Oh I want the spacetime to be an open submanifold of $\mathbb{R}^4$ (with the standard smooth structure)

Now that I think of it they will probably have to collapse because they will emit waves and so energy will be running away to infinity? Basically what people expected electrons to do when circling the nucleus (and what was later fixed with quantization).

Using light feels a little like a cheatcode. Can the objects be massive? Unfortunately I don't know anything about interpretations.

We would need a compactification of a "Kuga-Sato variety" (I'm not sure about the exact definition outside of the modular curve case).