L. Filipe O. Costa

Gravitoelectromagnetic analogy based on tidal tensors

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, which makes transparent both the similarities and key differences between the two interactions. The following realizations of the analogy are given. The first one matches linearized gravitational tidal tensors to exact electromagnetic tidal tensors in Minkowski spacetime. The second one matches exact magnetic gravitational tidal tensors for ultrastationary metrics to exact magnetic tidal tensors of electromagnetism in curved spaces. In the third we show that our approach leads to a two-step exact derivation of Papapetrous equation describing the force exerted on a spinning test particle. Analogous scalar invariants built from tidal tensors of both theories are also discussed.

Mathisson's helical motions for a spinning particle: Are they unphysical?

It has been asserted in the literature that Mathissons helical motions are unphysical, with the argument that their radius can be arbitrarily large. We revisit Mathissons helical motions of a free spinning particle, and observe that such statement is unfounded. Their radius is finite and confined to the disk of centroids. We argue that the helical motions are perfectly valid and physically equivalent descriptions of the motion of a spinning body, the difference between them being the choice of the representative point of the particle, thus a gauge choice. We discuss the kinematical explanation of these motions, and we dynamically interpret them through the concept of hidden momentum. We also show that, contrary to previous claims, the frequency of the helical motions coincides, even in the relativistic limit, with the zitterbewegung frequency of the Dirac equation for the electron.

Gravito-electromagnetic analogies

We reexamine and further develop different gravito-electromagnetic analogies found in the literature, and clarify the connection between them. Special emphasis is placed in two exact physical analogies: the analogy based on inertial fields from the so-called “1+3 formalism”, and the analogy based on tidal tensors. Both are reformulated, extended and generalized. We write in both formalisms the Maxwell and the full exact Einstein field equations with sources, plus the algebraic Bianchi identities, which are cast as the source-free equations for the gravitational field. New results within each approach are unveiled. The well known analogy between linearized gravity and electromagnetism in Lorentz frames is obtained as a limiting case of the exact ones. The formal analogies between the Maxwell and Weyl tensors are also discussed, and, together with insight from the other approaches, used to physically interpret gravitational radiation. The precise conditions under which a similarity between gravity and electromagnetism occurs are discussed, and we conclude by summarizing the main outcome of each approach.

Mathisson's helical motions demystified

The motion of spinning test particles in general relativity is described by Mathisson-Papapetrou-Dixon equations, which are undetermined up to a spin supplementary condition, the latter being today still an open question. The Mathisson-Pirani (MP) condition is known to lead to rather mysterious helical motions which have been deemed unphysical, and for this reason discarded. We show that these assessments are unfounded and originate from a subtle (but crucial) misconception. We discuss the kinematical explanation of the helical motions, and dynamically interpret them through the concept of hidden momentum, which has an electromagnetic analogue. We also show that, contrary to previous claims, the frequency of the helical motions coincides exactly with the zitterbewegung frequency of the Dirac equation for the electron.

The Coriolis field

We present a pedagogical discussion of the Coriolis field, emphasizing its not-so-well-understood aspects. We show that this field satisfies the field equations of the so-called Newton–Cartan theory, a generalization of Newtonian gravity that is covariant under changes of arbitrarily rotating and accelerated frames. Examples of solutions of this theory are given, including the Newtonian analogue of the Godel universe. We discuss how to detect the Coriolis field by its effect on gyroscopes, of which the gyrocompass is an example. Finally, using a similar framework, we discuss the Coriolis field generated by mass currents in general relativity, and its measurement by the gravity probe B and LAGEOS/LARES experiments.

TIDAL TENSOR APPROACH TO GRAVITOELECTROMAGNETISM

We summarize the gravito-electromagnetic analogy based on tidal tensors. This analogy leads to an exact and fully general form for the gravitational analogues of Maxwells equations, which allows for a transparent comparison between the two interactions. Special cases of matching between gravitational and electromagnetic tidal tensors are discussed.

Hidden Momentum in the Framework of Gravitoelectromagnetism

A still not well understood feature of extended bodies in general relativity is the fact that their momentum is not, in general, parallel to the center of mass 4-velocity—the body is said to have “hidden momentum”. It can be split in two main types, a physical one that is gauge invariant, and the pure gauge hidden momentum that arises from the spin supplementary condition. In this paper I focus on the latter, using the formalism of gravitoelectromagnetism, which yields an easy way of understanding it, and under which conditions it arises.

The gravitational force on a gyroscope and the electromagnetic force on a magnetic dipole as analogous tidal effects

We compare the covariant expression of the electromagnetic force exerted on a magnetic dipole with Papapetrous equation for the gravitational force exerted on a spinning test particle. We show that if Piranis supplementary spin condition holds, there is an exact, covariant, and fully general analogy relating these two forces: both are determined by a contraction of the spin 4-vector with a magnetic-type tidal tensor. Moreover, these tidal tensors obey strikingly analogous equations which are covariant forms for (some of) Maxwells and Einsteins field equations. These equations allow for an insightful comparison between the two interactions. It is shown that, in the special case that the gyroscope/dipole are at rest and far away from a stationary source, the two forces are similar (in accordance with the results known from linearized theory); but that for generic dynamics key differences arise. In particular we show that the time projection of the force on a dipole is the power transferred to it by Faradays induction, whereas the fact that the force on a gyroscope is spatial signals the absence of an analogous gravitational effect; that whereas the total work done on a magnetic dipole by a stationary magnetic field is zero, a stationary gravitomagnetic field, by contrast, does work on mass currents, which quantitatively explains the Hawking-Wald spin interaction energy.