A Kerr-Schild metric is a perturbation of a flat Minkowski metric η μν\eta_{\mu\nu} of the form

g μν=η μν+κϕk μk ν g_{\mu\nu} = \eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu}

where κ=32πG N\kappa=\sqrt{32\pi G_{\mathrm{N}}} is a constant with G NG_{\mathrm{N}} Newton's constant, ϕ\phi is a scalar field and k μk_\mu is a null covector satisfying the geodesic property, i.e.

η μνk μk ν=g μνk μk ν=0,(k⋅∂)k μ=0. \eta_{\mu\nu}k^\mu k^\nu = g_{\mu\nu}k^\mu k^\nu =0, \quad (k\cdot\partial)k^\mu =0.
The single copy gauge field (MOW 15) of this gravitational field is defined for any gauge group GG by

A μ=(c aT a)ϕk μA_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu

where c aT a∈𝔤c^a\mathbf{T}_a \in \mathfrak{g} is an arbitrary constant color charge, specified by a vector c ac^a in the basis {T a}\{\mathbf{T}_a\} of the Lie algebra 𝔤\mathfrak{g}.
Conversely, if we start from a gauge field of the form A μ=(c aT a)ϕk μA_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu for any constant color charge c aT a∈𝔤c^a\mathbf{T}_a \in \mathfrak{g} and null covector k μk_\mu satisfying the geodesic property, we can define its double copy gravitational field by the Kerr-Schild metric g μν=η μν+κϕk μk νg_{\mu\nu}=\eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu}.
Otherwise, if we repeat the procedure of replacing a covector k μk_\mu with any fixed color charge (c˜ bT˜ b)∈𝔤˜(\tilde{c}^b \tilde{\mathbf{T}}_b)\in\tilde{\mathfrak{g}} we can get a zeroth copy scalar field, defined by

Φ=(c aT a)⊗(c˜ bT˜ b)ϕ\Phi = (c^a \mathbf{T}_a)\otimes(\tilde{c}^b \tilde{\mathbf{T}}_b)\phi

where the new gauge group G˜\tilde{G} can be chosen different from the previous GG.

Field equations

By following (MOW 15) we have a comparison of the field equations. Assume without loss of generality that k 0=1k^0=1. We get the following:

The vacuum Einstein equations for the metric g μνg_{\mu\nu} are R=0R=0 (where RR is the Ricci curvature), which reduce to

R 0 0 =12∇ 2ϕ =0 R 0 i =12∂ j(∂ j(ϕk i)−∂ i(ϕk j)) =0 R j i =12∂ l(∂ i(ϕk lk j)+∂ j(ϕk lk i)−∂ l(ϕk ik j)) =0\begin{aligned} R^0_{\;0} &= \frac{1}{2}\nabla^2\phi && = 0 \\ R^i_{\;0} &= \frac{1}{2}\partial_j \left(\partial^j(\phi k^i)-\partial^i(\phi k^j)\right) && =0 \\ R^i_{\;j} &= \frac{1}{2}\partial_l \left(\partial^i(\phi k^l k^j)+\partial_j(\phi k^l k^i)-\partial^l(\phi k^i k_j)\right) && =0 \end{aligned}
The Maxwell equations for the gauge field AA are dF=0\mathrm{d} F = 0, which reduce to

(dF) 0 =∇ 2ϕ =0 (dF) i =∂ j(∂ j(ϕk i)−∂ i(ϕk j)) =0\begin{aligned} (\mathrm{d}F)^0 &= \nabla^2\phi && = 0 \\ (\mathrm{d}F)^i &= \partial_j \left(\partial^j(\phi k^i)-\partial^i(\phi k^j)\right) && = 0 \end{aligned}
The Klein-Gordon equation for the scalar field Φ\Phi are

∇ 2Φ=0\nabla^2\Phi = 0

Outlook

Summarizing, we have the following table:

zeroth copy	single copy	double copy
Φ=(c aT a)⊗(c˜ bT˜ b)ϕ\;\Phi = (c^a \mathbf{T}_a)\otimes(\tilde{c}^b \tilde{\mathbf{T}}_b)\phi\;	A μ=(c aT a)ϕk μ\;A_{\mu} = (c^a \mathbf{T}_a) \phi k_\mu\;	g μν=η μν+κϕk μk ν\;g_{\mu\nu} = \eta_{\mu\nu} + \kappa \phi k_{\mu}k_{\nu}\;

