Novel topological black holes from thermodynamics and deforming horizons

Two novel topological black hole exact solutions with unusual shapes of horizons in the simplest holographic axions model, the four-dimensional Einstein-Maxwell-axions theory, are constructed. We draw embedding diagrams in various situations to display unusual shapes of novel black holes. To understand their thermodynamics from the quasilocal aspect, we rederive the unified first law and the Misner-Sharp mass from the Einstein equation for the spacetime as a warped product ${\overline{\mathcal{M}}}_{(2)}\times {\widehat{\mathcal{M}}}_{(D-2)}$. The Ricci scalar $\widehat{R}$ of the submanifold ${\widehat{\mathcal{M}}}_{(D-2)}$ can be a nonconstant. We further improve the thermodynamics method based on the unified first law. Such a method simplifies constructing solutions and hints at generalization to higher dimensions. Moreover, we apply the unified first law to discuss black hole thermodynamics.

Figure 1

We set $r_g=8\pi G_{N}M/\mathrm{\Omega }$ and $G_N(4\pi Q/\mathrm{\Omega }{)}^2=0.09r_g^2$ to plot $f(r)=c-(8\pi G_{N}M/\mathrm{\Omega })/r+G_N(4\pi Q/\mathrm{\Omega }{)}^2/r^2$. The blue curve represents $f(r)$ when $c=1$. It has roots $r=0.9r_g$ indicating the black hole horizon location, and $r=0.1r_g$ corresponding to the inner Cauchy horizon; whereas the yellow curve is for the $c=1.5$ case, which has roots $r=0.5594r_g$ (black hole horizon) and $r=0.1073r_g$ (inner horizon).

Figure 2

We set $G_N(4\pi Q/\mathrm{\Omega }{)}^2=0.09r_g^2$ and $\mathrm{\Lambda }=0.09r_g^{-2}$ to plot $f(r)=c-(8\pi G_{N}M/\mathrm{\Omega })/r+G_N(4\pi Q/\mathrm{\Omega }{)}^2/r^2-\mathrm{\Lambda }{r}^2/3$ in which $r_g=8\pi G_{N}M/\mathrm{\Omega }$ is the unit of $r$. The blue curve is for the $c=1$ case. The largest root of $f(r)=0$ is $r=2.660r_g$, which represents the cosmic horizon, whereas other two roots are $r=1.000r_g$ (black hole horizon) and $r=0.1000r_g$ (inner horizon). As for the yellow curve, the case of $c=1.5$, roots are $r=3.707r_g$ (cosmic horizon), $r=0.5733r_g$ (black hole horizon), and $r=0.1072r_g$ (inner horizon).

Figure 3

We set $G_N(4\pi Q/\mathrm{\Omega }{)}^2=0.09r_g^2$ and $\mathrm{\Lambda }=-0.09r_g^{-2}$ to plot $f(r)=c-(8\pi G_{N}M/\mathrm{\Omega })/r+G_N(4\pi Q/\mathrm{\Omega }{)}^2/r^2-\mathrm{\Lambda }{r}^2/3$ in which $r_g=8\pi G_{N}M/\mathrm{\Omega }$. The blue curve is for $c=-1$. Roots of $f(r)=0$ are $r=3.743r_g$ (black hole horizon) and $r=0.08310r_g$ (inner horizon). The yellow curve is for $c=0$ with roots $r=2.201r_g$ (black hole horizon) and $r=0.09001r_g$ (inner horizon). The green curve is for $c=1$ with roots $r_H=0.8395r_g$ (black hole horizon) and $r=0.1000r_g$ (inner horizon).

Figure 4

The blue curve describes the function $(1-16\pi G_N{\alpha }^2\log \rho )/{\rho }^2$ when $16\pi G_N{\alpha }^2$ is taken as 10. It has a minimum value $-1.506$ at $\rho =1.822$. The yellow line $c=-1$ intersects the function at $\rho =1.313$ and $\rho =3.314$. Hence, we obtain a small branch $0<\rho <1.313$ and a large branch $\rho >3.314$ to ensure $c<(1-16\pi G_N{\alpha }^2\log \rho )/{\rho }^2$, namely, a positive ${\widehat{g}}_{\rho \rho }$.

