A new set of equations to describe the electrodynamics of superconductors at a macroscopic level is proposed. These equations resemble equations originally proposed by the London brothers but later discarded by them. Unlike the conventional London equations the new equations are relativistically covariant, and they can be understood as arising from the 'rigidity' of the superfluid wave function in a relativistically covariant microscopic theory. They predict that an internal 'spontaneous' electric field exists in superconductors, and that externally applied electric fields, both longitudinal and transverse, are screened over a London penetration length, as magnetic fields are. The associated longitudinal dielectric function predicts a much steeper plasmon dispersion relation than the conventional theory, and a blue shift of the minimum plasmon frequency for small samples. It is argued that the conventional London equations lead to difficulties that are removed in the present theory, and that the proposed equations do not contradict any known experimental facts. Experimental tests are discussed.