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 = dx^\mu / ds$ , then we have

$\displaystyle u^\mu \nabla_\mu u^\nu = f(s) u^\nu$

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

$\displaystyle t_1 = a t_2 + b$

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.



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As of this snapshot date, this entry was owned by rspuzio.