Finite Energy Superluminal Solutions of Maxwell Equations

We exhibit exact finite energy superluminal solutions of Maxwell equations in vacuum. e-mail: [email protected] e-mail: [email protected] or [email protected] 1 Recently, some papers [1,2] appeared in the literature showing that in some hypotetical media there is the possibility for the existence of superluminal electromagnetic pulses (solutions of Maxwell equations) such their fronts travel in the media travel with superluminal velocities. Now, the solutions discovered in [1,2], despite their theoretical interest have infinity energy and as such cannot be produced in the physical world. Only finite aperture approximations to these waves can eventually be produced (supposing the existence of the special media). The objective of this letter is to show that in contrast to the solutions discovered in [1,2] (that as already said have infinity energy), there exist vacuum solutions of Maxwell equations which are finite energy superluminal solutions. These solutions, as we shall see appear as solutions of Sommerfeld like problems [3,4] to be reported below. We also discusss if such solutions can be generated in the physical world. We start by recalling how to write electromagnetic field configurations in terms of Hertz potentials [5,6]. Suppose we have a Hertz potential ~ Πm of magnetic type. In what follows we use units such that the velocity of light c = 1. Then, the associated electromagnetic field is given by ~ E = − ∂ ∂t (∇× ~ Πm), ~ B = ∇×∇× ~ Πm. (1) Let us take ~ Πm = Φêz . Then, since the Hertz potential (in vacuum) satisfies a homogeneous wave equation, we have that Φ = 0. (2) The Sommerfled problem (not to be confused with a Cauchy problem) to be considered here is the following. In a given inertial frame (the laboratory) find a solution ΦX : (t, ~x) 7→ C (where C is the field of complex numbers) for eq.(2) satisfying the following boundary conditions at the z = 0 plane,