Figure 5

Plot for ${\widehat{g}}_{\rho \rho }-1$ in the case of $c=-1$ of $16\pi G_N{\alpha }^2=10$: ${\widehat{g}}_{\rho \rho }-1$ changes its sign at $\rho =1.138$ and $\rho =3.566$. It also blows up at $\rho =1.313$ and $\rho =3.314$. Hence, the positive ${\widehat{g}}_{\rho \rho }-1$ regions are $1.138<\rho <1.313$ and $3.314<\rho <3.566$; hence they can be embedded into a three-dimensional flat Euclidean space. The regions $0<\rho <1.313$ and $\rho >3.566$ can be embedded into a Minkowski space due to their negative ${\widehat{g}}_{\rho \rho }-1$.

Figure 6

The small branch for the case of $c=-1$ of $16\pi G_N{\alpha }^2=10$: The left figure corresponds to the $0<\rho <1.313$ region which is embedded into a Minkowski space; the middle one is for $1.138<\rho <1.313$ embedded in an Euclidean space; the right figure shows two regions joined together.

Figure 7

The small branch for the case of $c=-1$ of $16\pi G_N{\alpha }^2=10$: The left figure shows the function $l(\rho )$ in which the yellow curve is for another copy. We set $l=0$ at $\rho =1.313$, i.e., the maximum of $\rho$. The right figure is the complete embedded diagram.

Figure 8

The large branch for the case of $c=-1$ of $16\pi G_N{\alpha }^2=10$: The left figure corresponds to the $3.314<\rho <3.566$ region which is embedded into an Euclidean space; the middle one is for the $\rho >3.566$ region embedded in a Minkowski space; the right figure shows two regions joined together.

Figure 9

The large branch for the case of $c=-1$ of $16\pi G_N{\alpha }^2=10$: The left figure shows the function $l(\rho )$ in which the yellow curve is for another copy. We set $l=0$ at $\rho =3.566$, i.e., the minimum of $\rho$. The right figure is the complete embedded diagram.

Figure 10

The case of $c=-1.6$ and $16\pi G_N{\alpha }^2=10$: The upper plot shows that ${\widehat{g}}_{\rho \rho }-1$ is positive in $1.323<\rho <2.253$, but negative in $0<\rho <1.323$ and $\rho >2.253$. The lower figures from left to right are (1) the $0<\rho <1.323$ region embedded into a Minkowski space; (2) the $1.323<\rho <2.253$ region embedded into an Euclidean space; (3) the $\rho >2.253$ region embedded into a Minkowski space again; and (4) the whole $\rho >0$ region, respectively.

Figure 11

The case of $c=-2$ and $16\pi G_N{\alpha }^2=10$: The left plot shows ${\widehat{g}}_{\rho \rho }-1<0$. Hence, the whole ${\widehat{\mathcal{M}}}_2$ should be embedded into a Minkowski space, with the shape as the right figure shows.

Figure 12

The case of $c=1$ and $16\pi G_N{\alpha }^2=0.1$: The upper plot shows ${\widehat{g}}_{\rho \rho }-1$ with a root $\rho =0.3320$. The lower figures are embedded diagrams: the left figure is for $0.3320<\rho <1$ (Euclidean); the middle figure is for $0<\rho <0.3320$ (Minkowski); the right one is the joined figure.

Figure 13

The case of $c=1$ and $16\pi G_N{\alpha }^2=0.1$: The left figure shows the function $l(\rho )$. We set $l=0$ at $\rho =1$, i.e., the maximum of $\rho$. The right figure is the complete embedded diagram.

Figure 14

Embedding diagrams for $n=2$: from left to right, they are (1) $0<\rho <1$, the Minkowski region; (2) $1<\rho <\sqrt{3}$, the Euclidean region; (3) $\rho >\sqrt{3}$, the Minkowski region; (4) the whole embedding diagram.

Figure 15

Embedding diagrams for $n=0$: the left figure is for the Euclidean region $0\le \rho <\sqrt{2}$ containing the regular center $\rho =0$; the middle figure shows the $\rho >\sqrt{2}$ region embedded into a Minkowski space in which the $\rho =\infty$ is a singular direction. The right one is the whole embedding diagram.

Figure 16

Embedding diagrams for the $n=-2$ case: the left figure is for $0\le \rho <1$ (Euclidean); the middle one is for $\rho >1$ (Minkowski); the right figure is the whole embedding diagram.

Figure 17

The left embedding diagram is for $n=-4$, while the right one is for $n=-6$. There is no Euclidean region in these cases.

Figure 18

The time coordinate $v$ is the null retarded time, while $\overline{t}$ is given by $v-r$. Such a sketch describes a falling process changing the size of the trapping horizon. The red curve represents the evolving tapping horizon. Gray arrows portray the in-falling energy package. Both the initial and final moment are marked by gray dotted lines, while Killing horizons are indicated by black lines.