General Relativity: An Introduction for Physicists · Problem 12.12

Further spherically symmetric geometries — Hobson, Efstathiou & Lasenby

Physics · cosmic-string-uniform-density-line-element

Problem

Suppose that the matter distribution in a cosmic string has a uniform density across the string, such that

\rho(r) = \begin{cases} \rho_0 & \text{for } r \leq r_0, \\ 0 & \text{for } r > r_0. \end{cases}

By demanding that

g_{\phi\phi} \to -r^2

r \to 0

, so that the spacetime geometry is regular on the axis of the string, show that the line element for

r \leq r_0

ds^2 = c^2\,dt^2 - dr^2 - \left(\frac{\sin\lambda r}{\lambda r}\right)^2 d\phi^2 - dz^2,

where

\lambda = \sqrt{\kappa\rho_0 c^2}

. By demanding that

g_{\phi\phi}

and its derivative with respect to

r

are both continuous at

r = r_0

, show that the line element for

r > r_0

ds^2 = c^2\,dt^2 - dr^2 - \left[\frac{\sin\lambda r_0}{\lambda r} + (r - r_0)\cos\lambda r_0\right]^2 d\phi^2 - dz^2.

For the interesting case in which

\lambda r_0 \ll 1

, show that for

r \gg r_0

the line element takes the form

ds^2 = c^2\,dt^2 - dr^2 - \left(1 - \frac{8G\mu}{c^2}\right)r^2\,d\phi^2 - dz^2,

where

\mu = \pi r_0^2 \rho_0

is the 'mass per unit length' of the string. Interpret this line element physically.