On a recent explanation of the dynamics of the Meissner effect within the conventional theory of superconductivity

This paper refutes the arguments presented by Markos and Hlubina regarding the conventional theory's ability to describe Meissner effect dynamics, identifying flaws in their reasoning and proposing an experiment to further investigate the issue.

The Gist

The Big Picture: A Disagreement About a "Magic Trick"

Imagine a metal cylinder sitting in a magnetic field. When you cool this metal down to become a superconductor, it performs a "magic trick" called the Meissner effect: it suddenly kicks the magnetic field out of its interior, as if the field never belonged there.

For decades, physicists have used a standard set of rules (the "conventional theory") to explain how this happens. However, the author of this paper, J.E. Hirsch, has been arguing for years that the standard rules are missing a crucial piece of the puzzle. He claims the standard theory describes what happens, but not how it happens physically.

Recently, two other scientists, Markos and Hlubina, published a paper claiming they proved the standard theory is actually correct and that Hirsch's objections are wrong.

This paper is Hirsch's rebuttal. He argues that Markos and Hlubina made a fatal logical error, failed to explain the actual forces involved, and proposed a new experiment to settle the debate once and for all.

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The Core Conflict: The "Ghost Force"

To understand the argument, we need to look at how the two sides view the "magic trick."

1. The Standard View (Markos and Hlubina)

Markos and Hlubina say: "We can write down a mathematical equation that assumes the superconductor gradually 'turns on.' If we plug this into our math, the magnetic field naturally gets pushed out. Therefore, the standard theory works."

Hirsch's Analogy:
Imagine you are watching a magician make a rabbit disappear from a hat.

• Markos and Hlubina say: "I wrote a script that says, 'At time 0, the rabbit is there. At time 10, the rabbit is gone.' Since the script works on paper, the magic is explained."
• Hirsch says: "That's just a script! You haven't told me how the rabbit vanished. Did it jump? Did it shrink? Did it teleport? Your script assumes the rabbit disappears without explaining the physical mechanism. You are just describing the result, not the cause."

Hirsch argues that Markos and Hlubina's math relies on a "ghost force"—a mathematical term that makes the electrons move to kick out the magnetic field, but this force doesn't exist in the real world (like gravity or magnetism). It's a mathematical trick, not a physical reality.

2. Hirsch's View

Hirsch argues that for the magnetic field to be kicked out, electric charge must physically move outward (like water being pushed out of a pipe). He believes the standard theory ignores this necessary movement of charge. He also argues that for the electrons to stop moving and transfer their momentum to the metal body without creating heat (dissipation), the electrons must act like they have "negative mass," a concept the standard theory doesn't include.

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Why Hirsch Says Their Answers Don't Work

Markos and Hlubina tried to answer four specific questions Hirsch raised. Hirsch says their answers are like "sidestepping the question."

1. The Question: "What force pushes the electrons to start moving?"

   • Their Answer: "Our equation says they start moving."
   • Hirsch's Rebuttal: That's a tautology (saying the same thing twice). You can't just say "the math says it happens." You must identify the physical push (the force). They didn't.

2. The Question: "If the electrons move, they create an electric field that should stop them. Why don't they stop?"

   • Their Answer: "Our equation has a special term that overrides the stopping force."
   • Hirsch's Rebuttal: That special term is the "ghost force" again. It doesn't correspond to any real physical interaction.

3. The Question: "How does the metal body spin in the opposite direction without friction?"

   • Their Answer: "The electrons bounce off impurities to transfer momentum."
   • Hirsch's Rebuttal: Bouncing off impurities creates heat (friction). The process needs to be frictionless (reversible) to work thermodynamically. Their explanation violates the laws of thermodynamics.

4. The Question: "When the superconductor turns back into normal metal, where does the energy go?"

   • Their Answer: "We can't calculate the details, but we assume it works."
   • Hirsch's Rebuttal: They admitted they can't explain the "kinetics" (the step-by-step process). If you can't explain how the energy moves without heat, you haven't solved the problem.

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The Proposed Experiment: The "Hollow Cylinder" Test

To prove who is right, Hirsch proposes a specific experiment involving a hollow cylinder.

The Setup:
Imagine a solid metal cylinder with a tiny, empty hole (a cavity) right in the center. You place this in a magnetic field and cool it down to become a superconductor.

The Prediction (Standard Theory / Markos & Hlubina):
They predict the magnetic field will be kicked out of the entire cylinder, including the tiny empty hole in the middle. The field lines will curve around the outside, leaving the inside (and the hole) completely empty of magnetism.

• Analogy: It's like a force field that pushes water out of a bucket, even if the bucket has a tiny bubble of air inside. The bubble gets squeezed dry.

The Prediction (Hirsch's Theory):
Hirsch argues this is impossible. To kick the magnetic field out of a region, you need to push electric charge out of that region.

• The Problem: The tiny hole in the middle is empty. There is no charge inside the hole to push out.
• The Result: Because there is no charge to move, the magnetic field cannot be expelled from the hole. The field will remain trapped inside the hole, and the metal around it will stay in a "normal" (non-superconducting) state to accommodate it.
• Analogy: You can't push water out of a bubble of air because there's no water there to push. The "magic" of the superconductor fails in that specific spot.

The Stakes:

• If the experiment shows the field is expelled from the hole, Hirsch is wrong, and the standard theory is correct.
• If the experiment shows the field stays trapped in the hole, Hirsch is right, and the standard theory of superconductivity is fundamentally flawed and needs to be rewritten.

