Critical velocities $c/\sqrt 3$ and $c/\sqrt 2$ in general theory of relativity

TL;DR

This paper investigates critical velocities in general relativity, revealing specific thresholds like c/√3 outside celestial bodies and c/√2 for motion in a tower, affecting acceleration and travel time.

Contribution

It identifies new critical velocities in gravitational fields and analyzes their effects on particle acceleration and motion inside celestial bodies and towers.

Findings

Particles outside celestial bodies are accelerated or decelerated depending on their velocity relative to c/√3.

Inside constant density bodies, particles are always accelerated regardless of initial velocity.

Travel time symmetry occurs at initial velocity c/√2 in tower motion experiments.

Abstract

We consider a few thought experiments of radial motion of massive particles in the gravitational fields outside and inside various celestial bodies: Earth, Sun, black hole. All other interactions except gravity are disregarded. For the outside motion there exists a critical value of coordinate velocity ${\rm v}_c = c/\sqrt 3$: particles with ${\rm v} < {\rm v}_c$ are accelerated by the field, like Newtonian apples, particles with ${\rm v} > {\rm v}_c$ are decelerated like photons. Particles moving inside a body with constant density have no critical velocity; they are always accelerated. We consider also the motion of a ball inside a tower, when it is thrown from the top (bottom) of the tower and after classically bouncing at the bottom (top) comes back to the original point. The total time of flight is the same in these two cases if the initial proper velocity $v_0$ is equal to $c/\sqrt 2$.