Spinning bodies in curved space-time

TL;DR

This paper develops a covariant formalism for analyzing the motion of spinning bodies in curved space-time, revealing how spin affects orbital stability and the innermost stable circular orbit.

Contribution

It introduces a new covariant Poisson-Dirac bracket formulation for spinning bodies and derives novel conditions for constants of motion in curved space-times.

Findings

Spinning bodies cause periastron precession and radial variation in stable orbits.

The innermost stable circular orbit (ISCO) depends on spin, with a lower limit for prograde spins.

Equations of motion can be derived from Einstein's equations with an appropriate energy-momentum tensor.

Abstract

We study the motion of neutral and charged spinning bodies in curved space-time in the test-particle limit. We construct equations of motion using a closed covariant Poisson-Dirac bracket formulation which allows for different choices of the hamiltonian. We derive conditions for the existence of constants of motion and apply the formalism to the case of spherically symmetric space-times. We show that the periastron of a spinning body in a stable orbit in a Schwarzschild or Reissner-Nordstr{\o}m background not only precesses, but also varies radially. By analysing the stability conditions for circular motion we find the innermost stable circular orbit (ISCO) as a function of spin. It turns out that there is an absolute lower limit on the ISCOs for increasing prograde spin. Finally we establish that the equations of motion can also be derived from the Einstein equations using an appropriate energy-momentum tensor for spinning particles.