Double Kerr–Schild in Multigravity

• Double Kerr–Schild is a metric ansatz that deforms a background metric using two independent null terms, ensuring linear behavior in its curvature expressions.
• It enables the promotion of Einstein solutions, such as Taub–NUT–Kerr–(A)dS and Plebański–Demiański metrics, to multigravity settings with proportional ansatz.
• The inherent linearity of the ansatz simplifies the evaluation of matrix square roots in the interaction potentials and aids in classical double-copy constructions.

Double Kerr–Schild denotes a metric ansatz in which a background metric is deformed by two rank-one null terms rather than one. In one standard form,

$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu, \qquad l^2=0,\quad f^2=0,\quad l\cdot f=0$$

with respect to the background $\bar g_{\mu\nu}$; in multigravity it also appears as a proportional multi-metric ansatz in which all spin-2 fields share the same double Kerr–Schild directions and differ only by overall constants and profile functions. Recent work uses this structure to lift Taub–NUT–Kerr–$(A)dS$ and Plebański–Demiański solutions to multigravity, and to construct associated classical double copies (García-Compeán et al., 18 Feb 2026). A closely related formulation studies double Kerr–Schild spacetimes on maximally symmetric backgrounds with two null, geodesic, shear-free, mutually orthogonal congruences, especially for Kerr–Taub–NUT–$(A)dS$ (Farnsworth et al., 2023).

1. Definitions and null congruences

The single Kerr–Schild ansatz used in multigravity is

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$

with inverse

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$

The double Kerr–Schild generalization is

$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$

with

$l^2=0,\quad f^2=0,\quad l\cdot f=0,$

and inverse

$$g^{\mu\nu}=\bar g^{\mu\nu}-2S\,l^\mu l^\nu-2Q\,f^\mu f^\nu.$$

Thus the background is deformed by two independent null directions $l_\mu,f_\mu$ with two profiles $\bar g_{\mu\nu}$0 (García-Compeán et al., 18 Feb 2026).

A second formulation writes

$\bar g_{\mu\nu}$1

with $\bar g_{\mu\nu}$2 a maximally symmetric background, $\bar g_{\mu\nu}$3 and $\bar g_{\mu\nu}$4 null, geodesic, and shear-free with respect to $\bar g_{\mu\nu}$5, and mutually orthogonal,

$\bar g_{\mu\nu}$6

In that setting, the orthogonality condition forces them to work in Kleinian signature $\bar g_{\mu\nu}$7 (or complexified metrics), since two real, distinct null vectors in Lorentzian signature cannot be everywhere orthogonal (Farnsworth et al., 2023).

2. Algebraic structure and linearization

The defining technical advantage of double Kerr–Schild is that the inverse metric remains purely linear in the profile functions. In multigravity this linearity propagates to the matrix square roots entering the dRGT-type interaction potential. For the proportional ansatz,

$\bar g_{\mu\nu}$8

the powers of the square-root matrix satisfy

$\bar g_{\mu\nu}$9

Since the dRGT potential is a polynomial in $(A)dS$0, all interaction tensors remain linear combinations of $(A)dS$1, $(A)dS$2, and $(A)dS$3 (García-Compeán et al., 18 Feb 2026).

The same linearizing feature appears at the curvature level. For proportional double Kerr–Schild metrics with conserved matter, the multigravity equations collapse to

$(A)dS$4

or equivalently

$(A)dS$5

Because of the double Kerr–Schild structure, the Ricci tensor and Einstein tensor are linear in the profiles; schematically,

$(A)dS$6

This linearity is central both for exact solution generation and for double-copy constructions (García-Compeán et al., 18 Feb 2026).

In the Newman–Penrose formulation, each null, geodesic, shear-free congruence is encoded by a scalar function $(A)dS$7 obeying Kerr-like nonlinear PDEs; for Kerr–Taub–NUT–$(A)dS$8 these imply flat-space harmonicity of the relevant $(A)dS$9-functions. That harmonicity is what permits the construction of self-dual Maxwell fields on the background (Farnsworth et al., 2023).

