Spacetime metric from linear electrodynamics II

Following Kottler, É.Cartan, and van Dantzig, we formulate the Maxwell equations in a metric independent form in terms of the field strength F = (E,B) and the excitation H = (D,H). We assume a linear constitutive law between H and F . First we split off a pseudo-scalar (axion) field from the constitutive tensor; its remaining 20 components can be used to define a duality operator # for 2-forms. If we enforce the constraint ## = −1, then we can derive of that the conformally invariant part of the metric of spacetime. file weimar16a.tex, 1999-11-24 email: [email protected], [email protected] email: [email protected] email: [email protected] 1 AXIOMATICS OF METRIC-FREE ELECTRODYNAMICS 1 1 Axiomatics of metric-free electrodynamics We assume that spacetime is described by a smooth 4-dimensional manifold X and that it is possible to foliate X into 3-dimensional submanifolds which can be numbered by a monotonously increasing parameter σ. This could be called Axiom 0. Our formulation of classical electrodynamics [24, 18, 9] is based on four axioms, three of which being of general validity and independent of the metric and/or affine structures of spacetime. Only in the context of the fourth axiom, the constitutive relation, the metric comes into play. 1.1 Axiom 1: Electric charge conservation We assume the existence of a conserved electric current described by means of an odd 3-form J onX; for the exterior calculus involved, see [5]. Conservation of electric charge is a firmly established fact which basically can be verified by counting charged elementary particles inside a closed region. Mathematically, J is conserved when ∮ Ω3 J = 0 , ∂Ω3 = 0 , (1) where Ω3 is an arbitrary closed 3-dimensional submanifold of the 4-manifold X. By de Rham’s theorem, the current J is not only closed dJ = 0, but also exact, see [27, 23]. Thus the inhomogeneous Maxwell equation is a consequence of (1), J = dH , (2) with H as the odd electromagnetic excitation 2-form. Note that H has an independent operational interpretation, see [9]. 1.2 Axiom 2: Existence of the Lorentz force density We introduce a field of frames eα as reference system in X; by Greek letters α, β, . . . = 0, 1, 2, 3, we denote anholonomic or frame indices. The odd current 3-form J , together with the force density fα (odd covector-valued 4-form), the notion of which is assumed to be known from mechanics, allows us to formulate the Lorentz force density as fα = (eα⌋F ) ∧ J . (3) 1 AXIOMATICS OF METRIC-FREE ELECTRODYNAMICS 2 Thereby the electromagnetic field strength F is defined as an even 2-form. 1.3 Axiom 3: Magnetic flux conservation It is possible to count single quantized magnetic flux lines inside superconductors of type II. This suggests to take the conservation of magnetic flux as axiom 3, ∮ Ω2 F = 0 , ∂Ω2 = 0 , (4) for an arbitrary closed submanifold Ω2. As a consequence, we find the homogeneous Maxwell equation