Kerr-Schild Double Copy

• Kerr-Schild double copy is a classical mapping between gravity and gauge theory that expresses the spacetime metric as a linear deformation by a null, geodesic vector field.
• It relates Einstein’s equations and Yang–Mills theory across flat and curved backgrounds via a decomposed ansatz, enabling the construction of zeroth, single, and double copies.
• The framework extends to higher-spin, multi-Kerr–Schild, and bigravity scenarios, revealing structural insights into both classical and quantum gravity.

The Kerr-Schild double copy is a classical mapping between solutions of gauge theory and gravity, built upon the Kerr–Schild ansatz in which the spacetime metric is expressed as a linear deformation of a background metric by a term involving a null, geodesic vector field. This correspondence, originally observed in the context of scattering amplitudes, has been generalized to a map between exact solutions of Yang–Mills theory and Einstein’s equations across a wide array of spacetimes, including those with curved backgrounds, time-dependence, nontrivial matter content, higher dimensions, and higher-spin or bigravity sectors. The structure of the double copy formalism and its algebraic, geometric, and physical implications have been explored in depth, yielding new insights into the nature of classical and quantum gravity, as well as the limitations imposed by residual symmetry and cohomology structures.

1. Kerr–Schild Ansatz and Classical Double Copy Fundamentals

The Kerr–Schild ansatz posits the full spacetime metric as

$g_{\mu\nu} = \eta_{\mu\nu} + \phi(x)\,k_\mu k_\nu$

with $\eta_{\mu\nu}$ a background metric, $\phi(x)$ a scalar function (the "Kerr–Schild potential"), and $k_\mu$ a null, geodesic vector field (i.e., $k^\mu k_\mu = 0$ and $k^\nu \nabla_\nu k_\mu = 0$).

In the context of the double copy, this gravitational solution is mapped to a gauge field

$A_\mu^a = c^a \phi(x) k_\mu$

where $c^a$ is a color vector. Further, a "zeroth copy" denotes a biadjoint scalar field $\Phi^{aa'} = c^a \tilde{c}^{a'} \phi(x)$.

Linearization of field equations occurs because the Einstein tensor for Kerr–Schild metrics is linear in $\phi$. Thus, both the gravity and gauge equations reduce to linear equations in the respective sectors, enabling the double copy prescription to function straightforwardly for a large class of backgrounds and perturbations.

2. Generalization to Curved Backgrounds: Type A and Type B Double Copies

The extension of the Kerr-Schild double copy to curved spacetime proceeds via two complementary frameworks (Bahjat-Abbas et al., 2017):

• Type A Double Copy: The field is split into background and perturbation components. The background (not necessarily flat) is chosen to possess its own Kerr–Schild structure,

$$g_{\mu\nu} = \hat{g}_{\mu\nu} + \hat{h}_{\mu\nu},\qquad \hat{g}_{\mu\nu} = \eta_{\mu\nu} + \phi_1 k_\mu k_\nu$$

with $h_{\mu\nu} = \phi_2 k_\mu k_\nu$. The associated gauge field also splits as $A_\mu = \bar{A}_\mu + \hat{A}_\mu$, and both sets of background and perturbing fields are related through the customary double copy rules.

• Type B Double Copy: The gravitational perturbation is considered relative to a general curved (not necessarily Kerr–Schild) background, and its single copy is a gauge field defined on the same curved background, with all derivatives and contractions now evaluated with respect to the nontrivial background geometry. For instance, in black hole backgrounds:
  • For Schwarzschild: the single copy is a vacuum Maxwell field on the Schwarzschild background.
  • For de Sitter: the single copy features a source term representing the cosmological constant as a uniform charge density.
  • Type B constructions generally lack a natural zeroth copy—biadjoint scalar equations acquire source terms without gauge-theory interpretation.

These frameworks allow one to relate gravitational solutions in non-Minkowski backgrounds to Yang–Mills and biadjoint scalar fields in a controlled fashion, including for conformally flat backgrounds and multi-Kerr–Schild solutions.

3. Algebraic Structures and Symmetry Content

The symmetry structure of the Kerr–Schild double copy is markedly different in the gauge and gravity sectors (Holton, 28 Sep 2025, Holton, 30 Sep 2025):

• Residual Symmetries in Gauge Theory: The residual gauge transformations preserving the Kerr–Schild form are infinite-dimensional, corresponding e.g. to arbitrary "null functions" along $k^\mu$ (satisfying $k^\mu \partial_\mu \lambda(x) = 0$). In the non-Abelian case, these symmetries realize a current algebra $\mathfrak{g} \otimes C^\infty(\mathbb{R})$.
• Residual Diffeomorphisms in Gravity: Residual diffeomorphisms preserving the Kerr–Schild metric reduce (for the Schwarzschild solution) to the finite-dimensional global isometries ($\mathfrak{so}(3) \oplus \mathbb{R}$).

Including proper conformal Killing vectors (CKVs) for the angular coordinates formally yields an infinite-dimensional algebra of residual transformations (Holton, 30 Sep 2025), but these extra CKV-generated modes are shown to be BRST-exact and thus physically trivial at the quantum level when a Weyl compensator field is included. The physical spectrum, both classically and quantum-mechanically, is thus restricted to the finite-dimensional global isometries, strongly distinguishing the gravitational sector from the gauge sector.

4. Categorical Extensions: Higher Spins, Multi-Kerr–Schild, and Bigravity

Higher Spins and Multicopy Pattern: The double copy pattern extends to all symmetric-gauge (“higher-spin”) fields on (A)dS backgrounds (Didenko et al., 2022): $$\Phi_{\mu_1 \ldots \mu_s} = k_{\mu_1} \cdots k_{\mu_s} \phi$$ satisfies the Fronsdal equation with an appropriate mass-like term precisely tuned through the Kerr–Schild construction. The scalar (zeroth copy) mass is determined so that the multicopy (for all $s$) fits into a spectrum organized by higher-spin symmetry.

Multi-Kerr–Schild and Solutions with Multiple Null Directions: Metrics with sums over multiple, mutually orthogonal KS vectors generalize the procedure to more complicated spacetimes and allow decomposition into background and perturbation sectors, enabling richer double copy structures, including charged and rotating solutions.

Bigravity and Effective Metrics: In bigravity, a generalized double Kerr–Schild ansatz is imposed for both metrics, typically with proportional backgrounds and aligned null vectors. Matter couples to an "effective metric" $$g_{\mu\nu}^{\rm eff} = \alpha g_{\mu\nu} + \beta f_{\mu\nu}$$.

• For stationary solutions, the double copy yields two Maxwell and two massless scalar equations.
• For time-dependent backgrounds, linear interaction terms appear, but can be diagonalized into massless and massive fields, with the massive sector acquiring Fierz–Pauli mass proportional to the parameters of the bigravity potential (García-Compeán et al., 28 Mar 2024, García-Compeán et al., 22 Dec 2024).

5. Generalizations: Curved Backgrounds, Modified Theories, and Scalar-Gauge Matter

Curved backgrounds: The double copy formalism directly extends to curved and conformally flat backgrounds (Alkac et al., 2021). The presence of background curvature introduces an extra "deviation tensor" contributing a constant or spatially uniform source term to the Maxwell (single copy) equations—interpreted as a "cosmological charge density." Notably, a configuration with nontrivial matter (energy-momentum tensor) in gravity may correspond to a vacuum Maxwell solution if the matter and background curvature contributions cancel.

Modified Gravity and Universality Across Theories: In the context of Kundt type N spacetimes, the Kerr–Schild double copy extends universally to higher-order gravity and non-linear electromagnetic theories for null (single copy) fields. The universality is due to the immunity of these classes of spacetimes and fields from higher-order corrections (Ortaggio et al., 2023).

Three Dimensions and Matter Choices: In $d = 3$, the classical double copy yields non-vacuum black hole solutions—BTZ black holes with correct Newtonian limits—by suitably coupling Maxwell or Born–Infeld theories to gravity (possibly with ghost signs to ensure correct behavior), or via scalar field matter couplings. Regularity and horizon structure in the gravity sector correspond via the double copy to regular electric fields and stable orbits in the gauge sector (Alkac et al., 2021, Alkac et al., 2022).

6. Applications, Extensions, and Future Directions

• Black Hole Horizons: The presence and location of marginally trapped surfaces (horizons) in Kerr–Schild spacetimes follow directly from single and zeroth copy data through expansions and their derivatives (Chawla et al., 2023).
• Particle Dynamics: Probe particle (worldline) dynamics, governed by Wong’s equations in the single copy, are mapped to geodesic motion in the full gravitational (double copy) background, with explicit double copy rules for mapping conserved charges and orbital types (bound, unbound, plunge) (Gonzo et al., 2021).
• Solution Generating Techniques and Dualities: The Ehlers transformation and Buchdahl's reciprocal are directly realized as double copies of electromagnetic duality and charge conjugation, showing that a broad class of gravitational dualities have precise gauge-theoretic avatars (Alawadhi, 2023).
• Holonomy and Flow Equations: Holonomy groups and Ricci/Yang–Mills flow equations are related through the double copy, translating geometric evolution and global topology between the gravitational and gauge sectors.
• Limitations and Algebraic Obstacles: The structural mismatch of residual symmetry algebras (see Sections 3–4), demonstrated via explicit metric and PDE analyses, marks a boundary for the efficacy of the double copy at the symmetry level—a realization sharpened and formalized through BRST cohomology analyses (Holton, 28 Sep 2025, Holton, 30 Sep 2025). Infinite-dimensional residual symmetries in gauge theory become BRST-exact and thus nonphysical in gravity, leaving only global isometries.

Domain	Kerr–Schild Double Copy Structure	Notable Comments
Minkowski/Flat	Standard KS ansatz, linearization, full zeroth/single/double copy possible	Full background/residual symmetry on gauge side
Curved backgrounds (Type A/B)	Split into background/perturbation, single copy is gauge field on (possibly curved) background	Zeroth copy problematic in Type B; explicit examples include Schwarzschild, de Sitter
Higher-dimensions/Spins	Multicopy ansatz yields higher-spin Fronsdal fields with masses tuned by curvature	Spectrum matches singleton tensor products
Bigravity/effective metrics	Proportional backgrounds, effective metrics, coupled/double Maxwell and scalar sectors	Decoupling yields massive and massless fields in the double copy
Residual symmetries	Infinite-dimensional on gauge side; only finite isometry algebra in gravity; extra modes BRST-exact	Algebraic mismatch; quantum cohomology restricts to isometries

7. Outlook and Open Problems

The Kerr–Schild double copy framework, augmented by its generalizations to curved backgrounds, higher spin, bigravity, and modified gravity, continues to offer new perspectives on the relationship between classical solutions in gauge and gravitational theories. However, structural obstacles—most notably, the lack of symmetry preservation beyond global isometries—persist as fundamental limitations. Future directions include a more systematic exploration of the interplay between the BRST cohomology and double copy structure, consistency conditions in strongly curved or nontrivial topological backgrounds, and further connections to twistor-inspired and amplitude-based approaches for both classical and quantum gravity.

The universality of the double copy in the Kerr–Schild sector, its ability to encode broad classes of exact solutions and dualities, and its robust (but ultimately symmetry-restricted) map between gauge theory and gravity, mark it as a potent—though not unbounded—tool in the paper of gravitational physics, gauge/gravity duality, and the algebraic foundations of the classical and quantum field theories.