affine parameter

Given a geodesic curve, an affine parameterization for that curve is a parameterization by a parameter $t$ such that the parametric equations for the curve satisfy the geodesic equation.

Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter $s$ and sets $u^{\mu }=d{x}^{\mu }/ds$, then we have

$$u^{\mu }{\nabla }_{\mu }{u}^{\nu }=f(s)u^{\nu }$$

for some function $f$. In general, the right hand side of this equation does not equal zero — it is only zero in the special case where $t$ is an affine parameter.

The reason for the name “affine parameter” is that, if $t_1$ and $t_2$ are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants $a$ and $b$ such that

$$t_1=a{t}_2+b$$

Conversely, if $t$ is an affine parameter, then $at+b$ is also an affine parameter.

From this it follows that an affine parameter $t$ is uniquely determined if we specify its value at two points on the geodesic or if we specify both its value and the value of $d{x}^{\mu }/dt$ at a single point of the geodesic.