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What is a Null Geodesic? My textbook only explains it as the Minkowski metric which equals to zero, but I'd appreciate a more detailed explanation.

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A null geodesic is the path that a massless particle, such as a photon, follows. That's why it's called null, it's interval (it's "distance" in 4 D spacetime) is equal to zero and it does not have a proper time associated with it.

When they are drawn on a spacetime diagram, they are the edges of the light cones, as in the picture below, the lines at 45 degrees.

Light cones and Null Geodesics

It's also called a light-like geodesic, as opposed to time-like geodesics and space-like geodesics.

For two events separated by a time-like interval, sufficient time passes between them that there could be a cause–effect relationship between the two events. For a particle travelling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval.

When a space-like interval separates two events, insufficient time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past.

  Geodesics are the shortest path between two points, a straight line, in flat space at constant velocity, but in curved space (General Relativity) it won't normally be a straight line, as least not in the way we think of a straight line in the ordinary world around us.

Have a read of this link.

Null Geodesics

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A null geodesic is a geodesic (that is: with respect to length extremal line in a manifold), whose tangent vector is a light-like vector everywhere on the geodesic (that is $x(s)$ is a geodesic and $g_{\mu\nu} \frac{dx^\mu}{ds}\frac{dx^\nu}{ds} = 0$ for all $s$, where $s$ is an affine parameter along the curve).

The null geodesics are exactly the paths that light rays may take (while matter travels on geodesics with time-like tangent vectors).

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