= Five-dimensional Reissner--Nordstrom black hole
{c}
{title2=Five-dimensional Reissner–Nordström black hole}
In five spacetime dimensions, consider the static spherical charged metric
$$
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega_3^2,\qquad
f(r)=1-\frac{2M}{r^2}+\frac{Q^2}{r^4}.
$$
Here $M>0$ 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 give
$$
r_\pm^2=M\pm\sqrt{M^2-Q^2}.
$$
At $M=|Q|>0$ 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>.
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