PlanetPhysics/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 $$u^\mu \nabla_\mu u^\nu = f(s) u^\nu$$ 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 $$dx^\mu / dt$$ at a single point of the geodesic.