Affine parameter transformation

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 ${s}$ 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

$\displaystyle \frac{d^{2}x^{a}}{ds^{2}}+\frac{dx^{b}}{ds}\frac{dx^{c}}{ds}\Gamma_{cb}^{a}=0 \ \ \ \ \ (1)$

where ${\Gamma_{cb}^{a}}$ is the affine connection.

If we define a transformation of the parameter by

$\displaystyle s\rightarrow\sigma\left(s\right) \ \ \ \ \ (2)$

where ${\sigma}$ is some differentiable function of ${s}$, we can investigate the conditions on ${\sigma}$ such that it too is an affine parameter.

The terms in the equation above then transform as

 $\displaystyle \frac{d^{2}x^{a}}{ds^{2}}$ $\displaystyle =$ $\displaystyle \frac{d}{ds}\left(\frac{dx^{a}}{ds}\right)\ \ \ \ \ (3)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{d}{ds}\left(\frac{dx^{a}}{d\sigma}\sigma^{\prime}\right)\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{d^{2}x^{a}}{d\sigma^{2}}\left(\sigma^{\prime}\right)^{2}+\frac{dx^{a}}{d\sigma}\sigma^{\prime\prime}\ \ \ \ \ (5)$ $\displaystyle \frac{dx^{b}}{ds}$ $\displaystyle =$ $\displaystyle \frac{dx^{b}}{d\sigma}\sigma^{\prime} \ \ \ \ \ (6)$

Plugging these into the original equation we get

$\displaystyle \frac{d^{2}x^{a}}{d\sigma^{2}}\left(\sigma^{\prime}\right)^{2}+\frac{dx^{a}}{d\sigma}\sigma^{\prime\prime}+\frac{dx^{b}}{d\sigma}\frac{dx^{c}}{d\sigma}\left(\sigma^{\prime}\right)^{2}\Gamma_{cb}^{a}=0 \ \ \ \ \ (7)$

If ${\sigma}$ is to be an affine parameter, then we must have

$\displaystyle \frac{d^{2}x^{a}}{d\sigma^{2}}+\frac{dx^{b}}{d\sigma}\frac{dx^{c}}{d\sigma}\Gamma_{cb}^{a}=0 \ \ \ \ \ (8)$

which leaves us with

$\displaystyle \frac{dx^{a}}{d\sigma}\sigma^{\prime\prime}=0 \ \ \ \ \ (9)$

Since the tangent vector ${\frac{dx^{a}}{d\sigma}}$ will not be zero in general, we must have

$\displaystyle \sigma^{\prime\prime}=0 \ \ \ \ \ (10)$

from which we get by direct integration

$\displaystyle \sigma=\alpha s+\beta \ \ \ \ \ (11)$

for some constants ${\alpha}$ and ${\beta}$.