The classic way to represent a proton in flat spacetime is to use a simple point charge,
$$J = (e \delta(r), 0)$$
But if we're using an arbitrary metric (let's assume still globally hyperbolic orientable to avoid any issues), then the local conservation of charge is expressed via
\begin{eqnarray} \nabla_\mu J^\mu &=& \partial_\mu J^\mu + \frac{\partial_\mu \sqrt{g}}{\sqrt{g}} J^\mu\\ &=& \dot{\rho} + \vec{\nabla} \cdot \vec{J} + \frac{\partial_t \sqrt{g}}{\sqrt{g}} \rho + \frac{\vec{\nabla} \sqrt{g}}{\sqrt{g}} \cdot \vec{J}\\ &=& 0 \end{eqnarray}
Obviously if our spacetime isn't static, we get that
\begin{eqnarray} \frac{d}{dt} Q = - \int_\Sigma \left[ \vec{\nabla} \cdot \vec{J} + \frac{\partial_t \sqrt{g}}{\sqrt{g}} \rho + \frac{\vec{\nabla} \sqrt{g}}{\sqrt{g}} \cdot \vec{J} \right] \sqrt{q} d^3x \end{eqnarray}
With $q$ the metric on the Cauchy surface $\Sigma$. Given enough conditions on the metric, we could probably get back conservation of total charge, using Stokes theorem, but for a non-static one, it seems a forlorn hope.
What would then be the most appropriate for the notion of a point charge then? Do we simply pick $\rho = e \delta(r)$, and have the total charge vary, or do we pick a charge density such that the total charge is conserved? I'm not sure which one would approximate the behaviour of a real charge best here. This may also make it depend on the choice of foliation so I'm not sure if the notion of total charge is even that decent in this case.
