Still The New Classical Relativistic Equation of Charge Motion in an Electromagnetic Field

This paper proposes a covariant generalization of the non-relativistic Goedecke equation to derive a new classical relativistic equation for point charge motion that eliminates runaway solutions and encompasses the Abraham-Lorentz-Dirac and Mo-Papas equations as approximate consequences.

The Gist

Here is an explanation of Anatoliy V. Sermyagin's paper, translated into simple, everyday language with creative analogies.

The Big Picture: The "Ghost" in the Machine

Imagine you are driving a car. You press the gas pedal, and the car speeds up. But there's a catch: as the car speeds up, it creates a wake of wind and turbulence behind it. This turbulence pushes back against the car, making it harder to accelerate.

In physics, an electrically charged particle (like an electron) is like that car. When it moves, it creates an electromagnetic "wake" (radiation). This wake pushes back on the particle. This is called radiation reaction.

For decades, physicists have struggled to write the perfect math equation to describe this. The old equations had a major problem: they predicted that if you gave a particle a tiny push, it would suddenly accelerate to infinite speed on its own, out of nowhere. Physicists call this a "runaway solution." It's like saying your car, once you press the gas, would suddenly fly off the road into space without you doing anything else. Obviously, that doesn't happen in real life.

This paper proposes a new equation that fixes this problem. It says: "No runaway solutions. The particle behaves normally, but with a slight delay in how it reacts to its own wake."

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The Problem with the Old Map

The author starts by looking at an older, non-relativistic equation (from 1975 by Goedecke). This equation worked well for slow-moving particles and didn't have the "runaway" bug.

However, when you try to upgrade this equation to work for fast-moving particles (near the speed of light), things get messy.

• The Issue: In the old math, the "force" pushing the particle and the "acceleration" of the particle didn't line up correctly when you switched reference frames (like looking at the car from a helicopter vs. from the ground).
• The Analogy: Imagine trying to measure the speed of a runner while you are running alongside them, and then trying to measure it again while you are standing still. If you don't adjust your math for the difference in your own speed, your measurements will contradict each other. The old equations were like a map that worked for a walking pace but fell apart when you started flying.

The Solution: The "Time-Traveling" Mirror

The author's solution is clever. He realizes that the particle's "wake" (the radiation reaction) depends on where the particle was a tiny fraction of a second ago, not where it is right now.

To fix the math, he uses a Lorentz Transformation.

• The Analogy: Think of the particle's past self and its present self as two different people standing on two different moving walkways at an airport.
  • Person A is the particle at time $t$ (Present).
  • Person B is the particle at time $t - \text{delay}$ (Past).
  • Because they are on different walkways, they are moving at different speeds relative to the ground.
• The Fix: You can't just compare Person A and Person B directly. You have to use a "magic mirror" (the Lorentz Transformation) to translate Person B's view into Person A's frame of reference. Once you do that, the math lines up perfectly.

The paper presents two new, equivalent ways to write this equation (Equations 14 and 15 in the text). They are essentially the same recipe written in two different languages.

Why This Matters: No More "Runaway" Cars

The most exciting part of this paper is that the new equation does not predict runaway solutions.

• Old Math: "If you push the particle, it might spontaneously explode into infinite speed." (Bad physics).
• New Math: "If you push the particle, it accelerates smoothly, taking into account the drag of its own wake, and stops accelerating when you stop pushing." (Good physics).

The "Special Case" vs. The "General Rule"

The author admits that the specific formulas he wrote down (Equations 14 and 15) are a "special case."

• The Analogy: Imagine he built a perfect bridge, but he only showed you the blueprint for the part that goes straight across a river. He hasn't shown you the blueprint for the part that curves around a mountain yet.
• The formulas in the paper work perfectly for particles moving in a straight line (1D motion). The author promises that the "general" version (for particles turning corners and moving in 3D) will be published separately.

The "Grandparents" of the Theory

The paper also shows that two famous, older equations (the Abraham-Lorentz-Dirac equation and the Mo-Papas equation) are actually just approximations of this new, more accurate theory.

• The Analogy: Think of the new equation as a high-definition 4K video. The old equations are like a blurry, low-resolution photo of the same scene. If you zoom in too much on the blurry photo, you see artifacts and errors (like the runaway solutions). But if you look at the 4K video, you see the true picture. The old equations are still useful for rough estimates, but the new one is the real deal.

Summary in a Nutshell

1. The Problem: Old equations for charged particles predicted impossible "runaway" speeds.
2. The Cause: The math didn't correctly account for the delay between a particle's current position and its past position when moving near light speed.
3. The Fix: The author used a "physical" method to translate the particle's past self into its present frame of reference using a specific type of mathematical mirror (Lorentz transformation).
4. The Result: A new, clean equation that behaves like real life (no runaway speeds) and explains why the older, famous equations were just "blurry approximations" of this new truth.

It's a bit like finding the missing piece of a puzzle that finally makes the picture of the universe snap into place, removing the weird, impossible glitches that had been there for 50 years.

Technical Summary

Here is a detailed technical summary of the paper "Still The New Classical Relativistic Equation of Charge Motion in an Electromagnetic Field" by Anatoliy V. Sermyagin.

1. Problem Statement

The paper addresses the long-standing issue of formulating a consistent relativistic equation of motion for a point charge that accounts for radiation reaction (self-force) without suffering from unphysical "runaway" solutions.

