Ricci-curvature

We derive the wave equation obeyed by electromagnetic fields in curved spacetime. We find that there are Riemann and Ricci curvature coupling terms to the photon polarisation which result in a polarisation dependent deviation of the photon trajectories from null geodesics. Photons are found to have an effective mass in an external gravitational field and their velocity in an inertial frame is in general less than c. A physically relevant consequence of the analysis is that the curvature corrections to the propagation of electromagnetic radiation (in a homogenous and isotropic spacetime) keep the velocities subluminal provided the strong energy condition is satisfied. We further show that the claims of superluminal velocities in higher derivative gravity theories are erroneous and arise due to the neglect of Riemann and Ricci coupling terms in the wave equation, of Einstein gravity.

On a complete Riemannian manifold M with Ricci curvature satisfying Ric(∇r, ∇r) ≥ −Ar 2 (log r) 2 (log(log r)) 2 • • • (log k r) 2 for r ≫ 1, where A > 0 is a constant, and r is the distance from an arbitrarily fixed point in M. we prove some Liouville-type theorems for a C 2 function f : M → R satisfying ∆f ≥ F (f) for a function F : R → R. As an application, we obtain a C 0 estimate of a spinor satisfying the Seiberg-Witten equations on such a manifold of dimension 4. We also give applications to the Yamabe problem and isometric immersions of such a manifold.