Proposition 5.5 (Paramaterization by Arc Length)

Let C be a non-null path xⁱ = xⁱ(t) in M. Fix a point t = a on this path, and define a new function s (arc length) by s(t) = L(a, t) = length of path from t = a to t. Then s is an invertible function of t, and, using s as a parameter, ||dxⁱ/ds||² is constant, and equals 1 if C is space-like and -1 if it is time-like.

Conversely, if t is any parameter with the property that ||dxⁱ/dt||² = ±1, then, choosing any parameter value t = a in the above definition of arc-length s, we have

t = ±s + C

for some constant C. (In other words, t must be, up to a constant, arc length.

Physicists call the parameter = s/c, where c is the speed of light, proper time for reasons we shall see below.)

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