Primordial Magnetogenesis from Killing Vector Fields
Papapetrou showed that the covariant derivative of a Killing vector field satisfies Maxwell's equations in vacuum. We extend Papapetrou's result to show that the covariant derivative of a Killing vector field satisfies Maxwell's equations in non-vacuum backgrounds as well, if we allow electromagnetic currents of purely geometric origin. We postulate that every Killing vector field gives rise to a physical electromagnetic field and, in a non-vacuum background, a physical electromagnetic current---hereafter called Killing electromagnetic field and Killing electromagnetic current, respectively. We show that the Killing electromagnetic field of the flat FLRW universe comprises a Killing magnetic field and a rotational Killing electric field; we derive an upper bound on the Killing magnetic field and find that the upper bound is consistent with the current upper bound on the cosmic magnetic field. We show that the time-like Killing vector of the Schwarzschild spacetime gives rise to a radial Killing electric field. We also show that in the weak field regime---and far from the matter distribution---the back reaction of the radial Killing electric field changes the Schwarzschild metric to Reissner-Nordstr$\ddot{\mbox{o}}$m metric; our calculation yields the converse of Wald's result. Next, drawing upon Rainich's work on Rainich-Riemann manifolds, we discuss the etiological question of how a physical electromagnetic field can arise out of geometry; we also argue that detection of the Killing electric field of flat FLRW spacetime may be within current experimental reach. Finally, we discuss the relevance of Killing electromagnetic currents and the aforementioned transmutation of Schwarzschild spacetime to Reissner-Nordstrom spacetime, to Misner and Wheeler's program of realizing ``charge without charge''.