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}.

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