Stability of Lovelock Black Holes under Tensor Perturbations

Abstract

We study the stability of static black holes in the third order Lovelock theory. We derive a master equation for tensor perturbations. Using the master equation, we analyze the stability of Lovelock black holes mainly in seven and eight dimensions. We find there are cases where the linear analysis breaks down. If we restrict ourselves to the regime where the linear analysis is legitimate, black holes are always stable in seven dimensions. However, in eight dimensions, there exists a critical mass below which black holes are unstable. Combining our result in the third order Lovelock theory with the previous one in Einstein-Gauss-Bonnet theory, we conjecture that small black holes are unstable in any dimensions. The instability found in this paper will be important for the analysis of black holes at the LHC.

Figures (7)

• Figure 1: The intersection between the solid curve and thin horizontal line determines the solution $\psi = \psi (r)$ for the case $n=4$. Apparently, the infinity $r=\infty$ corresponds to $\psi =0$. The intersection between solid and dashed curve gives a horizon $r_H$.
• Figure 2: The same method as $n=4$ case is illustrated for $n=5$ case.
• Figure 3: The behavior of $l(x)$ and $k(x)$ in seven dimensions is shown. Both the lines cross the $x$-axis. Notice that $l(x)$ crosses earlier than $k(x)$.
• Figure 4: The behavior of $l(x)$ and $k(x)$ in ten dimensions is shown. Apparently, $l(x)$ has no positive root. Only $k(x)$ crosses the $x$-axis.
• Figure 5: The graphs of $a(t)$ and $b(t)$ are shown for $t>3/2$ in $D=7$. We clearly see the relation $a(t)< b(t)$.