The self-dual classical double copy, and the Eguchi-Hanson instanton

TL;DR

This work investigates the double copy in the self-dual sector of gauge and gravity by formulating a Kerr-Schild operator $\\hat{k}_\mu$ that acts on a harmonic function $\Phi$ to generate a biadjoint scalar, a gauge field $A_\mu$, and a graviton $h_{\mu\nu}$ with $h_{\mu\nu} = \\hat{k}_\mu \\hat{k}_\nu \Phi$. Applying this framework to the Eguchi-Hanson gravitational instanton yields a single copy gauge field that is abelian-like with dipole behavior and vanishing electric and magnetic charges, while the gravity solution captures the non-trivial topology after removing the bolt singularity. The analysis reinforces that the zeroth copy $\Phi$ satisfies a linearized biadjoint equation, aligning with the self-dual amplitude double copy, and offers insights into the nonperturbative aspects and potential non-abelian counterparts of the double copy. Overall, the paper clarifies how self-dual Kerr-Schild double copy operates for exact classical solutions and informs prospective nonperturbative classifications and mappings between gauge and gravity theories.

Abstract

The double copy is a map from non-abelian gauge theories to gravity, that has been demonstrated both for scattering amplitudes and exact classical solutions. In this study, we reconsider the double copy for exact solutions that are self-dual in either the gauge or gravity theory. In this case, one may formulate a general double copy in terms of a certain differential operator, which generates the gauge and gravity solutions from a harmonic function residing in a biadjoint scalar theory. As an illustration, we examine the single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge field thus obtained represents an abelian-like object whose field is dipole-like at large distances, and which has no magnetic or electric charge.

Figures (4)

• Figure 1: The function of eq. (\ref{['fdef']}) entering the gauge transformation of eqs. (\ref{['AmuN']}, \ref{['alphadef']}).
• Figure 2: Northern and southern hemispheres for fixed $r$ and $\psi$, and the equator $C$.
• Figure 3: The real part of the electric field of the Eguchi-Hanson single copy, using Cartesian coordinates in Minkowski signature: (a) for $(t,z)=(0,0)$; (b) for $(t,z)=(1,0)$. The imaginary part can be obtained by rotating the figures by $90^\circ$.
• Figure 4: The real part of the electric field of the Eguchi-Hanson single copy, using Cartesian coordinates in Minkowski signature: (a) for $(t,y)=(0,0)$; (b) for $(t,y)=(1,0)$. The imaginary part is zero.