Spacetime structure of static solutions in Gauss-Bonnet gravity: Charged case

We have studied spacetime structures of static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-Maxwell-$\Lambda$ system. Especially we focus on effects of the Maxwell charge. We assume that the Gauss-Bonnet coefficient $\alpha$ is non-negative and $4\tilde{\alpha }/{\ell }^2\le 1$ in order to define the relevant vacuum state. Solutions have the $(n-2)$-dimensional Euclidean submanifold whose curvature is $k=1$, 0, or $-1$. In Gauss-Bonnet gravity, solutions are classified into plus and minus branches. In the plus branch all solutions have the same asymptotic structure as those in general relativity with a negative cosmological constant. The charge affects a central region of a spacetime. A branch singularity appears at the finite radius $r=r_b>0$ for any mass parameter. There the Kretschmann invariant behaves as $O((r-r_b{)}^{-3})$, which is much milder than the divergent behavior of the central singularity in general relativity $O(r^{-4(n-2)})$. In the $k=1$ and 0 cases plus-branch solutions have no horizon. In the $k=-1$ case, the radius of a horizon is restricted as $r_h<\sqrt{2\tilde{\alpha }}$ ($r_h>\sqrt{2\tilde{\alpha }}$) in the plus (minus) branch. Some charged black hole solutions have no inner horizon in Gauss-Bonnet gravity. There are topological black hole solutions with zero and negative mass in the plus branch regardless of the sign of the cosmological constant. Although there is a maximum mass for black hole solutions in the plus branch for $k=-1$ in the neutral case, no such maximum exists in the charged case. The solutions in the plus branch with $k=-1$ and $n\ge 6$ have an inner black hole and inner and outer black hole horizons. In the $4\tilde{\alpha }/{\ell }^2=1$ case, only a positive mass solution is allowed, otherwise the metric function takes a complex value. Considering the evolution of black holes, we briefly discuss a classical discontinuous transition from one black hole spacetime to another.

Figure 1

The conformal diagrams around a singularity. Wavy lines denote singularities. There are time-reversed diagrams for (c) and (d).

Figure 2

The $\tilde{M}\mathrm{\text{−}}r$ diagrams for the static solutions in the four-dimensional Einstein-Maxwell-$\Lambda$ system. The diagrams in the upper, middle, and lower rows are the $1/{\ell }^2=0$ (zero cosmological constant), $1/{\ell }^2=-1$ (positive cosmological constant), and $1/{\ell }^2=1$ (negative cosmological constant) cases, respectively. The diagrams in the left, the middle, and the right columns are the $k=1$, 0, and $-1$ cases, respectively. The $\tilde{M}\mathrm{\text{−}}{r}_h$ relations (thick curves) and the $\tilde{M}\mathrm{\text{−}}{r}_{\mathrm{ex}}$ relations (dotted curves) are plotted. The dots with characters “E” and “D” imply the degenerate horizon and the doubly degenerate horizon, respectively. The region with a star which is bounded by the $\tilde{M}\mathrm{\text{−}}{r}_h$ curve represents an untrapped region. It changes from an untrapped region to a trapped region or vice versa by crossing the $\tilde{M}\mathrm{\text{−}}{r}_h$ curve. In cases with more than four dimensions, the $\tilde{M}\mathrm{\text{−}}r$ diagrams are qualitatively the same.

Figure 3

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the six-dimensional Einstein-Gauss-Bonnet-Maxwell-$\Lambda$ system with $4\tilde{\alpha }/{\ell }^2\ne 1$. The diagrams in the upper, middle, and lower rows are the $1/{\ell }^2=0$ (zero cosmological constant), $1/{\ell }^2=-1$ (positive cosmological constant), and $1/{\ell }^2=1$ (negative cosmological constant) cases, respectively. The diagrams in the columns from the left-hand side are the $k=1$ (the minus branch), 0 (the minus branch), $-1$ (the minus branch), and $-1$ (the plus branch) cases, respectively. We show the $\tilde{M}\mathrm{\text{−}}{r}_h$ relations (thick solid curves) and the $\tilde{M}\mathrm{\text{−}}{r}_b$ relations (dot-dashed curves). We set $\tilde{\alpha }=0.2$ and $\tilde{Q}=0.1$ ($k=1,0$) and $\tilde{Q}={10}^{-3}$ ($k=-1$). We choose the value of the charge as $|\tilde{Q}|<|{\tilde{Q}}_d|$ in the $k=-1$ case. Otherwise, the two extreme solutions in the plus branch coincide, and the $\tilde{M}\mathrm{\text{−}}{r}_h$ relation becomes single valued. The dotted curve represents a sequence of the degenerate horizon ($\tilde{M}\mathrm{\text{−}}{r}_{\mathrm{ex}}$ relations) by varying $\tilde{Q}$. Below the $\tilde{M}\mathrm{\text{−}}{r}_b$ line, there are no solutions. The dots labeled “E,” “D,” and “B” imply the degenerate horizon, the doubly degenerate horizon, and the branch point, respectively. See Fig. 2 for the meanings of the stars. The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the higher-dimensional solutions with $n\ge 6$ have similar configurations to these.

