Velocity and velocity bounds in static spherically symmetric metrics

TL;DR

This paper derives simple formulas for particle velocities in static spherically symmetric spacetimes, analyzing their behavior across various metrics, including black holes and naked singularities, with implications for understanding relativistic motion.

Contribution

It provides new, explicit expressions for velocities in these metrics and characterizes their behavior, especially in naked singularity cases, including noncommutative geometry inspired metrics.

Findings

Massless particle velocities can exceed light speed near naked singularities.

Radial velocity in Reissner-Nordström-de Sitter metrics can be algebraically characterized.

Velocity profiles in polytropic interior solutions show bounded, oscillating behavior.

Abstract

We find simple expressions for velocity of massless particles in dependence of the distance $r$ in Schwarzschild coordinates. For massive particles these expressions put an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordstr\"om with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there exists always a region where the massless particle moves with a velocity bigger than the velocity of light in vacuum. In the case of Reissner-Nordstr\"om-de Sitter we completely characterize the radial velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.