Spherically symmetric solutions of modified field equations in $f(R)$ theories of gravity

Spherically symmetric static empty space solutions are studied in $f(R)$ theories of gravity. We reduce the set of modified Einstein’s equations to a single equation and show how one can construct exact solutions in different $f(R)$ models. In particular, we show that for a large class models, including e.g. the $f(R)=R-{\mu }^4/R$ model, the Schwarzschild-de Sitter metric is an exact solution of the field equations. The significance of these solutions is discussed in light of solar system constraints on $f(R)$ theories of gravity.