[Paper Review] Relativistic Dynamics of a Charged Particle in an Electroscalar Field

D. V. Podgainy, O.A. Zaimidoroga|arXiv (Cornell University)|Mar 12, 2012

Quantum and Classical Electrodynamics13 references3 citations

TL;DR

This paper extends a non-relativistic theory of electroscalar fields to relativistic dynamics, introducing a new relativistic force from a longitudinal electroscalar field and deriving a relativistically invariant Lagrangian for charged particles. The key contribution is a force law that includes velocity-dependent corrections via the scalar potential λ, unifying longitudinal electroscalar and transverse electromagnetic fields in a gauge-invariant framework that preserves charge conservation and enables a natural connection to scalar gravity theories.

ABSTRACT

This article devoted to relativistic dynamics of a charged massive particle in an electroscalar field. It represents a continuation of paper [1] where the authors constructed a non-relativistic theory which describes transverse electromagnetic waves along with longitudinal electroscalar ones, responsible for the wave transport of the Coulomb field. A new type of relativistic force exerted by electroscalar field on an electrically charged particle and the relativistic law of superposition of electromagnetic transverse and electroscalar longitudinal fields are established. Also, a relativistically invariant form of a Lagrangian describing the interaction between an electroscalar field and massive electrically charged particle is defined.

Motivation & Objective

• To develop a relativistic extension of a non-relativistic theory that includes longitudinal electroscalar waves as carriers of the Coulomb field.
• To resolve causality and quantization issues in Maxwell electrodynamics by introducing a scalar potential λ independent of the vector potential.
• To derive a relativistically invariant Lagrangian describing the interaction between a massive charged particle and an electroscalar field.
• To establish a relativistic law of superposition combining transverse electromagnetic and longitudinal electroscalar fields.
• To explore connections between electroscalar dynamics and scalar gravity theories, particularly in the context of dark matter and dark energy.

Proposed method

• Derives the relativistic force on a charged particle from the 4-scalar potential λ, incorporating velocity-dependent corrections via the total time derivative of λ.
• Constructs a relativistically invariant Lagrangian (equation 43) that generalizes the free-particle Lagrangian by including a field-dependent effective mass.
• Applies the principle of least action to the derived Lagrangian to obtain the relativistic equations of motion, including corrections of order (v/c)².
• Introduces the field variables E|| = ∇λ and W = −(1/c)∂λ/∂t to describe the longitudinal electroscalar field and derives their relativistic wave equations.
• Demonstrates that the magnetic field is induced not only by transverse electric fields but also by the cross product [v × E||], indicating a kinematic induction effect.
• Establishes a formal analogy between electroscalar electrodynamics and scalar gravity by combining the electroscalar Lagrangian with the scalar gravity Lagrangian (equation 57).

Experimental results

Research questions

• RQ1 How does the relativistic force exerted by an electroscalar field on a charged particle differ from its non-relativistic counterpart?
• RQ2 Can a gauge-invariant relativistic theory of electroscalar fields be formulated without violating electric charge conservation?
• RQ3 What is the relativistically invariant form of the Lagrangian describing the interaction between a massive charged particle and an electroscalar field?
• RQ4 How do transverse electromagnetic and longitudinal electroscalar fields superpose in a relativistically consistent manner?
• RQ5 What are the cosmological implications of a unified electroscalar-gravity framework based on 4-scalar fields?

Key findings

• The relativistic force on a charged particle includes a velocity-dependent correction proportional to (1−v²/c²)⁻¹/² (v/c²)(dλ/dt), directed along the particle's velocity, which is absent in the non-relativistic limit.
• The derived force law (equation 39) explicitly depends on the total time derivative of the scalar potential λ, distinguishing it from the non-relativistic force −q∇λ.
• A relativistically invariant Lagrangian is constructed (equation 43) that describes a free particle with a field-dependent effective mass, given by m₀(1−v²/c²)⁻¹/² exp(qλ/m₀c²).
• The relativistic superposition law (equations 37–38) shows that the magnetic field is induced by both transverse electric fields and the vector product [v × E||], indicating a kinematic induction effect.
• A unified Lagrangian (equation 57) is proposed that combines electroscalar and scalar gravity interactions, with the exponent representing the ratio of total interaction energy to rest energy.
• The theory provides a natural framework for addressing dark matter and dark energy, as the 4-scalar field structure allows for energy-mass coupling analogous to scalar gravity models.