%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: affine parameter
%%% Primary Category Code: 02.40.Hw
%%% Filename: AffineParameter.tex
%%% Version: 1
%%% Owner: rspuzio
%%% Author(s): rspuzio
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\begin{document}
Given a \htmladdnormallink{geodesic}{http://planetphysics.us/encyclopedia/GeodesicEquation.html} curve, an \emph{affine parameterization} for that curve is a parameterization by a \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} $t$ such that the parametric equations for the curve satisfy the \htmladdnormallink{geodesic equation}{http://planetphysics.us/encyclopedia/GeodesicEquation.html}.
Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter $s$ and sets $u^\mu = dx^\mu / ds$, then we have
$$u^\mu \nabla_\mu u^\nu = f(s) u^\nu$$
for some \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $f$. 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 $dx^\mu / dt$ at a single point of the geodesic.
\end{document}
