Rindler limit of a Schwarzschild horizon

Outside a Schwarzschild metric horizon $r_s$, write $r-r_s={\rho }^2/(4r_s)$. To leading order the radial metric isIt 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 $d{s}^2\simeq d{\rho }^2-[c\rho /(2r_s){]}^{2}d{t}^2$, 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.