Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it.

Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null hypersurface, by which we mean its normal is null.

It is common in GR to argue the following:

  1. Since the normal vector $\zeta$ to $\Sigma$ is null, it is orthogonal to itself. Hence the normal to $\Sigma$ also lives in its tangent bundle.

  2. In that case, one can consider its integral lines. One works formally to show that $\zeta$ satisfies the autoparallel equation $$\zeta^\mu \nabla_\mu\zeta_\nu=\alpha\zeta_\nu.$$

Now, I have one problem here. It seems intuitively clear to me that being a geodesic on a submanifold is not the same as being a geodesic on the ambient space.

One example that comes to my mind is in $\mathbb{R}^3$. The geodesics are straight lines. If one looks to the submanifold $S^2$, the geodesics are great circles which as seen on curves on $\mathbb{R}^3$ are not geodesics.

The central question seems to be: what is the connection with respect to which geodesics on $\Sigma$ are being considered, and how it connects to the Levi-Civita connection on $M$?

Is it the pullback connection? Or is it some sort of "projected" connection, as when we "project the covariant derivative to the tangent space of $S^2$" in $\mathbb{R}^3$? How it relates to the induced metric $\phi^\ast g$?

In short: when physicists consider these generators of the null surface, which are integral lines of the normal as seen as a tangent vector, what is the appropriate way to make sense of the resulting geodesics? How to make this whole discussion rigorous?


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