Photon propagation 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. 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. 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 paper 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) implies that there is a polarisation dependence in the gravitational red shift, bending, and Shapiro time delay of light even in classical Einsteins gravity. A curious phenomena discovered by Drummond and Hathrell [4] is that in higher derivative gravity which arises by QED radiative corrections, the photon velocity in local inertial frame can exceed the velocity of light in the Minkowski space. Due to the coupling of the electromagnetic fields with the Riemann and Ricci tensors in the Lagrangian, it was claimed that the photon velocity in the Schwarzschild, RobertsonWalker, gravitational wave and deSitter backgrounds is larger than the 2 flat space velocity c. This result was extended by Daniels and Shore to charged [5] and rotating blackholes [6] with the same conclusions. Latorre et al [7] have shown a universal relation between the velocity shift of photons to the energy density which generates the background metric, and Shore [8] has related the velocity shift to the coefficients of conformal anomaly. Lafrance and Myers [9] interpret these results as the breakdown of the Equivalence principle in higher derivative gravity and Dolgov and Khriptovich [10,11] derive this result from field theoretic dispersion relations. Finally Mende [12] has proposed this effect as a test for string theories of quantum gravity. In this paper we show that claims of superluminal photon velocity are due to the neglect of the Riemann coupling terms in the wave equation which arises from the minimal FμνF μν Lagrangian. We find that in Einstein’s gravity the photon velocity in a Schwarzschild blackhole metric and in the Friedman-Robertson-Walker metric is less than c. In other words the photon trajectories are always inside the null cone. We find that for the photon trajectories not to go out of the null cone the background matter should satisfy the strong energy condition ρ ≥ 3p. This provides us with an answer to the question raised by Zeldovich and Novikov [13] What law of physics would be violated if the strong energy condition is not satisfied ? Our answer is that special relativity in the free fall inertialframe demands that the strong energy condition be satisfied. We also derive the wave in higher derivative gravity and show that Riemann coupling terms in the lagrangian of the higher derivative gravity are always smaller in magnitude than the Riemann term that already exists in Einstein’s gravity. This analysis shows that the photon velocity does not exceeds c, even by the inclusion of the radiative correction higher derivative terms in the Lagrangian contrary to the claims [4-12]. The absence of superluminal propagation is not dependent on the eikonal 3 ansatz but is a consequence of the fact that the wave-equation in for the electromagnetic fields in Einstein and higher derivative gravity is hyperbolic. Consequently the field solution at a given point can only depend upon sources inside the past nullcone. In references [4-12] the vector potential solution is assumed to be of the eikonal form, and it is shown that the vector potential propagates outside the null-cone. The eikonal approximation for the vector potential does not always correctly describe the propagation of electromagnetic waves since the vector potential is not gauge invariant and it can be non-zero even in the acausal regions where the electromagnetic field is zero. The interaction of electromagnetic fields with gravity is given by the action S = ∫ dx √ −gFμνF μν (3) (We use the convention c = 1, signature −2 and Greek letters denote spacetime indices 0− 3). From (3) we obtain the equations of motion ∇Fμν = 0 (4) Equation (4) with the Bianchi identity ∇μFνλ +∇νFλμ +∇λFμν = 0 (5) gives the Maxwell’s equation in curved background. Operating on equation (4) by the ∇λ and using the Bianchi identity (5) the wave equation may be obtained as ∇∇μFνλ + ∇λ (∇Fνμ) + [∇,∇ν ]Fλμ − [∇λ,∇]Fμν = 0 (6) The second term vanishes owing to (4). Using the identity for the commutator of covariant derivatives: [∇,∇ν ]Fαμ = RρανμF ρμ +RρλF ρ ν (7) 4 and the circular identity Rρλνμ +Rρνμλ +Rρμλν = 0 (8) the wave equation (6) reduces to the form ∇∇μFνλ +RρμνλF ρμ +RρλFνρ − RνFλρ = 0 (9) The Riemann and Ricci curvature coupling terms to the photon polarisation (or spin) give rise to the polarisation dependent deviation of photon orbits from null geodesics. Photon trajectories are described by the eikonal solutions of the wave equation (9) of the form Fμν = e fμν (10) where the phase S varies more rapidly in spacetime than the amplitude fμν . The wavenumber of the photon trajectories is given by the gradient of the phase Kμ = ∇μS (11) In the geometrical optics approximation where ∇μFαβ = iKμFαβ (12) the wave equation (9) can be written in the form −KKμfνλ +R νλfρμ − Rρλfνρ +Rνfλρ = 0 (13) The dispersion relation may be written as K = ( R νλfρμ −R ρ λfνρ +R ρ νfλρ ) f νλ (fαβf) (14) Operating on (14) by ∇α, the L.H.S. is ∇K = 2K∇μK (15) 5 where we have used the identity ∇Kμ = ∇μK which follows from the definition of Kμ as a gradient. Light rays are defined as the integral curves of the wave vector Kμ i.e. the curves xμ(s) for which dXμ ds = Kμ. Substituting Kμ = dXμ/ds in (15) we have for the L.H.S. ∇K = 2 μ ds ∇μ ( dX ds ) = 2 ( dX ds2 + Γμν dX ds dX ds ) (16) Using (14), (15) and (16) we obtain the modified geodesic equation for photon trajectories dX ds2 + Γβγ dX ds dX ds = 1 2 ∇ (R νλ · fρμ − Rρλfνρ +Rνfλρ)× f νλ (ffαβ) (17) To obtain the photon velocity in the curved space, we can use the dispersion relation (13) directly. Of the six components of Fμν in equation (13), only three are independent owing to the Bianchi identity (5). Choosing the components of the electric field vector Ei = foi as the independent components we have from the Bianchi identity (5) Kofij +Kifjo +Kjfoi = 0 (18) Using (18) to substitute for fij in terms of the electric field components fjo and foi in the wave equation (13) we obtain