Double-Copy Replica Approach

• Double-copy Replica Approach is a method that constructs gravitational fields by squaring gauge theory data through precise color–kinematics and algebraic duality.
• It employs techniques such as the Kerr–Schild ansatz and Weyl double copy to map radiative and source terms between gauge and gravity, ensuring consistency from classical amplitudes to quantum loops.
• Integrating homotopy L∞ frameworks, the approach enables accurate modeling of scattering, radiation, and quantum corrections across diverse gravitational scenarios.

The double-copy replica approach is a family of methodologies that systematically relate gauge theories (typically Yang–Mills) and gravity (including general relativity and their classical solutions) by “squaring” gauge-theoretic data to produce gravitational field content. Born out of insights from perturbative scattering amplitudes, the approach extends through Lagrangian, algebraic, and geometric formulations, and it reveals a deep underlying color–kinematics (and, more generally, algebraic) duality between the two kinds of theories. This unification is expressed at multiple structural levels—including source terms for classical field equations, curvature tensors, BRST/BV quantized actions, and even at the level of exact amplitudes and integrable model data—allowing profound cross-pollination of techniques and solutions between gauge and gravitational physics.

1. Key Principles: Kerr–Schild, Weyl, and Homotopy Algebra Frameworks

The foundational insight of the double-copy replica approach is that gravitational fields and associated observables can be constructed from a “single copy” gauge theory via structure-preserving maps. In classical situations, the Kerr–Schild ansatz introduces a reduction of the Einstein field equations for certain exact solutions:

$$g_{\mu\nu} = \eta_{\mu\nu} - (\kappa^2/2)\, \phi\, k_\mu k_\nu,$$

with the scalar function $\phi$ (encoding mass/radiation profile) and a null geodesic vector $k_\mu$ constructed from the source's worldline dynamics. The gauge-theory single copy is

$A_\mu = g\, \phi\, k_\mu,$

corresponding to a “square root” of the gravitational solution, and transferring the tensor (or spinor) structure from gravity to gauge theory.

In the spinorial Weyl double copy, the gravitational curvature spinor $C_{ABCD}$ is equated to a quadratic in the Maxwell curvature spinor $f_{AB}$, scaled by a “zeroth copy” scalar: $C_{ABCD} = \frac{1}{S} f_{(AB} f_{CD)}.$ The compatibility of these constructions with the classical and BCJ double copy for amplitudes (Luna et al., 2016, Luna et al., 2018) reveals a canonical procedure where the color algebra of the gauge theory is replaced with an additional kinematic structure that is isomorphic in a strong sense.

From a mathematical perspective, the double-copy replica approach can be formulated at the Lagrangian level using homotopy algebra. The full BRST and auxiliary field content of the gauge theory is strictified to yield a cyclic $L_\infty$-algebra, which factorizes as

$$L_{\text{YM}}^{st} \simeq \mathfrak{g} \otimes (\mathcal{K}^{st} \otimes_\tau \mathfrak{S}),$$

with $\mathfrak{g}$ the color Lie algebra and $\mathcal{K}^{st}$ the (off-shell) kinematic algebra. The double copy replaces $\mathfrak{g}$ by a second copy of $\mathcal{K}^{st}$ (Borsten et al., 2021), thereby lifting the double-copy structure to the quantum and loop levels using homotopy transfer.

2. Classical Double Copy with Accelerating/Radiating Sources

The double-copy prescription maintains validity in dynamically radiating settings, such as accelerating black holes or point particles. The Kerr–Schild construction for an accelerating source yields a generalized Schwarzschild solution. The scalar $\phi$ is promoted to depend on proper time via the source's worldline $y(\tau)$, and the null vector field $k_\mu(x)$ is constructed in retarded coordinates: $$k_\mu(x) = \frac{x - y(\tau)}{r},\quad r = \lambda \cdot (x - y),$$ with $\lambda_\mu = dy_\mu / d\tau$ the four-velocity.

Plugging the ansatz into Einstein’s equations, the emergent stress-energy tensor reflects radiative content (the “Bremsstrahlung” field):

$$T_{\mathrm{KS}}^{\mu\nu} = \frac{3M}{4\pi} \frac{k \cdot \dot{\lambda}}{r^2}\, k^\mu k^\nu,$$

where $\dot{\lambda} = d\lambda/d\tau$. The gauge-theory single copy yields an analogous radiative current: $$j_{\mathrm{KS}}^\mu = \frac{2g}{4\pi} \frac{k \cdot \dot{\lambda}}{r^2}\, k^\mu.$$

In momentum space, for Bremsstrahlung radiation (as in a sudden velocity change $u^\mu \to u^{\prime\mu}$), the gauge current Fourier transforms to: $$\tilde{j}_{\mathrm{KS}}^\nu(k) = - ig \left( \frac{u^{\prime\nu}}{u' \cdot k} - \frac{u^\nu}{u \cdot k} \right),$$ producing an amplitude on contraction with a polarization vector $\epsilon^\nu(k)$: $$\mathcal{A}_{\text{gauge}} = - i g \left( \frac{\epsilon \cdot u'}{u' \cdot k} - \frac{\epsilon \cdot u}{u \cdot k} \right).$$ The gravitational amplitude (upon “squaring”): $$\tilde{T}_{\mathrm{KS}}^{\mu\nu}(k) = - i M \left( \frac{u^{\prime\mu} u^{\prime\nu}}{u' \cdot k} - \frac{u^\mu u^\nu}{u \cdot k} \right),$$ and

$$\mathcal{A}_{\mathrm{grav}} = - i M \left( \frac{(\epsilon \cdot u')^2}{u' \cdot k} - \frac{(\epsilon \cdot u)^2}{u \cdot k} \right).$$

This is fully consistent with the classical double copy as an avatar of the BCJ procedure at amplitude level (Luna et al., 2016).