Figure 4

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the five-dimensional Einstein-Gauss-Bonnet-Maxwell-$\Lambda$ system with $4\tilde{\alpha }/{\ell }^2\ne 1$. The diagrams in the upper, middle, and lower rows are the $1/{\ell }^2=0$ (zero cosmological constant), $1/{\ell }^2=-1$ (positive cosmological constant), and $1/{\ell }^2=1$ (negative cosmological constant) cases, respectively. The diagrams in the columns from the left-hand side are the $k=1$ (the minus branch), 0 (the minus branch), $-1$ (the minus branch), and $-1$ (the plus branch) cases, respectively. We show the $\tilde{M}\mathrm{\text{−}}{r}_h$ relations (thick lines) and the $\tilde{M}\mathrm{\text{−}}{r}_b$ relations (dot-dashed lines). We set $\tilde{\alpha }=0.2$ and $\tilde{Q}=0.1$. Below the $\tilde{M}\mathrm{\text{−}}{r}_b$ line, there are no solutions. The dotted curve represents a sequence of the degenerate horizon ($\tilde{M}\mathrm{\text{−}}{r}_{\mathrm{ex}}$ relations) by varying $\tilde{Q}$. See Figs. 2, 3 for the meanings of the dots and the stars.

Figure 5

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the six-dimensional Einstein-Gauss-Bonnet-Maxwell-$\Lambda$ system with $4\tilde{\alpha }/{\ell }^2=1$. From the left-hand side, the diagrams are the $k=1$ (the minus branch), $k=0$ (the minus branch), $k=-1$ (the minus branch), and $k=-1$ (the plus branch) cases. We show the $\tilde{M}\mathrm{\text{−}}{r}_h$ relations (thick solid curves) and the $\tilde{M}\mathrm{\text{−}}{r}_b$ relations (dot-dashed curves). We set $\tilde{\alpha }=0.25$ and $\tilde{Q}=0.1$ for (a) and (b), $\tilde{Q}=0.2$ for (c), and $\tilde{Q}={10}^{-3}$ for (d). We choose the value of the charge as $|\tilde{Q}|<|{\tilde{Q}}_d|$ for (d). Otherwise, the two extreme solutions in the plus branch coincide, and the $\tilde{M}\mathrm{\text{−}}{r}_h$ relation becomes single valued. The dotted curve represents a sequence of the degenerate horizon ($\tilde{M}\mathrm{\text{−}}{r}_{\mathrm{ex}}$ relation) by varying $\tilde{Q}$. Below the $\tilde{M}\mathrm{\text{−}}{r}_b$ line or the negative mass regions, there are no solutions. See Figs. 2, 3 for the meanings of the dots and the stars.

Figure 6

The conformal diagrams of black hole solutions. bI, bII, bIII, bIV, bVII, and bVIII have the same structure as those of the Schwarzschild, Schwarzschild-dS, Schwarzschild-adS, RN-adS, RN, and RN-dS spacetimes, respectively.

Figure 7

The conformal diagrams of black hole solutions with degenerate horizons. The degenerate horizon is drawn with a double or triple line according to its degeneracy. By passing across the triple (double) line, the trapped region (does not) changes to an untrapped one. dbI, dbII, and dbIII have the same structure as those of the extreme RN-adS, extreme RN, and extreme RN-dS spacetimes with degenerated horizons of the inner and black hole horizons, respectively. dbIV is the case with triple degenerate horizons. Although the horizontal lines in dbIII are the infinities, they show distinct infinities by a single line depending on how to approach (from above or from below) them.

Figure 8

The conformal diagrams of the solutions with a globally naked singularity. There are time-reversed diagrams for sVII and sVIII.

Figure 9

The conformal diagrams of the solutions with a globally naked singularity and degenerate horizons. The timelike singularities in dsII show distinct singularities by a single wavy line depending on how to approach (from the right- or left-hand side) to them. There is a time-reversed diagram for dsI.