Conformal Killing Gravity Overview

• Conformal Killing Gravity is a modified theory that reformulates third-order field equations into a second-order Einstein-like system using a divergence-free conformal Killing tensor.
• The theory produces rich static solutions—including charged black holes, black bounces, and modified TOV-like equilibrium—highlighting distinct compact-object phenomenology.
• In cosmology, CKG implies an emergent cosmological constant and effective dark energy behavior (ω = -5/3), influencing late-time expansion while preserving early-universe dynamics.

Conformal Killing Gravity (CKG) is a modified gravitational theory in which Harada’s original third-order field equations, written in terms of a totally symmetric rank-3 tensor built from covariant derivatives of the Ricci tensor and an analogous matter tensor, can be rewritten as Einstein equations supplemented by a divergence-free conformal Killing tensor $K_{\mu\nu}$. In that reformulation, the metric equations become second order, the cosmological constant appears as an integration constant, and every Einstein solution remains a solution of the theory (Mantica et al., 2023, Mantica et al., 2024). CKG is therefore distinct both from general relativity and from Weyl gravity: it is not defined by local conformal invariance of the metric dynamics, but by the insertion of conformal-Killing-type geometric structure into the gravitational field equations (Harada, 2023).

1. Definition and geometric framework

In the reformulation due to Mantica and Molinari, the central field equation is

$R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$

with $K_{jk}$ a divergence-free gradient conformal Killing tensor obeying

$$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$

where $K=g^{jk}K_{jk}$ (Mantica et al., 2023). This converts Harada’s third-order system into a second-order Einstein-like system while preserving the statement that all Einstein manifolds solve CKG (Mantica et al., 2024).

A recurrent motivation in the CKG literature is that the cosmological constant emerges as an integration constant rather than as a parameter inserted by hand, that local energy-momentum conservation follows from the gravitational equations, and that conformally flat metrics need not be vacuum solutions (Junior et al., 1 Feb 2025). In static and cosmological sectors this structure is especially tractable because $K_{\mu\nu}$ can be written explicitly in symmetry-adapted form, so that the higher-order content is absorbed into a constrained auxiliary tensor rather than into higher derivatives of the metric (Mantica et al., 2024).

The terminology also sits inside a broader geometric tradition. Conformal Killing vectors, conformal Killing tensors, and conformal Killing–Yano forms are long-established symmetry objects in gravitational theory; in higher-dimensional Einstein gravity, a principal rank-2 closed conformal Killing–Yano tensor generates towers of Killing tensors and Killing vectors and controls integrability and separability in Kerr–NUT–(A)dS spacetimes (Yasui et al., 2011). In the modified-gravity usage, however, “Conformal Killing Gravity” denotes the Harada–Mantica–Molinari theory rather than that broader symmetry framework.

2. Static spherical sector and compact objects

The static spherically symmetric sector is the best-developed part of CKG. Harada’s vacuum solution has metric function

$$A(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^2-\frac{\lambda}{5}r^4,$$

where $M$ is the Schwarzschild mass, $\Lambda$ is an integration constant interpretable as the cosmological constant, and $\lambda$ is a new CKG parameter controlling quartic growth at large $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$0 (Junior et al., 1 Feb 2025). In the Einstein-plus-$R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$1 reformulation, Mantica and Molinari identified an anisotropic divergence-free conformal Killing tensor for static spherical spacetimes and proved equivalence between the known third-order equations and the second-order equations obtained in the conformal Killing parametrization (Mantica et al., 2024).

Coupling to linear electrodynamics yields the Reissner–Nordström-like static solution

$R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$2

while coupling to nonlinear electrodynamics supports broader black-hole families (Chen et al., 4 Jun 2025). In a systematic analysis of charged solutions, three nonlinear-electrodynamic models were constructed. These solutions exhibit up to three horizons, transitions between extreme and non-extreme configurations, central curvature singularities diagnosed by the Kretschmann scalar, and nonlinear Lagrangians that reduce to Maxwell theory as $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$3 (Junior et al., 1 Feb 2025). The same work computed photon propagation in the effective NED metric and reported shadow sizes compatible with EHT constraints for Sgr A* over nontrivial parameter ranges (Junior et al., 1 Feb 2025).

CKG has also been used to construct regular black-bounce geometries. With areal radius $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$4, coupled nonlinear electrodynamics, and scalar fields, the theory admits generalized Simpson–Visser-type and Bardeen-type black bounces, together with explicit NED Lagrangian densities and scalar potentials (Junior et al., 2024). The paper emphasizes that in CKG a Simpson–Visser-type black bounce can be modeled by nonlinear electrodynamics alone, whereas corresponding GR constructions typically require a phantom scalar field (Junior et al., 2024).

