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## Main Question or Discussion Point

Hi,

How can null geodesics be defined?

Obviously the concept of parallel-transport, of the tangent to the curve, applies equally well to null curves as to time/space-like curves. Technically this is only the definition for an "auto-parallel", not for a "geodesic". For example in Einstein-Cartan theory, where the connection (e.g. the covariant derivative) is permitted to have nonzero torsion, the autoparallels and the geodesics each follow slightly different trajectories.

Usually spacelike and timelike geodesics are defined as extremum of the distance between two points. (E.g. using a Euler-Lagrange approach they make the integral of the metric line-element be stationary to first order with respect to any infinitesimal variations of the path coordinates.) Can null geodesics be defined in the same way? GR for physicists seems to think so, but doesn't there seem to be counterexamples in Minkowski space of paths with both timelike and spacelike deviations, that still have the same (zero) total integrated length between the endpoints? I think I've also heard of null geodesics being defined as a limit between a series of timelike and spacelike geodesics, is that unnecessarily cumbersome?

How can null geodesics be defined?

Obviously the concept of parallel-transport, of the tangent to the curve, applies equally well to null curves as to time/space-like curves. Technically this is only the definition for an "auto-parallel", not for a "geodesic". For example in Einstein-Cartan theory, where the connection (e.g. the covariant derivative) is permitted to have nonzero torsion, the autoparallels and the geodesics each follow slightly different trajectories.

Usually spacelike and timelike geodesics are defined as extremum of the distance between two points. (E.g. using a Euler-Lagrange approach they make the integral of the metric line-element be stationary to first order with respect to any infinitesimal variations of the path coordinates.) Can null geodesics be defined in the same way? GR for physicists seems to think so, but doesn't there seem to be counterexamples in Minkowski space of paths with both timelike and spacelike deviations, that still have the same (zero) total integrated length between the endpoints? I think I've also heard of null geodesics being defined as a limit between a series of timelike and spacelike geodesics, is that unnecessarily cumbersome?

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