• The Context: The standard Abraham-Lorentz-Dirac (ALD) equation, derived from the non-relativistic Abraham-Lorentz (AL) equation, is known to possess "runaway" solutions where a particle's acceleration grows exponentially even in the absence of external forces.
• The Starting Point: The author utilizes the non-relativistic Goedecke equation (1975), which describes a point charge with a time-delayed acceleration term. Unlike the AL equation, the Goedecke equation naturally avoids runaway solutions but exhibits "pre-acceleration."
• The Challenge: Directly covariantizing the Goedecke equation by simply replacing 3-vectors with 4-vectors fails. Specifically, the 4-acceleration at a retarded time ($\dot{v}(\tau - \tau_0)$) and the 4-velocity at the current time ($v(\tau)$) are not orthogonal in the general case because they belong to different instantaneous rest frames. This violates the fundamental constraint of relativistic dynamics where 4-acceleration must be orthogonal to 4-velocity ($\dot{v} \cdot v = 0$).

2. Methodology

Sermyagin proposes a "physical" method of covariant generalization based on Lorentz transformations rather than simple algebraic orthogonalization (projection).

• Physical Orthogonalization: Instead of mathematically projecting the 4-acceleration onto the hyperplane orthogonal to the current 4-velocity (which lacks clear physical meaning and alters norms), the author argues that the non-orthogonality arises because the acceleration and velocity vectors are defined in different inertial frames.
• Lorentz Transformation Approach: The methodology involves transforming the 4-acceleration vector from the retarded frame (velocity $u = v(\tau - \tau_0)$) to the current frame (velocity $v = v(\tau)$) using a specific Lorentz transformation $\Lambda$.
  • The transformation $\Lambda$ maps $u \to v$ without spatial rotations.
  • The core equation is formulated as: $\Lambda \dot{u} = \frac{e}{m} F \cdot v$, where $\dot{u}$ is the acceleration in the retarded frame.
• Derivation of Forms: Using index-free notation and explicit matrix representations of the Lorentz transformation (derived from the work of Krause and Bazanski), the author derives two equivalent forms of the relativistic equation.

3. Key Contributions

The paper makes several distinct theoretical contributions:

1. New Covariant Equation: Derivation of a new classical relativistic equation of motion that is a strict covariant generalization of the Goedecke equation.
2. Two Equivalent Forms: The presentation of the equation in two mathematically equivalent forms (Equations 14 and 15 in the text):
   • Form A (Implicit): $m \dot{u} - m (\dot{u} \cdot v) \frac{u + v}{1 + u \cdot v} = e F \cdot v$
   • Form B (Explicit): $m \dot{u} = e F \cdot v - e (F \cdot v \cdot u) \frac{u + v}{1 + u \cdot v}$
   • Note: Here $u = v(\tau - \tau_0)$ and $v = v(\tau)$.
3. Resolution of Runaway Solutions: The author demonstrates that for a free particle ($F=0$), the equation reduces to $\dot{u} = 0$ (constant velocity), thereby proving the absence of runaway solutions in the general form of the theory.
4. Unification of Existing Theories: The paper establishes that the well-known Abraham-Lorentz-Dirac (ALD) and Mo-Papas (MP) equations are not fundamental laws but rather approximate consequences of this new theory, valid only when expanding in powers of the delay parameter $\tau_0$ and discarding higher-order terms.

4. Results

• Recovery of ALD: By expanding the terms dependent on the delay $\tau - \tau_0$ in a Taylor series and keeping only terms linear in $\tau_0$, the author recovers the standard ALD equation:
  $$\dot{v} - \tau_0 (\ddot{v} + \dot{v} \cdot \dot{v} v) = \frac{e}{m} F \cdot v$$
• Recovery of MP: By applying similar approximations to the second form of the new equation, the Mo-Papas equation is derived:
  $$\dot{v} = \frac{e}{m} F \cdot v + \tau_0 \frac{e}{m} (F \cdot \dot{v} + F \cdot v \cdot \dot{v} v)$$
• One-Dimensional Limit: The author notes that the specific forms (14) and (15) derived using Lorentz transformations without spatial rotations strictly describe one-dimensional motion. However, the general tensor forms (Equations 10 and 13) are applicable to general 3D motion.

5. Significance

• Theoretical Consistency: The work provides a physically motivated path to relativistic generalization that preserves the "no runaway" property of the original non-relativistic Goedecke equation, a property lost in the standard ALD formulation.
• Hierarchy of Equations: It recontextualizes the ALD and MP equations, positioning them as low-order approximations of a more fundamental, delay-based relativistic theory. This suggests that the pathologies of the ALD equation (runaways) may be artifacts of the approximation process rather than inherent features of classical electrodynamics.
• Methodological Innovation: The use of "physical orthogonalization" via Lorentz transformations between instantaneous rest frames offers a novel approach to handling delayed arguments in relativistic dynamics, distinct from the standard projection methods used in the literature.

Conclusion:
Sermyagin presents a revised classical relativistic framework for charged particle motion that eliminates runaway solutions by rigorously treating the time-delay between the particle's state and its self-interaction through Lorentz transformations. While the specific closed-form solutions (14) and (15) are limited to 1D motion, the general tensor formulation offers a promising alternative to the ALD equation, with the latter emerging only as a first-order approximation.