= Rindler limit of a Schwarzschild horizon
{c}
Outside a <Schwarzschild metric> horizon $r_s$, write $r-r_s=\rho^2/(4r_s)$. To leading order the radial metric is
$$
ds^2\simeq\left(\frac{c\rho}{2r_s}\right)^2dt^2-d\rho^2.
$$
It has <Rindler coordinates> with $\alpha=c/(2r_s)$. The proper-distance-like coordinate $\rho$ has null paths exponential in $\pm ct/(2r_s)$; the Schwarzschild coordinate difference $r-r_s\propto\rho^2$ is exponential in $\pm ct/r_s$.
Inside, writing $r_s-r=\rho^2/(4r_s)$ instead gives $ds^2\simeq d\rho^2-[c\rho/(2r_s)]^2dt^2$, where the radial coordinate is timelike. The negative-$x$ Rindler wedge is therefore not the black-hole interior: its time coordinate retains the same causal type as in the positive-$x$ wedge. The flat radial limit does not remove the genuine curvature of the full Schwarzschild spacetime.
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