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Gravitational Chern-Simons Lagrangian terms and spherically symmetric spacetimes

We show that for general spherically symmetric configurations, contributions of general gravitational and mixed gauge-gravitational Chern-Simons terms to the equations of motion vanish identically in $D>3$ dimensions. This implies that such terms in the action do not affect Birkhoff's theorem or any previously known spherically symmetric solutions. Furthermore, we investigate the thermodynamical properties using the procedure described in an accompanying paper. We find that in $D>3$ static spherically symmetric case Chern-Simons terms do not contribute to the entropy either. Moreover, if one requires only for the metric tensor to be spherically symmetric, letting other fields unrestricted, the results extend almost completely, with only one possible exception --- Chern-Simons Lagrangian terms in which the gravitational part is just the $n=2$ irreducible gravitational Chern-Simons term.