For self-gravitating matter, the static reformulation yields an analog of the Tolman–Oppenheimer–Volkoff equation. In the perfect-fluid case,

$R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$5

so equilibrium differs from GR by dark radial-pressure terms sourced by the conformal Killing tensor (Mantica et al., 2024). The same framework supports junction conditions to the Harada vacuum and an extension of Buchdahl’s critical compactness relation for uniform-density spheres (Mantica et al., 2024).

3. Cosmology and the effective dark sector

In homogeneous and isotropic cosmology, CKG admits exact Friedmann-type solutions without inserting dark energy as an independent fluid. Mantica and Molinari showed that Harada’s equations are equivalent to Einstein equations extended by an arbitrary conformal Killing tensor and used this to derive the modified Friedmann equations in a direct way (Mantica et al., 2023). In the Robertson–Walker sector, the conformal-Killing contribution is encoded by a scalar $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$6 satisfying

$R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$7

with $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$8 an integration constant and $R_{jk}-\frac{1}{2}g_{jk}R = T_{jk}+K_{jk},$9 another integration constant fixed by boundary data (Mantica et al., 2023).

A later cosmological analysis writes the exact background evolution as

$K_{jk}$0

where the last term is the distinctive CKG contribution (Harada, 2023). Interpreting it as a conserved effective fluid forces a unique equation-of-state parameter,

$K_{jk}$1

so the extra term behaves as effective dark energy in the phantom regime (Harada, 2023). The paper argues that this component is negligible at high redshift and becomes relevant only at low redshift, so it can modify late-time expansion while leaving early-universe behavior essentially unchanged (Harada, 2023).

This cosmological branch is one of the main reasons CKG has been discussed as a possible response to the cosmological constant problem and to late-time acceleration. Harada’s theory was explicitly presented as allowing a transition from cosmological deceleration to accelerated expansion without assuming dark energy or a non-zero cosmological constant (Barnes, 2024). The cosmological literature then recasts that claim in the second-order conformal-Killing-tensor language, where the modification is geometrically transparent (Mantica et al., 2023).

4. Exact solution landscape beyond spherical symmetry

Beyond spherical symmetry, CKG admits a notably large exact-solution space. In the Kundt class, the vacuum equations reduce, under constant scalar invariants, to

$K_{jk}$2

and for Kundt metrics this becomes a linear third-order partial differential equation for the profile $K_{jk}$3 (Hervik et al., 2024). The resulting families include Einstein metrics, symmetric spaces, and many non-Einstein, non-symmetric spacetimes in four and higher dimensions, among them Siklos waves, product-space waves, and solvegeometry waves (Hervik et al., 2024). The same paper stresses that the theory then contains so many vacuum solutions that “a selection criterium is needed” to explain why the observed universe is described by homogeneous cosmology rather than one of these many alternatives (Hervik et al., 2024).

Wave sectors make the contrast with GR particularly sharp. For pp-waves in Harada’s theory, the most general exact solution differs from the GR result by an extra non-propagating term $K_{jk}$4; the metrics remain Petrov type $K_{jk}$5 or $K_{jk}$6, and the Ricci tensor is either zero or of Segre type $K_{jk}$7 with zero eigenvalue (Barnes, 2024). In a broader analysis of Cotton gravity and CKG wave metrics, pp-wave spacetimes with non-flat wave surfaces were shown to be exact solutions of both theories, with the field equations reducing to inhomogeneous Laplace or Helmholtz equations depending on the curvature of the two-dimensional wave surface (Gürses et al., 2024). Those non-flat-surface pp-waves are absent in classical GR, which the paper identifies as a crucial distinction (Gürses et al., 2024).

The three-dimensional theory is also unusually rich. In sourceless three-dimensional CKG with two Killing vectors, all stationary, axisymmetric solutions were classified: besides singular solutions and BTZ black holes, the theory admits regular warped $K_{jk}$8 black holes and wormholes (Clement et al., 7 Aug 2025). In that sector the reduced field equations imply that the superspace trajectory $K_{jk}$9 is quadratic in $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$0, which fully integrates the stationary problem and exposes regular non-Einstein branches beyond the BTZ sector (Clement et al., 7 Aug 2025).

