On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-Dimensional Spacetime

TL;DR

This paper analyzes the stability of monatomic fluid spheres across different spacetime dimensions, revealing how cosmological constants influence gravitational stability and collapse, with implications for black hole formation and cosmic censorship.

Contribution

It extends the Chandrasekhar instability analysis to (N+1)-dimensional spacetimes, highlighting the unique stability of (3+1) dimensions and the effects of cosmological constants.

Findings

(3+1)-dimensional spacetime uniquely stable for fluid spheres.

Positive cosmological constant destabilizes, negative stabilizes.

No stable configuration larger than 10^{21} solar masses with current cosmological constant.

Abstract

In this note, I derive the Chandrasekhar instability of a fluid sphere in ($N$+1)-dimensional Schwarzschild-Tangherlini spacetime and take the homogeneous (uniform energy density) solution for illustration. Qualitatively, the effect of positive (negative) cosmological constant tends to destabilize (stabilize) the sphere. In the absence of cosmological constant, the privileged position of (3+1)-dimensional spacetime is manifest in its own right. As it is the marginal dimensionality in which a monatomic ideal fluid sphere is stable but not too stable to trigger the onset of gravitational collapse. Furthermore, it is the unique dimensionality that can accommodate stable hydrostatic equilibrium with positive cosmological constant. However, given the current cosmological constant observed no stable configuration can be larger than $10^{21}~{\rm M}_\odot$. On the other hand, in (2+1) dimensions it is too stable either in the context of Newtonian Gravity (NG) or Einstein's General Relativity (GR). In GR, the role of negative cosmological constant is crucial not only to guarantee fluid equilibrium (decreasing monotonicity of pressure) but also to have the Ba{\~n}ados-Teitelboim-Zanelli (BTZ) solution. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk with mass $0<\mathcal{M}\leq0.5$ to collapse into a naked singularity, which supports the Cosmic Censorship Conjecture. However, the relativistic instability can be triggered for a homogeneous disk with mass $0.5<\mathcal{M}\lesssim0.518$ under causal limit, which implies that BTZ holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emerge from collapsing fluid disks under proper conditions. The implicit assumptions and implications are also discussed.