# A GR question about null surfaces, vectors and coordinates

qtm912
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

Mentor
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
qtm912
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.

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

qtm912
Ok many thanks to you both for clarifying.

By the way, in case you wanted an example, consider the surface ##\Sigma## given by ##r = 2M## in Schwarzschild space-time. The vector field ##\nabla r## is a normal vector field to ##\Sigma##. Now ##\nabla r = g^{\mu r}\partial_{\mu} = (1 - \frac{2M}{r})\partial_r## so ##g(\nabla r, \nabla r) = (1 - \frac{2M}{r})##. On ##\Sigma## then, ##g(\nabla r, \nabla r) = 0## so ##\nabla r## is null on ##\Sigma## meaning ##\Sigma## is a null hypersurface ("hyper" because it is of codimension 1); this is of course just the event horizon of the Schwarzschild black hole.

Note that for any tangent vector ##v \in T_p \Sigma##, ##v \perp \nabla r## implies that ##v## is itself null so this is equivalent to Peter's definition above.

qtm912
Hi again, I am a bit unfamiliar with the notation here, how is g(del-r, del-r) defined?

Abbas Sherif
Hi again, I am a bit unfamiliar with the notation here, how is g(del-r, del-r) defined?
That's the scalar product (the dot product) of the vector field del-r with itself.