Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not necessarily imply $v=0$.

Also, and additional hypothesis (*) is: if $x\in A$, $g_x$ defines an inner product on $T_xM$.

I have the following question:

is it possible to prove a tubular neighborhood theorem is this case? I mean, find an open neighborhood of the zero section in the normal bundle of $A$ in $M$, and a diffeomorphism to an open neighborhood of $A$ in $M$. For example, I was thinking on using the exponential map, but I really don't know much about semi-riemannian metrics, and if the geodesics and the exponential are well defined, and if they could be used to prove what i want in this case.

Notice that: the additional hyp (*) allow us to define the normal bundle as the vectors of $T_xM$ wich are orthogonal to $T_xA$