Abstract
We extend the Kerr-Schild double copy to the case of a probe particle moving in the Kerr-Schild background. In particular, we solve Wong's equations for a test color charge in a Coulomb non-Abelian potential ($\sqrt{\text{Schw}}$) and on the equatorial plane for the potential generated by a rotating disk of charge known as the single copy of the Kerr background ($\sqrt{\text{Kerr}}$). The orbits, as the corresponding geodesics on the gravity side, feature elliptic, circular, hyperbolic and plunge behavior for the charged particle. We then find a new double copy map between the conserved charges on the gauge theory side and the gravity side, which enables us to fully recover geodesic equations for Schwarzschild and Kerr. Interestingly, the map works naturally for both bound and unbound orbits.
Highlights
The double copy relation between gauge and gravity theories was first discovered for quantum scattering amplitudes [1,2,3], and in recent years, it quickly became a formidable tool to tame the complexity of perturbative gravity calculations [4]
We extend the Kerr-Schild double copy to the case of a probe particle moving in the Kerr-Schild backgrounpd.ffiffiIffinffiffiffiffipffiffiffiaffi rticular, we solve Wong’s equations for a test color charge in a Coulomb non-Abelian potential ( Schw) and on the equatorial plane forptffihffiffieffiffiffiffipffiffiotential generated by a rotating disk of charge known as the single copy of the Kerr background ( Kerr)
In the extreme limit, where one mass is much bigger than the other, i.e., at leading order in the expansion in the mass ratio, the problem is equivalent to a light particle following geodesics in the background sourced by the other heavy particle
Summary
The double copy relation between gauge and gravity theories was first discovered for quantum scattering amplitudes [1,2,3], and in recent years, it quickly became a formidable tool to tame the complexity of perturbative gravity calculations [4]. The single-copy version of the Kerr metric can be found, and it corresponds to the potential field generated by a rotating disk of color charge as explained inp[3ffiffi4ffiffi]ffiffiffi(ffiffisffiffiee alspo ffiAffiffiffiffipffiffipffiffiendix A). We denote those solutions as Schw and Kerr, respectively. We find a direct map of the conserved charges for probe particles in Schwarzschild and Kerr which makes possible to recover geodesic equations
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