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 .