Universal black hole stability in four dimensions

We show that four-dimensional black holes become stable below a certain mass when the Einstein-Hilbert action is supplemented with higher-curvature terms. We prove this to be the case for an infinite family of ghost-free theories involving terms of arbitrarily high order in curvature. The new black holes, which are nonhairy generalizations of Schwarzschild’s solution, present a universal thermodynamic behavior for general values of the higher-order couplings. In particular, small black holes have infinite lifetimes. When the evaporation process makes the semiclassical approximation break down (something that occurs after a time which is usually infinite for all practical purposes), the resulting object retains a huge entropy, in stark contrast with Schwarzschild’s case.

Figure 1

Metric function $f(r)$ for Schwarzschild’s solution (red) and for the new higher-order black holes (blue), with ${\lambda }_3={\lambda }_4={\lambda }_5={\lambda }_6=1$, ${\lambda }_{n>6}=0$, and $GM{M}_c=0.5$, 0.2, 0.1, respectively (from left to right).

Figure 2

Black hole temperature as a function of the mass for Schwarzschild’s solution (red) and for the higher-order black holes with ${\lambda }_3={\lambda }_4={\lambda }_5={\lambda }_6=1$, ${\lambda }_{n>6}=0$ (note that the blue line is valid for any $M_c$). The higher-order black holes become stable below $M_{\max }\sim M_P^2/M_c$. The shape of this curve is qualitatively the same for any other choice of couplings (except ${\lambda }_n=0$ for all $n$).

Figure 3

We plot $a_2/M_c^2$ as a function of the mass (interpolation) for ${\lambda }_{\ge 4}=0$, ${\lambda }_3=1$ and for the Schwarzschild solution, $a_2=-(2GM{)}^{-2}$ (red).