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One-loop graviton corrections to Maxwell's equations

We compute the graviton induced corrections to Maxwell's equations in the one-loop and weak field approximations. The corrected equations are analogous to the classical equations in anisotropic and inhomogeneous media. We analyze in particular the corrections to the dispersion relations. When the wavelength of the electromagnetic field is much smaller than a typical length scale of the graviton two-point function, the speed of light depends on the direction of propagation and on the polarisation of the radiation. In the opposite case, the speed of light may also depend on the energy of the electromagnetic radiation. We study in detail wave propagation in two special backgrounds, flat Robertson-Walker and static, spherically symmetric spacetimes. In the case of a flat Robertson-Walker gravitational background we find that the corrected electromagnetic field equations correspond to an isotropic medium with a time-dependent effective refractive index. For a static, spherically symmetric background the graviton fluctuations induce a vacuum structure which causes birefringence in the propagation of light.