Analytic solutions for the longitudinal and the transverse components of the vector potential in the Lorenz gauge

This paper derives analytic solutions for the longitudinal and transverse components of the vector potential in the Lorenz gauge applicable to arbitrary time-dependent charge-current distributions.

The Gist

Imagine you are trying to describe the weather in a city. You could describe it by looking at the wind blowing in straight lines (like a river flowing down a street) or by looking at the swirling eddies and vortices (like a whirlpool in a drain). In physics, electric and magnetic fields are similar: they can be broken down into "straight-line" parts and "swirling" parts.

This paper is about solving a specific puzzle regarding how these parts of the electromagnetic field behave when things are moving and changing over time.

Here is the breakdown of the paper in simple terms:

1. The Setting: The "Lorenz Gauge"

In physics, when we calculate how electric and magnetic fields move, we use "potentials" (think of them as the invisible blueprints that create the fields). There are different ways to set up these blueprints, called "gauges."

The authors are working in the Lorenz gauge. Think of this as a specific rulebook for how to draw the blueprint. It's a popular rulebook because it treats space and time very symmetrically, making the math look like waves rippling across a pond.

2. The Mystery: The "Longitudinal" vs. "Transverse" Split

The authors are looking at the Vector Potential (the main blueprint, let's call it A). They want to split this blueprint into two pieces:

• The Longitudinal Part ($A_\ell$): The part that points in the same direction as the flow of charge (like water flowing straight down a pipe).
• The Transverse Part ($A_{tr}$): The part that swirls or moves sideways (like the ripples on a pond).

The Problem:
A few years ago, a physicist named Hnizdo noticed a small confusion in a famous textbook by Jackson. The textbook seemed to suggest that the "Longitudinal" part of the blueprint might be doing something weird: it seemed to be reacting instantly to changes in charge, as if it could travel faster than the speed of light.

But we know from Einstein that nothing travels faster than light. So, is the math broken? Or is the textbook just slightly unclear?

3. The Solution: Three Different Paths to the Same Destination

The authors, Kuo-Ho Yang and Robert Nevels, decided to solve this mystery by deriving the exact mathematical formulas for these two parts ($A_\ell$ and $A_{tr}$) from scratch. They didn't just guess; they used three different mathematical "methods" to prove they get the same answer every time.

Think of it like trying to find the shortest path from your house to a park.

• Method 1: You take a map of the "Coulomb gauge" (a different, simpler rulebook) and translate it into the Lorenz rulebook.
• Method 2: You treat the "Longitudinal" part as a hill and calculate the slope directly.
• Method 3: You treat the "Transverse" part as a spinning top and calculate its rotation.

The Result:
All three methods led to the exact same conclusion. The "Longitudinal" part does not travel faster than light.

4. The Big Reveal: How It Actually Works

The paper shows that the "Longitudinal" part of the field is actually a cancellation effect.

Imagine two runners:

1. Runner A (The Instantaneous Part): This runner represents the "Coulomb" potential. It reacts instantly to charge changes, but it's not a real physical wave; it's just a mathematical snapshot.
2. Runner B (The Retarded Part): This runner represents the "Lorenz" potential. It travels at the speed of light, taking time to get from point A to point B.

The "Longitudinal" component is essentially Runner A minus Runner B.

• Because Runner A is "instant" and Runner B is "delayed," their difference creates a term that looks like it might be doing something weird.
• However, when you do the full math, the "instant" part and the "delayed" part cancel each other out perfectly in a way that ensures no information travels faster than light.

5. Why This Matters

The authors have cleared up a "mystery" that confused physicists for a while. They proved that:

1. The math in the famous textbook (Jackson) wasn't wrong, just perhaps slightly hard to interpret.
2. The "Longitudinal" and "Transverse" parts of the electromagnetic field are well-behaved.
3. Even though the math involves complex integrals and derivatives, the physical reality remains consistent: nothing breaks the speed of light.

In a Nutshell:
The paper is a rigorous "math detective story." The authors investigated a suspicious clue (a potential faster-than-light signal), ran the numbers using three different investigative techniques, and proved that the suspect was innocent. The electromagnetic field behaves exactly as nature intended, with no super-speedy secrets hidden in the math.

Technical Summary

1. Problem Statement

The paper addresses a specific theoretical ambiguity regarding the decomposition of the vector potential $\mathbf{A}$ in the Lorenz gauge into its longitudinal ($\mathbf{A}^{(L)}_\ell$) and transverse ($\mathbf{A}^{(L)}_{tr}$) components for arbitrary time-dependent charge-current distributions.

• Context: In a previous analysis (Ref. [3]), it was suggested that the longitudinal component $\mathbf{A}^{(L)}_\ell$ contained a term propagating faster than the speed of light ($c$) when derived from physical charge and current densities.
• Motivation: A recent note by Hnizdo [1] pointed out a minor mis-statement in J.D. Jackson's classic paper [2] concerning these components. This prompted the authors to re-examine the problem to resolve the "mystery" surrounding the propagation speeds and mathematical consistency of these components.
• Goal: To derive rigorous, analytic solutions for $\mathbf{A}^{(L)}_\ell$ and $\mathbf{A}^{(L)}_{tr}$ that are valid for arbitrary time-dependent sources and expressed in familiar electrodynamical quantities.

2. Methodology

The authors employ three distinct mathematical approaches to derive and verify the solutions, ensuring robustness through cross-validation. They work within Gaussian units and consider localized charge ($\rho$) and current ($\mathbf{J}$) densities turned on at time $t_0$.

