Solution of Maxwell’s equations on a de Sitter background

Abstract

The Maxwell equations for the electromagnetic potential, supplemented by the Lorenz gauge condition, are decoupled and solved exactly in de Sitter spacetime studied in static spherical coordinates. There is no source besides the background. One component of the vector field is expressed, in its radial part, through the solution of a fourth-order ordinary differential equation obeying given initial conditions. The other components of the vector field are then found by acting with lower-order differential operators on the solution of the fourth-order equation (while the transverse part is decoupled and solved exactly from the beginning). The whole four-vector potential is eventually expressed through hypergeometric functions and spherical harmonics. Its radial part is plotted for given choices of initial conditions. We have thus completely 123 52 D. Bini et al. succeeded in solving the homogeneous vector wave equation for Maxwell theory in the Lorenz gauge when a de Sitter space-time is considered, which is relevant both for inflationary cosmology and gravitational wave theory. The decoupling technique and analytic formulae and plots are completely original. This is an important step towards solving exactly the tensor wave equation in de Sitter space-time, which has important applications to the theory of gravitational waves about curved backgrounds.

Maxwell wave packets in de Sitter expanding universe

“…In all these models the photons are considered as classical massless particles moving along null geodesics since we do not have yet a complete quantum theory of light propagating in curved space-times, despite much progress in studying the Maxwell equations in different manifolds including the de Sitter one. On such space-times of various dimensions, the Maxwell equations were studied either in local charts (called here frames) with static coordinates [41][42][43][44][45] or in comoving frames [46,47]. Other studies were devoted to the Maxwell field involved in the cosmological particle creation in Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes [48][49][50][51][52][53][54][55][56][57].…”

Polarized vector bosons on the de Sitter expanding universe

“…The isometry group of the dS manifold is just the group SO(1, 4) of the pseudoorthogonal transformations in M 5 . For this reason, the basis-generators of the SO (1,4) algebra are associated to ten independent Killing vectors, K (AB) = −K (B A) , which give rise to the basis-generators X (AB) of the vector representation of the SO (1,4) group carried by the space of the vector fields A. In what follows we focus only on the Hamiltonian (or energy) operator H = ωX (05) , the momentum components P i = ω(X (5i) − X (0i) ) and those of the total angular momentum [15].…”

“…The free scalar and Dirac fields minimally coupled to gravity were studied in static charts as well as in the (co)moving charts with proper or conformal times. The free massive vector field is considered only in static charts [3,4] with spherical coordinates and, therefore, the problem of its quantum modes in moving charts is still open.…”

Maxwell field in spatially flat FLRW space-times

“…Other solutions of the Maxwell equations were studied in the de Sitter manifolds, either in local frames with static coordinates [4][5][6][7][8] or in co-moving frames of the expanding portions of arbitrary dimensions [9]. In the (1 + 3)-dimensional de Sitter expanding portion, known as the de Sitter expanding universe, we performed some time ago the canonical quantization of the Maxwell free field in the Coulomb gauge [10], which was the starting point in studying the de Sitter QED in the first order of perturbations [11], the quantum theory of redshift [12] and the propagation of the Maxwell wave packets in this manifold [13].…”