Schwarzschild-Tangherlini metric ($f(r)=1-\mu /r^{D-3}$)

For $D\ge 4$, the static spherical vacuum metric $d{s}^2=-fd{t}^2+f^{-1}d{r}^2+r^{2}d{\Omega }_{D-2}^2$ has an event horizon at $r_h={\mu }^{1/(D-3)}$. With time normalized at infinity its ADM mass is $M=(D-2){\Omega }_{D-2}\mu /(16\pi G_D)$, where ${\Omega }_{D-2}$ is the unit-sphere area. For $D=5$ and $\mu =2m$, $r_h=\sqrt{2m}$ and $M=3\pi m/(4G_5)$. The spherical orbits are three-spheres, even though the radial-time Penrose diagram has the same causal pattern as its four-dimensional counterpart.

The tortoise coordinate is $r_{\ast }=r+(r_h/2)\log |(r-r_h)/(r+r_h)|$. Set $u=t-r_{\ast }$ and $v=t+r_{\ast }$, then $U=-e^{-u/r_h}$ and $V=e^{v/r_h}$ in the right exterior. The extended radial metric is $-r_h^2(r+r_h{)}^2{e}^{-2r/r_h}dUdV/r^2$, regular at $r=r_h$. The horizons are $UV=0$, while $r=0$ gives $UV=1$ and is a spacelike curvature singularity. Compactification by $\arctan U,\arctan V$ displays two exteriors, a future black hole and a past white hole.