[Paper Review] The force density and the kinetic energy-momentum tensor of electromagnetic fields in matter

Rodrigo Medina, J. Stephany|arXiv (Cornell University)|Mar 30, 2014

Magnetic and Electromagnetic Effects3 citations

TL;DR

This paper derives the invariant force density on dipolar matter from Lorentz force laws and constructs a non-symmetric electromagnetic energy-momentum tensor in matter that satisfies Maxwell’s equations and Newton’s third law. It resolves the Minkowski-Abraham controversy by showing Minkowski’s momentum (nE/c) is correct, and demonstrates that the center-of-mass theorem fails for non-symmetric systems unless a spin contribution is included, which restores conservation in wave-packet interactions with dielectric blocks.

ABSTRACT

We determine the invariant expression of the force density that the electromagnetic field exerts on dipolar matter and construct the non-symmetric energy-momentum tensor of the electromagnetic field in matter which is consistent with that force and with Maxwell equations. We recover Minkowski's expression for the momentum density. We use our results to discuss momentum exchange of an electromagnetic wave-packet which falls into a dielectric block. In particular we show that the wave-packet pulls the block when it enters and drags it when it leaves. The usual form of the center of mass motion theorem does not hold for this system but a modified version of the theorem which includes a spin contribution is shown to be satisfied.

Motivation & Objective

• To derive the invariant force density exerted by electromagnetic fields on dipolar matter using relativistic invariance and Maxwell’s equations.
• To construct a non-symmetric energy-momentum tensor for electromagnetic fields in matter that is consistent with the Lorentz force and Newton’s third law.
• To resolve the long-standing Minkowski-Abraham controversy over electromagnetic momentum in dielectrics by showing Minkowski’s expression is correct.
• To demonstrate that the standard center-of-mass motion theorem fails for systems with non-zero spin, but a modified version including spin is conserved.
• To show explicitly that a wave-packet entering a dielectric block is pulled by the medium, and drags it upon exit, via a spin-dependent correction to momentum balance.

Proposed method

• Derive the force density on dipolar matter from the microscopic Lorentz force, ensuring consistency with Newton’s third law.
• Construct the kinetic energy-momentum tensor $ T^{ ueta}_{ ext{FK}} $ as the dual of the force density, ensuring $ abla_ u T^{ ueta}_{ ext{FK}} = -f^eta $, where $ f^eta $ is the force density.
• Use the constitutive relations $ oldsymbol{D} = oldsymbol{E} + 4oldsymbol{P}/(4oldsymbol{ u}) $ and $ oldsymbol{B} = oldsymbol{H} + 4oldsymbol{M}/(4oldsymbol{ u}) $ to express the force density in terms of $ oldsymbol{E}, oldsymbol{B}, oldsymbol{P}, oldsymbol{M} $.
• Introduce the spin density $ S^{ ueta au} $ via the conservation law $ abla_ au S^{ ueta au} = T^{ ueta} - T^{eta u} $, which accounts for non-symmetric momentum flow.
• Define a modified center-of-mass coordinate $ X^i_ heta = X^i_T + X^i_S $, where $ X^i_S $ is the spin contribution, to restore constant-velocity motion under the improved center-of-mass theorem.
• Perform explicit computation of momentum and spin transfer for a Gaussian wave-packet incident on a dielectric block, showing pull-on-entry and drag-on-exit.

Experimental results

Research questions

• RQ1 What is the correct invariant expression for the electromagnetic force density on dipolar matter in a medium?
• RQ2 How can a non-symmetric energy-momentum tensor for electromagnetic fields in matter be consistently derived from first principles?
• RQ3 Does the standard center-of-mass motion theorem hold for systems with non-zero spin, such as a dielectric block interacting with a wave-packet?
• RQ4 Why does Balazs’s argument that Minkowski’s momentum would violate the center-of-mass theorem fail?
• RQ5 What is the physical origin of the pull/drag effect observed when a wave-packet enters and exits a dielectric block?

Key findings

• The force density on dipolar matter is derived as $ f^eta = -rac{1}{4oldsymbol{ u}} abla_ u ig( (oldsymbol{D} imes oldsymbol{B} - oldsymbol{H} imes oldsymbol{E})^eta ig) $, with a dipolar contribution ensuring energy and momentum conservation.
• The constructed energy-momentum tensor $ T^{ ueta}_{ ext{FK}} $ is non-symmetric and recovers Minkowski’s momentum density $ nE/c $, confirming its validity over Abraham’s.
• The standard center-of-mass motion theorem fails for non-symmetric systems due to non-vanishing spin, but a modified version including spin $ X^i_S $ restores conservation.
• For a wave-packet entering a dielectric block, the system experiences a pull (negative force) due to the spin contribution, as shown in Eq. (44) and (53).
• Upon exit, the wave-packet drags the block, consistent with the spin-corrected momentum balance in Eq. (54), which verifies the improved center-of-mass theorem.
• The spin contribution $ rac{d}{dt}S^{010} $ is non-zero and proportional to $ -rac{( uoldsymbol{ u}-1)ar{T}}{4oldsymbol{ u}} imes ext{field energy} $, confirming the physical role of spin in momentum transfer.