Electrodynamics and Spacetime Geometry: Foundations

Abstract

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We briefly review the foundations of electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations which introduce the spacetime metric. We then proceed with the tensor formulation by assuming local, linear, homogeneous and isotropic constitutive relations, and explore the physical, observable consequences of Maxwell's equations in curved spacetime. The field equations, charge conservation and the Lorentz force are explicitly expressed in general (pseudo) Riemanian manifolds. The generalized Gauss and Maxwell-Ampère laws, as well as the wave equations, reveal potentially interesting astrophysical applications. In all cases new electromagnetic couplings and related phenomena are induced by spacetime curvature. The implications and possible applications for gravity waves detection are briefly addressed. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry (and causal structure) of spacetime.

An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time

“…They obtain necessary and sufficient conditions for a static space-time to be locally conformally flat. Cabral and Lobo [10] discuss the connection between electrodynamics and the geometry and causal structure of space-time. Schwarz [11] discusses the AdS/CFT correspondence and generalizations to + 1 dimensions.…”

On the Fundamental Physical Constants: II. Field Coupling Geometry

“…Although it is quite obvious, it might still be relevant to mention that only during literature search for compilation of the draft did we become aware of the significant development in the subject of "Quantum Geometrodynamics" QGD, Wheeler (1957), Misner and Wheeler (1957), Anderson (2004), Pitkanen (2016), Cabral & Lobo (2017). Our approach has been developed quite independent of the QGD; indeed our tag "electro-geometrics" intends shorthand for "electricity is geometrical", any bearing of the EGM concept to QGD is incidental.…”

Electrodynamics and spacetime geometry: Astrophysical applications

“…Different electromagnetic models arriving from such changes are those of Heisenberg-Euler non-linear electrodynamics [8] or Mashhoon-Volterra non-local theory [9], for example. In a previous work [10], we briefly mentioned the idea of letting go the requirement of homogeneity and isotropy, which can be understood as a consequence of assuming that the electromagnetic properties of spacetime (or electrovacuum) have symmetries which follow the spacetime isometries (locally).…”

“…In Section II, we briefly review the foundations of electrodynamics and its deep relation to spacetime geometry [10], presenting the basic equations to be explored in the applications. In Section III, we explore some physical implications for two important geometrical backgrounds, namely the Schwarzschild metric and the exterior geometry of a stationary spherical gravitational mass with slow rotation in the weak field limit.…”