Chern-Simons Modification of General Relativity
R. Jackiw, S. -Y. Pi
TL;DR
The paper introduces a four-dimensional Chern-Simons deformation of general relativity by coupling to an external time-like embedding coordinate $v_\mu$, yielding a modified field equation $G^{\mu\nu}+C^{\mu\nu}=-8\pi G T^{\mu\nu}$ with a Cotton tensor $C^{\mu\nu}$ and a consistency condition involving the Pontryagin density ${^*RR}$. The authors derive the CS gravity action, analyze stationary and linear regimes, and show that the Schwarzschild solution persists while Kerr is generally not a solution; in the linear theory, gravitational waves retain two polarizations that propagate at speed $c$, but their intensities exhibit parity-violating differences due to the CS extension. They also construct a conserved symmetric gravitational energy-momentum pseudotensor, illustrating that diffeomorphism symmetry breaking is effectively hidden in the dynamics. Although measuring polarization differences in gravitational waves is not yet feasible, the work provides a controlled theoretical framework linking topological CS terms to observable gravitational phenomena and highlights potential cosmological implications for dynamical $ heta$ scenarios.
Abstract
General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate v_μ-- an external quantity, which we fix to be a non-vanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension.