Stability Analysis in N dimensional Gravitational collapse with equation of state

TL;DR

This paper investigates the stability of black hole and naked singularity formation in higher-dimensional gravitational collapse with a linear equation of state, showing that such outcomes are stable under small initial data variations.

Contribution

It demonstrates that the final states of gravitational collapse in N dimensions are stable and form open sets in initial data space, extending stability analysis to higher dimensions and specific equations of state.

Findings

Collapse outcomes form open sets in initial data space.

Black hole and naked singularity formations are stable.

Results support the cosmic censorship hypothesis.

Abstract

We study stability of occurrence of black holes and naked singularities that arise as a final state for a complete gravitational collapse of type I matter field in a spherically symmetric $N$ dimensional spacetime with equation of state $p = k \rho$, $0 \leq k \leq 1$. We prove that for a regular initial data comprising of pressure (or density) profiles at an initial surface $t = t_{i}$, from which the collapse evolves, there exists a large class of the velocity functions and classes of solutions of Einstein equations, such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity. We further prove that in an infinite dimensional separable Banach space, the set of regular initial data leading the collapse to a black hole or a naked singularity, forms an open subset of the set of all regular initial data. In this sense, the gravitational collapse leading to either a black hole or a naked singularity is stable. These results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics.