**Required math: algebra, calculus**

**Required physics: none**

Reference: d’Inverno, Ray, *Introducing Einstein’s Relativity* (1992), Oxford Uni Press. – Section 6.4; Problem 6.9.

An affine parameter is a parameter that can be used to define a curve in such a way that the tangent vector to the curve remains constant as it is transported along the curve using parallel transport. This condition is expressed as

where is the affine connection.

If we define a transformation of the parameter by

where is some differentiable function of , we can investigate the conditions on such that it too is an affine parameter.

The terms in the equation above then transform as

Plugging these into the original equation we get

If is to be an affine parameter, then we must have

which leaves us with

Since the tangent vector will not be zero in general, we must have

from which we get by direct integration

for some constants and .

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