Spherically symmetric solutions in $f\left(R\right)\mathrm{}$ theories of gravity obtained using the first order formalism

The solutions of a class of theories obtained when we apply the first order formalism are studied. In the linear approximation we obtain the Green function and we prove that the field is independent of the size and internal stresses of the source. We show that the solutions of the field equations for a mass point are also the exterior solutions for an arbitrary spherically symmetric mass distribution. We construct the solutions of the field equation, without any approximation, for the spherically symmetric matter distribution, and prove that the exterior solutions match correctly with the interior solutions. We also prove that one of the exterior solutions is always the Schwarzschild solution. Finally, in the same case, we show that Birkhoff’s theorem is satisfied. All the above results are quite similar to general relativity but are very different from the results of the fourth order theories; then we have shown that the first order formalism theories have better classical properties than fourth order theories.