Black hole for the Einstein-Chern-Simons gravity

We consider a five-dimensional action which is composed of a gravitational sector and a sector of matter, where the gravitational sector is given by a Einstein-Chern-Simons gravity action instead of the Einstein-Hilbert action. We obtain the Einstein-Chern-Simons (EChS) field equations together with their spherically symmetric solution, which lead, in certain limit, to the standard five-dimensional solution of the Einstein-Cartan field equations. The conditions are found under which the EChS field equations admit black hole type solutions. The maximal extension and conformal compactification are also studied.

Figure 1

Space-time diagram in Schwarzschild-like coordinates for $l^2=2$ and ${\kappa }_{EC}(6{\pi }^2{)}^{-1}M=20$, so that $r_0=3$. Some future light cone has been drawn.

Figure 2

Space-time diagram in advanced Eddington-Finkelstein coordinates for $l^2=2$ and $r_0=3$. Some future light cone has been drawn.

Figure 3

Space-time diagram in retarded Eddington-Finkelstein coordinates. $\overline{t}\equiv u+r$.

Figure 4

Space-time diagram in Kruskal-Szekeres coordinates that shown its maximal analytic extension.

Figure 5

Penrose diagram for $\alpha >0$ with $\frac{{\kappa }_E}{6{\pi }^2}M>l^2$.

Figure 6

Graph for $r^{*}(r)$ with $l^2=2$ and $m=1/2$.

Figure 7

Penrose diagram for the case $\alpha >0$ with $\frac{{\kappa }_E}{6{\pi }^2}M\le l^2$.

Figure 8

Space-time diagram in Schwarszchild-like coordinates for $r_m=2$ and $r_0=6$.

Figure 9

Space-time diagram in advanced Eddington-Finkelstein coordinates.

Figure 10

Space-time diagram in retarded Eddington-Finkelstein coordinates.

Figure 11

Graph for $F_{\alpha <0}(r)$ with $l=1$, $r_m=\sqrt{5}$, and $r_0=\sqrt{13}$.

Figure 12

Kruskal-Szekeres diagram.

Figure 13

Penrose diagram for $\alpha <0$ with $\frac{{\kappa }_E}{6{\pi }^2}M>l^2$.

Figure 14

Graph for $r^{*}(r)$ with $\frac{{\kappa }_{EC}}{6{\pi }^2}M=1$ and $l=4$, so that $r_m=2$ and $r_0=\sqrt{\frac{17}{2}}\approx 2.92$.

Figure 15

Penrose diagram for $\alpha <0$ and $\frac{{\kappa }_E}{6{\pi }^2}M<l^2$.