Spinning black holes in Einstein–Gauss-Bonnet–dilaton theory: Nonperturbative solutions

We present an investigation of spinning black holes in Einstein–Gauss-Bonnet–dilaton (EGBd) theory. The solutions are found within a nonperturbative approach, by directly solving the field equations. These stationary axially symmetric black holes are asymptotically flat. They possess a nontrivial scalar field outside their regular event horizon. We present an overview of the parameter space of the solutions together with a study of their basic properties. We point out that the EGBd black holes can exhibit some physical differences when compared to the Kerr solution. For example, their mass is always bounded from below, while their angular momentum can exceed the Kerr bound. Also, in contrast to the Kerr case, the extremal solutions are singular, with the scalar field diverging on the horizon.

Figure 1

The metric functions $f$, $l$, $m$, and $\omega$ are shown as for a typical EGBd spinning black hole with the input parameters $r_{\mathrm{H}}=0.35$, ${\mathrm{\Omega }}_{\mathrm{H}}=0.15$, and $\alpha =0.5$. This solution has $M=2.472$, $J=5.886$, $D=0.231$, $T_{\mathrm{H}}=.00737$, $A_{\mathrm{H}}=193.401$, and $S=55.751$.

Figure 2

The scalar field $\varphi$ is shown together with the scalar curvature $R$ and the Gauss-Bonnet term (2) for the same solution as in Fig. 1.

Figure 3

The value of the scalar field at the poles of the horizon is shown as a function of the scaled temperature for two values of the scaled angular velocity ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }$.

Figure 4

The domain of existence of the horizon area, entropy, temperature and dilaton charge is shown vs $\alpha$ (left panels) and vs $J$ (right panels). Here and in Fig. 5, all quantities are normalized with respect to the mass of the solutions.

Figure 5

Domain of existence of EGBd spinning BHs in an angular momentum vs $\alpha$ diagram.

Figure 6

The ratio $L_e/L_p$ is shown for two sets of EGBd solutions with fixed values of the dimensionless parameter ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }$, which extend from the critical configurations to the extremal ones. Also shown is this ratio for a set of Kerr solutions close to extremality.

Figure 7

The shape of the horizon is shown for the EGBd black hole in Figs. 1 and 2 and for a Kerr black hole with the same values of the mass and angular momentum. The solid lines indicate an embedding in pseudo-Euclidean space, the dashed lines in Euclidean space.

Figure 8

The scalar field (left panel) and its component $T_t^{t(\varphi )}$ of the energy momentum tensor (multiplied with a factor of ${10}^3$; right panel) on the horizon, are shown for the typical EGBd spinning solution of Figs. 1 and 2.

Figure 9

A cross section of the ergosurface in the $x-z$ plane (left panel) and the metric function $g_{tt}$ in the equatorial plane (right panel) for the EGBd spinning solution in Figs. 1 and 2. In the right panel, the inset shows the near horizon behavior of $g_{tt}(r,\pi /2)$ for this solution and for a Kerr black hole with the same values of the mass and angular momentum.

Figure 10

The magnitude of the scaled quadrupole moment $QM/J^2$ is shown versus the scaled angular momentum $J/M^2$ for several values of the scaled horizon angular velocity ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }$. The shaded area represents the domain of existence of the EBGd black holes.

Figure 11

The scaled moment of inertia $J/({\mathrm{\Omega }}_{\mathrm{H}}{M}^3)$ is shown versus the magnitude of the scaled quadrupole moment $QM/J^2$ for several values of the scaled angular momentum $J/M^2$.

Figure 12

The circumferential ISCO radius $R_{\mathrm{ISCO}}$ (given in units of the mass) is shown versus the reduced angular momentum $J/M^2$ for several values of the scaled event horizon velocity ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }=0.055$ (the curve closest to the GR one), 0.082, 0.106, and 0.11.

Figure 13

The inverse of the orbital frequency ${\mathrm{\Omega }}_c$ is shown versus the angular momentum for coupling constant $\beta =0$ and $\beta =0.5$ for several values of the scaled event horizon velocity ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }$. Both ${\mathrm{\Omega }}_c$ and $J$ are given in units set by the mass.

Figure 14

The quantity $|p^2/(4q)|-1$ which enters the discriminant $\mathrm{\Delta }$ of the equation (a1) is shown as a function of $\alpha /M^2$ for several values of the scaled angular velocity ${\mathrm{\Omega }}_{\mathrm{H}}\sqrt{\alpha }$.