Summary

Hirsch is essentially saying: "Markos and Hlubina are good at doing the math, but they are ignoring the physics. They are describing a magic trick without explaining the mechanism. I have a different theory that relies on real forces and charge movement. Let's test it with a hollow cylinder to see if the 'magic' actually works the way the textbooks say it does."

Technical Summary

Technical Summary: Critique of Markos-Hlubina's Explanation of Meissner Effect Dynamics

Problem Statement
This paper addresses a recent challenge by Markos and Hlubina (MH) to the author's long-standing argument that the conventional theory of superconductivity fails to describe the dynamics of the Meissner effect. The author, J. E. Hirsch, previously argued that the expulsion of magnetic fields from a metal transitioning to a superconducting state requires radial charge flow, a mechanism absent in conventional theory. MH countered this by modeling the transition using a generalized time-dependent London equation, claiming that the conventional theory correctly describes the dynamics without requiring radial charge flow or "exotic" new physics. This paper aims to demonstrate that MH's arguments contain fatal flaws and to propose an experiment to distinguish between the conventional view and the author's alternative theory.

Methodology and Analysis
The author employs a critical theoretical analysis of MH's mathematical framework and physical assumptions. The methodology involves:

1. Re-examination of MH's Equations: The author analyzes MH's generalized London equation (Eq. 2 in this paper), which introduces a time-dependent superconducting fraction $f(r,t)$ into the relation between supercurrent density $\mathbf{j}_s$ and the vector potential $\mathbf{A}$.
2. Physical Consistency Check: The author evaluates whether MH's mathematical solutions correspond to physical reality, specifically focusing on momentum conservation, the nature of forces acting on electrons, and the reversibility of thermodynamic processes.
3. Thought Experiment Design: The author constructs a specific geometric scenario—a superconducting cylinder with a small, empty cylindrical cavity in its interior—to test the predictions of MH's theory against the author's own theory regarding charge expulsion.

Key Contributions and Arguments
The paper identifies four specific "problems" raised by the author in previous works that MH claimed to have solved, arguing that MH's solutions are invalid:

• Problem (i) - Initiation of Current: MH claims that a finite supercurrent flows immediately as the superconducting fraction $f$ becomes non-zero. The author argues this is a tautology that fails to identify the physical force required to accelerate electrons from rest to generate momentum, violating momentum conservation.
• Problem (ii) - Faraday Induction: MH suggests that a term in their acceleration equation (proportional to $\partial f/\partial t$) allows current to accelerate against the induced electric field. The author contends that there is no physical force proportional to the vector potential $\mathbf{A}$; the only physical forces are the Lorentz force terms. Thus, MH's mechanism is physically unfounded.
• Problem (iii) - Momentum Transfer to Lattice: MH proposes that momentum is transferred to the lattice via scattering on impurities/phonons. The author argues this is a dissipative process, whereas the Meissner effect and the Becker-London effect (rotating superconductors) require reversible, dissipationless momentum transfer.
• Problem (iv) - Dissipationless Transition: MH admits their formalism cannot study the kinetics of the conversion from superfluid to normal electrons without dissipation. The author emphasizes that the conventional theory lacks a mechanism for this reversible conversion.

The author further refutes MH's claim that the formulation of constitutive equations (using $\mathbf{A}$ vs. $\mathbf{B}$) is the source of the difference. The author demonstrates that a similar "non-physical force" term arises even when using the magnetic field formulation, indicating the flaw lies in the assumption of the equation's validity rather than its mathematical form.

Proposed Experiment and Predicted Results
To resolve the theoretical impasse, the author proposes an experiment involving a superconducting cylinder with a small, empty cylindrical cavity (inclusion) in its interior, cooled in a uniform magnetic field.

• MH/Conventional Prediction: The theory predicts that the superconducting fraction $f(r,t)$ can evolve arbitrarily (as an external parameter). Consequently, the system will reach the global energy minimum where the magnetic field is completely expelled from the entire volume, including the empty cavity (Fig. 2).
• Author's Prediction: Based on the theory of hole superconductivity, the expulsion of a magnetic field requires the outward motion of electric charge. Since the cavity contains no charge, the magnetic field within the cavity cannot be expelled. Consequently, the system cannot reach the global energy minimum. Instead, it will settle into a metastable state where the magnetic field remains trapped in the cavity, and the superconducting state is restricted to regions where charge expulsion is possible (Fig. 4).

Significance and Conclusion
The paper concludes that the Markos-Hlubina theory fails to address the fundamental physical mechanisms (forces and momentum transfer) required for the Meissner effect, relying instead on a mathematical formalism that assumes the outcome rather than deriving it from first principles. The author asserts that the conventional theory does not explain the dynamics of the Meissner effect because it lacks the mechanism of radial charge flow driven by quantum pressure.

The significance of the proposed experiment lies in its ability to empirically test these competing views. If the magnetic field is expelled from the empty cavity, the author's objections are unfounded, and the conventional theory is vindicated. If the field remains trapped in the cavity, it would strongly support the author's argument that charge expulsion is indispensable for magnetic field expulsion, necessitating a revision or discard of the conventional theory of superconductivity. The author notes that F. London himself regarded the London equations as phenomenological assumptions derived from experimental consequences rather than first principles, a view the author argues MH misinterprets by attributing explanatory power to the equations that London explicitly denied.