Null Hypersurface: Building a Tangent Surface in 2 Dimensions

[tommyj]

Hi

Why is it possible to be able to pick a spacelike 2 surface S that lies in a null hypersurface N? We know that all the tangents vectors to N are either spacelike or parrelel to the normal vector. I imagine we want to build up S as the surface that is tangent to all the spacelike vectors in N, but I'm not quite sure how to build this up.. A hint rather than the exact answer would be much appreciated (although I will be grateful for either!)

Thanks

 

[robphy]

The light cone is an example of a null hypersurface.

 

[George Jones]

Let $\left\{ e_0, e_1, e_2, e_3 \right\}$ be an orthonormal basis. WLOG, let $n = e_0 + e_1$ be a null vector. What spacetime vectors are orthogonal to $n$?

 

[PeterDonis]

We know that all the tangents vectors to N are either spacelike or parrelel to the normal vector.

This is true. Now ask yourself: how many linearly independent tangent vectors to N are there? (In other words, if I want to construct a basis of tangent vectors to N at a particular point, how many vectors will I need?) Of this total number of linearly independent tangent vectors, how many are spacelike?

 

[tommyj]

Using notation as George above, the basis for T_pN would be (n, e_2, e_3 ) .The two spacelike vectors would be e_2 , e_3, not e_1 since this is not orthogonal to n. Hence, at this particular point p where we have this ortho basis, I would like to have T_pS = sp(e_2, e_3) for the desired surface S. This would induce a Riemannian metric at p for S. I would like to do this for all p\in N in a smooth way so as to build up a spacelike surface S. But how would this be done? I'm aware that the orthormal basis we picked only holds at the point p so the question is how to combine this method together over all of N .I'm still not entirely sure either how I would get a form for S from this anyway. I'm guessing somehow using this operation can help me to define S as the level set of some function f:N\rightarrow \mathbb{R} ?

Thanks for your comments!

 

[PeterDonis]

I would like to do this for all $p \in N$ in a smooth way so as to build up a spacelike surface $S$

You've got the right idea, but note that if you do it (i.e., construct an induced Riemannian metric at each point $p$ using the two spacelike basis vectors $e_2$ and $e_3$) for all $p \in N$, you won't get one spacelike surface; you will get an infinite family of them. You might want to think about how this infinite family of spacelike surfaces can be parametrized.

 

[George Jones]

Maybe you should work this out for the example that robphy suggested, a future lightcone in Minkowski spacetime. It seems to me (without thinking much, or putting pen to paper, so I could be wrong) that, in this case, each member of (as PeterDonis pointed out) the family of 2-dimensional hypersurfaces is (diffeomorphic to) $S^2$.

I base this on analogy. Consider a 3-dimesional version of Minkowski spacetime that has one timelike dimension and two spacelike dimensions. A lightcone is a 2-dimensional surface in this spacetime. There is a family of 1-dimensional spacelike hypersurfaces, i.e., the closed circles that result when $t$ is held constant.

 

[tommyj]

Yes I did that example just now, I must apologize to robphy I thought he was just giving an example of a null hypersurface. Indeed, doing it for the Minkowski light cone things follow through as I had hoped (in this case it was clear that the spacelike surface would be 2 spheres, but it was good to see that the tangent space was indeed spanned by the two "angle" spacelike vectors). These 2 spheres are also the level set of f:N\rightarrow\mathbb{R}, f(X) = r on the light cone, parametrised by r (=t). Maybe I need to generalise this so that my (I'll be really sketchy here) "null" coordinate is the one that defines by function f (so that the spacelike surface is given by f = constant . But I also notice that r appears when we redefine the angle coordinates to have unit length ( i.e choosing our ortho basis) so maybe this is another thing I need to try and generalize? This is what I'll try to do now (unless I've got it horribly wrong!) just thought I would give an update as to my progress!

 

[tommyj]

okay a sort of different approach but how about this. I'll start by the motivation with the light cone for 3D Minkowski (in polar coordinates). The light cone has normal n^a = (\frac{\partial}{\partial t} + \frac{\partial}{\partial r} )^a . We know that the integral curves of n^a are null geodesics that lie in N . It's clear from say drawing a picture that these generate all of the light cone. Let's label each null geodesic as X^a(\lambda) where \lambda is an affine parameter. It is clear that for fixed \lambda = \lambda _0, X^a(\lambda _0) defines a circle, which is spacelike. Hence, for the light cone the spacelike surfaces (which are circles) are parametrised by the affine parameter.

Let's do this for our general null hypersurface N . With the same notation as above, set S = X^a(\lambda _0). Pick p\in S. Then knowing what we have above from our orthonormal representation, we have T_pS = span(e_2. e_3) since n^a = (\frac{d}{d\lambda})^a vanishes as \lambda = \lambda _0 is constant. Thus we have an induced Riemannian metric for S at p and since p was arbitrary we are done. Then the spacelike hypersurfaces would be parametrised by the affine parameter \lambda along the generators of N.

This seems right to me. The only issue is whether S as defined above actually defines a surface. What do people think?

 

[PeterDonis]

The only issue is whether $S$ as defined above actually defines a surface.

If we pick a point $p$ in a given $S$, do the integral curves of $e_2$ and $e_3$, starting from $p$, lie in $S$?