[Paper Review] Elimination of High-Energy Divergence in Relativistic Lagrangean Formulation of Gravitating Particle Dynamics

Anatoli Vankov|ArXiv.org|Feb 23, 2004

Cosmology and Gravitation Theories5 references8 citations

TL;DR

This paper proposes a relativistic Lagrangian formulation where the proper mass of a particle depends on the gravitational field strength, eliminating the high-energy divergence caused by the 1/r potential singularity. By treating mass as a dynamical variable, the theory avoids the need for renormalization and predicts superluminal motion for ultra-high-energy particles, offering a testable alternative to general relativity in strong gravitational fields.

ABSTRACT

The nonrenormalizable singularity of the gravitational 1/r potential at ralativistic and quantum levels is a longstanding problem of modern physics. The problem is discussed in Relativistic Lagrangean framework with the variable proper mass. It is shown that the so-called self-energy divergence for the 1/r graviational potntial can be eliminated within the variable proper mass concept. The problem has many aspects outlined, which should be further investigated before the final conclusion could be made.

Motivation & Objective

• To resolve the high-energy divergence in relativistic gravitational theories caused by the 1/r potential singularity.
• To address the incompatibility of Newtonian gravity with special relativity by reformulating particle dynamics with field-dependent proper mass.
• To provide a covariant, relativistic alternative to general relativity that avoids curvature-based explanations of light bending and gravitational redshift.
• To eliminate the need for artificial renormalization in quantum field theories by grounding the theory in a dynamical mass concept.
• To propose a testable prediction of superluminal motion for ultra-high-energy cosmic rays via backward-directed Cherenkov radiation.

Proposed method

• Formulates a relativistic Lagrangian $ L = m - W $, where $ m $ is the field-dependent proper mass and $ W $ is the potential energy, in 4-dimensional spacetime with proper time $ au $.
• Derives equations of motion via the Hamilton principle in the form $ rac{d}{ds}(mu^ u) = K^ u $, where $ u^ u $ is the proper velocity and $ K^ u $ is the Minkowski force 4-vector.
• Introduces the proper mass as a dynamical variable that changes under field influence, ensuring energy conservation and eliminating the $ 1/r $ singularity.
• Applies the formalism to spherically symmetric gravitational fields, comparing predictions with general relativity for photon and massive particle trajectories.
• Predicts that particles with Lorentz factor $ ho_{ ext{thr}} > 2 \times 10^4 $ would become superluminal, detectable via backward-directed Cherenkov radiation.
• Uses the concept of gravitational refraction (not spacetime curvature) to explain light bending and redshift, consistent with photon energy and frequency conservation.

Experimental results

Research questions

• RQ1 Can the $ 1/r $ singularity in the gravitational potential be eliminated in a relativistic Lagrangian formulation without introducing spacetime curvature?
• RQ2 Does a field-dependent proper mass lead to a consistent relativistic dynamics that avoids divergences and renormalization?
• RQ3 How do predictions for photon deflection and redshift differ from general relativity when mass is treated as a dynamical field-dependent variable?
• RQ4 What are the implications for massive particle motion in strong gravitational fields under the proposed dynamical mass model?
• RQ5 Can superluminal motion for ultra-high-energy particles be a physical prediction of this theory, and is it testable via Cherenkov radiation?

Key findings

• The $ 1/r $ potential singularity is eliminated by treating the proper mass as a field-dependent dynamical variable, removing the high-energy divergence in the Lagrangian formulation.
• The theory predicts that massive particles with initial Lorentz factor $ \gamma_{\text{thr}} > 2 \times 10^4 $ would become superluminal in strong gravitational fields, violating general relativity’s speed limit.
• Photon deflection and gravitational redshift are explained via gravitational refraction rather than spacetime curvature, consistent with energy and frequency conservation.
• The model predicts backward-directed Cherenkov radiation from ultra-high-energy cosmic rays, with photon flashes appearing to originate from Earth and durations in the ms range.
• The theory reproduces weak-field effects (e.g., light bending, redshift) identically to general relativity, but diverges in strong fields, offering a falsifiable test.
• The Minkowski force 4-vector $ K^ u $ provides a consistent relativistic generalization of Newton’s second law, with $ d(mu^ u)/ds = K^ u $, replacing the fixed-mass assumption.