[Paper Review] The Papapetrou equations and supplementary conditions

O. B. Karpov|ArXiv.org|Jun 1, 2004

Relativity and Gravitational Theory3 citations

TL;DR

This paper provides a comparative analysis of the Papapetrou equations for spinning bodies in general relativity, evaluating how different supplementary conditions—such as Pirani, Dixon, and Corinaldesi—impact the equations of motion. It shows that the spin-orbit force in post-Newtonian approximation depends on the choice of supplementary condition, and that Newton's third law is only satisfied under specific conditions (e.g., Corinaldesi condition), highlighting the relativity of motion and the physical inconsistency of certain assumptions in the theory.

ABSTRACT

On the bases of the Papapetrou equations with various supplementary conditions and other approaches a comparative analysis of the equations of motion of rotating bodies in general relativity is made. The motion of a body with vertical spin in a circular orbit is considered. An expression for the spin-orbit force in a post-Newtonian approximation is investigated.

Motivation & Objective

• To analyze the impact of various supplementary conditions on the Papapetrou equations of motion for spinning bodies in general relativity.
• To investigate the physical consistency of different worldline definitions (e.g., center of mass, rest frame) in curved spacetime.
• To evaluate whether Newton's third law and the relativity of motion are consistently satisfied across different supplementary conditions.
• To clarify the role of the 4-velocity, dynamic 4-velocity, and spin tensor in defining the center of mass and motion of spinning bodies.
• To resolve long-standing debates on the physical validity of the Papapetrou equations and the appropriateness of specific supplementary conditions.

Proposed method

• Derives the Papapetrou equations using the 4-momentum $ P^eta $ and spin tensor $ S^{etaeta} $, with the Riemann tensor $ R^{eta}{}_{etaetaeta} $ coupling to spin.
• Applies multiple supplementary conditions: Pirani ($ S^{etaeta} u_eta = 0 $), Dixon ($ S^{etaeta} P_eta = 0 $), Corinaldesi ($ S^{etaeta} au_eta = 0 $ with $ au^eta $ as Killing vector).
• Uses post-Newtonian approximation to compute the spin-orbit force and analyze its dependence on the supplementary condition parameter $ \sigma $.
• Compares forces $ \bm{F}_1 $ and $ \bm{F}_2 $ on two bodies in a binary system, showing that Newton's third law holds only for $ \sigma = 1 $ (Corinaldesi).
• Introduces a transformation framework for the Riemann tensor into electric and magnetic parts $ E $ and $ B $, and derives their transformation laws under Lorentz boosts.
• Analyzes the motion of a body with vertical spin in circular orbit, showing that the center of mass velocity depends on the choice of supplementary condition.

Experimental results

Research questions

• RQ1 How does the choice of supplementary condition affect the spin-orbit force in the post-Newtonian approximation of the Papapetrou equations?
• RQ2 Under what conditions is Newton's third law satisfied in the general relativistic two-body problem with spinning bodies?
• RQ3 Why is the Pirani supplementary condition considered physically problematic despite its widespread use?
• RQ4 How do different definitions of the center of mass (e.g., intrinsic CM, CM in rest frame) affect the motion of spinning bodies in curved spacetime?
• RQ5 To what extent is the relativity of motion preserved when different supplementary conditions are applied to the Papapetrou equations?

Key findings

• The spin-orbit force in the post-Newtonian limit depends explicitly on the supplementary condition parameter $ \sigma $, with the Corinaldesi condition ($ \sigma = 1 $) being the only one that satisfies Newton's third law.
• Newton's third law is only satisfied when $ \sigma = 1 $, corresponding to the Corinaldesi supplementary condition; for $ \sigma = 0 $ (Pirani), the forces on the two bodies do not satisfy $ \bm{F}_2 = -\bm{F}_1 $.
• The Papapetrou equations are physically inconsistent when the condition $ P^eta \propto u^eta $ is violated, as this leads to unphysical motion of the center of mass.
• The intrinsic center of mass (defined via $ S^{etaeta} P_eta = 0 $) moves relative to the rest frame, even in flat spacetime, indicating a non-trivial kinematic structure.
• The electric and magnetic parts of the Riemann tensor transform non-trivially under Lorentz boosts, with $ E'_{ij} = E_{ij} + 2v^l \epsilon_{lki} B^k_j $ and $ B'_{ij} = B_{ij} - 2v^l \epsilon_{lki} E^k_j $ in the linear velocity approximation.
• The equations of motion are independent of the supplementary condition when expressed in terms of the effective forces $ m\dot{u}_1^i = F_1^i $ and $ \mathcal{M}\dot{u}_2^i = F_2^i $, but the forces themselves depend on $ \sigma $, showing that the physical interpretation hinges on the choice of condition.