Static spherically symmetric perfect fluid solutions in $f(R)$ theories of gravity

Static spherically symmetric perfect fluid solutions are studied in metric $f(R)$ theories of gravity. We show that pressure and density do not uniquely determine $f(R)$ i.e. given a matter distribution and an equation state, one cannot determine the functional form of $f(R)$. However, we also show that matching the outside Schwarzschild–de Sitter metric to the metric inside the mass distribution leads to additional constraints that severely limit the allowed fluid configurations.