Forces on Fields Charles T. Sebens University of California, San Diego 5th International Summer School in Philosophy of Physics Saig, Germany July 21, 2017

Newton’s Third Law Whenever one body exerts a force on a second, the second body exerts an equal and opposite force on the first. Example: Electrostatic Coulomb repulsion of two equal and opposite charges

Apparent Violation of the Third Law in Electromagnetism Example discussed in Lange (2002).

The Physicist’s Response “…the proof of conservation of momentum rests on the cancellation of internal forces, which follows from the third law. When you tamper with the third law, you are placing the conservation of momentum in jeopardy, and there is no principle in physics more sacred than that. Momentum conservation is rescued in electrodynamics by the realization that the fields themselves carry momentum.” -Griffiths (1999, pg. 351); see also Feynman (1964, sec. 26-2 & 27-6) True, but… Does Newton’s third law hold in electromagnetism?

Quick Case for “No” “Newton used the third law to derive the law of conservation of momentum; from a deeper perspective, however, conservation of momentum is the more fundamental idea (derived via Noether’s theorem from Galilean invariance), and holds in cases where Newton’s third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.” -Wikipedia (2017) A footnote to the conservation of energy and momentum: “However, Newton’s third Law (‘Every action is accompanied by an equal and opposite reaction’) is still violated, even if fields are real. Bodies do not exert forces on fields; bodies alone feel forces. Newton’s third law was thus abandoned before relativity theory came on the scene.” -Lange (2002, pg. 163)

Quick Case for “yes” Force is simply the rate of change of momentum. Because the amount of momentum in the field is changing, forces must be acting on the field (presumably from matter as it is the only other actor on the scene). Because momentum is conserved, changes in momentum must cancel and thus forces must balance. Problem: This argument begs the question against someone who thinks that the conservation of momentum is a deeper principle than Newton’s third law as it makes obedience of the third law an immediate consequence of the conservation of momentum.

Strategy Outline of the Talk Show that the electromagnetic field really obeys a force law, not just a law of conservation of momentum. Outline of the Talk Apparent Violation of the Third Law Fluid Properties and Field Properties Eulerian Dynamics Lagrangian Dynamics

Fluid Properties the (relativistic) mass density the velocity field the momentum density the momentum flux density tensor the stress tensor All of these quantities are functions of space and time.

Mass By mass-energy equivalence, we can assign a relativistic mass density of to the electromagnetic field. (This is just the energy density divided by c2.) You must attribute the above mass to the electromagnetic field to ensure that the total relativistic mass of matter and field is conserved and that the center of relativistic mass for a closed system moves inertially (Einstein, 1906). It has often been noted that the field around or inside a body can make it harder to accelerate that body. But, this is an indirect way to get at the inertial role played by the mass of the field. We will see that the mass of the field quantifies the resistance to acceleration of the field itself, just as the mass of a fluid quantifies its resistance to acceleration.

Velocity The velocity which describes the flow of this mass is equal to the field momentum density divided by the mass density or, equivalently, the energy flux density (Poynting vector) divided by the energy density. The magnitude of the velocity is maximized at c when the electric and magnetic fields are perpendicular and equal in magnitude.

Fluid Properties of the EM Field the (relativistic) mass density the velocity field the momentum density the momentum flux density tensor (the Maxwell stress tensor) the true stress tensor (not the Maxwell stress tensor)

Historical Fluid Analogies Poincaré Maxwell In “On Faraday’s Lines of Force” (1890), Maxwell drew a different analogy between hydrodynamics and electromagnetism, introducing (as “a purely imaginary substance”) an incompressible fluid that flows along the electric lines of force and a second fluid that flows along magnetic lines of force. Poincaré’s analogy is closer to the one explored here. He examined how the electric and magnetic fields together resemble a single compressible fluid which flows in the direction of energy flux. In his 1900 paper, “La Théorie de Lorentz et le Principe de Réaction,” Poincaré uses the fact that “electromagnetic energy behaves as a fluid which has inertia” to understand the status of Newton’s third law in electromagnetism.

Relativistic Continuum Mechanics In a relativistic description of a continuous distribution of matter, there are two equations that might deserve to be called “the force law.” Eulerian Force Law Lagrangian Force Law is the force per unit volume on the matter. If the source of this force is the electromagnetic field, this is the Lorentz force. D/Dt is the material derivative (a.k.a. comoving or Lagrangian derivative). The surface integral in the first equation is of the momentum flux density tensor. In the second equation the surface integral is of the stress tensor. We will see that both force laws are obeyed by the electromagnetic field.

The Eulerian Force Law: Matter the change in the momentum of the matter in a fixed volume per time is equal to (2) the force exerted on the matter in that volume plus (3) the flux of momentum into that volume from matter outside

The Eulerian Force Law: Matter (1) (2) (3) the change in the momentum of the matter in a fixed volume per time is equal to (2) the force exerted on the matter in that volume plus (3) the flux of momentum into that volume from matter outside

The Eulerian Force Law: Field (1) (2) (3) the change in the momentum of the field in a fixed volume per time is equal to (2) the force exerted on the field in that volume by matter (equal and opposite the Lorentz force exerted on matter by the field) plus (3) the flux of momentum into that volume from field outside

The Eulerian Force Law: Field the change in the momentum of the field in a fixed volume per time is equal to (2) the force exerted on the field in that volume by matter (equal and opposite the Lorentz force exerted on matter by the field) plus (3) the flux of momentum into that volume from field outside

The Lagrangian Force Law: Matter the change in the momentum of the matter which happens to be at this moment inside of a given volume is equal to (2) the force exerted on the matter in that volume plus (3) the force on the matter in that volume from matter outside

The Lagrangian Force Law: Matter (1) (2) (3) the change in the momentum of the matter which happens to be at this moment inside of a given volume is equal to (2) the force exerted on the matter in that volume plus (3) the force on the matter in that volume from matter outside

The Lagrangian Force Law: Field (1) (2) (3) the change in the momentum of the field mass which happens to be at this moment inside of a given volume is equal to (2) the force exerted on the field in that volume plus (3) the force on the field in that volume from the field outside

The Lagrangian Force Law: Field the change in the momentum of the field mass which happens to be at this moment inside of a given volume is equal to (2) the force exerted on the field in that volume plus (3) the force on the field in that volume from the field outside

Conclusion Newton’s third law holds in electromagnetism because the force from the electromagnetic field on matter is balanced by an equal and opposite force from matter on the field. The response of the field to this force can be given in Eulerian or Lagrangian form. These equations perfectly match those that govern a relativistic fluid. From examination of the analogy it is clear that the mass of the field plays exactly the same inertial role as the mass of a fluid. It is also clear that the Maxwell stress tensor is really a momentum density tensor, not a stress tensor. Further questions addressed in the paper: Can we replace Maxwell’s equations with fluid equations? To what extent can we understand the classical electromagnetic field as composed of photons? Is it possible to define a proper mass density for the field? The paper is available at: arxiv.org/abs/1707.04198

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