On manifolds with positive definite metrics (e.g. dot product), any localized embedding of an \(n-1\) dimensional submanifold will have tangent spaces that partition those of the original manifold into an \(n-1\) dimensional subspace (that of the submanifold) and a \(1\) dimensional subspace (the part of the tangent space orthogonal to the submanifold).
The black vector here denotes the subspace orthogonal to the tangent space of the submanifold, and the red is the tangent space of the submanifold.
If we remove our postitive-definite condition on our metric, however, we obtain a much more peculiar structure.
For our manifold \(M\), say we chose our metric \(\eta\) to be the standard Lorentzian metric used in Minkowski spacetime: [ ds^2 = dt^2 - dx^2 - dy^2 - dz^2]. We no longer have our positive definite condition. Let’s choose some \(3\) dimensional embedded hypersurface \(S\), and consider it’s tangent space at a point \(p \in S, T_p S\). Our embedding allows us to identify \(T_p S\) as a sub-vector space of \(T_p M\), the tangent space at \(p\) in the original manifold.
So what do different potential configurations of \(T_p S\) look like? We know they will form a \(3\) dimensional subspace. For illustration’s sake, we move to a 2-dimensional analogue, supressing the \(z\) coordinate. We first look at the case when \(T_p S\) is directly orthogonal to the local \(t\) coordinate:
Just like in the positive-definite case, our “normal” subspace is independent of the subspace of \(T_p S\), shown in red. We have some new structure, however. The blue cone shown (which was previously just the origin) is the space of “null vectors”, or vectors \(X\) for which \(\eta(X, X) = 0\). Interestingly, even though we have a different structure for these null vectors, we still end up with the same kind of orthogonality we would have previously. When we start to “tilt” \(T_p S\), the way in which our normal subspace changes is quite different:
Here we see that as the subspace tilts, the normal tilts towards the subspace, and at the point of the cone, they are equivalent. This means that when \(T_p S\) intersects with this null cone, the normal subspace to \(T_p S\) is it’s intersection with the light cone!
We can pull this back to a property of \(S\), defining \(S\) as a null hypersurface if at every \(p \in S\), the normal subspace to \(T_p S\) lies in the normal cone.
Physically, our manifold is spacetime, with time having the positive sign in our metric. Particles are curves in spacetime with velocities that lie somewhere on the interior or the surface of the null cone in every tangent space. A tangent vector lying on the null cone corresponds to a velocity of a particle traveling at the speed of light.
Imagine converting a point-like piece of matter to photons with them scattering in all directions. The path these photons take will trace out a surface in our spacetime in the exact shape of the null cone, meaning this surface is a null hypersurface! In general these surfaces are very useful for performing computations and describing certain submanifolds of spacetime.