General Relativity: An Introduction for Physicists · Problem 3.14

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · parallel-transport-geodesic-character

Problem

If the two vectors with contravariant components

v^a

and

w^a

are each parallel-transported along a curve, show that

v^a w_a

remains constant along the curve. Hence show that if a geodesic is timelike (or null or spacelike) at some point, it is timelike (or null or spacelike) at all points.

Related theory

Parallel Transport And Holonomy

Browse the problem library