Black holes and global structures of spherical spacetimes in Horava-Lifshitz theory

We systematically study black holes in the Horava-Lifshitz theory by following the kinematic approach, in which a horizon is defined as the surface at which massless test particles are infinitely redshifted. Because of the nonrelativistic dispersion relations, the speed of light is unlimited, and test particles do not follow geodesics. As a result, there are significant differences in causal structures and black holes between general relativity (GR) and the Horava-Lifshitz theory. In particular, the horizon radii generically depend on the energies of test particles. Applying them to the spherical static vacuum solutions found recently in the nonrelativistic general covariant theory of gravity, we find that, for test particles with sufficiently high energy, the radius of the horizon can be made as small as desired, although the singularities can be seen, in principle, only by observers with infinitely high energy. In these studies, we pay particular attention to the global structure of the solutions, and find that, because of the foliation-preserving-diffeomorphism symmetry, $\mathrm{Diff}(M,\mathcal{F})$, they are quite different from the corresponding ones given in GR, even though the solutions are the same. In particular, the $\mathrm{Diff}(M,\mathcal{F})$ does not allow Penrose diagrams. Among the vacuum solutions, some give rise to the structure of the Einstein-Rosen bridge, in which two asymptotically flat regions are connected by a throat with a finite nonzero radius. We also study slowly rotating solutions in such a setup, and obtain all the solutions characterized by an arbitrary function $A_0(r)$. The case $A_0=0$ reduces to the slowly rotating Kerr solution obtained in GR.

Figure 1

(a) The light cone of the event $p$ in special relativity. (b) The causal structure of the point $p$ in Newtonian theory.

Figure 2

The Penrose diagram for $N^r=0$, $C>0$, and ${\Lambda }_g=0$. The double vertical solid lines represent the center ($r=0$), at which the spacetime is singular. This singularity is clearly naked. Note that the restricted diffeomorphisms (1.2) do not allow for the transformations needed in order to draw Penrose diagrams. Therefore, these diagrams cannot be used to study the global structures of spacetimes in the HL theory but are included only for comparison.

Figure 3

The global structure of the spacetime in the $(\overline{t},{\overline{r}}^{*})$ plane for $N^r=0$, $C>0$, and ${\Lambda }_g=0$. The double vertical solid lines represent the center ($r=0$), at which the spacetime is singular. The vertical line $AB$ represents the spatial infinity $r=\infty$, while the horizontal line $i^{+}A(i^{-}B)$ is the line where $t=\infty$ ($t=-\infty$). The lines $t=\mathrm{\text{Const}}$ are the straight lines parallel to $OC$, while the ones $r=\mathrm{\text{Const}}$ are the straight lines parallel to $i^{-}{i}^{+}$. The lines $BP$, $B0$, $BQ$, $PA$, $0A$, and $QA$ represent the radial null geodesics.

Figure 4

The function $r$ defined in (a) by Eq. (4.24) and in (b) by Eq. (4.31).

Figure 5

The global structure of the spacetime for $N^r=0$, $C=-2M<0$, and ${\Lambda }_g=0$. The vertical line $i^{+}{i}^{-}$ represents the Einstein-Rosen throat ($r=r_g\equiv 2M$), which is nonsingular and connects the two asymptotically flat regions $I$ and $I^{\prime }$. The horizontal line $AB(CD)$ is the line where $t=-\infty (\infty )$, while the vertical lines $CA$ and $DB$ are the lines where $r=\infty$. The lines $t=\mathrm{\text{Const}}$ are the straight lines parallel to $i^0{i}^0$, while the ones $r=\mathrm{\text{Const}}$ are the straight lines parallel to $i^{-}{i}^{+}$. The curved dotted lines $AD$ and $BC$, as well as the straight solid lines $AD$ and $BC$, are the radial null geodesics.

Figure 6

The Penrose diagram for $N^r=0$, $C=-2M<0$, and ${\Lambda }_g=0$. The straight lines $i^{+}{i}^0$ represent the future null infinities at which we have $r=\infty$ and $t=\infty$, while the ones $i^{-}{i}^0$ represent the past null infinities where $r=\infty$ and $t=-\infty$. The vertical line $i^{+}{i}^{-}$ represents the Einstein-Rosen throat ($r=2M$), which is nonsingular and connects the two asymptotically flat regions.

Figure 7

The Penrose diagram for $N^r=0$, $C=0$, and ${\Lambda }_g>0$, which is the Einstein static universe. The curves $i^{-}{i}^0$ and $i^{+}{i}^0$ are, respectively, the lines where $t=-\infty$, $x=\pi$, and $t=+\infty$, $x=\pi$.

Figure 8

The function $F(r)\equiv r{e}^{-2\nu }$ for $N^r=0$, $C>0$, and ${\Lambda }_g>0$, where $r_m=1/\sqrt{{\Lambda }_g}$.

Figure 9

(a) The spacetime in the $(t,x)$ plane, where $x_0\equiv \sqrt{r_g}$. (b) The Penrose diagram for $N^r=0$, $C>0$, ${\Lambda }_g>0$. The curves $i^{-}{i}^{+}$ are the lines where $r=0$, at which the spacetime is singular. The straight line $i^{-}{i}^{+}$ represents the surface $r=r_g$.

Figure 10

The function $F(r)=r{e}^{-2\nu }$ for $C<0$ and ${\Lambda }_g>0$, where $r_m\equiv 1/\sqrt{{\Lambda }_g}$. $F(r)=0$ has two positive roots $r_{\pm }$ only for ${\Lambda }_g<4/(9C^2)$. When ${\Lambda }_g\ge 4/(9C^2)$, $F(r)$ is always nonpositive for any $r>0$.

Figure 11

(a) The function $r$ vs $x$ given by Eq. (4.60), where $x_0\equiv \sqrt{r_{+}-r_{-}}$. (b) The function $r$ vs $x$ given by Eq. (4.61), where $x_1\equiv x_0+\sqrt{x_0}$.

Figure 12

The global structure of the spacetime for $C<0$, ${\Lambda }_g>0$, and ${\Lambda }_g<4/(9C^2)$. The vertical line $i^{+}{i}^{-}$ is the one where $r=r_{-}$, and the ones $AC$ and $BD$ represent the lines where $r=r_{+}$, while on the lines $EG$ and $FH$ we have $r=r_{-}$. The spacetime repeats itself infinitely many times in both directions of the $x$ axis.

Figure 13

The Penrose diagram for $C<0$, ${\Lambda }_g>0$, and ${\Lambda }_g<4/(9C^2)$.

Figure 14

The function $F(r)=r{e}^{-2\nu }$ for $C<0$ and ${\Lambda }_g<0$, where $r_g$ is the only positive root of $F(r)=0$.

Figure 15

The global structure of the de Sitter solution $N^2=f=1$, $N^r=-\sqrt{r/\ell }$ in the HL theory with the restricted diffeomorphisms (1.2) for the region $t^{\prime }\ge 0$. The horizontal line $AB$ corresponds to $t^{\prime }=\infty$ (or $t=-\infty$), while the vertical line $BD$ to $r^{\prime }=\infty$ (or $r=\infty$).