On the catastrophe time of fluids under the action of a gravitational field

TL;DR

This paper analyzes the timing of catastrophic gravitational collapse in fluids, using perturbative methods in Newtonian and Schwarzschild spacetimes, revealing the influence of a dimensionless parameter on collapse dynamics.

Contribution

It introduces a perturbative approach to determine catastrophe times in gravitational fluids, highlighting the role of a specific dimensionless parameter controlling the expansion's validity.

Findings

Perturbative expansion remains valid even under strong gravity.

The dimensionless parameter α governs the perturbative approach.

Extension of analysis to Schwarzschild spacetime for radial geodesics.

Abstract

Motivated by the central role of the Zel'dovich approximation in the description of cosmic structure formation through gravitational collapse, we investigate Burgers-type dynamics in a spherically symmetric gravitational field. In the Newtonian setting, we derive perturbatively the catastrophe time for radial motion by imposing the loss of invertibility of the Lagrangian map. We show that the perturbative expansion is controlled by the dimensionless parameter $ \alpha=\mu/{r_0^3 v_0(r_0)'^2}, $ rather than by the local gravitational acceleration alone. Hence, the expansion remain valid even when gravity is strong. We then extend the analysis to radial geodesic motion in Schwarzschild spacetime.