Asymptotic flatness, reflection symmetry, and Birkhoff's theorem forR+R2actions containing quadratic torsion terms

The proof of the Birkhoff-type theorem: For asymptotically flat O(3) spherically symmetric spacetime the unique solution of the vacuum field equations is the Schwarzschild solution with vanishing torsion, is extended to $R+R^2$ actions containing a quadratic torsion term. The simplifying power of the assumption of asymptotic flatness is severely degraded when reflection symmetry is not assumed, further underscoring the sensitivity of Riemann-Cartan theories to reflection symmetry, and inhibiting the proof of a similar theorem for SO(3) spherical symmetry.