3. Proportional double Kerr–Schild in multigravity

In multigravity the proportional double Kerr–Schild ansatz is

$(A)dS$0

with the same background $(A)dS$1 and the same null directions $(A)dS$2 in every sector. “Proportional” means that each metric differs from a single seed double Kerr–Schild metric only by a constant conformal factor and possibly by sector-dependent profiles $(A)dS$3, but all share the same background and null directions (García-Compeán et al., 18 Feb 2026).

For sector $(A)dS$4, the field equations initially contain aligned $(A)dS$5 and $(A)dS$6 interaction terms. Imposing covariant conservation $(A)dS$7 and using the Bianchi identities requires

$(A)dS$8

so all the Kerr–Schild-aligned off–cosmological-constant pieces vanish. The equations then reduce to Einstein equations with effective cosmological constants,

$(A)dS$9

Thus, for proportional double Kerr–Schild metrics with conserved matter, each sector obeys Einstein equations with a cosmological constant; the interaction potential fixes the $$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$0 and relates them to the coupling parameters and conformal factors (García-Compeán et al., 18 Feb 2026).

This reduction gives a direct lifting strategy. One starts with a general-relativistic double Kerr–Schild Einstein solution on a constant-curvature background, promotes it to proportional metrics,

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$1

and then chooses the multigravity parameters so that $$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$2 and the algebraic $$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$3-relations are satisfied. Any GR double Kerr–Schild Einstein metric with constant Ricci scalar can then be lifted to a multi-metric proportional solution (García-Compeán et al., 18 Feb 2026).

4. Exact solution families

Two principal double Kerr–Schild families are promoted from general relativity to multigravity. The first is the Taub–NUT–Kerr–$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$4 class, written in double Kerr–Schild form as

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$5

with background

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$6

and null vectors

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$7

in $$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$8 coordinates. Its multigravity extension is

$$(g_j)_{\mu\nu}=C^2(j)\,\big[\bar g_{\mu\nu}+2S(j)\,l_\mu l_\nu\big], \qquad l^2=\bar g^{\mu\nu}l_\mu l_\nu=0,$$9

with $$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$0 and a common cosmological constant fixed algebraically (García-Compeán et al., 18 Feb 2026).

The second is the Plebański–Demiański family, whose double Kerr–Schild form is

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$1

Its multigravity extension is

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$2

again with $$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$3. Each metric carries its own Plebański–Demiański parameters $$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$4 (García-Compeán et al., 18 Feb 2026).

Both families have constant Ricci scalar and are Einstein spaces with

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$5

Horizons and singularities are inherited from the underlying Taub–NUT/Plebański–Demiański geometry; the multigravity extension keeps the same causal structure in each sector up to constant scalings (García-Compeán et al., 18 Feb 2026).

5. Double-copy and Newman–Penrose constructions

For proportional double Kerr–Schild solutions in multigravity, the single-copy gauge fields and zeroth-copy scalars are obtained by replacing each spin-2 sector with fields on the common background. For Multi–Taub–NUT,

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$6

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$7

For Multi–Plebański–Demiański,

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$8

$$(g_j)^{\mu\nu}=\frac{1}{C^2(j)}\Big[\bar g^{\mu\nu}-2S(j)\,l^\mu l^\nu\Big].$$9

For vacuum Einstein double-KS seeds with constant $$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$0, the sources vanish and the single and zero copies satisfy

$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$1

with common mass

$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$2

The single-copy sector is identified as a quadratic $$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$3 Proca theory, and the zero-copy sector as a quadratic multi-scalar theory with $$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$4 symmetry (García-Compeán et al., 18 Feb 2026).

