Minimum lifetime of a black hole

Abstract

We derive bounds on the lifetime of an evaporating black hole. The bound follows from energy conservation and purification, within the framework of `asymptotically semiclassical spacetimes'. We use the recently derived expression for the Bondi flux of Hawking radiation, together with the expression for the entanglement entropy of Hawking radiation at null infinity, to investigate the purification phase after the last semiclassical ray. We discuss the energy-cost of entanglement purification and we find a lower bound on the purification time of the black hole, which scales as $M_0^4/\hbar^{3/2}$, where $M_0$ is the initial black hole mass. Additionally, motivated by quantum gravity considerations, we include the additional assumption that a Planck mass black hole is metastable. With this assumption, we find that the the purification time is extended to be exponential in the square of the initial black hole mass, i.e. in its initial area. We find that the redshift exponent is negative in this purification phase, which indicates the existence of a white-hole remnant which releases information slowly. We comment on phenomenological implications for primordial black hole remnants.

Figures (4)

• Figure 1: The Penrose diagram for an evaporating black hole. The shaded grey region represents the bulk which is left arbitrary. The colored asymptotic regions represent the three different phases of evaporation. Assuming information is recovered, we constrain how long the purification phase C must last.
• Figure 2: Minimal remnant scenario. Top: a plot of the redshift exponent $k$ as a function of retarded time $u$. Bottom: plots of the entanglement entropy of radiation $S_{\rm rad}$ and Bondi mass $M$ as a function of retarded time $u$. The purification time is $\tau_C \sim M_0^4/\hbar^{3/2}$.
• Figure 3: Metastable remnant scenario. Top: a plot of the redshift exponent $k$ as a function of retarded time $u$. Bottom: plots of the entanglement entropy of radiation $S_{\rm rad}$ and Bondi mass $M$ as a function of retarded time $u$. The purification time is $\tau_C \sim \sqrt{\hbar}\, \mathrm{e}^{\gamma M_0^2/\hbar}$.
• Figure 4: The moving mirror trajectory for the bulk spacetime motivated from our asymptotic considerations for spherically-symmetric black hole evaporation. The ray tracing function $p(u)$ determines the trajectory. It is thermal during phase A and highly constrained during phases B, C, and afterwards.