# Bound on Rindler trajectories in Black Hole spacetime

## TL;DR

This paper derives bounds on the acceleration of radially moving Rindler trajectories in Schwarzschild black hole spacetime, showing conditions for escape or capture and the minimal approach distance.

## Contribution

It provides the first explicit bounds on Rindler acceleration in black hole spacetime, linking acceleration magnitude to trajectory behavior and closest approach distance.

## Key findings

- Bound on acceleration: |a| ≤ 1/(√27 M) for Schwarzschild black holes.
- Trajectories with acceleration above the bound fall into the black hole.
- Minimum approach distance approaches 3M when the bound is saturated.

## Abstract

We investigate radial Rindler trajectories in a static spherically symmetric black hole spacetime. We assume the trajectory to remain linearly uniformly accelerated throughout its motion, in the sense of the curved spacetime generalisation of the Letaw-Frenet equations. For the Schwarzschild spacetime, we arrive at a bound on the magnitude of the acceleration $|a|$ for radially inward moving trajectories, in terms of the mass $M$ of the black hole given by $|a| \leq 1/(\sqrt{27} M)$ for a particular choice of asymptotic initial data $h$, such that, for acceleration $|a|$ greater than the bound value, the linearly uniformly accelerated trajectory always falls into the black hole. For $|a|$ satisfying the bound, there is a minimum radius or the distance of closest approach for the radial linearly uniformly accelerated trajectory to escape back to infinity. However, this distance of closest approach is found to approach its lowest value of $r_b = 3M $, greater than the Schwarzschild radius of the black hole, when the bound, $|a| = 1/( \sqrt{27}M)$ is saturated. We further show that a finite bound on the value of acceleration, $ |a| \leq B(M,h)$ and a corresponding distance of closest approach $r_{b} > 2M$ always exists, for all finite asymptotic initial data $h$.