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

We study the spacetime structures of the static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-$\Lambda$ system systematically. We assume the Gauss-Bonnet coefficient $\alpha$ is non-negative and a cosmological constant is either positive, zero, or negative. The solutions have the $(n-2)$-dimensional Euclidean submanifold, which is the Einstein manifold with the curvature $k=1$, 0, and $-1$. We also assume $4\tilde{\alpha }/{\ell }^2\le 1$, where $\ell$ is the curvature radius, in order for the sourceless solution ($M=0$) to be defined. The general solutions are classified into plus and minus branches. The structures of the center, horizons, infinity, and the singular point depend on the parameters $\alpha$, ${\ell }^2$, $k$, $M$, and branches complicatedly so that a variety of global structures for the solutions are found. In our analysis, the $\tilde{M}\mathrm{\text{−}}r$ diagram is used, which makes our consideration clear and enables easy understanding by visual effects. In the plus branch, all the solutions have the same asymptotic structure at infinity as that in general relativity with a negative cosmological constant. For the negative-mass parameter, a new type of singularity called the branch singularity appears at nonzero finite radius $r=r_b>0$. The divergent behavior around the singularity in Gauss-Bonnet gravity is milder than that around the central singularity in general relativity. There are three types of horizons: inner, black hole, and cosmological. In the $k=1,0$ cases, the plus-branch solutions do not have any horizon. In the $k=-1$ case, the radius of the horizon is restricted as $r_h<\sqrt{2\tilde{\alpha }}$ ($r_h>\sqrt{2\tilde{\alpha }}$) in the plus (minus) branch. The black hole solution with zero or negative mass exists in the plus branch even for the zero or positive cosmological constant. There is also the extreme black hole solution with positive mass. We briefly discuss the effect of the Gauss-Bonnet corrections on black hole formation in a collider and the possibility of the violation of the third law of the black hole thermodynamics.

Figure 1

The conformal diagrams around a regular center and a singularity. Thick and wavy lines denote a regular center and a singularity, respectively. There are time-reversed diagrams for (c), (d), (g), and (h).

Figure 2

The conformal diagrams of the region of $r\to \infty$. There are time-reversed diagrams for (c) and (d).

Figure 3

The $\tilde{M}\mathrm{\text{−}}r$ diagrams for the static solutions in the Einstein-$\Lambda$ system. The diagrams in the upper, middle, and lower rows are $1/{\ell }^2=0$ (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, middle, and right columns are the $k=1$, $0$, and $-1$ cases, respectively. The $\tilde{M}\mathrm{\text{−}}{r}_h$ relations in $n=4$ (thick solid curve), in $n=5$ (thick dashed curve), and in $n=6$ (thick dot-dashed curve) are shown. There is a central singularity at $r=0$ except for $\tilde{M}=0$. The dots with character “E” imply the degenerate horizons. The region with a star which is bounded by the $\tilde{M}\mathrm{\text{−}}{r}_h$ curve is an untrapped region. It changes from untrapped to trapped region or vice versa by crossing the $\tilde{M}\mathrm{\text{−}}{r}_h$ curve.

Figure 4

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the six-dimensional Einstein-Gauss-Bonnet-$\Lambda$ system. The diagrams in the upper, the middle, and the lower rows express the cases $1/{\ell }^2=0$ (zero cosmological constant), $1/{\ell }^2=-1$ (positive cosmological constant), and $1/{\ell }^2=1$ (negative cosmological constant), respectively. The diagrams in the columns on the left-hand side express 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 (thin dot-dashed curves). The thin dotted curves indicate the sequence of the degenerate horizons produced by varying $\tilde{\alpha }$. We also plot the $\tilde{M}\mathrm{\text{−}}{r}_h$ relations in general relativity in $n=6$ (thin solid curves) for comparison. We set $\tilde{\alpha }=0.2$. The dots with characters E and “B” imply the degenerate horizon and the branch point, respectively. Below the $\tilde{M}\mathrm{\text{−}}{r}_b$ line, there are no solutions. The region with a star that is bounded by the $\tilde{M}\mathrm{\text{−}}{r}_h$ and $\tilde{M}\mathrm{\text{−}}{r}_b$ curves is the untrapped region. It changes from an untrapped to a trapped region or vice versa by crossing the $\tilde{M}\mathrm{\text{−}}{r}_h$ curve. The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the higher-dimensional solutions where $n\ge 6$ have qualitatively similar configurations to these.

