Spacetime structure of an evaporating black hole in quantum gravity

The impact of the leading quantum gravity effects on the dynamics of the Hawking evaporation process of a black hole is investigated. Its spacetime structure is described by a renormalization group improved Vaidya metric. Its event horizon, apparent horizon, and timelike limit surface are obtained by taking the scale dependence of Newton’s constant into account. The emergence of a quantum ergosphere is discussed. The final state of the evaporation process is a cold, Planck size remnant.

Figure 1

The ratio $M/M_{\mathrm{cr}}$ as a function of $v/r_{\mathrm{cr}}$ for various initial masses.

Figure 2

Time dependence of the Bekenstein-Hawking temperature during the evaporation process for the same initial masses as in Fig. 1.

Figure 3

The black hole’s luminosity as a function of $v/r_{\mathrm{cr}}$, for the same initial masses as in Figs. 1, 2.

Figure 4

Light rays of the outgoing null congruence for $M/M_{\mathrm{cr}}=2$. The thick solid line is the EH determined numerically, the dashed lines are the outer and inner $\mathrm{TLS}=\mathrm{AH}$, and the thin solid lines are generic null rays, inside and outside the EH.

Figure 5

Light rays of the outgoing null congruence with $\ddot{r}>0$, $\ddot{r}=0$, and $\ddot{r}<0$ at $v_1$. Those with $\ddot{r}>0$ are most likely to eventually cross the TLS.

Figure 6

The EH determined numerically (solid thick line) and by York’s approximate criterion (dashed line) for $M/M_{\mathrm{cr}}=2$. The solid thin line is the AH.

Figure 7

The conformal diagram of the evaporating quantum black hole: region I is a flat spacetime, and region II is the evaporating BH spacetime. EH is the event horizon, CH is the inner (Cauchy) horizon, and $\mathrm{A}$ is the apparent horizon.