- #1

- 8,938

- 2,945

##s = A \tau + B##

for constants ##A## and ##B##. Or you can write it as a differential equation:

##\dfrac{d^2 \tau}{ds^2} = 0##

This implies a differential equation for ##x^m##, via ##d \tau = \sqrt{|g_{\mu \nu} dx^\mu dx^\nu|}##:

##\frac{1}{2} \partial_\lambda g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds} \dfrac{dx^\lambda}{ds} + g_{\mu \nu} \dfrac{d^2 x^\mu}{ds^2} \dfrac{dx^\nu}{ds} = 0##

This equation doesn't assume that ##x^\mu(s)## is a geodesic, it only assumes that it is either always timelike or always spacelike.

Now, my question is: Can we formulate a similar differential equation for deciding whether ##s## is affine when we relax that constraint on ##x^\mu(s)##? That is, is there a differential equation describing an affine parameter for a path ##x^\mu(s)## that allows the path to be lightlike at points?