General Relativity: An Introduction for Physicists · Problem 5.21

Special relativity revisited — Hobson, Efstathiou & Lasenby

Physics · free-particle-spherical-polar-coordinates

Problem

In Minkowski spacetime, the coordinates

(cT, X, Y, Z)

correspond to a Cartesian inertial frame. The coordinates

(ct, r, \theta, \phi)

are related to them by

X = r\sin\theta\cos\phi, \qquad Y = r\sin\theta\sin\phi, \qquad Z = r\cos\phi.

Obtain the special-relativistic equations of motion of a free particle in the

(ct, r, \theta, \phi)

coordinate system, and interpret these equations physically.