[Paper Review] Leading anomalies, the drift Hamiltonian and the relativistic two-body system

Bernard Michael Nabet, Barak Kol|arXiv (Cornell University)|Aug 12, 2014

Pulsars and Gravitational Waves Research30 references3 citations

TL;DR

This paper introduces a novel approach to solving the relativistic two-body problem in General Relativity by focusing on the leading-order violations of Newtonian conserved quantities—termed 'relativistic anomalies'—rather than computing full solutions to a given post-Newtonian order. It derives a drift Hamiltonian as the time average of the perturbation Hamiltonian, enabling efficient computation of relativistic effects such as periapsis shift and spin-orbit precession, with results matching known expressions in the test mass limit.

ABSTRACT

We suggest to solve for the motion of the two body problem in General Relativity by identifying the leading violation of conserved quantities, referred to as (relativistic) anomalies, ordered by the post-Newtonian order at which they appear. This differs from the standard procedure of obtaining the full solution up to a prescribed order. We find that the reduced Hamiltonian which describes the drift in the space of conserved quantities is given by the average of the perturbation Hamiltonian. Using this approach the averaging is done prior to the derivation of time evolution, thereby economizing the computation. The computations become similar to those in the Hamilton-Jacobi method, while staying in the more comfortable setting of the Hamiltonian formulation. We apply this approach of leading anomalies and the drift Hamiltonian to the binary problem and treat several perturbations: 1PN, spin-orbit and spin-spin. On the way we discuss the interpretation of the Laplace-Runge-Lenz vector as a generator of scale-preserving conformal transformations in momentum space.

Motivation & Objective

• To address the relativistic two-body problem in General Relativity by focusing on leading-order violations of Newtonian conserved quantities, termed 'relativistic anomalies'.
• To develop a computationally efficient method for post-Newtonian dynamics by deriving a drift Hamiltonian that governs the slow evolution of conserved quantities.
• To provide a new derivation of the 1PN periapsis shift using the Laplace-Runge-Lenz vector anomaly, avoiding full equations of motion.
• To treat spin-orbit and spin-spin interactions systematically within the anomaly framework, computing their associated anomalies.
• To clarify the geometric and symmetry interpretation of the Laplace-Runge-Lenz vector as a generator of scale-preserving conformal transformations in momentum space.

Proposed method

• Identify the leading-order post-Newtonian violation of each Newtonian conserved quantity as a relativistic anomaly, ordered by PN order of appearance.
• Define the drift Hamiltonian as the time-average of the perturbation Hamiltonian, which governs the slow evolution of conserved quantities in the reduced space of constants of motion.
• Use analytic continuation and residue calculus (e.g., contour integration in the complex plane) to compute the required time averages efficiently.
• Apply the drift Hamiltonian formalism to compute anomalies for 1PN, spin-orbit, and spin-spin interactions in the two-body system.
• Utilize the Poisson bracket structure to derive phase space generators of symmetries associated with the Laplace-Runge-Lenz vector and interpret them as conformal transformations in momentum space.
• Confirm results by matching known expressions in the test mass limit, particularly for spin-orbit and spin-spin precession frequencies.

Experimental results

Research questions

• RQ1 How can the relativistic two-body problem be reinterpreted through the lens of leading-order anomalies in conserved quantities rather than full equations of motion?
• RQ2 What is the mathematical and physical role of the time-averaged perturbation Hamiltonian in describing the slow drift of conserved quantities?
• RQ3 How does the Laplace-Runge-Lenz vector’s symmetry relate to conformal transformations in momentum space, and what is its physical interpretation in the relativistic regime?
• RQ4 What are the explicit expressions for the leading anomalies in the 1PN, spin-orbit, and spin-spin interactions, and how do they compare to known results?
• RQ5 Can the periapsis shift be derived as the anomaly of the LRL vector using this drift Hamiltonian approach, and is the derivation more efficient than standard methods?

Key findings

• The drift Hamiltonian is shown to be the time-average of the perturbation Hamiltonian, providing an economical method to compute relativistic corrections without solving full equations of motion.
• The 1PN periapsis shift is derived as the anomaly of the Laplace-Runge-Lenz vector, yielding the expression $\Omega_{\text{1PN}} = \frac{3}{2} \frac{(-2E)^{3/2}}{L^4} \mu \mu_2$.
• For spin-orbit coupling, the leading anomalies are computed as $\Omega_{\text{SO}} = \frac{3}{2} \frac{(-2E)^{3/2}}{L^4} (\mathbf{S}_1 \cdot \mathbf{\hat{L}}) \mu \mu_2$ and $\Omega_{\text{SO}} = \frac{3}{2} \frac{(-2E)^{3/2}}{L^4} (\mathbf{S}_2 \cdot \mathbf{\hat{L}}) \mu \mu_2$.
• For spin-spin coupling, the anomalies are derived as $\Omega_{\text{SS}} = \frac{3}{2} \frac{(-2E)^{3/2}}{L^4} \left[ (\mathbf{S}_1 \cdot \mathbf{\hat{L}}) \mathbf{S}_2 + (\mathbf{S}_2 \cdot \mathbf{\hat{L}}) \mathbf{S}_1 + \left( (\mathbf{S}_1 \cdot \mathbf{S}_2) - 5(\mathbf{S}_1 \cdot \mathbf{\hat{L}})(\mathbf{S}_2 \cdot \mathbf{\hat{L}}) \right) \mathbf{\hat{L}} \right]$.
• All computed anomalies in the test mass limit agree with known results from Barker (1970), confirming the validity of the method.