Preliminaries

The authors define the decomposition of the current density $\mathbf{J}$ into longitudinal ($\mathbf{J}_\ell$) and transverse ($\mathbf{J}_{tr}$) parts:

• $\mathbf{J}_\ell = \frac{1}{4\pi}\nabla \frac{\partial}{\partial t} \Phi^{(C)}$ (related to the Coulomb potential).
• $\mathbf{J}_{tr} = \mathbf{J} - \mathbf{J}_\ell$.

They establish the governing wave equations for the Lorenz gauge potentials $\Phi^{(L)}$ and $\mathbf{A}^{(L)}$:
$$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) \mathbf{A}^{(L)} = -\frac{4\pi}{c}\mathbf{J}$$
This is split into two separate wave equations for the components:

1. Longitudinal: $\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) \mathbf{A}^{(L)}_\ell = -\frac{4\pi}{c}\mathbf{J}_\ell = -\frac{1}{c}\nabla \frac{\partial}{\partial t}\Phi^{(C)}$
2. Transverse: $\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) \mathbf{A}^{(L)}_{tr} = -\frac{4\pi}{c}\mathbf{J}_{tr} = -\nabla \times (\nabla \times \mathbf{A}_\infty)$

Three Derivation Methods

Method 1: Relation to the Coulomb Gauge

• Utilizes a known identity relating the vector potential in an arbitrary gauge to the Coulomb gauge potential $\mathbf{A}^{(C)}$.
• By comparing the Lorenz and Coulomb gauges, they derive:
  $$\mathbf{A}^{(L)}_{tr} = \mathbf{A}^{(C)} = \mathbf{A}^{(L)} + c\nabla \int (\Phi^{(L)} - \Phi^{(C)}) dt$$
  $$\mathbf{A}^{(L)}_\ell = c\nabla \int (\Phi^{(C)} - \Phi^{(L)}) dt$$
• They verify this solution by applying the wave operator to the integral expression and confirming it satisfies the source equation for $\mathbf{A}^{(L)}_\ell$.

Method 2: Direct Solution via Jackson's Equation (Longitudinal)

• Assumes $\mathbf{A}^{(L)}_\ell = \nabla \Psi$ and substitutes this into the longitudinal wave equation.
• Differentiates the resulting equation with respect to time to transform it into a standard wave equation for the quantity $(\frac{\partial \Psi}{\partial t} - c\Phi^{(C)})$.
• Solves using the retarded potential method (propagation at speed $c$), yielding:
  $$\Psi = c \int (\Phi^{(C)} - \Phi^{(L)}) dt \implies \mathbf{A}^{(L)}_\ell = c\nabla \int (\Phi^{(C)} - \Phi^{(L)}) dt$$

Method 3: Direct Solution via Jackson's Equation (Transverse)

• Assumes $\mathbf{A}^{(L)}_{tr} = \nabla \times \mathbf{V}$ and further $\mathbf{V} = \nabla \times \mathbf{W}$.
• Derives a wave equation for $\mathbf{W}$ involving the instantaneous potential $\mathbf{A}_\infty$.
• By differentiating twice with respect to time and utilizing the wave equation for $\mathbf{A}_\infty$, they isolate the second time derivative of the transverse potential.
• The result confirms that $\frac{\partial^2 \mathbf{A}^{(L)}_{tr}}{\partial t^2}$ is consistent with the expression derived in Method 1.

3. Key Results

The paper successfully derives exact analytic expressions for the components of the Lorenz-gauge vector potential:

1. Longitudinal Component:
   $$\mathbf{A}^{(L)}_\ell(\mathbf{r}, t) = c\nabla \int_{-\infty}^{t} \left[ \Phi^{(C)}(\mathbf{r}, t') - \Phi^{(L)}(\mathbf{r}, t') \right] dt'$$
   Where $\Phi^{(C)}$ is the instantaneous Coulomb potential and $\Phi^{(L)}$ is the retarded Lorenz scalar potential.

2. Transverse Component:
   $$\mathbf{A}^{(L)}_{tr}(\mathbf{r}, t) = \mathbf{A}^{(C)}(\mathbf{r}, t) = \mathbf{A}^{(L)}(\mathbf{r}, t) - \mathbf{A}^{(L)}_\ell(\mathbf{r}, t)$$
   Notably, the transverse component in the Lorenz gauge is identical to the vector potential in the Coulomb gauge.

3. Propagation Speed Resolution:
   The derivation clarifies that while $\mathbf{A}^{(L)}_\ell$ is constructed from the difference between an instantaneous potential ($\Phi^{(C)}$) and a retarded potential ($\Phi^{(L)}$), the resulting field does not violate causality. The "superluminal" terms cancel out or are mathematically structured such that the physical observables remain consistent with relativity. The solution is expressed entirely in terms of standard electrodynamical quantities.

4. Significance

• Resolution of Ambiguity: The paper resolves the confusion regarding the "faster-than-light" propagation of the longitudinal component. It demonstrates that the mathematical structure of the solution is sound and does not imply a physical violation of causality.
• Correction of Literature: It corrects and clarifies the discussion found in Jackson's seminal text and the subsequent analysis by Hnizdo, providing a rigorous derivation for equations (2.9) and (2.10) in Jackson's paper.
• Methodological Robustness: By solving the problem via three independent methods (gauge transformation, direct scalar potential integration, and vector potential decomposition), the authors provide a comprehensive proof that the derived solutions are unique and correct.
• General Applicability: The solutions are valid for arbitrary time-dependent charge-current distributions, making them applicable to a wide range of problems in classical electrodynamics, from antenna theory to radiation from moving charges.

In summary, Yang and Nevels provide a definitive mathematical framework for decomposing the Lorenz gauge potential, proving that the longitudinal and transverse components can be expressed analytically using the difference between Coulomb and retarded potentials, thereby demystifying the behavior of these fields.