A distinct but related construction extends the Newman–Penrose map to double Kerr–Schild spacetimes. There the gauge potential is built from two spin-raising operators,

$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$5

and for Kerr–Taub–NUT–$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$6 the real part of the resulting complex self-dual gauge potential matches the Kerr–Schild classical double copy up to an overall constant factor of $$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$7 and pure gauge terms. The field strength is self-dual and obeys the vacuum Maxwell equations on the maximally symmetric background; the corresponding field also exhibits a discrete electric–magnetic duality acting on the charge pair $$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$8, distinct from Hodge self-duality (Farnsworth et al., 2023).

These two constructions emphasize different aspects of the same class of geometries. The proportional multigravity double copy exploits the linearity of the interaction tensors and the constant-curvature reduction to Einstein sectors, whereas the Newman–Penrose map exploits harmonic scalar functions associated with each congruence and a self-dual Maxwell field. For Kerr–Taub–NUT–$$g_{\mu\nu}=\bar g_{\mu\nu}+2S\,l_\mu l_\nu+2Q\,f_\mu f_\nu,$$9, the two descriptions are explicitly compatible (García-Compeán et al., 18 Feb 2026).

6. Terminology, variants, and open problems

The phrase “double Kerr–Schild” is not used uniformly across the literature. In some works it denotes the genuine two-term metric ansatz with two null directions; in other works the word “double” refers to the double copy rather than to a two-congruence metric. The distinction is substantive rather than stylistic.

Work	Meaning of “double”
(García-Compeán et al., 18 Feb 2026)	Two null Kerr–Schild directions; also proportional multi-metric ansatz
(Farnsworth et al., 2023)	Two null, geodesic, shear-free, mutually orthogonal congruences
(Alkac et al., 2021)	Not two Kerr–Schild terms; standard single KS in curved backgrounds
(Berman et al., 2020)	Double copy, not a multi- or double-Kerr–Schild metric
(Lee, 2018)	DFT generalized KS with a pair of null generalized vectors

A related generalization is the “double-extended Kerr–Schild” structure in five-dimensional Einstein–Maxwell–Chern–Simons theory,

$l^2=0,\quad f^2=0,\quad l\cdot f=0,$0

which extends the ordinary double Kerr–Schild form by terms linear in a unit vector $l^2=0,\quad f^2=0,\quad l\cdot f=0,$1. In that class, fulfillment of Einstein equations constrains the Chern–Simons coupling through

$l^2=0,\quad f^2=0,\quad l\cdot f=0,$2

a value determined by the trace of the energy-momentum tensor of the electromagnetic configuration (Arcodía et al., 2021).

Several limitations are explicitly noted in the recent multigravity treatment. The analysis restricts to proportional branches, in which all metrics share the same background and Kerr–Schild directions. Non-proportional double Kerr–Schild solutions are left for future work. Stability analysis and cosmological implications of the multi–double-KS solutions are not addressed. Extending the Weyl double copy to these multigravity double Kerr–Schild solutions is proposed as an interesting direction, especially for Petrov type D and type N wave solutions (García-Compeán et al., 18 Feb 2026).

Within the Newman–Penrose framework, the extension is demonstrated explicitly for Kerr–Taub–NUT–$l^2=0,\quad f^2=0,\quad l\cdot f=0,$3, but not proved for all double Kerr–Schild spacetimes. The authors state that they do not prove the harmonicity of the relevant $l^2=0,\quad f^2=0,\quad l\cdot f=0,$4-functions for arbitrary double Kerr–Schild metrics, and identify generic double Kerr–Schild spacetimes and their twistor description as open problems (Farnsworth et al., 2023).

Double Kerr–Schild therefore occupies a precise position in current research. It is a special metric ansatz with two null congruences, a multigravity solution-generating mechanism, and a natural setting for several classical double-copy constructions. At the same time, the topic remains sharply bounded by background choice, proportionality assumptions, and the unresolved status of more general multi-congruence extensions.