General Relativity: An Introduction for Physicists · Problem 9.29

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · cosmic-string-line-element

Problem

Consider a static, infinitely long, cylindrically symmetric matter distribution of constant radius that is invariant to Lorentz boosts along the symmetry axis (a 'cosmic string'). Show that the line element outside the body can be written as

ds^2 = c^2\,dt^2 - d\rho^2 - (\alpha + \beta\rho)^2\,d\phi^2 - dz^2,

where

\alpha

and

\beta

are constants. For the case

\alpha = 0

, consider the spacelike surfaces defined by

t = \text{constant}

and

z = \text{constant}

and calculate the circumference of a circle of constant coordinate radius

\rho

in such a surface. Hence show that, for

\beta < 1

, the geometry on the spacelike surface is that of a two-dimensional cone embedded in three-dimensional Euclidean space.

Related theory

Cosmic String Network

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