Junction conditions and thin-shells in perfect-fluid $f\left(R,T\right)$ gravity

In this work we derive the junction conditions for the matching between two spacetimes at a separation hypersurface in the perfect-fluid version of $f\left(R,T\right)$ gravity, not only in the usual geometrical representation but also in a dynamically equivalent scalar-tensor representation. We start with the general case in which a thin-shell separates the two spacetimes at the separation hypersurface, for which the general junction conditions are deduced, and the particular case for smooth matching is considered when the stress-energy tensor of the thin-shell vanishes. The set of junction conditions is similar to the one previously obtained for $f\left(R\right)$ gravity but features also constraints in the continuity of the trace of the stress-energy tensor $T_{ab}$ and its partial derivatives, which force the thin-shell to satisfy the equation of state of radiation $\sigma=2p_t$. As a consequence, a necessary and sufficient condition for spherically symmetric thin-shells to satisfy all the energy conditions is the positivity of its energy density $\sigma$. For specific forms of the function $f\left(R,T\right)$, the continuity of $R$ and $T$ ceases to be mandatory but a gravitational double-layer arises at the separation hypersurface. The Martinez thin-shell system and a thin-shell surrounding a central black-hole are provided as examples of application.