Five-dimensional Reissner--Nordstrom black hole (Five-dimensional Reissner–Nordström black hole)

The static spherically symmetric electrically charged black hole in five spacetime dimensions has $f(r)=1-2M/r^2+Q^2/r^4$. When $M>Q$ it has outer and inner horizon radii $r_{\pm }^2=M\pm \sqrt{M^2-Q^2}$; for $M=Q$ it is an extremal black hole, while $M<Q$ exposes a naked singularity.

For $V=1-2M/r^2+Q^2/r^4$, an orthonormal-frame curvature calculation gives $(V^{\prime \prime }{)}^2+6(V^{\prime }/r{)}^2+12((1-V)/r^2{)}^2$. Expansion gives the displayed invariant, diverging at zero radius. It is finite at every positive root of V. Thus horizon coordinate singularities are distinct from the central curvature singularity.

In the regular metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a future causal tangent has $\dot{v}\ge 0$. For $\dot{v}>0$, causality gives $\dot{r}\le f\dot{v}/2-r^2|\dot{\Omega }{|}^2/(2\dot{v})$. If $f<0$ between two horizons, the areal radius must decrease. An inner event cannot reach the original exterior future null infinity without traversing that band in the forbidden direction.

For distinct horizon radii $a>b>0$, integrating $d{r}_{\star }/dr=f^{-1}$ gives $r_{\star }=r+\frac{a^3}{2(a^2-b^2)}\log |(r-a)/(r+a)|-\frac{b^3}{2(a^2-b^2)}\log |(r-b)/(r+b)|+C$. The tortoise coordinate defines constant retarded or advanced coordinates on radial null geodesics.

For $f=1-s/r^2+q/r^4$ and the spacetime orientation $-dt\wedge dr\wedge d{\Omega }_3$, the Hodge star operator gives ${\int }_{S^3}\star d{k}^{♭}=2{\pi }^2{r}^3{f}^{\prime }\to 4{\pi }^{2}s$. A prefactor $1/(16\pi )$ produces $\pi s/4$, whereas the dimension-adjusted ADM mass normalization produces $3\pi s/8$. The normalization and orientation must be specified rather than imported from four dimensions.