Gravitational Chern-Simons Lagrangian terms and spherically symmetric spacetimes

TL;DR

This paper demonstrates that in dimensions greater than three, gravitational and mixed gauge-gravitational Chern-Simons terms do not influence spherically symmetric solutions or their thermodynamics, except possibly for a specific gravitational Chern-Simons term.

Contribution

It proves the vanishing of Chern-Simons contributions to equations of motion and entropy in spherically symmetric spacetimes for D>3, clarifying their limited physical impact.

Findings

Chern-Simons terms do not affect Birkhoff's theorem in D>3

Chern-Simons terms do not contribute to entropy in static spherically symmetric spacetimes

Results extend to cases with unrestricted matter fields, with a specific exception

Abstract

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.