5. Horizons, thermodynamics, and observational programs

Thermodynamic work in CKG has proceeded along two partly distinct lines. Within Harada’s modified theory, a topology-based classification of static black-hole thermodynamics uses the generalized off-shell Helmholtz free energy and Duan’s $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$1-mapping method. In that framework, static black holes fall into four universal thermodynamic topological classes, $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$2, $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$3, $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$4, and $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$5, with the sign of the CKG parameter $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$6 determining the class (Chen et al., 4 Jun 2025). For charged AdS black holes in CKG, $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$7 gives $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$8 and $$\nabla_j K_{kl} + \nabla_k K_{lj} + \nabla_l K_{jk} = \frac{1}{6}\big( g_{kl}\nabla_j K + g_{lj}\nabla_k K + g_{jk}\nabla_l K \big),$$9 gives $K=g^{jk}K_{jk}$0; for Schwarzschild-like solutions, $K=g^{jk}K_{jk}$1 gives $K=g^{jk}K_{jk}$2 and $K=g^{jk}K_{jk}$3 gives $K=g^{jk}K_{jk}$4 (Chen et al., 4 Jun 2025).

A related but separate literature studies conformal Killing symmetry itself as the organizing principle of horizon thermodynamics in dynamical spacetimes. For the charged Vaidya metric, a conformal Killing vector exists only when the mass and charge functions are proportional and linear in advanced time, yielding a homothetic Killing vector and three conformal Killing horizons (Koh et al., 2023). Under a suitable conformal transformation, those horizons map to Killing horizons of an associated static spacetime, and the conformally invariant surface gravity $K=g^{jk}K_{jk}$5 gives a Hawking temperature

$K=g^{jk}K_{jk}$6

which matches the static-horizon result (Koh et al., 2023). This use of “conformal Killing gravity” is symmetry-based rather than tied to Harada’s field equations, but it has influenced the thermodynamic vocabulary around conformal-Killing structures.

Observationally, the static CKG literature has emphasized compact-object phenomenology. The shadow program in charged CKG black holes coupled to nonlinear electrodynamics computes light propagation in the effective metric rather than the background metric and compares the resulting shadow radius to EHT bounds (Junior et al., 1 Feb 2025). The compact-star program modifies pressure balance, surface matching, and Buchdahl-type bounds (Mantica et al., 2024). Black-bounce constructions extend the available regular geometries and suggest further tests through lensing, shadows, and causal structure (Junior et al., 2024).

6. Conserved currents, consistency debates, and open problems

The sharpest controversy surrounding CKG concerns conserved quantities and theoretical consistency. Altas and Tekin argue that field equations based on a tensor with rank greater than $K=g^{jk}K_{jk}$7 have consistency problems because integration constants such as the Schwarzschild mass parameter $K=g^{jk}K_{jk}$8 do not admit a clear interpretation as conserved charges in the theory (Altas et al., 10 Feb 2025). In their account, CKG does not arise from a diffeomorphism-invariant action, lacks the off-shell generalized Bianchi identities needed for standard charge constructions, gives no clear notion of black-hole energy or angular momentum, and does not support a conventional quadrupole formula for gravitational radiation (Altas et al., 10 Feb 2025).

A partially contrasting response has been developed in the static Einstein-plus-$K=g^{jk}K_{jk}$9 reformulation. There, conserved currents are constructed explicitly for static spherical symmetry with anisotropic matter or coupled electromagnetic sectors, and a conserved current proposed by Altas and Tekin is evaluated and found nonzero for Harada’s vacuum solution (Mantica et al., 17 Apr 2025). In this reformulation, both the ordinary matter current $K_{\mu\nu}$0 and the geometric current $K_{\mu\nu}$1 are conserved for a static Killing vector $K_{\mu\nu}$2, and the Harada vacuum admits a nontrivial quasi-local geometric energy contribution (Mantica et al., 17 Apr 2025).

These two strands show that CKG is still under active conceptual negotiation. On one side stand the exact-solution, cosmology, compact-object, and thermodynamic programs, all of which exploit the second-order conformal-Killing-tensor parametrization and treat the theory as a workable extension of Einstein gravity (Mantica et al., 2023, Mantica et al., 2024). On the other side stand objections centered on action principles, conserved charges, and the interpretation of integration constants in the original third-order formulation (Altas et al., 10 Feb 2025). The current literature therefore treats CKG simultaneously as a productive exact-solution framework and as a theory whose foundational status remains unsettled.

Open problems follow directly from that split. The literature repeatedly identifies the need for an action principle, a general conserved-charge formalism, rotating and dynamical solutions beyond special ansätze, stability analyses, and a physically motivated selection criterion within the very large solution space (Hervik et al., 2024, Junior et al., 1 Feb 2025). Within that still-evolving landscape, CKG remains one of the more distinctive recent attempts to combine Einstein-like second-order dynamics, emergent cosmological-constant structure, and conformal-Killing geometry in a single gravitational framework.