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

In five spacetime dimensions, consider the static spherical charged metricHere $M>0$ and $Q$ are geometric parameters, with dimensions of squared length; the physical ADM mass and electric charge depend on the gravitational and electromagnetic normalization. The spherical sections are three-spheres.

Putting $y=r^2$ reduces $f=0$ to $y^2-2My+Q^2=0$. For $0<|Q|<M$, the two positive roots giveAt $M=|Q|>0$ they coincide, giving an extremal black hole; for $M<|Q|$ no positive horizon root hides the central curvature singularity. The sign of electric charge cannot change these metric conditions because only $Q^2$ occurs. When $Q=0$, the positive horizon is $r_{+}=\sqrt{2M}$ and the other formal root is the singular centre, rather than a second regular horizon. This uncharged limit is the five-dimensional Schwarzschild-Tangherlini metric.

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.