3. Weyl Double Copy and Type D Spacetimes

The Weyl double copy operates at the level of curvature invariants, especially in algebraically special spacetimes. For Petrov type D solutions, the Weyl spinor admits a factorization: $C_{ABCD} = \frac{1}{S} f_{(AB} f_{CD)},$ where $f_{AB}$ is the (complexified) Maxwell field strength spinor, and $S$ is a scalar field solving the (possibly curved) massless wave equation. This is uniquely determined by the algebraic structure of the spacetime (i.e., the existence of a rank-2 Killing spinor).

For the C-metric, which describes uniformly accelerating black holes, the Weyl double copy unequivocally maps the gravitational solution to a pair of Liènard–Wiechert fields for accelerating charges. This is achieved through a double Kerr–Schild form and suitable scalings. The resulting gauge potential matches the Liènard–Wiechert form up to a gauge transformation, demonstrating a covariant, geometric double-copy structure for radiating, non-stationary, and algebraically general (in the bulk) backgrounds (Luna et al., 2018).

Further, for self-dual spacetimes (e.g., Eguchi–Hanson instanton), both “pure” and “mixed” Weyl double copy formulations are possible, by decomposing the Weyl and Maxwell spinors across different cohomology representations or with different principal null directions.

4. BRST/L∞ Algebraic Double Copy and Quantum Loops

The replica approach, using the homotopy ($L_\infty$) algebraic framework, goes beyond on-shell amplitudes or classical solutions. By strictifying the BRST-extended Yang–Mills Lagrangian (which includes an infinite tower of auxiliary fields resolving contact terms into cubic vertices), the action is reorganized to manifest color–kinematics duality off-shell:

$$L_{\text{YM}}^{st} \simeq \mathfrak{g} \otimes (\mathcal{K}^{st} \otimes_\tau \mathfrak{S}),$$

where $\mathfrak{g}$ is the color Lie algebra, $\mathcal{K}^{st}$ encodes the off-shell kinematics, and $\mathfrak{S}$ encodes the cubic “skeleton.”

The double-copy is then achieved by the purely algebraic step: $$L_{\text{SUGRA}}^{st} = \mathcal{K}^{st} \otimes_\tau (\mathcal{K}^{st} \otimes_\tau \mathfrak{S}),$$ and similarly for the BRST operator. This construction is functorial in $L_\infty$-algebras, implying that once a tree-level color–kinematics duality exists, the structure carries to loop orders as well (Borsten et al., 2021). Minimal models (encoding amplitudes) are obtained via quasi-isomorphism, so the double-copy persists naturally in quantum regimes.

5. Applications: Scattering, Radiation, and Source Terms

The methodology has direct consequences for the computation of radiative processes, gravitational waveforms, and classical scattering:

• Radiation from acceleration (Bremsstrahlung) is encoded fully in the extra source/stress–energy term in the Kerr–Schild solution, mapping to a gauge-theory current under the double copy.
• Upon Fourier transform, classical expressions for radiated power, angular distributions, and deflection angles recover their BCJ amplitude equivalents; the gravitational observables are directly the “squares” (at the level of numerators) of the gauge-theory expressions.
• The classical double copy at the level of sources (currents and stress–energy tensors) is structurally exact, and the correspondences persist when the nontrivial stress–energy is due strictly to radiation (non-stationary, time-dependent sources).

A summary table relating key structural elements:

Gravity (Kerr–Schild)	Gauge Theory (Single Copy)	Mapping/Prescription
$h_{\mu\nu} = -(\kappa/2) \phi k_\mu k_\nu$	$A_\mu = g \phi k_\mu$	Strip one $k_\mu$
$T_{\text{KS}}^{\mu\nu}$	$j_{\text{KS}}^\mu$	Replace $M \to g$, $k^\mu \to k^\mu k^\nu$
$\tilde{T}^{\mu\nu}(k)$	$\tilde{j}^\nu(k)$	Numerator “squaring”
Gravitational amplitude	Gauge amplitude	$A_{\text{grav}} = (A_{\text{gauge}})^2$

6. Extensions and Generalizations

The double-copy replica approach is highly extensible:

• Algebraic general Petrov type spacetimes (not merely type D): Weyl double copy and its asymptotic and local variants extend the scope of double-copy mapping into time-dependent and algebraically general regions.
• Loop-level validity, effective field theories, and integrability: The $L_\infty$ algebraic construction ensures persistence to quantum corrections, higher-derivative effective actions, and even allows mapping between integrability structures (Lax pairs, Wilson lines) under the double copy.
• Matter sources: Inclusion of additional fields (e.g., dilaton, axion, higher spins) and nontrivial backgrounds is natural in the replica framework.
• Cohomology and twistor formulations: Alternative formalisms (e.g., twistor space, Dolbeault/Čech cohomology) underlie rigorous, unambiguous double-copy representations, emphasizing the utility of harmonic representatives for uniqueness.

7. Impact and Implications

The double-copy replica approach demonstrates, for a wide class of gravitational phenomena—including time-dependent and radiative classical solutions—a structurally precise correspondence to gauge theory sourced solutions and amplitudes. It provides rigorous algebraic and geometric frameworks (Kerr–Schild, spinor/twistor, and $L_\infty$-algebraic) for deriving and classifying solutions, enables the mapping of field content and sources, preserves structure at the quantum/loop level, and unifies computational and conceptual tools across gauge and gravitational settings. The approach also suggests new lines of research into generalized color–kinematics dualities, new bi-colored scalar theories, extensions to curved backgrounds, and connections to string theory and holography.

References: (Luna et al., 2016, Luna et al., 2018, Borsten et al., 2021)