Barrow Entropy and Quantum-Induced Fractality of Black Hole Horizons

At quantum scales, the classical description of spacetime no longer holds: the metric tensor develops intrinsic quantum fluctuations and cannot be treated as a smooth continuous field. Consequently, a black hole horizon also fails to remain an ideal smooth null hypersurface. Instead, it acquires quantum “wiggles,” corresponding to soft degrees of freedom—low-energy excitations capable of storing information as soft hair. These irregularities prevent the horizon from acting as a smooth two-dimensional surface at the Planck scale, giving it a highly jagged or fractal character. To model this deviation from smoothness, Barrow [Phys. Lett. B 808 (2020) 135643] proposed that the horizon is described by a fractal dimension, d=2+Δ , 0<Δ<1, with Δ quantifying the geometric deformation. For such a surface, the effective area scales as Aeff∝Rgd, instead of the classical A_{Cls}∝(R_g)^d. Based on the Bekenstein–Hawking argument that entropy counts horizon-covering Planck cells, this fractal surface increases the microscopic degrees of freedom, leading to the modified Barrow entropy S_B∝(A / A_{Pl})(1+Δ/2). Using the first law dM=T_BdS_B​, the corresponding temperature becomes a multiple of T_H. The heat capacity remains negative for all 0<Δ<1, so the thermodynamic instability of the Schwarzschild black hole persists. However, the reduced temperature suggests a slower evaporation process and the possible formation of a long-lived remnant. Geometrothermodynamics is reconstructed using this new fractalized entropy.