Reference request: Energy-momentum tensor for antimatter in general relativity

Question

The intended setting is non-quantum general relativity.

My question: What is the relation between the energy-momentum tensor $T^{\mu\nu}$ and the baryon four-current $J^\mu$ in the case of antimatter, or in the case where the baryon density $J^0$ is zero but there's a non-zero baryon flux $J^i$? I'd be thankful to anyone who could share references about this.

To make the question clear:

In the simplified case of matter without internal forces ("dust"), the twice-contravariant energy-momentum for matter can be connected to the baryon four-current in several ways, eg:
$$ T^{\mu\nu} = \rho\, u^\mu \, u^{\nu}
\qquad\text{with}\quad
u^\mu = c\,J^\mu/\sqrt{\lvert J^\alpha g_{\alpha\beta} J^{\beta}\rvert}$$
where $\rho$ is the (rest) mass density (mass divided by volume), or as
$$ T^{\mu\nu} = c\, m\, J^{\mu} \, J^{\nu}/\sqrt{\lvert J^\alpha g_{\alpha\beta} J^{\beta}\rvert}$$
where $m$ is the (rest) mass per baryon (or molar mass density, if we measure $J$ in moles).

It seems to me that both expressions could be used in the case of antimatter: irrespective of whether $J^0 \gtreqless 0$, we would still have $T^{00} \ge 0$ as confirmed by the Alpha-g experiments. But I'd be happy if anyone could share some references that discuss this kind of situations.

(Note: zero baryon density but a non-zero baryon current can occurr, for instance, if at an event there is a flux of baryons in one direction and a flux of antibaryons in the opposite direction – similarly to what can happen with electric current:zero charge density but non-zero current)