Conformal coupling of gravitational wave field to curvature

Abstract

Conformal properties of the equations for weak gravitational waves in a curved space–time are investigated. The basic equations are derived in the linear approximation from Einstein’s equations. They represent, in fact, the equations for the second-rank tensor field hαβ, restricted by the auxiliary conditions hαβ;α =0, h≡γαβhαβ=0, and embedded into the background space–time with the metric tensor γαβ. It is shown that the equations for hαβ are not conformally invariant under the transformations γ̂αβ =e2σγαβ and ĥαβ =eσhαβ, except for those metric rescalings which transform the Ricci scalar R̂ of the original background space–time into e−2σR, where R is the Ricci scalar of the conformally related background space–time. The general form of the equations for hαβ which are conformally invariant have been deduced. It is shown that these equations cannot be derived in the linear approximation from any tensor equations which generalize the Einstein equations.