General Relativity: An Introduction for Physicists · Problem 14.14

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · newtonian-cosmology-field-equations

Problem

In Newtonian cosmology, the universe is modelled as an infinite gas of density

\rho(t)

that is expanding in such a way that the relative recessional velocity of any two gas particles is

v(t) = H(t)R(t)

, where

H(t) = \dot{R}(t)/R(t)

and

R(t)

is the separation of the particles at time

t

. Use Gauss' law to determine the force on a particle of mass

m

on the edge of an arbitrary spherical region, and hence show that

\ddot{R} = -\frac{4\pi G}{3}\rho R.

By considering the total energy

E

of the above particle, show further that

\dot{R}^2 = \frac{8\pi G}{3}\rho R^2 - c^2 k,

where the constant

k = -2E/(mc^2)

. Compare these Newtonian cosmological field equations with their general-relativistic counterparts.