[Paper Review] Relativistic Physics in Arbitrary Reference Frames

N. V. Mit︠s︡kevich|arXiv (Cornell University)|Feb 7, 2005

Cosmology and Gravitation Theories31 references25 citations

TL;DR

This paper develops a relativistic framework for describing physics in arbitrary reference frames using the monad formalism, enabling a unified treatment of gravitational and electromagnetic phenomena through quasi-Maxwellian equations. It derives key results such as gravitomagnetic effects, photon redshift, and orbit shifts in curved spacetimes like Kerr and Taub-NUT, revealing deep analogies between gravity and electromagnetism via Noether symmetries and monad decompositions of field equations.

ABSTRACT

Preface Introduction A general characterisation of the subject A synopsis of notations of Riemannian geometry The Noether theorem: space-time invariance The Noether densities transformation laws Reference frames calculus The monad formalism and its place in the description of reference frames in relativistic physics Reference frames algebra Geometry of congruences. Acceleration, rotation, expansion and shear of a reference frame Differential operations and identities of the monad formalism Equations of motion of test particles The electric field strength and magnetic displacement vectors Monad description of the motion of a test charged mass in gravitational and electromagnetic fields Motion of photons, the redshift and Doppler effects The dragging phenomenon Dragging in circular equatorial orbits in the Kerr space-time An orbit shift in the TaubNUT space-time Dragging in the space-time of a pencil of light Other dragging effects More general gravitoelectromagnetic and gravitoelectric phenomena The Maxwell field equations The four-dimensional Maxwell equations The electromagnetic stress-energy tensor and its monad decomposition Monad representation of Maxwells equations A charged fluid without electric field An Einstein-Maxwell field with kinematic magnetic charges The Einstein field equations The four-dimensional Einstein equations Monad representation of Einsteins equations The geodesic deviation equation and a new level of analogy between gravitation and electromagnetism New quasi-Maxwellian equations of the gravitational field Remarks on classification of intrinsic gravitational fields Example of the Taub-NUT field Example of the spinning pencil-of-light field Gravitational fields of the G..odel universe Perfect fluids Introductive remarks Rank 2 and 3 fields Free rank 2 field Free rank 3 field Rotating fluids Special relativistic theory Additional remarks Mechanics versus field theory Canonical approach to field theory Canonical formalism and quantisation Concluding remarks References Index.

Motivation & Objective

• To provide a general formalism for relativistic physics in arbitrary, non-inertial reference frames, extending beyond standard inertial or comoving frames.
• To unify the description of gravitational and electromagnetic fields using a monad-based decomposition of field equations, drawing analogies with Maxwell's theory.
• To derive exact solutions for dragging effects, redshift, and orbit shifts in specific spacetimes such as Kerr and Taub-NUT, validating the formalism.
• To establish a four-dimensional, covariant formulation of Einstein-Maxwell equations using the monad formalism, enabling kinematic and dynamic decomposition of fields.
• To explore the role of Noether symmetries in linking spacetime invariance to conserved currents and field equations in arbitrary frames.

Proposed method

• Employing the monad formalism to decompose four-dimensional tensors into spatial and temporal components relative to an arbitrary timelike congruence, enabling observer-dependent descriptions of fields.
• Applying the Noether theorem to derive conserved currents from spacetime symmetries, linking invariance under diffeomorphisms to energy-momentum and angular momentum conservation.
• Deriving the four-dimensional Einstein and Maxwell equations in a monad-representation, allowing explicit decomposition into kinematic (acceleration, rotation, shear, expansion) and dynamic (field strength, stress-energy) components.
• Using the geodesic deviation equation to model tidal forces and establish a new analogy between gravity and electromagnetism via the curvature of spacetime and field strength tensors.
• Constructing quasi-Maxwellian equations for gravity by mapping the Riemann curvature tensor to gravitational field strength and gravitomagnetic analogs.
• Analyzing specific spacetimes—Kerr, Taub-NUT, G"odel, and pencil-of-light fields—using the formalism to compute dragging, redshift, and orbit shifts.

Experimental results

Research questions

• RQ1 How can relativistic physics be consistently formulated in arbitrary, non-inertial reference frames using a geometric and covariant formalism?
• RQ2 To what extent do the Einstein and Maxwell equations admit a decomposition into observer-dependent components that resemble Maxwell's equations in flat spacetime?
• RQ3 What are the exact expressions for gravitomagnetic effects such as frame-dragging in the Kerr and Taub-NUT spacetimes using the monad formalism?
• RQ4 How do the kinematic quantities (acceleration, rotation, shear, expansion) of a reference frame influence the observed electromagnetic and gravitational fields?
• RQ5 Can the gravitational field be described via a quasi-Maxwellian formalism that reveals analogies with electromagnetism, particularly in rotating or cosmological spacetimes?

Key findings

• The monad formalism successfully decomposes the Einstein and Maxwell equations into observer-dependent components, enabling a kinematic and dynamic interpretation of gravitational and electromagnetic fields in arbitrary frames.
• In the Kerr spacetime, the formalism reproduces the frame-dragging effect for circular equatorial orbits, confirming the rotational influence of the black hole on test particles and photons.
• For the Taub-NUT spacetime, the method predicts a non-zero orbit shift due to the NUT charge, consistent with known solutions in general relativity.
• The dragging of photons and the resulting redshift and Doppler shifts are derived from the monad decomposition of the geodesic equation, showing agreement with standard relativistic optics.
• The formalism reveals that the gravitational field in the G"odel universe exhibits a non-zero gravitomagnetic component, leading to closed timelike curves and rotational dragging effects.
• A new set of quasi-Maxwellian equations for gravity is derived, where the Riemann curvature tensor is decomposed into gravitational field strength and gravitomagnetic analogs, enabling electromagnetic-like intuition for gravitational phenomena.