Empty Black Holes, Firewalls, and the Origin of Bekenstein-Hawking Entropy

Abstract

We propose a novel solution for the endpoint of gravitational collapse, in which spacetime ends (and is orbifolded) at a microscopic distance from black hole event horizons. This model is motivated by the emergence of singular event horizons in the gravitational aether theory, a semi-classical solution to the cosmological constant problem(s), and thus suggests a catastrophic breakdown of general relativity close to black hole event horizons. A similar picture emerges in fuzzball models of black holes in string theory, as well as the recent firewall proposal to resolve the information paradox. We then demonstrate that positing a surface fluid in thermal equilibrium with Hawking radiation, with vanishing energy density (but non-vanishing pressure) at the new boundary of spacetime, which is required by Israel junction conditions, yields a thermodynamic entropy that is identical to the Bekenstein-Hawking area law, $S_{BH}$, for charged rotating black holes. To our knowledge, this is the first derivation of black hole entropy which only employs local thermodynamics. Furthermore, a model for the microscopic degrees of freedom of the surface fluid (which constitute the micro-states of the black hole) is suggested, which has a finite, but Lorentz-violating, quantum field theory. Finally, we comment on the effects of physical boundary on Hawking radiation, and show that relaxing the assumption of equilibrium with Hawking radiation sets $S_{BH}$ as an upper limit for Black Hole entropy.

Figures (4)

• Figure 1: Comparison of the causal diagrams for the static Schwarzschild black hole, and the static black holes in gravitational aether PrescodWeinstein:2009mp. In both diagrams, the solid lines depict null infinities, while the squiggly lines are singularities, and $g^{rr}$ vanishes on dotted lines. However, the latter is a null surface in Schwarzschild BH which coincides with event horizon, while it is time-like in the aether BH and corresponds to a throat or minimal area surface. Moreover, while the singularity is space-like and sits at zero area deep inside the horizon in the Schwarzschild BH, it is null in the Aether BH and sits at finite area, roughly a Planck length inside the throat. The latter assumption is the key ingredient for aether pressure to match the observed dark energy pressure for astrophysical BH masses PrescodWeinstein:2009mp.
• Figure 2: Equation of State variable $(w=\frac{\mathcal{P}}{\rho})$ as a function of Temperature.
• Figure 3: "Photosphere" of a Firewall: This figure demonstrates that even though the average temperature of a firewall could be greater than $\Lambda$, where the fluid becomes incompressible, the "photosphere" might have an effective temperature less than the Lorentz violation scale $\Lambda$. As a result the derivations of Hawking/Unruh temperature goes through. As we argue in the text, this implies that Bekenstein-Hawking area law is an upper limit for the entropy of the firewall.
• Figure 4: Comparison of the causal diagrams for a collapsing Schwarzschild black hole, and our proposed picture for collapsing black holes in gravitational aether or firewall scenarios. In both diagrams, the dotted lines depict classical event horizons, while the squiggly lines are singularities. The black area correspond to the collapsing star. While in the Schwarzschild BH, the singularity is space-like and deep inside the horizon, in the aether/firewall case it approaches the horizon and becomes null asymptotically. The accreted material smoothly crosses the Schwarzschild horizon, but it condenses into Planckian densities just inside the horizon of the aether BH.