Classical Electrodynamics in Quasi-Metric Space-Time

Abstract

The quasi-metric manifold $\cal N$ is equipped with two one-parameter families of metric tensors ${\bf {\bar g}}_t$ and ${\bf g}_t$, each parametrized by the global time function $t$. Moreover, in $({\cal N},{\bf {\bar g}}_t)$ one must define two different electromagnetic field tensor families corresponding to the active electromagnetic field tensor family ${\bf {\tilde F}}_t$ and the passive electromagnetic field tensor family ${\bf {\bar F}}_t$, respectively. The active electromagnetic field tensor family ${\bf {\tilde F}}_t$ couples to gravity. By construction, the norm of the passive electromagnetic field tensor family ${\bf {\bar F}}_t$ experiences a secular decrease, defining a global cosmic attenuation (not noticeable locally) of the electromagnetic field. Local conservation laws for passive electromagnetism imply that ${\bf {\bar {\nabla}}}{\cdot} {\bf {\bar F}}_t=0$ in electro-vacuum, ensuring that photons move on null geodesics of $({\cal N},{\bf {\bar g}}_t)$. From ${\bf {\bar F}}_t$, one may construct the passive electromagnetic field tensor family ${\bf F}_t$ in $({\cal N},{\bf g}_t)$ in the same way as ${\bf g}_t$ is constructed from ${\bf {\bar g}}_t$. This ensures that photons move on null geodesics of $({\cal N},{\bf g}_t)$ as well. As a simple example, the (exact) quasi-metric counterpart to the Reissner-Nordstr\"{o}m solution in General Relativity is calculated. Besides, it is found that a classical charged test particle electromagnetically bound to a central charge will participate in the cosmic expansion. But since quantum-mechanical states should be unaffected by the expansion, this classic calculation is hardly relevant for quantum-mechanical systems such as atoms, so there is no reason to think that the cosmic expansion should apply to them. Finally, it is shown that the main results of geometric optics hold in quasi-metric space-time.