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

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 α = μ/r03v0(r0)′2, rather than by the local gravitational acceleration alone. Hence, the expansion remains valid even when gravity is strong. We then extend the analysis to radial geodesic motion in Schwarzschild spacetime.