This question is cross-posted at the physics stack exchange at
https://physics.stackexchange.com/q/488358/83357
Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. Let $N$ be the section of $T\Sigma ^g$. Physicists often talk about the evolution of $q$ along $N$ as $\mathcal{L}_Nq$. But this expression makes no sense as $N$ does not belong to $\mathfrak{X}(\Sigma)$. As such, Lie derivatives are defined using flows of vector fields; I don't see any natural way of extending it to arbitrary vector bundles$^{[1]}$.
What is happening here? What do physicists mean when they construct quantities like these$^{[2]}$? Even a link to a reference that treats this on a mathematically justifiable level is welcome.
[1] Naively, I would even expect that one would require some sort of a connection on the vector bundle to make this question tractable.
[2] The only argument I can think of is that the operation is actually being performed on the ambient manifold. Say $tan:\mathfrak{X}(M)\to \mathfrak{X}(\Sigma)$ is the canonical projection operation associated to the embedding ($tan:=q^{\sharp}\circ\iota \circ g^\flat$). Now, $tan^*(q)\in \Omega^2(M)$, so $\mathcal{L}_N(tan^*(q))\in\Omega^2(M)$ is well defined. But this feels like an incomplete picture, and possibly even wrong.
Best Answer
I am no physicist, but I have thought about connections associated to submanifolds. I don't know how to describe the specific construction in your question, but it reminds me of the Bott connection associated to the foliation. When $M$ is pseudo-Riemannian and $\Sigma$ is nondegenerate, $\Sigma$ is locally the leaf of a foliation via the Riemannian structure. So, I will just deal with foliations $\mathcal{F}$ where $\Sigma$ is a leaf.
Connections associated to a foliation:
There is the Bott connection on the normal bundle of $\mathcal{F}$. The normal bundle $N_{M/\mathcal{F}}$ is defined in the same way as the normal bundle of a submanifold, but is on $M$: $$N_{M/\mathcal{F}} = T_M/T_\mathcal{F}.$$ On the leaves of the foliation, the normal bundle has a canonical connection given by the Lie bracket: $$\nabla^{Bott}: N_{M/\mathcal{F}} \otimes \Gamma(T_\mathcal{F})\to N_{M/\mathcal{F}},\\ \nabla^{Bott}_X Y = [X,Y] = \mathcal{L}_X(Y).$$ It is a nice exercise to check that this gives a well-defined connection. This restricts to a connection on $N_{M/\mathcal{F}}|_\Sigma$. To compute $\nabla_X Y$ for $X \in \Gamma(T_\Sigma)$, first extend $X$ to $\tilde X$ tangent to the foliation in a neighborhood of $\Sigma$, extend $Y$, and then compute with the Bott connection. There is also a dual connection on the conormal bundle $N^\vee_{M/\mathcal{F}} = (T_\mathcal{F}^\perp \subseteq \Omega^1_M)$ over the foliation, again given by the Lie derivative.
I suspect that your object $\mathcal{L}_Nq$ is defined similarly, but using the orthogonal complement under the Riemannian structure instead of the normal or conormal bundles. It should be well-defined in the same way the Bott connection is well-defined.