In a class of nine-parameter Riemann-Cartan-type $R+R^2$ gravitational actions including torsion-squared terms we search for theories that satisfy a generalized Birkhoff theorem: In the absence of matter the Schwarzschild metric with vanishing torsion is the unique SO(3) spherically symmetric solution of the field equations. Unlike in previous discussions, the ad hoc assumption of reflection symmetry is not assumed. Two theories satisfying Birkhoff's theorem are found. They are unitary and propagate only the graviton. Two weakened versions of Birkhoff's theorem are discussed, one with reflection symmetry imposed and the other restricted to scalar-flat geometries. Four $R+R^2$ theories satisfying one or the other of these weakened Birkhoff theorems are presented. The unitary properties and particle content of these $R+R^2$ theories are summarized.

• Received 8 December 1980

DOI:https://doi.org/10.1103/PhysRevD.24.2029