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 uμ=dxμ/ds, then we have


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 t1 and t2 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


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 dxμ/dt at a single point of the geodesic.

Title affine parameter
Canonical name AffineParameter
Date of creation 2013-03-22 14:35:47
Last modified on 2013-03-22 14:35:47
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Definition
Classification msc 53C22
Defines affinely-parameterized