[Paper Review] Generalization of regular solutions of Einstein's gravity equations and Maxwell's equations for point-like charge

Yi-Shi Duan|arXiv (Cornell University)|May 19, 2017

Geophysics and Gravity Measurements3 citations

TL;DR

This paper generalizes regular solutions to Einstein's gravity and Maxwell's equations for a point-like charge by introducing an arbitrary function $ f(r) $ that ensures finiteness and regularity at $ r=0 $. It shows that the total mass of the particle is finite and proportional to the charge, with the classical electron radius emerging naturally as $ r_0 = e^2/(mc^2) $, unifying gravitational and electromagnetic contributions to mass without singularities.

ABSTRACT

In the Born, Infeld, Bopp, Podolsky and Dirac theories the electron mass is finite (or zero), but gravity effects have not been considered. Shirokov and Fisher showed that in studying of origin of elemental particle masses we can not neglect these effects. Gravity field plays an essential role in interpretation of particle mass. It has been shown that for point-like charge the general solution of gravity and electromagnetic (EM) equations that contains arbitrary function $μ(r)$, can in special case of this function, describe gravity and EM fields that do not have singularities in full range of $r$ from $0$ to $\infty$. In this paper we are trying to find more general solution.

Motivation & Objective

• To generalize existing regular solutions of Einstein's gravity and Maxwell's equations for point-like charges beyond specific cases.
• To ensure the metric and physical quantities remain finite and non-singular at $ r=0 $, resolving the issue of curvature and field singularities.
• To derive a finite, self-consistent expression for the particle's total mass as the sum of gravitational and electromagnetic contributions.
• To show that the classical electron radius emerges naturally from the theory when the arbitrary function $ f(r) $ satisfies specific boundary conditions at $ r=0 $.

Proposed method

• Introduces an arbitrary, non-singular function $ f(r) $ via $ \mu = -2\ln f(r) $, which parameterizes the metric components and ensures regularity.
• Derives the metric components $ g_{11}, g_{22}, g_{33}, g_{44} $ in terms of $ f(r) $ and its derivative $ f'(r) $, ensuring finiteness at $ r=0 $.
• Uses the energy-momentum tensor $ T^i_k $ for the electromagnetic field and the gravitational energy-momentum pseudotensor to compute the total mass via integration.
• Applies boundary conditions at $ r=0 $: $ f(0)=0 $, $ f'(0)=\alpha \neq 0 $, $ f''(0)=\beta \neq 0 $, which guarantee regular metric components.
• Evaluates the mass integral $ m = 2\int \sqrt{-g} T^4_4 \, d\tau $, yielding $ m = 4\pi \varepsilon^2 \alpha $, with $ \alpha $ related to the classical electron radius.
• Demonstrates that the classical electron radius $ r_0 = e^2/(mc^2) $ arises naturally when $ \alpha = 1/r_0 $, linking the theory to experimental particle parameters.

Experimental results

Research questions

• RQ1 Can a general class of regular solutions to Einstein's and Maxwell's equations be constructed for a point-like charge without singularities at $ r=0 $?
• RQ2 How does the introduction of an arbitrary function $ f(r) $ affect the finiteness and regularity of the metric and physical fields?
• RQ3 What is the total mass of the system, and how is it composed of gravitational and electromagnetic contributions?
• RQ4 Does the classical electron radius emerge as a natural outcome of the theory under appropriate boundary conditions on $ f(r) $?
• RQ5 Can the mass be finite and non-zero even when the charge is point-like, without requiring ad hoc mass renormalization?

Key findings

• The metric components remain finite and non-singular at $ r=0 $ when $ f(r) $ satisfies $ f(0)=0 $, $ f'(0)=\alpha \neq 0 $, and $ f''(0)=\beta \neq 0 $, ensuring regular spacetime geometry.
• The total mass of the particle is finite and given by $ m = 4\pi \varepsilon^2 \alpha $, with $ \alpha $ determined by the behavior of $ f(r) $ near the origin.
• The classical electron radius $ r_0 = e^2/(mc^2) $ emerges naturally when $ \alpha = 1/r_0 $, linking the theoretical framework to experimental particle parameters.
• The energy-momentum tensor and metric components are explicitly expressed in terms of $ f(r) $, showing that the theory remains regular for any smooth, non-singular $ f(r) $ satisfying the boundary conditions.
• The mass is shown to be the sum of gravitational and electromagnetic field contributions, with no need for additional mass terms or renormalization.
• The solution generalizes previous results by Einstein and Shirokov, extending them to a broader class of functions $ f(r) $, while preserving finiteness and regularity.