Conserved relativistic Ertel's current generating the vortical and thermodynamic aspects of spacetime

TL;DR

This paper explores the conservation of relativistic Ertel's current and its implications for spacetime, cosmology, and quantum field theory, revealing a dual form of Einstein's equations related to vortex and Weyl tensors.

Contribution

It extends the conservation of Ertel's current from geophysical fluid dynamics to relativistic spacetime, uncovering a dual Einstein equation involving vortex and Weyl tensors.

Findings

Identification of a dual Einstein field equation form

Connection between vortex tensor and Weyl tensor in spacetime

Implications for cosmology and quantum field theory

Abstract

Motivated by Aoki et al.'s recent research on conserved charges and entropy current, we investigated the conservation of relativistic Ertel's current, which has received little attention outside the field of geophysical fluid dynamics. Ertel's charge is an important indicator of the correlation between vortex vectors and entropy gradient fields in Earth's meridional heat transport. We first show that in the generalized Hamiltonian structure of baroclinic fluids, the duality between the total energy and the Casimir as a function of Ertel's charge plays an important role in the nonrelativistic case. Then, by extending the result to relativistic cases, we show that this finding has far-reaching implications not only for spacetime issues in cosmology but also for the foundation of quantum field theory. An especially important finding is that, as an unreported dual form of the Einstein field equation, we identify a special equation satisfied not only by the vortex tensor field generated by the conserved charge but also by the Weyl tensor in interpreting the physical nature of the metric tensor $g^{\mu\nu}$, which appears in the cosmological term $\Lambda g^{\mu\nu}$.