On the static Lovelock black holes

Abstract

We consider static spherically symmetric Lovelock black holes and generalize the dimensionally continued black holes in such a way that they asymptotically for large $r$ go over to the d-dimensional Schwarzschild black hole in dS/AdS spacetime. This means that the master algebraic polynomial is not degenerate but instead its derivative is degenerate. This family of solutions contains an interesting class of pure Lovelock black holes which are the $N$th order Lovelock $\Lambda$-vacuum solutions having the remarkable property that their thermodynamical parameters have the universal character in terms of the event horizon radius. This is in fact a characterizing property of pure Lovelock theories. We also demonstrate the universality of the asymptotic Einstein limit for the Lovelock black holes in general.

Appendix A

Here we study some features of Eq. (16) for $d$ even $(d=2(N+1))$ and odd $(d=2N+1)$, with $\mu >0$.

1.1 A.1 $\alpha >0,\,d$ even

For the background to have physical sense, $V$ must be positive at the maximum $\displaystyle r_0= \left( \frac{\mu }{2 N \alpha }\right) ^{\frac{1}{1+2 N}}$, which implies for given $\alpha$ an upper limit for the mass of the BH,

$$\begin{aligned} \mu ^{2 N} \alpha < \frac{(2N)^{2N}}{(2N+1)^{2N+1}} \end{aligned}$$

(29)

The horizon of the BH is located at a value for the radial coordinate, $0<r_h<r_0$ for which $V(r_h)=0$. In terms of the BH parameters, the allowed values are

$$\begin{aligned} 0<r_h-\mu < \frac{\mu }{2N}. \end{aligned}$$

(30)

From $r_0$ onwards, $V$ decreases until it reaches again the zero value ar $r=r_s>r_0$, which is a coordinate singularity similar to the one present in the de Sitter background in static coordinates (see for instance [40]).

1.2 A.2 $\alpha >0,\,d$ odd

To make sense of the metric, we need $\mu <1$. There is only the coordinate singularity at $\displaystyle r_s= \left( \frac{1-\mu }{\alpha }\right) ^{\frac{1}{2 N}}$ but no BH horizon. Notice from Eq. (19) that precisely $\mu >1$ is required for a positive temperature. We conclude that this case must be excluded.

1.3 A.3 $\alpha <0,\,d$ even

We write $\alpha =-l^2$. For $r$ small, $r\rightarrow 0$, $V$ is large and negative. It increases with increasing $r$ and its vanishing defines the BH horizon. From $\displaystyle r_1 = \left( \frac{\mu }{l^2}\right) ^\frac{1}{2 N+1}$ onwards we will need $N$ odd to keep $V$ real. Thus $d$ is $d= 2(N+1)=\dot{4}$.

1.4 A.4 $\alpha <0,\, d$ odd

For small $r$ the quantity $\displaystyle (-l^2\, r^{2N}+\mu )$ is positive. If $\mu <1$ there will be no BH horizon. If $\mu >1$ there is a BH horizon at $\displaystyle r_h= \left( \frac{\mu -1}{l^2 }\right) ^{\frac{1}{2 N}}$. From $\displaystyle r_1 = \left( \frac{\mu }{l^2}\right) ^\frac{1}{2 N}$ onwards we will need $N$ odd to keep $V$ real. This means that $d= 2N+1 =2(N+1)-1 =\dot{4} +3$.

Note from the analysis above that for $\alpha <0$ one always needs $N$ to be odd, in agreement with previous findings in Sect. 5 (see (27)).