Geodesic Deviation of Photons in Einstein and Higher Derivative Gravity

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. The effective photon mass in the Schwarzschild metric is mγ = (2GM/r ) 1/2 and near the horizon it is larger than the Hawking temperature of the blackhole. Our result implies that Hawking radiation of photons would not take place. We also conclude that there is no superluminal photon velocity in higher derivative gravity theories (arising from QED radiative corrections), as has been claimed in literature. We show that these erroneous claims are due to the neglect of the Riemann and Ricci coupling terms which exist in Einstein’s gravity. A standard result of Einstein’s gravity is that the trajectories of all massless particles are null geodesics. A question worth examining is whether there is a deviation from the null geodesics if the particles have a spin i.e. due to the interaction of spin with the Riemann and Ricci curvatures of the gravitational fields. In this letter we have studied this question for the case of photon propagation in a curved background. Starting with the action −gFμν F μν for electromagnetic fields in a gravitational field, we derive the wave equation for electromagnetic field tensor Fμν , which turns out to be of the form (Eddington [1] , Noonan [2]) ∇∇μFνλ +RρμνλF ρμ −RρλFνρ +RνFλρ = 0 (1) We see that the photon propagation depends upon the coupling between the Riemann and Ricci curvatures and the photon polarisation. This leads to a deviation of the photon trajectories from the null geodesic by amounts proportional to the Riemann and Ricci curvatures. The photon trajectories in the geometrical optics limit are described by the following generalisation of the geodesic equation dX ds2 + Γβγ dX ds dX ds = 1 2 ∇ [R νλfρμ +Rρλfνρ − Rνfλρ]× f νλ | f 2 | (2) The nonzero right hand side of the modified geodesic equation (2) leads to a polarisation dependence in the gravitational red shift, bending, and Shapiro time delay of light. The redshift in a photon’s wavelength propagating in the gravitational field of a spherical mass (or a blackhole) is given by λ2 λ1 = ( 1− 2GM/r2 1− 2GM/r1 1/2 