Entanglement entropy of a near-extremal black hole

Abstract

We study how the entanglement entropy of the Hawking radiation derived using the island recipe for the Reissner–Nordström black hole behaves as the black hole mass decreases. A general answer to the question essentially depends not only on the character of the mass decrease but also on the charge decrease. We assume a specific relation between the charge and mass $Q^2=GM^2[1-(M/\mu)^{2\nu}]$, which we call the constraint equation. We discuss whether it is possible to have a constraint such that the entanglement entropy does not blow up at the end of evaporation, as happens in the case of thermodynamic entropy and the entanglement entropy for the Schwarzschild black hole. We show that for some special scaling parameters, the entanglement entropy of radiation does not blow up if the mass of the evaporating black hole exceeds the Planck mass.

Appendix Approximations for entanglement entropy near the extremal regime

We study how closely the extremal case can be approached such that the approximations under which the entropy with an island (2.5) was derived remain valid. Our analysis shows that for fixed parameters $b$, $c$, $G$, $\mu$, and $\nu$, there exists $M_{\mathrm{cr}}(b,c,G,\mu,\nu)$ such that all approximations are valid for $M>M_{\mathrm{cr}}$.

For this purpose, we sketch the main steps of the derivation of the entropy with an island, Eq. (2.5). From the general formula (2.1), we obtain the entropy with an island for the configuration presented in Fig. 2:

$$\begin{aligned} \, S_{\mathcal I}(a,t_a)={}&\frac{2\pi a^2}{G} +\frac{c}{6}\log\biggl(\frac{16f(a)f(b)}{\kappa^4}\cosh^2\kappa t_a\cosh^2\kappa t_b\biggr)+{} \nonumber \\ &+\frac{c}{3} \log\biggl|\frac{\cosh\kappa(r_\ast(a)-r_\ast(b))-\cosh\kappa(t_a-t_b)} {\cosh\kappa(r_\ast(a)-r_\ast(b))+\cosh\kappa(t_a+t_b)}\biggr|, \end{aligned}$$

(A.1)

The following notation is used here:

$$r_*(r)=r+\frac{r_+^2}{r_+-r_-}\log\biggl|\frac{r-r_+}{r_+}\biggr| -\frac{r_-^2}{r_+-r_-} \log\biggl|\frac{r-r_-}{r_-}\biggr|,\qquad \kappa=\frac{r_+-r_-}{2 r_+}. $$

(A.2)

It is assumed that entropy (A.1) is extremized with respect to the coordinates $(a,t_a)$.

Assuming that $a=r_++X$, $X\ll r_+$, we use the approximation

$$\cosh\kappa(r_\ast(a)-r_\ast(b))\simeq\frac{1}{2}e^{\kappa(r_\ast(b)-r_\ast(a))} =\frac{1}{2}e^{\kappa(b-a)} \biggl(\frac{b-r_+}{a-r_+}\biggr)^{1/2} \biggl(\frac{a-r_-}{b-r_-}\biggr)^{r^2_-/(2r^2_+)}, $$

(A.3)

which is satisfied if

$$e^{2\kappa(r_\ast(b)-r_\ast(a))}\gg 1. $$

(A.4)

The late-time approximation is also used:

$$\cosh\kappa(t_a+t_b)\gg\frac{1}{2}e^{\kappa(r_\ast(b)-r_\ast(a))},\qquad \kappa t_{a,b}\gg 1. $$

(A.5)

It can then be seen that extremization over $t_a$ gives $t_a=t_b$.

We expand as

$$\log[1-2e^{\kappa(r_\ast(a)-r_\ast(b))}] \simeq -2e^{\kappa(r_*(a)-r_*(b))}, $$

(A.6)

which is valid if the inequality

$$\mathbf Y_1\colon\; 2e^{\kappa(r_*(a)-r_*(b))}\ll 1 $$

(A.7)

holds. Inequality (A.7) is sufficient for inequality (A.4) to hold. Then the approximate entropy with an island has the form

$$S_{\mathcal I}=\frac{2\pi a^2}{G} +\frac{c}{6}\log\biggl(\frac{f(a)f(b)}{\kappa^4}\biggr) +\frac{c}{3}\kappa(r_\ast(b)-r_\ast(a)) -\frac{2c}{3}e^{\kappa(r_*(a)-r_*(b))}. $$

(A.8)

The derivative of (A.8) with respect to $a=r_++X$, $X\ll r_+$, is expanded with respect to $X$. In particular, we expand as

$$\frac{1}{X+r_+-r_-}\simeq\frac{1}{(r_+-r_-)} \biggl(1-\frac{X}{r_+-r_-}\biggr), $$

(A.9)

which is valid if the inequality

$$\mathbf Y_2\colon\; X\ll r_+-r_- $$

(A.10)

holds. Then the extremization of (A.8) with respect to $a$ gives

$$X=\frac{c^2G^2r^2_+(r_+-r_-)^2 e^{-(b-r_+)(r_+-r_-)/r^2_+} ((b-r_-)/(r_+-r_-))^{r^2_-/r^2_+}} {(b-r_+)[cG(r_+-2r_-)+12\pi r^2_+(r_--r_+)]^2}. $$

