# Generalized Carter constant for quadrupolar test bodies in Kerr spacetime

## TL;DR

This paper derives a generalized conserved quantity, akin to the Carter constant, for spinning test bodies with quadrupole moments in Kerr spacetime, valid up to second order in spin, with implications for understanding compact object dynamics.

## Contribution

It introduces a perturbed Carter constant for spinning bodies with quadrupole moments in Kerr spacetime, extending previous conserved quantities to more realistic astrophysical objects.

## Key findings

- Conservation holds perturbatively up to second order in spin.
- Explicit construction of the generalized Carter constant.
- No conserved quantity exists for generic quadrupole couplings.

## Abstract

We establish the existence of a deformation of the usual Carter constant which is conserved along the motion in a fixed Kerr background of a spinning test body possessing the spin-induced quadrupole coupling of a black hole. The conservation holds perturbatively up to second order in the test body's spin. This constant of motion is obtained through the explicit resolution of the conservation constraint equations, employing covariant algebraic and differential relations amongst covariant building blocks of the Kerr background. For generic spin-induced quadrupole couplings, which describe compact objects such as neutron stars, we obtain a no-go result on the existence of such a conserved quantity.