New framework for studying spherically symmetric static solutions in $f(R)$ gravity

We develop a new covariant formalism to treat spherically symmetric spacetimes in metric $f(R)$ theories of gravity. Using this formalism we derive the general equations for a static and spherically symmetric metric in a general $f(R)$ gravity. These equations are used to determine the conditions for which the Schwarzschild metric is the only vacuum solution with vanishing Ricci scalar. We also show that our general framework provides a clear way of showing that the Schwarzschild solution is not a unique static spherically symmetric solution, providing some insight into how the current form of Birkhoff’s theorem breaks down for these theories.