Examples

Examples of classical double copy of gauge fields:

gauge theory solution	gravity solution	ref.
electric monopole	Schwarzschild spacetime	(MOW 15)
magnetic monopole	massless Taub-NUT spacetime	(LMOW 15, Kim 24b)
self-dual dyon	self-dual Taub-NUT space	(Kim 24a)
planar wave	pp-wave	(MOW 15)
planar shockwave	Aichelburg-Sexl shockwave	(BSW 20)

Double copy and topology

From (LMOW 15) we know that in terms of charges we have the following correspondence:

gauge theory solution	gravity solution
electric charge	mass
magnetic charge	NUT charge

The topological consequences were explored by (AWW 20):

A magnetic monopole is geometrically a principal bundle of the form

ℝ 1⏟worldline×(ℝ 3−{0})⏟transverse space≃ diffℝ 1⏟worldline×ℝ +⏟radial dir.×S 2⏟angular dir.⟶BU(1)\underset{ \text{worldline} }\underbrace{\mathbb{R}^{1}} \times \underset{ \text{transverse space} }\underbrace{ (\mathbb{R}^3-\{0\}) } \;\,\simeq_{\mathrm{diff}}\;\, \underset{ \text{worldline} }\underbrace{\mathbb{R}^{1}} \times \underset{ \text{radial dir.} }\underbrace{\mathbb{R}^+} \times \underset{ \text{angular dir.} }\underbrace{S^2} \longrightarrow B U(1)

which is trivial only on the worldline ℝ 1\mathbb{R}^{1} of the monopole. Therefore, since we have the homotopy ℝ 3−{0}≃S 2\mathbb{R}^3-\{0\} \simeq S^2, the first Chern class of the bundle will be an element [F]∈H 2(S 2,ℤ)≅ℤ[F] \in H^2(S^2,\mathbb{Z})\cong \mathbb{Z}. In other words we have

[F]=g˜[Vol S 2] [F] = \tilde{g} [\mathrm{Vol}_{S^2}]

where Vol S 2\mathrm{Vol}_{S^2} is the volume form of S 2S^2 and g˜∈ℤ\tilde{g}\in\mathbb{Z} is the quantized magnetic charge.
The massless Taub-NUT spacetime with NUT charge NN is a circle bundle too. In fact it is diffeomorphic to the manifold ℝ +×L(1,N)\mathbb{R}^+\times L(1,N), where ×L(1,N)\times L(1,N) is the 33-dimensional Lens space with quantized first Chern class N∈ℤ≅H 2(S 2,ℤ)N\in\mathbb{Z}\cong H^2(S^2,\mathbb{Z}). In this case the S 1S^1 fiber has the interpretation of time direction, which is periodic and non-trivially fibrated on the sphere S 2S^2 of the angular directions.

Therefore the double copy procedure exchange the first Chern class of the magnetic monopole with the one of Taub-NUT spacetime, i.e.

g˜↦N. \tilde{g} \mapsto N.

Double copy and Wilson lines

The classical double copy of Wilson lines was introduced by (AWW 20). We can use as gravitational Wilson lines on spacetime MM the action functional e iS kin:[S 1,M]→U(1)e^{i S_{\mathrm{kin}}}: [S^1, M]\rightarrow U(1) of a test particle. For any loop γ∈[S 1,M]\gamma\in[S^1, M] we can then write

W grav(γ)=e iS kin(γ)=exp(im∫ γds(g μνdx μdsdx νds) 12)∈U(1) W_{\mathrm{grav}}(\gamma) = e^{i S_{\mathrm{kin}}}(\gamma) = \exp \left(i m \int_\gamma \mathrm{d}s \left(g_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s}\right)^{\frac{1}{2}} \right) \;\in\, U(1)

If we assume that the metric is of the form g μν=η μν+κh μνg_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}, we can expand W grav(γ)W_{\mathrm{grav}}(\gamma) at first order in κ\kappa and obtain

