General Relativity: An Introduction for Physicists · Problem 2.3

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · metric-tensor-non-orthogonal-coordinates

Problem

In three-dimensional Euclidean space a coordinate system

x'^a

is related to the Cartesian coordinates

x^a

x^1 = x'^1 + x'^2, \qquad x^2 = x'^1 - x'^2, \qquad x^3 = 2x'^1 x'^2 + x'^3.

Describe the coordinate surfaces in the primed system. Obtain the metric functions

g'_{ab}

in the primed system and hence show that these coordinates are not orthogonal. Calculate the volume element

dV

in the primed coordinate system.