Arc Length - Generalization To (pseudo-)Riemannian Manifolds

Generalization To (pseudo-)Riemannian Manifolds

Let M be a (pseudo-)Riemannian manifold, γ : → M a curve in M and g the (pseudo-) metric tensor.

The length of γ is defined to be

where γ'(t) ∈ T_γ(t)M is the tangent vector of γ at t. The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves.

In theory of relativity, arc-length of timelike curves (world lines) is the proper time elapsed along the world line.