Figure 5

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the five-dimensional Einstein-Gauss-Bonnet-$\Lambda$ system. The diagrams in the upper, middle, and lower rows express the cases where $1/{\ell }^2=0$ (zero cosmological constant), $1/{\ell }^2=-1$ (positive cosmological constant), and $1/{\ell }^2=1$ (negative cosmological constant), respectively. The diagrams in the columns on 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 (thin dot-dashed curves). The thin dotted curves indicate the sequence of the degenerate horizons produced by varying $\tilde{\alpha }$. We also plot the $\tilde{M}\mathrm{\text{−}}{r}_h$ relations in general relativity in $n=5$ (thin solid curves) for comparison. We set $\tilde{\alpha }=0.2$. See Fig. 4 for the meanings of the dots and the stars.

Figure 6

The $\tilde{M}\mathrm{\text{−}}r$ diagrams of the static solutions in the Einstein-Gauss-Bonnet-$\Lambda$ system where $4\tilde{\alpha }/{\ell }^2=1$. The diagrams are (a) the minus branch and (b) the plus branch for $k=-1$. The solid and dashed curves are for $n=6$ and $n=5$, respectively. In (a), both curves overlap each other. See Figs. 3, 4 for the meanings of the dots and the stars.

Figure 7

The $\tilde{M}\mathrm{\text{−}}S$ (solid curve) and $\tilde{M}\mathrm{\text{−}}T$ (dashed curves) diagram of the black hole solutions with $1/{\ell }^2=1$ and $k=-1$ in the six-dimensional Einstein-Gauss-Bonnet-$\Lambda$ system. We set $\tilde{\alpha }=0.2$. The thick solid curve ending at point ${\mathrm{E}}_{\mathrm{m}}$ shows the entropy of the black hole in the minus branch, and the thick solid curve connecting points B and ${\mathrm{E}}_{\mathrm{p}}$ shows it in the plus branch. Points ${\mathrm{E}}_{\mathrm{m}}$ and ${\mathrm{E}}_{\mathrm{p}}$ indicate the extreme solutions. The thin solid curves between ${\mathrm{E}}_{\mathrm{m}}$ and B, and ${\mathrm{E}}_{\mathrm{p}}$ and O are the “entropy” of the inner horizon calculated by the analogy of the black hole horizon case. Although the radius of the black hole horizon decreases as the mass becomes large in the plus branch, its entropy increases. This behavior is consistent with the second law of the black hole thermodynamics. The dashed curve with ${\mathrm{E}}_{\mathrm{m}}$ (${\mathrm{E}}_{\mathrm{p}}$) at the end shows the temperature of the black hole in the minus (plus) branch. At ${\mathrm{E}}_{\mathrm{m}}$ and ${\mathrm{E}}_{\mathrm{p}}$, the temperature becomes zero, while at point B, it diverges.

Figure 8

The conformal diagrams of regular solutions. The region with a star is the untrapped region where the Killing vector ${\partial }_t$ is timelike. It changes from the untrapped to the trapped region or vise versa by crossing the horizon. In the following figures, we use the same mark. rI, rV, and rVII are Minkowski, dS, and adS spacetimes, respectively. There are time-reversed diagrams for rIII, rIV, and rVI.

Figure 9

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

Figure 10

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

Figure 11

The conformal diagrams of (dbI) the black hole solution with degenerate horizons and (dsI) the solutions with a globally naked singularity and degenerate horizons. The degenerate horizon is drawn with a double line. By passing across the double line, the trapped (untrapped) region does not change to an untrapped (trapped) one. dbI has the same structure as that of the extreme RN-adS spacetimes. There is a time-reversed diagram for dsI.