Proportional Kerr–Schild Metrics

• Proportional Kerr–Schild metrics are defined as a background geometry plus a null rank-one deformation with a fixed proportionality factor, preserving relations across sectors.
• They enable simplification of Lovelock, quadratic-curvature, and multigravity field equations by truncating nonlinearities and yielding exact linearizations under unique-vacuum or specific symmetry conditions.
• Applications include unique-vacuum Lovelock black holes, AdS-wave geometries, and double-copy constructions that extend classical Kerr–Schild solutions to diverse spin-2 fields.

Proportional Kerr–Schild denotes a class of Kerr–Schild deformations in which a metric, or a collection of metrics, is written as a background geometry plus a null rank-one deformation carrying either a free proportionality parameter or a profile constrained to preserve a proportional relation among sectors. In the simplest Lovelock form one writes

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$

with $\bar g_{ab}$ a fixed background solution, $\lambda$ a real Kerr–Schild parameter, and $k^a$ null with respect to the background (Ett et al., 2011). In multigravity the same idea appears as a family of metrics sharing a common null direction and differing by constant conformal factors, for example

$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$

in bigravity (García-Compeán et al., 18 Feb 2026). Closely related constructions also occur in higher-curvature gravity, where Kerr–Schild–Kundt metrics exhibit an exact proportionality between the full and linearized field equations, and in separability-preserving deformations of Benenti–Francaviglia metrics, where a proportional replacement of a single structure function keeps the metric within the same degenerate class (Gurses et al., 2012, Nozawa et al., 8 Oct 2025).

1. Algebraic definition and canonical ansätze

The common algebraic core is a background metric plus a null-square deformation. In the Lovelock setting, the proportional Kerr–Schild ansatz is

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$

with $\bar g_{ab}k^a k^b=0$, inverse metric

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$

and $k^a$ taken geodesic in the background,

$k^b\bar\nabla_b k^a=\alpha(x)\,k^a$

(Ett et al., 2011). In the Benenti–Francaviglia construction one instead uses

$\bar g_{ab}$0

or equivalently $\bar g_{ab}$1, with $\bar g_{ab}$2; because $\bar g_{ab}$3 is null and geodesic with respect to the seed metric, it remains null and affine-geodesic in the full metric (Nozawa et al., 8 Oct 2025). In multigravity the proportional sector is formulated by taking all metrics to share the same background and null field,

$\bar g_{ab}$4

or, in the simplest bigravity case, the pair $\bar g_{ab}$5 above (García-Compeán et al., 18 Feb 2026).

Context	Ansatz	Proportional element
Lovelock gravity	$\bar g_{ab}$6	Free parameter $\bar g_{ab}$7 multiplies the deformation
Bigravity / multigravity	$\bar g_{ab}$8	Constant conformal factors $\bar g_{ab}$9 and shared null direction $\lambda$0
Degenerate BF metrics	$\lambda$1	Special case $\lambda$2 gives a proportional deformation

These ansätze are technically useful because the inverse metric remains linear in the deformation, the Riemann tensor truncates at finite order in the Kerr–Schild parameter in Lovelock theory, and the null congruence retains strong integrability properties in the BF setting (Ett et al., 2011, Nozawa et al., 8 Oct 2025). In higher-curvature AdS Kerr–Schild–Kundt geometries, an even stronger simplification occurs: all non-trivial curvature becomes linear in the profile $\lambda$3 (Gurses et al., 2012).

2. Reduction of Lovelock field equations

For Lovelock gravity of maximal order $\lambda$4, the proportional Kerr–Schild ansatz leads to a finite expansion of the Riemann tensor,

$\lambda$5

with all higher orders vanishing because $\lambda$6 is linear in $\lambda$7 (Ett et al., 2011). The Lovelock tensor correspondingly expands as

$\lambda$8

and in a constant-curvature background the coefficients with $\lambda$9 collapse, leaving only the $k^a$0 and $k^a$1 terms.

The decisive distinction is between unique-vacuum and non-unique-vacuum theories. In a unique-vacuum theory, where $k^a$2 and

$k^a$3

the double-null contraction $k^a$4 has no contribution beyond order $k^a$5, and its $k^a$6 piece is killed by taking $k^a$7 geodesic. For geodesic $k^a$8, the entire $k^a$9 tensor then vanishes identically by antisymmetrizations, so the full Lovelock equations reduce to a single $$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$0th-order equation,

$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$1

(Ett et al., 2011).

In non-unique-vacuum theories, extra lower-order equations appear. The Gauss–Bonnet case ($$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$2) yields a particularly explicit contrast: after imposing the geodesic condition, one must solve both an order-$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$3 equation, which is the linearized Einstein-tensor condition on $$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$4, and an order-$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$5 equation quadratic in $$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$6. The paper emphasizes that these equations are not obviously compatible (Ett et al., 2011).

