General Relativity: An Introduction for Physicists · Problem 2.10

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · non-euclidean-space-geometry

Problem

Consider the three-dimensional space with line element

ds^2 = \frac{dr^2}{1-2\mu/r} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2),

and calculate the following quantities: ((a)) the area of a sphere of coordinate radius

r = R

; ((b)) the 3-volume of a sphere of coordinate radius

r = R

; ((c)) the radial distance between the sphere

r = 2\mu

and the sphere

r = 3\mu

; ((d)) the 3-volume contained between the two spheres in part (c). Verify that your answers reduce to the usual Euclidean results in the limit

\mu \to 0