Static Schwarzschild clocks at 2s and 4s, and the near-horizon acceleration limit

Let $(M,g,N)$ be the adopted Schwarzschild exterior datum with $s>0$ and $f(r)=1-s\mathrm{/}r$. Compare ideal static clocks at radii $r_E=2s$ and $r_R=4s$ over a common Schwarzschild coordinate-time interval of duration $\mathrm{\Delta }t>0$. For the frequency ratio, choose a specified smooth, affinely parametrized future-directed null geodesic $\eta :J\to M$ in the exterior with endpoint parameters ${\lambda }_E<{\lambda }_R$ on those static worldlines, and use the ideal geometric-optics frequencies assigned by the endpoint static observers. Here $f(2s)=1\mathrm{/}2$ and $f(4s)=3\mathrm{/}4$.

1. The exact model ratios are

   $\frac{{\omega }_R}{{\omega }_E}=\sqrt{\frac{f(2s)}{f(4s)}}=\sqrt{\frac{2}{3}},\frac{\mathrm{\Delta }{\tau }_R}{\mathrm{\Delta }{\tau }_E}=\sqrt{\frac{f(4s)}{f(2s)}}=\sqrt{\frac{3}{2}}\mathrm{.}$

   The static proper acceleration at $2s$ is $a(2s)=c^2\mathrm{/}(4\sqrt{2}s)$.

2. Write $r=s(1+ϵ)$ for $ϵ>0$. The exact static proper acceleration is

   $a(r)=\frac{c^2}{2s\sqrt{ϵ}(1+ϵ{)}^{3\mathrm{/}2}},$

   which tends to $+\mathrm{\infty }$ as $ϵ\downarrow 0$. In contrast, the curvature scalar has limit $K\to 12\mathrm{/}{s}^4$ by Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric, and the advanced extension chart is smooth through $r=s$ by Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart. The acceleration divergence is therefore a property of maintaining a static worldline near the boundary, not evidence of a curvature blow-up there.