General Relativity: An Introduction for Physicists · Problem 15.25

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · einstein-static-universe-stability

Problem

Show that for a physically reasonable perfect fluid (i.e. density >0> 0 and pressure ≥0\geq 0) there is no static isotropic homogeneous solution to Einstein's equations with Λ=0\Lambda = 0. Show that it is possible to obtain a static zero-pressure solution by the introduction of a cosmological constant Λ\Lambda such that Λc2=4πGρm,0=c2kR02.\Lambda c^2 = 4\pi G\rho_{m,0} = \frac{c^2 k}{R_0^2}. Show that this solution is unstable, however.

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