General Relativity: An Introduction for Physicists · Problem 8.11

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · newtonian-limit-cosmological-constant

Problem

In the Newtonian limit of weak gravitational fields, for a slowly moving perfect fluid with pressure

p \ll \rho c^2

show that the 00-component of the Einstein field equations with a non-zero cosmological constant

\Lambda

reduces to

\nabla^2 \Phi = 4\pi G\rho - \Lambda c^2,

where

\nabla^2 = \delta^{ij}\partial_i\partial_j

and

\rho

is the proper density of the fluid. Hence show that the corresponding Newtonian gravitational potential of a spherically symmetric mass

M

centred at the origin can be written as

\Phi = -\frac{GM}{r} - \frac{\Lambda c^2 r^2}{6}

where

r^2 = \delta^{ij}x_i x_j

. Give a physical interpretation of this result.