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
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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
. 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
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
under causal limit, which implies that BTZ holes of mass
could emerge from collapsing fluid disks under proper conditions. The implicit assumptions and implications are also discussed.
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