Spacetime and Geometry · Problem 4

Gravitation — Sean Carroll

Physics — General Relativity · paraboloidal-coordinates

Problem

In Euclidean three-space, we can define paraboloidal coordinates (u,v,ϕ)(u, v, \phi) via x=uvcos⁡ϕy=uvsin⁡ϕz=12(u2−v2).x = uv\cos\phi \quad y = uv\sin\phi \quad z = \tfrac{1}{2}(u^2 - v^2). (a) Find the coordinate transformation matrix between paraboloidal and Cartesian coordinates ∂xα/∂xβ′\partial x^\alpha/\partial x^{\beta'} and the inverse transformation. Are there any singular points in the map? (b) Find the basis vectors and basis one-forms in terms of Cartesian basis vectors and forms. (c) Find the metric and inverse metric in paraboloidal coordinates. (d) Calculate the Christoffel symbols. (e) Calculate the divergence ∇μVμ\nabla_\mu V^\mu and Laplacian ∇μ∇μf\nabla_\mu \nabla^\mu f.

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