General Relativistic version of unorthodox two classical point particles collision problem?

Question

So I was playing around with an idea but got stuck.

Premise

Imagine there are two classical point particles. I was wondering how would one talk about a collision and came up with something interesting. The normal equation of motions would be:

$$ m_1 \ddot{x_1} = 0 $$ $$ m_2 \ddot{x_2} = 0 $$

However, this does not talk about the collision about these point particles. Hence, consider:

$$ m_1 \ddot{x_1} = c_{12} \delta(x_1 - x_2) $$ $$ m_2 \ddot{x_2} = c_{21} \delta(x_2 - x_1) $$

where something "bad" happens upon collision which is shown by a dirac $\delta$ function. A little bit of playing around one can show the Hamiltonian can be:

$$ H = \frac{1}{2} m_1 \dot x_1^2 + \frac{1}{2} m_1 \dot x_2^2 + c_{12} \theta(x_1 - x_2) $$

with $c_{21} = - c_{12}$ and $\theta$ as the Heaviside function.

Question

I was then wondering how this would be modelled in General Relativity (and seeing the effects on curvature). Usually one never does collisions in GR because they involve potentials which are usually associated with fields. I was wondering if allowing discontinuities and distributions gave a bit room to wiggle this in? If yes, feel free to show the GR version?

Answer 1

The notion of a massive pointlike particle doesn't make sense in classical relativity. A particle is surrounded by field (an electric field if it's charged, and in any case a gravitational field, if it has mass). $E=mc^2$, so this field carries mass. Therefore the particle isn't pointlike. This is why, for example, we have the classical electron radius.

Even if you could evade this issue somehow, the particle would then be a naked singularity. GR doesn't predict what happens at a naked singularity.