(A.11)

Expanding (A.11) with respect to $G$ gives

$$X=\frac{c^2G^2e^{2\kappa(r_+-b)}((b-r_-)/(r_+-r_-))^{r^2_-/r^2_+}} {144\pi^2r^2_+(b-r_+)}+ O(G^3), $$

(A.12)

which is valid (see the denominator in (A.11)) if the inequality

$$\mathbf Y_3\colon\;\frac{cG|r_+-2 r_-|}{12\pi r^2_+}\ll r_+-r_- $$

(A.13)

holds. The two inequalities in (A.5) can be rewritten as

$$\begin{aligned} \, &t_b\gg\frac{1}{2}(b-r_+)+\frac{r^2_+}{2(r_+-r_-)} \log\biggl[\frac{b-r_+}{X} \biggl(\frac{X+r_+-r_-}{b-r_-}\biggr)^{r^2_-/r^2_+}\biggr], \\ &t_b\gg\frac{2r^2_+}{r_+-r_-}. \end{aligned} $$

(A.14)

The right-hand sides of inequalities (A.14) diverge in the extremal case, but are limited from below in the near-extremal case, at least due to (A.13).

To control the degree of smallness of the left-hand sides of (A.7), (A.10), and (A.13) we set

$$\begin{aligned} \, &Y_1(\rho)=\rho2 e^{\kappa (r_*(a)-r_*(b))}-1, \\ &Y_2(\rho)=\rho X-r_++r_-, \\ &Y_3(\rho)=\rho\frac{cG|r_+-2r_-|}{12\pi r^2_+}-r_++r_-. \end{aligned} $$

(A.15)

We take some particular $\rho>1$, for instance $\rho=5$, and see when $Y_{1,2,3}(\rho)<0$. Under constraint equation (1.1), the quantities in (A.15) become

$$\begin{aligned} \, &Y_1(\rho)|_{\text{on constr.}}=\rho 2 e^{\kappa (G M (1+\alpha)+ X-b)} \biggl(\frac{X}{b-G M(1+\alpha)}\biggr)^{1/2}\times{} \\ &\hphantom{Y_1(\rho)|_{\text{on constr.}}={}}\times\biggl(\frac{b-G M(1-\alpha)} {X+2\alpha GM}\biggr)^{(1-\alpha)^2/(2(1+\alpha)^2)}-1, \\ &Y_2(\rho)|_{\text{on constr.}}=\rho X-2\alpha GM, \\ &Y_3(\rho)|_{\text{on constr.}} =\rho \frac{c (1-3\alpha)}{12\pi M(1+\alpha)^2}-2\alpha GM, \end{aligned} $$

(A.16)

where $X$ in Eq. (A.12), under constraint equation (1.1), is

$$X=\frac{c^2e^{2\kappa(G M (1+\alpha)-b)} ((b-GM(1-\alpha))/(2\alpha GM))^{(1-\alpha)^2/(1+\alpha)^2}} {144\pi^2 M^2 (1+\alpha)^2(b-GM(1+\alpha))}. $$

(A.17)

Numerical analysis of the inequalities $Y_{1,2,3}(\rho)<0$, Eq. (A.16), under constraint equation (1.1) on the $(M,\mu)$ plane for $\nu=1.5$ and $\nu=2$, various $b=10,100,1000$, and $\rho=5$ is presented in Fig. 13. It can be seen that the condition $Y_3<0$ is the strongest in most cases, except at relatively small $b$ and large $\mu$, as shown in Fig. 13a and Fig. 13d. The larger $\nu$ and $\mu$ are, the larger the value of the critical mass $M_{\mathrm{cr}}$ up to which $M>M_{\mathrm{cr}}$ the evaporation process can be considered. We also see that there is practically no dependence of $M_{\mathrm{cr}}$ on $b$.

Under constraint equation (1.1), the inequality ${\mathbf Y}_3$, Eq. (A.13), can be written as

$${\mathbf Y}_3\colon\frac{c(1-3\alpha)}{12\pi M(1+\alpha)^2}\ll 2\alpha GM, $$

(A.18)

which for small $\alpha$ can be simplified to

$$\mathbf Y_3\colon\frac{c}{24\pi GM^2}\ll\alpha. $$

(A.19)

Summarizing the above consideration, we can say that the approximations that ensure the validity of representation (2.5) for not too small $b$ simply reduce to one inequality (A.19). Recalling the requirement of closeness to the extremal case, $\alpha\ll1$, we obtain the inequality

$$\frac{c}{24\pi GM^2}\ll\alpha\ll 1. $$

(A.20)

We note that as can be seen from (A.20), inequality (A.20) may not be satisfied at sufficiently small $\alpha$ (large $\mu$ and/or $\nu$) and large $c$.