Diffeomorphism Radiative Degrees of Freedom of Thomas-Whitehead Gravity

The geometric action of the semi-direct product of the Kac-Moody and Virasoro algebras contains the WZW action equipped with a background vector potential $A$ associated to a coadjoint element of the Kac-Moody algebra as well as the 2D gravitational Polyakov action and an accompanying background field, $\mathcal{D}$, called the diffeomorphism field. Just as the coadjoint element, $A$, is related to a gauge fixed Yang-Mills vector potential $A_a$, the diffeomorphism field, $\mathcal{D}$, is related to a component, $\mathcal{D}_{a b}$ of the projectively invariant connection called the Thomas Operator. The Yang-Mills action provides dynamics for the vector potential $A_a,$ while the Thomas-Whitehead (TW) gravitational action, provides dynamics to $\mathcal{D}_{ab}$. The TW action embeds the projectively invariant connection into a gravitational theory that contains general relativity. In this work, the diffeomorphism field $\mathcal{D}_{a b}$ is examined in Minkowski space where salient features of this field can be explored. In particular, we study the radiative degrees of freedom of this field while in a Minkowski space background. We show that it can be decomposed into irreducible representations, corresponding to tensor, vector, and scalar radiating solutions. Furthermore we examine geodesic deviation in the context of TW gravity about a Minkowski space background. We do this both at zeroth and first order in metric fluctuations $h_{ab}$. We discuss that response of a gravitational wave antennae to the geodesics deviations.