Known static, spherically symmetric Lovelock black holes fit this structure. For Gauss–Bonnet in the unique-vacuum case one has

$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$7

with $$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$8 mass appearing as the free Kerr–Schild parameter multiplying the null-square term. By contrast, in the general two-vacua Gauss–Bonnet case the metric function contains a square root,

$$g_{\mu\nu}=\bar g_{\mu\nu}+H\,k_\mu k_\nu,\qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\widetilde H\,k_\mu k_\nu$$9

so there is no simple overall factor multiplying $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$0 (Ett et al., 2011). A central consequence is that the proportional Kerr–Schild strategy is especially natural in unique-vacuum Lovelock theories and significantly more restrictive otherwise.

3. Exact linearization in quadratic-curvature Kerr–Schild–Kundt metrics

A related but distinct structural result appears in quadratic-curvature gravity on AdS backgrounds. Gürses, Şişman, and Tekin consider the Kerr–Schild form

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$1

with $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$2 null, geodesic, non-expanding, shear-free, and twist-free, together with the gauge condition $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$3 (Gurses et al., 2012). In this class all non-trivial curvature is linear in $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$4. The Ricci tensor takes the form

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$5

while the scalar curvature remains constant,

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$6

(Gurses et al., 2012).

When this ansatz is inserted into the most general quadratic-curvature vacuum action,

$g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$7

all nonlinear terms in $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$8 cancel. The full nonlinear equations reduce to a single fourth-order linear PDE, equivalently a linear operator acting on $g_{ab}=\bar g_{ab}+\lambda\,k_a k_b,$9 (Gurses et al., 2012). For both the AdS-wave (Siklos) and spherical-AdS-wave families, the equation factorizes as

$\bar g_{ab}k^a k^b=0$0

Two explicit solution classes are given. The AdS-wave metric is

$\bar g_{ab}k^a k^b=0$1

while the spherical-AdS-wave metric is

$\bar g_{ab}k^a k^b=0$2

(Gurses et al., 2012). At the critical point $\bar g_{ab}k^a k^b=0$3, the spherical family admits logarithmic modes.

The paper formulates the resulting statement as a proportionality property: because every curvature invariant and every tensor in the quadratic-gravity action is linear in $\bar g_{ab}k^a k^b=0$4 on this Kerr–Schild–Kundt class, the exact field equations coincide with the linearized ones around AdS, and any solution of the linearized Einstein or higher-derivative equations in the profile $\bar g_{ab}k^a k^b=0$5 automatically solves the full nonlinear theory (Gurses et al., 2012). This is not identical to the free-parameter notion of proportional Kerr–Schild in Lovelock gravity, but it is a closely related exact-linearity phenomenon within the Kerr–Schild framework.

4. Proportional Kerr–Schild sectors in bigravity and multigravity

In ghost-free bigravity, the proportional Kerr–Schild ansatz uses a common background $\bar g_{ab}k^a k^b=0$6 and a single null, geodesic vector $\bar g_{ab}k^a k^b=0$7,

$\bar g_{ab}k^a k^b=0$8

with inverse metrics

$\bar g_{ab}k^a k^b=0$9

(García-Compeán et al., 18 Feb 2026). The Hassan–Rosen equations reduce to

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$0

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$1

where $g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$2 are algebraic functions of the interaction coefficients $g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$3 and of $g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$4 (García-Compeán et al., 18 Feb 2026).

Imposing that $g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$5 be null and geodesic with respect to both metrics, together with conservation, forces the off-diagonal terms to vanish: $g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$6 Asymptotic flatness or matching at infinity sets the constant to zero, so

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$7

The two metrics then become Einstein spaces,

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$8

with

$g^{ab}=\bar g^{ab}-\lambda\,k^a k^b,$9

(García-Compeán et al., 18 Feb 2026). The conformal factor is fixed algebraically by

$k^a$0

This sector lifts ordinary single-metric Kerr–Schild Einstein solutions to multimetric ones. The explicit examples listed are multi-Schwarzschild with

$k^a$1

multi-Kerr with

$k^a$2

and multi-Schwarzschild–AdS on a constant-curvature background (García-Compeán et al., 18 Feb 2026). In each case the profile is the same as in the corresponding GR Kerr–Schild solution, while the constants $k^a$3 or $k^a$4 are fixed by the dRGT couplings. The GR limit is $k^a$5, with $k^a$6.

A 2024 bigravity study extends this perspective to a formalism for a proportional generalized double Kerr–Schild ansatz in bigravity with both metrics coupled to matter. It examines time-dependent AdS waves and stationary Plebański–Demiański-type solutions, including configurations with different masses, NUT parameters, electric and magnetic charges, the same kinematical parameters, and related cosmological constants. The solutions are presented in Plebański coordinates, where the classical double copy equations simplify and admit a clearer interpretation in terms of the defined fields; some cases are also interpreted for the separate matter sector and by using the effective metric (García-Compeán et al., 2024).

