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## Main Question or Discussion Point

The geodesic equation for a path [itex]X^\mu(s)[/itex] is:

[itex]\frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0[/itex]

where [itex]U^\mu = \frac{d}{ds} X^\mu[/itex]

But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter [itex]s[/itex] must be linearly related to the proper time [itex]\tau[/itex]:

[itex]s = A + B \tau[/itex]

But what is the constraint on the parameter [itex]s[/itex] when the path [itex]X^\mu(s)[/itex] is a null path (that is, [itex]g_{\mu \nu}U^\mu U^\nu = 0[/itex])?

[itex]\frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0[/itex]

where [itex]U^\mu = \frac{d}{ds} X^\mu[/itex]

But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter [itex]s[/itex] must be linearly related to the proper time [itex]\tau[/itex]:

[itex]s = A + B \tau[/itex]

But what is the constraint on the parameter [itex]s[/itex] when the path [itex]X^\mu(s)[/itex] is a null path (that is, [itex]g_{\mu \nu}U^\mu U^\nu = 0[/itex])?