[Paper Review] Variational generalization of free relativistic top

Roman Matsyuk|arXiv (Cornell University)|Jul 25, 2014

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper establishes the variational equivalence between first-order relativistic top equations and third-order Mathisson equations on the Mathisson-Pirani constraint surface. By constructing a Lagrangian for the third-order system in flat spacetime, the authors derive a continuous spectrum of effective proper masses dependent on spin, revealing that all extremals correspond to spinning particles with spin-dependent inertial mass, generalizing the standard free relativistic top model via variational principles with higher derivatives.

ABSTRACT

We prove that well known first-order (in spin, momentum, and space-time coordinates) equations of motion of relativistic top are equivalent to the third-order equations of Mathisson on the surface of the Mathisson-Pirani auxiliary constraint. We then consider these third-order equations in flat space-time with constant spin 4-vector and invent a Lagrange function for them. Allowing physical interpretation to be applied to the complete set of extremals yields a whole spectrum of spin-dependent effective 'proper mass' of the relativistic top.

Motivation & Objective

• To prove the equivalence between first-order relativistic top equations and third-order Mathisson equations on the Mathisson-Pirani constraint surface.
• To construct a variational Lagrangian for the third-order equations of motion in flat spacetime with constant spin 4-vector.
• To interpret the full set of extremals of the variational system as physical trajectories of spinning particles with effective mass dependent on spin orientation.
• To generalize the standard free relativistic top model by allowing a continuous spectrum of proper masses, parameterized by spin, through relaxation of the standard mass constraint.

Proposed method

• Derive the third-order equation of motion from the first-order system (1) and (2) under the Mathisson-Pirani constraint (4), using algebraic and differential manipulations.
• Use the constraint $ u_j S^{ij} = 0 $ and its derivative to eliminate momentum $ P^i $, leading to a third-order equation in $ u^i $.
• Construct a Lagrangian function $ L_{(k)} $ for the third-order system in 3+1-dimensional flat spacetime, parameterized by a pseudo-Euclidean 3-frame $ \{ \mathbf{e}_{(k)} \} $.
• Apply the Ostrogradsky formalism to handle the higher-derivative dynamics, ensuring variational consistency.
• Use the integral of motion $ \mathbf{s} \cdot \mathbf{u} / \| \mathbf{u} \| $ to define the effective mass $ m = \mu \left(1 - \frac{(\mathbf{s} \cdot \mathbf{u})^2}{\|\mathbf{s}\|^2 \|\mathbf{u}\|^2} \right)^{3/2} $.
• Demonstrate that all extremals of the variational system correspond to trajectories of spinning particles with this effective mass, forming a continuous spectrum.

Experimental results

Research questions

• RQ1 Are the first-order equations of motion for a relativistic top equivalent to the third-order Mathisson equations on the Mathisson-Pirani constraint surface?
• RQ2 Can a variational Lagrangian be constructed for the third-order system of a free relativistic top in flat spacetime?
• RQ3 What physical interpretation arises from the complete set of extremals of the variational system, particularly regarding the effective mass of the spinning particle?
• RQ4 How does relaxing the standard mass constraint lead to a continuous spectrum of effective masses parameterized by spin?

Key findings

• The first-order equations of motion for the relativistic top are mathematically equivalent to the third-order Mathisson equations on the surface of the Mathisson-Pirani constraint $ u_j S^{ij} = 0 $.
• A Lagrangian function for the third-order system is explicitly constructed in flat spacetime, parameterized by a choice of pseudo-Euclidean 3-frame, confirming the variational origin of the dynamics.
• All extremals of the variational system correspond to physical trajectories of spinning particles with an effective proper mass $ m = \mu \left(1 - \frac{(\mathbf{s} \cdot \mathbf{u})^2}{\|\mathbf{s}\|^2 \|\mathbf{u}\|^2} \right)^{3/2} $, which depends continuously on the spin orientation relative to the 4-velocity.
• The full solution space of the variational system includes a continuous spectrum of effective masses, generalizing the standard relativistic top model beyond the fixed mass case.
• The effective mass formula ensures that the spin 4-vector $ \mathbf{s} $ forms a constant angle with the 4-velocity $ \mathbf{u} $, consistent with the conservation of the integral $ \mathbf{s} \cdot \mathbf{u} / \| \mathbf{u} \| $.
• The variational generalization is equivalent to 'freezing' the integral of motion $ \mathbf{s} \cdot \mathbf{u} / \| \mathbf{u} \| $, and this relaxation leads to a physically consistent, spin-dependent mass spectrum.