hole and cosmological horizons
• Apparent Dark Matter Inspired by Einstein Equation of State
• Barrow entropy and stochastic gravitational wave background generated from cosmological QCD phase transition
• The $D$-dimensional charged AdS black holes solutions in polytropic dark energy from Barrow entropy
• Do Black Holes With Generalized Entropy Violate Bekenstein Bound?
• Mass-to-Horizon Relation and Entropy Beyond the Bekenstein-Hawking Limit
• Modified Cosmology from Mass-to-Horizon Relation: Background Evolution