Massive Conformal Gravity

• Massive Conformal Gravity is a conformally invariant gravitational theory integrating a Weyl-squared term with a dilaton-coupled Einstein–Hilbert framework.
• It exhibits a mixed spectrum of massless and massive modes, offering insights into weak-field gravity, renormalizability, and the challenge of ghost states.
• MCG is applied to study black hole stability, cosmological dynamics, and gravitational-wave signatures, paving the way for alternative gravity models.

Massive Conformal Gravity (MCG) is a conformally invariant gravitational theory built from a Weyl-squared term together with an Einstein-Hilbert–type term conformally coupled to a scalar dilaton field. In the formulation introduced as a “massive theory of gravity that is invariant under conformal transformations,” the fundamental fields are the metric $g_{\mu\nu}$ and a scalar field $\varphi$, and the theory is designed to combine local conformal symmetry with a finite mass scale in the gravitational sector (Faria, 2013). Across the literature, MCG is studied as a higher-derivative theory with a massless spin-2 sector, a massive spin-2 sector, and a scalar sector; as a candidate power-counting renormalizable quantum gravity theory with ghost states; and as a framework for modified weak-field gravity, black-hole physics, and cosmology (Faria, 2015).

1. Action, conformal symmetry, and reduced field equations

In the original construction, the gravitational action is written as

$$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$

with $k=16\pi G/c^4$ and $\lambda=\hbar/(mc)$, so that the mass parameter enters through the graviton Compton wavelength (Faria, 2013). The defining conformal transformations are

$$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$

and the Weyl tensor $C^\lambda{}_{\mu\nu\kappa}$ provides the purely metric conformal invariant in four dimensions (Faria, 2013).

A widely used later normalization writes the action as

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$

or with equivalent sign conventions for the scalar sector depending on the paper (Faria, 2014). In these formulations, $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$ is rewritten through curvature invariants, and variation yields a metric equation involving the Bach tensor $W_{\mu\nu}$ or $\varphi$0 together with the conformally coupled scalar equation $\varphi$1 (Faria, 2016).

Below the Planck scale, several papers assume that the dilaton acquires a constant vacuum expectation value $\varphi$2. In that regime the field equations reduce to

$\varphi$3

in one common normalization, or to closely related forms differing by overall factors fixed by convention (Faria, 2014). This reduction is central to both the weak-field and cosmological analyses, because with $\varphi$4 the MCG line element effectively reduces to the usual metric form and the trace condition $\varphi$5 becomes a persistent structural constraint (Faria, 2016).

2. Linearized spectrum, propagating modes, and the massless limit

The weak-field theory is obtained by expanding about Minkowski spacetime and a constant scalar background. In the original formulation,

$\varphi$6

and the linearized equations reduce, after gauge fixing, to

$\varphi$7

showing a massless spin-2 sector, a massive spin-2 sector, and a massive scalar sector (Faria, 2013).

A canonical second-order reduction makes this mode content explicit. In one quantum treatment, the higher-derivative action is rewritten in terms of ordinary second-order fields $\varphi$8, $\varphi$9, and $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$0, obeying

$$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$1

The interpretation given there is that $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$2 is a massless spin-2 mode, $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$3 a massive spin-2 mode, and $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$4 a massive spin-0 mode; the sign structure of the reduced action identifies the massive tensor and massive scalar as ghost sectors (Faria, 2015).

In the gravitational-wave analysis of the linearized field equations in unitary gauge, the vacuum equation becomes

$$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$5

with plane-wave solution

$$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$6

where $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$7 and $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$8. After gauge reduction, the massless sector carries two independent components and the massive sector five, so MCG is said to have seven physical plane-wave degrees of freedom, $$S_g = \frac{1}{2kc} \int d^4x\,\sqrt{-g}\, \left[ \alpha\, C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa} - \beta\lambda^{-2}\left( \varphi^2 R + 6\,g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi \right) \right],$$9 massless plus $k=16\pi G/c^4$0 massive (Faria, 2020).

