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.
There are other questions, such as this, related to the same general topic, but not about this specific relationship.
(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)
