Junction conditions and thin shells in perfect-fluid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity
João Luís Rosa
Abstract
In this work we derive the junction conditions for the matching between two spacetimes at a separation hypersurface in the perfect-fluid version of $f(R,T)$ 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 $f(R)$ 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 $\ensuremath{\sigma}=2{p}_{t}$. 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 $\ensuremath{\sigma}$. For specific forms of the function $f(R,T)$, 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.