The massless limit has been analyzed in detail in the context of the van Dam–Veltman–Zakharov discontinuity. In the diagonalized linearized theory coupled to a traceless conformal source, the vector and scalar modes do not couple to the source in the $k=16\pi G/c^4$1 limit, and the theory is therefore argued to be free of the vDVZ discontinuity (Faria, 2017). In the same analysis, the gravitational potential of a point source is

$k=16\pi G/c^4$2

which is finite at $k=16\pi G/c^4$3; this nonsingular Newtonian potential is presented as consistent with the smooth sourced massless limit (Faria, 2017).

3. Renormalizability, higher derivatives, and ghost states

The quantum literature on MCG emphasizes that the Weyl-squared sector produces fourth-order propagators with improved ultraviolet behavior. Using the Lanczos identity, one common form of the action is

$k=16\pi G/c^4$4

and expansion around flat space yields propagators whose asymptotic momentum dependence is $k=16\pi G/c^4$5 (Faria, 2016). The same ultraviolet falloff appears in the canonical quantization analysis, where the original higher-derivative propagator behaves as $k=16\pi G/c^4$6 for $k=16\pi G/c^4$7, leading to the statement that the theory is power-counting renormalizable (Faria, 2015).

The principal difficulty is that the residues at the massive poles have the wrong sign. In the canonical treatment, the creation–annihilation algebra for the massive tensor and scalar fields carries minus signs, and the Hamiltonian analysis shows the usual higher-derivative dilemma: one may preserve a lower-bounded energy spectrum at the price of negative-norm states, or avoid ghosts at the price of an unbounded Hamiltonian (Faria, 2015). In that sense, renormalizability is established in the power-counting sense, while unitarity is not established in the same paper.

A central controversy concerns whether renormalizability becomes “meaningless” if the massive ghost states are unstable. In the reply to Myung’s comment, the answer given is negative: MCG can still be renormalizable with unstable ghost states, because renormalization is controlled by the ultraviolet behavior of propagators rather than by the asymptotic-state stability of ghost poles (Faria, 2016). The proposed treatment replaces bare propagators by dressed propagators,

$k=16\pi G/c^4$8

while excluding self-energy insertions on unstable ghost lines; the key claim is that such dressing modifies finite-width properties but does not alter the $k=16\pi G/c^4$9 ultraviolet falloff relevant to power-counting renormalizability (Faria, 2016).

A related one-loop perspective appears in the singularity analysis, where MCG is treated as effectively one-loop renormalizable even though a trace anomaly appears. There the on-shell effective action is used as the diagnostic quantity for quantum consistency near singularities, and the renormalizability discussion is presented as a prerequisite for making that on-shell analysis meaningful (Faria, 2023).

4. Weak-field gravity, solar-system bounds, and gravitational radiation

The static weak-field phenomenology of MCG is typically expressed through two potentials in isotropic coordinates,

$\lambda=\hbar/(mc)$0

for a point mass source (Faria, 2016). After linearization and use of the reduced field equations with $\lambda=\hbar/(mc)$1, the potential $\lambda=\hbar/(mc)$2 obeys a fourth-order Yukawa-type equation and admits the explicit solution

$\lambda=\hbar/(mc)$3

with $\lambda=\hbar/(mc)$4 interpreted as the mass of a massive spin-2 ghost mode (Faria, 2016). In this solution, $\lambda=\hbar/(mc)$5 remains finite as $\lambda=\hbar/(mc)$6, whereas $\lambda=\hbar/(mc)$7 diverges, which is taken to indicate breakdown of the linear approximation near the origin (Faria, 2016).

Solar-system tests then constrain the Yukawa scale strongly. Using light deflection by the Sun, radar echo delay, and Mercury’s perihelion advance, one study finds consistency with general relativistic observations only if the graviton mass is sufficiently large, with the strongest quoted lower bound

$\lambda=\hbar/(mc)$8

and weaker compatible bounds $\lambda=\hbar/(mc)$9 and $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$0 from the other two tests (Faria, 2016). That analysis also notes that such a large lower bound makes it difficult for MCG to explain galaxy-scale phenomena without additional ingredients (Faria, 2016).

