Question

Let x
µ = x
µ
(u) be a parametric equation for
the curve in pseudo-Riemannian four-space connecting points P and Q. Here u is any,
not necessarily affine parameter. Assume that the derivative satisfies
Update: Let x
µ = x^µ(u) be a parametric equation for
the curve in pseudo-Riemannian four-space connecting points P and Q. Here u is any,
not necessarily affine parameter. Assume that the derivative satisfies
(a) u^µ(P) is in the future light-cone;
(b) u^µ /= 0 everywhere along the curve;
(c) u^µuµ ≥ 0 everywhere along the curve.
Prove that u^µ
is everywhere along the curve in the future light-cone.