5. Separability-preserving proportional deformations of Benenti–Francaviglia metrics

The Benenti–Francaviglia family provides a different realization of proportional Kerr–Schild structure. In four dimensions, the degenerate BF seed metric in coordinates $k^a$7 is

$k^a$8

with two commuting Killing vectors $k^a$9, $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$0 and an irreducible Killing tensor arising from Hamilton–Jacobi separability (Nozawa et al., 8 Oct 2025). Null geodesics with $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$1, separation constant $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$2, and $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$3 const lie along

$k^b\bar\nabla_b k^a=\alpha(x)\,k^a$4

and these congruences are shear-free; in the Petrov type D Carter subclass they define the repeated principal null directions of the Weyl tensor (Nozawa et al., 8 Oct 2025).

Requiring the deformed metric to preserve the same commuting Killing vectors and circularity fixes the allowed profile almost completely. The profile can depend only on $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$5, and the unique solution of the linearized Einstein equations, or equivalently of the circularity conditions, is

$k^b\bar\nabla_b k^a=\alpha(x)\,k^a$6

with $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$7 an arbitrary new radial structure function (Nozawa et al., 8 Oct 2025). After the coordinate shifts

$k^b\bar\nabla_b k^a=\alpha(x)\,k^a$8

all $k^b\bar\nabla_b k^a=\alpha(x)\,k^a$9 and $\bar g_{ab}$00 cross terms are removed, and the deformed metric is again a degenerate BF metric of the same form, with the single replacement rule

$\bar g_{ab}$01

(Nozawa et al., 8 Oct 2025).

The proportional case is the special choice $\bar g_{ab}$02. Then

$\bar g_{ab}$03

and

$\bar g_{ab}$04

(Nozawa et al., 8 Oct 2025). The new metric components coincide with those of the BF form with $\bar g_{ab}$05.

The same replacement-rule mechanism extends to five dimensions, under the condition $\bar g_{ab}$06. In that case the null vector remains geodesic but is no longer shear-free, and the allowed profile becomes

$\bar g_{ab}$07

(Nozawa et al., 8 Oct 2025). The 4D and 5D constructions are applied respectively to a dyonic generalization of the Chong–Cvetič–Lü–Pope rotating black hole and to the minimal 5D gauged supergravity black hole of Chong–Cvetič–Lü–Pope.

6. Classical double copy, interpretation, and common restrictions

In the multigravity setting, proportional Kerr–Schild solutions admit a direct classical double-copy interpretation. For each spin-2 field written as

$\bar g_{ab}$08

the single copy and zeroth copy are defined by

$\bar g_{ab}$09

The linearized trace-reversed Ricci equation around $\bar g_{ab}$10 becomes a Proca equation, or Maxwell when $\bar g_{ab}$11, for $\bar g_{ab}$12, and a Klein–Gordon equation for $\bar g_{ab}$13: $\bar g_{ab}$14 with $\bar g_{ab}$15 (García-Compeán et al., 18 Feb 2026). The gauge-theory realization is a $\bar g_{ab}$16 Proca theory on the common background, and the scalar sector is an $\bar g_{ab}$17-invariant multi-scalar theory. In the special case $\bar g_{ab}$18, the equations reduce to Maxwell and Laplace equations (García-Compeán et al., 18 Feb 2026).

The 2024 bigravity study places time-dependent and stationary proportional generalized double Kerr–Schild solutions directly in the Kerr–Schild classical double-copy framework and reports the classical Kerr–Schild for the double, single and zeroth copy equations (García-Compeán et al., 2024). In that work, Plebański coordinates are singled out because they simplify the copy equations for stationary bigravity solutions of Plebański–Demiański type.

Several restrictions emphasized in the literature delimit what proportional Kerr–Schild does and does not imply. First, the simplification is not generic for arbitrary null deformations: geodesicity of the Kerr–Schild vector is essential in Lovelock gravity and in BF constructions (Ett et al., 2011, Nozawa et al., 8 Oct 2025). Second, higher-curvature truncation to a single equation is not universal: in Lovelock gravity it is tied to the unique-vacuum condition, while non-unique-vacuum theories produce additional lower-order equations that may be incompatible (Ett et al., 2011). Third, shear-free behavior is dimension- and ansatz-dependent: the four-dimensional BF congruence is shear-free, whereas in the five-dimensional extension the null vector is geodesic but no longer shear-free (Nozawa et al., 8 Oct 2025). Fourth, in bigravity and multigravity the profile functions are not freely independent once conservation and asymptotic matching are imposed; the off-diagonal sector forces $\bar g_{ab}$19 and, in the asymptotically matched case, $\bar g_{ab}$20 (García-Compeán et al., 18 Feb 2026).

Taken together, these results show that proportional Kerr–Schild is less a single ansatz than a recurrent algebraic mechanism. In Lovelock theory it isolates a finite-order polynomial sector controlled by a free null-square parameter; in quadratic-curvature AdS Kerr–Schild–Kundt geometries it yields exact linearization; in bigravity and multigravity it collapses coupled spin-2 systems to Einstein spaces with algebraically fixed proportional factors; and in BF geometries it preserves hidden symmetries through a one-function replacement rule (Ett et al., 2011, Gurses et al., 2012, García-Compeán et al., 18 Feb 2026, Nozawa et al., 8 Oct 2025).