Black holes in higher dimensional gravity theory with corrections quadratic in curvature

Static spherically symmetric black holes are discussed in the framework of higher dimensional gravity with quadratic in curvature terms. Such terms naturally arise as a result of quantum corrections induced by quantum fields propagating in the gravitational background. We focus our attention on the correction of the form ${\mathcal{C}}^2=C_{\alpha \beta \gamma \delta }{C}^{\alpha \beta \gamma \delta }$. The Gauss-Bonnet equation in four-dimensional spacetime enables one to reduce this term in the action to the terms quadratic in the Ricci tensor and scalar curvature. As a result the Schwarzschild solution which is Ricci flat will be also a solution of the theory with the Weyl scalar ${\mathcal{C}}^2$ correction. An important new feature of the spaces with dimension $D>4$ is that in the presence of the Weyl curvature-squared term a necessary solution differs from the corresponding “classical” vacuum Tangherlini metric. This difference is related to the presence of secondary or induced hair. We explore how the Tangherlini solution is modified by “quantum corrections,” assuming that the gravitational radius $r_0$ is much larger than the scale of the quantum corrections. We also demonstrated that finding a general solution beyond the perturbation method can be reduced to solving a single third order ordinary differential equation (master equation).

Figure 1

Iterations of the mass function $M$ up to the second order. The lines 0, 1, and 2 show the functions $M_0(x)$, $M_1(x)$, and $M_2(x)$, respectively. The plot is constructed for $n=3$ and $a=0.03$.

Figure 2

Iterations of $K/K_0$ up to the second order. Here $K_0=n(n+1)x$ is the value of $K$ for $p=0$. The line 0 is zero order in the $p$ result. Lines 1 and 2 show the functions $K_1(x)/K_0(x)$ and $K_2(x)/K_0(x)$, respectively. The plot is constructed for $n=3$ and $a=0.03$.

Figure 3

Iterations of $C(x)$ up to the second order. The functions $C_0$ and $C_1$ vanish, so that the plot shows only $C_2$ function.

Figure 4

Function $Q(x,p)$ in the domain $x\in [0,1]$, $p\in [0,-1/12]$. It is clear from this plot that this function is positive everywhere in this domain.