A gravito-electromagnetic analogy based on tidal tensors

TL;DR

This paper introduces a new covariant approach to the gravito-electromagnetic analogy using tidal tensors, providing novel formulations, proofs, and clarifications on the analogy's validity and physical interpretation.

Contribution

It presents a covariant formalism based on tidal tensors, offering new realizations of the analogy, a derivation of the Papapetrou force, and insights into invariants and the analogy's scope.

Findings

Matched linearized gravitational tidal tensors to electromagnetic ones in Minkowski spacetime

Derived the Papapetrou force on a gyroscope using the new approach

Established invariants of tidal tensors analogous to electromagnetic invariants

Abstract

We propose a new approach to a physical analogy between General Relativity and Electromagnetism, based on tidal tensors of both theories. Using this approach we write a covariant form for the gravitational analogues of the Maxwell equations. The following realisations of the analogy are given. The first one matches linearised gravitational tidal tensors to exact electromagnetic tidal tensors in Minkwoski spacetime. The second one matches exact magnetic gravitational tidal tensors for ultra-stationary metrics to exact magnetic tidal tensors of electromagnetism in curved spaces. In the third we show that our approach leads to two-step exact derivation of the Papapetrou force on a gyroscope. We then establish a new proof for a class of tensor identities that define invariants of the type $\vec{E}^2-\vec{B}^2$ and $\vec{E}\cdot\vec{B}$, and we exhibit the invariants built from tidal tensors in both gravity and electromagnetism. We contrast our approach with the two gravito-electromagnetic analogies commonly found in the literature, and argue that it sheds light on the debate about the limit of validity of one of the analogies, and clarifies issues concerning the physical interpretation of the other.