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

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 a positive (negative) cosmological constant tends to destabilize (stabilize) the sphere. In the absence of a 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 a positive cosmological constant. However, given the current cosmological constant observed, no stable configuration can be larger than 1021M⊙. 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 Bañados–Teitelboim–Zanelli (BTZ) black hole solution. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk with mass 0<M≤0.5 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<M≲0.518 under causal limit, which implies that BTZ holes of mass MBTZ>0 could emerge from collapsing fluid disks under proper conditions. The implicit assumptions and implications are also discussed.