Null geodesics and null curves

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Null geodesics are specific paths in spacetime where the tangent vector is parallel transported, while null curves do not have this requirement. An example of a null curve is a circular light-speed path in polar coordinates, represented as θ = ct/R, which is not a geodesic. Such paths indicate that light would not naturally follow these trajectories in a vacuum. The distinction lies in the properties of parallel transport along the paths. Understanding these differences is crucial in the study of general relativity and the behavior of light in various spacetime geometries.
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What is the exact difference between null geodesics and null curves? Please explain both qualitatively and quantitatively.
 
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Null geodesics are parametrised such that the tangent vector is parallel transported along it. This is not necessarily the case for any null curve.
 
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Consider the path, in polar coordinates: θ = ct/R

A circular light speed path of radius R. This is null path that is not geodesic. Light would not take such a path in vaccuo.
 
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ok..thanks Orodruin and PAllen.
 

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