Imagine a metric space to be discontinuous: a Schwarzschild outer metric that changes to a flat FLRW metric once inside the boundary of a $2$-sphere of fixed $r$. If an object (or ray) just hitting the boundary at $r=R$ from the outside has its own four velocity, then in order to continue it, one would have to find the FLRW version of the vector at $r=$R. So the trajectory would be discontinuous. What I want to know is how to find that corresponding vector.
All I know is that it involves the tensor $h_{ab}$, which I don't know the name of. But I would appreciate any help.
