# A GR question about null surfaces, vectors and coordinates

## Main Question or Discussion Point

I wondered anyone can explain the significance of the above as applied to metrics in the context of general relativity. This came up when the video lecturer in GR mentioned that r for example, was null or this or that vector or surface was null, say in the context of the eddington finkelstein coordinate system. I am unable to grasp the meaning or significance of the word null. I think null geodesics are clear however but even here not sure if there is any connection with null vectors etc.

## Answers and Replies

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PeterDonis
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2019 Award
A null vector is a vector whose length is zero; that is, a vector ##k^a## such that ##g_{ab} k^a k^b = 0##, where ##g_{ab}## is the metric. One of the things that makes spacetime different from an ordinary Riemannian space is that you can have vectors with zero length that are not the zero vector.

A null surface is a surface that has null tangent vectors; in other words, there are null vectors that lie completely within the surface. For example, the event horizon of a black hole is a null surface.

A null geodesic is just a geodesic whose tangent vector is null; for example, the worldline of a light ray traveling through a vacuum is a null geodesic. So there is indeed a close connection between null vectors and null geodesics.

bcrowell
Ok thanks I especially found helpful your comment about 4 vectors that have zero length but that are not the zero vector. A follow on question if ok : what then is a null coordinate.

WannabeNewton
It means the vector field associated with the coordinate is null.

Ok many thanks to you both for clarifying.

WannabeNewton