[Paper Review] Resolution of the Mansuripur Paradox

Daniel J. Cross|arXiv (Cornell University)|May 23, 2012

Experimental and Theoretical Physics Studies4 citations

TL;DR

This paper resolves the Mansuripur paradox by showing that the apparent contradiction in torque on a magnetic dipole in different inertial frames arises from neglecting hidden momentum in the 3-vector Lorentz force formulation. Using the covariant 4-vector formulation of electrodynamics, the paper demonstrates that the torque in the moving frame is not a violation but a necessary balance of changing hidden angular momentum, with the time-space components of the torque tensor accounting for the relativistic transformation of momentum and angular momentum.

ABSTRACT

The interaction of a magnetic dipole with a point charge leads to an apparent paradox when analyzed using the 3-vector formulation of the Lorentz force. Specifically, the dipole is subject to a torque in some frames and not in others. We show that when analyzed according to the covariant 4-vector formulation the paradox disappears. The torque that arises in certain frames is connected to the time-space components of the torque in the rest frame, giving rise to "hidden" momentum.

Motivation & Objective

• To resolve the apparent contradiction in torque on a magnetic dipole when viewed from different inertial frames.
• To show that the Lorentz force law is consistent with relativity when analyzed using the covariant 4-vector formulation.
• To clarify the physical origin of 'hidden momentum' and its role in maintaining angular momentum balance across frames.
• To demonstrate that the Einstein-Laub force law is not required to resolve the paradox, as the Lorentz law is fully consistent when properly formulated.

Proposed method

• Formulates the electromagnetic force and torque using the covariant 4-vector formalism with the Faraday tensor $F^{\alpha\beta}$ and current 4-vector $j^\beta = \partial_\alpha M^{\alpha\beta}$.
• Represents electric and magnetic dipole moments as components of an antisymmetric second-rank tensor $M^{\alpha\beta}$, with $M = (\mathbf{P}, -\mathbf{M})$.
• Derives the four-force $f^\alpha = F^{\alpha\beta} j_\beta$, showing it vanishes in both rest and moving frames for the point dipole.
• Analyzes the antisymmetric torque tensor $t^{\alpha\beta} = x^\alpha f^\beta - x^\beta f^\alpha$, including non-vanishing time-space components.
• Uses Lorentz transformation of the torque tensor to show that the spatial torque in the moving frame arises from the time-space components in the rest frame.
• Connects the rest-frame time-space torque components to the hidden momentum $\mathbf{p}_{\text{hid}} = -\boldsymbol{\mu} \times \mathbf{E}$, which evolves into hidden angular momentum under boost.

Experimental results

Research questions

• RQ1 Why does the Lorentz force law appear to violate relativity in the Mansuripur thought experiment?
• RQ2 What is the physical origin of the apparent torque on a magnetic dipole in a moving frame?
• RQ3 How is angular momentum conserved when the torque is non-zero in one frame but zero in another?
• RQ4 Can the paradox be resolved without abandoning the Lorentz force law?
• RQ5 What role does 'hidden momentum' play in maintaining relativistic consistency in electromagnetic systems?

Key findings

• The torque observed in the moving frame is not a failure of the Lorentz force law but a necessary consequence of the relativistic transformation of the torque tensor.
• The non-zero torque in the moving frame arises from the time-space components of the torque tensor in the rest frame, specifically $T^{yt} = -\gamma v R^y$.
• In the rest frame, the spatial torque has a non-zero time-space component $R^y = -mE$, corresponding to a constant hidden momentum $\mathbf{p}_{\text{hid}} = -mE\hat{\mathbf{y}}$.
• Upon boosting to the moving frame, this hidden momentum becomes time-dependent, leading to a changing hidden angular momentum $\mathbf{J}' = \gamma v m E \tau \hat{\mathbf{x}}'$.
• The resulting rate of change of angular momentum $d\mathbf{L}/d\tau = \gamma v m E \hat{\mathbf{x}}'$ exactly matches the observed torque, confirming conservation.
• The Einstein-Laub force law predicts zero torque in all frames, but this is not a physical necessity—instead, the Lorentz law with proper covariant treatment is fully consistent with relativity.