= Reissner-Nordstrom Kretschmann scalar
{c}
{title2=$\mathcal K=48M^2/r^6-96MQ^2/r^7+56Q^4/r^8$}
For the <Reissner-Nordstrom metric> with $F=1-2M/r+Q^2/r^2$, the <Kretschmann scalar> is
$$
\mathcal K=(F'')^2+4\left(\frac{F'}r\right)^2+4\left(\frac{1-F}{r^2}\right)^2
=\frac{48M^2}{r^6}-\frac{96MQ^2}{r^7}+\frac{56Q^4}{r^8}.
$$
The first expression follows by squaring the independent <Riemann curvature tensor> components in an <orthonormal frame>. This <Kretschmann scalar> diverges at $r=0$ for any nonzero $M$ or $Q$, but is finite at every positive root of $F$. Regular <Eddington-Finkelstein coordinates> show that those positive roots are <coordinate singularities> of the static chart. In the extremal case $Q^2=M^2>0$, its horizon value is $8/M^4$.
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