W grav(γ)=exp(iκ2∫ γdsh μνdx μdsdx νds)∈U(1) W_{\mathrm{grav}}(\gamma) = \exp \left(\frac{i \kappa}{2} \int_\gamma \mathrm{d}s\, h_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s} \right) \;\in\, U(1)

where the mass mm is absorbed into the new parameter ss. If now we write the holonomy of the single copy gauge field along the same path γ\gamma we get

W gauge(γ)=𝒫exp(ig∫ γdsA μ adx μdsT a)∈G W_{\mathrm{gauge}}(\gamma) = \mathcal{P}\exp \left(i g \int_\gamma \mathrm{d}s\, A_{\mu}^a \frac{\mathrm{d}x^\mu}{\mathrm{d}s}\mathbf{T}_a \right) \;\in\, G

Thus we immediately see that the double copy rules for a Wilson line are the following:

T a↦dx νds,g↦κ2 \mathbf{T}_a \mapsto \frac{\mathrm{d}x^\nu}{\mathrm{d}s}, \quad\; g \mapsto \frac{\kappa}{2}

Notice that they precisely mirror the BCJ prescription of double copy for scattering amplitudes by exchanging color data with kinematic data and gauge coupling constant with its gravitational analogue.

This suggests that this formulation can be a bridge to formally connect classical double copy with double copy for scattering amplitudes.

Double copy and S-duality

In (ABSP 19) it was proved that an electric-magnetic duality (i.e. S-duality) transformation on the single copy gauge fields corresponds to an Ehlers transformation on the double copy gravitational field. In other words the following ideal diagram commutes:

electricmonopole →doublecopy Schwarzschildblackhole S−duality↓ ↓ Ehlerstransformation magneticmonopole →doublecopy NUT−chargedspacetime\array{{electric \; monopole} & \overset{{\;\; double \; copy \;\;}}{\to} & {Schwarzschild \; black \; hole}\\ & \\ ^{{S-duality}}\downarrow && \downarrow^{{Ehlers \; transformation}}\\ & \\ {magnetic \; monopole}& \underset{{\;\; double \; copy \;\;}}{\to} & {NUT-charged \; spacetime}}

KLT relations
Schwarzschild spacetime, Taub-NUT space
duality in physics
string theory results applied elsewhere, open/closed string duality
magic pyramid, magic supergravity
Kaluza-Klein mechanism, Kaluza-Klein monopole
effective QFT incarnations of open/closed string duality,

relating (super-)gravity to (super-)Yang-Mills theory:

References

Fundamental bibliography:

Ricardo Monteiro, Donal O’Connell, Chris D. White, Black holes and the double copy (arXiv:1410.0239)
Andrés Luna, Ricardo Monteiro, Donal O’Connell, Chris D. White, The classical double copy for Taub-NUT spacetime (arXiv:1507.01869)
Chris D. White, The double copy: gravity from gluons (arXiv:1708.07056)
David Berman, Erick Chacón, Andrés Luna, Chris D. White, The self-dual classical double copy, and the Eguchi-Hanson instanton (arXiv:1809.04063)
Kwangeon Kim, Kanghoon Lee, Ricardo Monteiro, Isobel Nicholson, David Peinador Veiga, The Classical Double Copy of a Point Charge (arXiv:1912.02177)
Nadia Bahjat-Abbas, Ricardo Stark-Muchão, Chris D. White, Monopoles, shockwaves and the classical double copy (arXiv:2001.09918)
Joon-Hwi Kim, Single Kerr-Schild Metric for Taub-NUT Instanton (arXiv:2405.09518)
Joon-Hwi Kim, Newman-Janis Algorithm from Taub-NUT Instantons (arXiv:2412.19611)

Foundational issues:

Chris D. White, A Twistorial Foundation for the Classical Double Copy (arXiv:2012.02479)

Some global aspects of the classical double copy were explored in the following paper:

Luigi Alfonsi, Chris D. White, Sam Wikeley, Topology and Wilson lines: global aspects of the double copy (arXiv:2004.07181)

In the following paper it is shown that a S-duality on a gauge field corresponds to an Ehlers transformation on its double copy:

Rashid Alawadhi, David Berman, Bill Spence, David Peinador Veiga, S-duality and the Double Copy (arXiv:1911.06797)

The following paper is a proposal of extension of classical double copy to double field theory:

Kanghoon Lee, Kerr-Schild Double Field Theory and Classical Double Copy (arXiv:1807.08443)