Black holes in Thomas-Whitehead projective gravity

The Thomas-Whitehead projective gravity theory has it origins in string theory. The identified correspondence between the of the coadjoint elements of the Virasoro algebra and Sturm-Liouville operators. This identification of the projective structure in one dimension allows for relating the Virasoro algebra and projective geometry in higher dimensions. We've examined how the Thomas Whitehead projective connection can be used to form curvature tensor, that are then used to build an action for Thomas Whitehead projective gravity. The fields Pi and D of the Thomas-Whitehead connection source gravitation through an associated energy momentum tensor.We consider the Pi and Diffeomorphism fields as providing a source for the supermassive black hole. The Pi and Diffeomorphism fields then contribute to the energy momentum tensor, and so should modify the Schwarzschild and Kerr solutions for their respective symmetries. We propose to investigate this possibility the energy momentum tensor contributions from the emanated Pi and D fields corresponding to supermassive black hole solutions could supply the dark matter contribution to galaxy rotation curves. Finster, Smoller, Yau showed that there were no spinor solutions to the Dirac-Einstein-Yang-Mills equations. We propose investigating with Dirac-Yang-Mills-Thomas-Whitehead action with our spinors from the Thomas cone to examine whether the spinors on the Thomas cone could source a black hole. The projective geodesic deviation equation result allows for us to identify the possibility of gravitational radiation independent of the metric degrees of freedom. We would like to explore these features to make predictions about the possible gravitational radiation or tidal effects from Black Holes in Thomas-Whitehead gravity.