The gravitational-wave literature sharpens the same tension. In the binary-radiation analysis, the theory with large graviton mass can reproduce the orbital decay of binaries by gravitational-wave emission, while the small-mass regime predicts radiation much smaller than in general relativity (Faria, 2020). A related fourth-order treatment in Weyl gauge reaches the same conclusion in a more comparative form: MCG reduces to GR in the large-$$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$1 limit, but for small graviton masses the emitted gravitational radiation is too weak to account for observed binary-pulsar orbital decay (1804.01876). The combined implication is that the parameter regime compatible with compact-binary timing is the one in which MCG is effectively close to GR rather than the one invoked to mimic dark-matter-free galactic dynamics (1804.01876).

5. Black holes, instability, and singularities

The Schwarzschild black hole is a natural test background because it solves the reduced MCG equations for $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$2, $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$3, and constant background scalar. In the Jordan-frame analysis of MCG, the linearized system is simplified by the relation

$$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$4

which decomposes the perturbations into a trace sector and a traceless sector (Myung, 2014). The trace equation becomes

$$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$5

or equivalently $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$6, and for $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$7 this sector is found to be stable because the associated potential is positive outside the horizon (Myung, 2014).

The instability appears in the traceless massive spin-2 sector. There the principal equation is

$$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$8

which is formally identical to the equation governing the Gregory–Laflamme $$\tilde g_{\mu\nu}(x)=e^{2\theta(x)} g_{\mu\nu}(x), \qquad \tilde\varphi=e^{-\theta(x)}\varphi,$$9-mode instability of the five-dimensional black string (Myung, 2014). Numerical analysis yields an instability band

$C^\lambda{}_{\mu\nu\kappa}$0

and the interpretation given is that in MCG the graviton mass plays the role otherwise played by the extra dimension in the black-string problem (Myung, 2014). The paper further argues that MCG in the Jordan frame is conformally equivalent to Einstein–Weyl gravity in the Einstein frame, so the Schwarzschild instability is frame-independent (Myung, 2014).

A distinct line of work studies whether classical singularities remain problematic after quantum effects are taken into account. Using the on-shell effective action as the criterion, one paper concludes that both the big bang singularity and a black-hole singularity are “harmless” in MCG because the relevant on-shell gravitational action does not diverge and in the examples studied actually vanishes (Faria, 2023). For the cosmological case, with

$C^\lambda{}_{\mu\nu\kappa}$1

the on-shell gravitational action is found to be zero, and the matter contribution also vanishes for relativistic matter with $C^\lambda{}_{\mu\nu\kappa}$2 (Faria, 2023). For the black-hole case, the analysis uses a numerical static, spherically symmetric solution singular at $C^\lambda{}_{\mu\nu\kappa}$3, but again the on-shell gravitational action is finite and vanishes in the calculation presented (Faria, 2023). The finite-action conclusion is explicitly limited to the singular solutions examined and does not claim a nonperturbative resolution of all possible MCG singularities (Faria, 2023).

6. Cosmological background, thermodynamics, nucleosynthesis, and structure formation

The late-time cosmology developed for MCG uses the reduced equations with $C^\lambda{}_{\mu\nu\kappa}$4 together with a conformally invariant matter sector containing a Higgs-like scalar and fermions (Faria, 2014). After symmetry breaking, the effective matter stress tensor becomes

$C^\lambda{}_{\mu\nu\kappa}$5

because the vacuum-energy term drops out of the final dynamical equations after taking the trace and using $C^\lambda{}_{\mu\nu\kappa}$6 (Faria, 2014). In FLRW geometry the resulting generalized Friedmann equation implies

$C^\lambda{}_{\mu\nu\kappa}$7

so $C^\lambda{}_{\mu\nu\kappa}$8 (Faria, 2014).

In that late-universe analysis, only the open universe with $C^\lambda{}_{\mu\nu\kappa}$9 is argued to be compatible with observed $$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$0 and the age of the universe. For $$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$1, the solution is

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$2

the expansion is decelerating, and the best-fit supernova values reported from the Pantheon compilation are

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$3

with

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$4

and age

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$5

(Faria, 2014). The paper explicitly presents this model not as an accelerating $$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$6CDM analogue but as a decelerating negative-curvature cosmology in which vacuum energy does not gravitate in the final equations (Faria, 2014).

Early-universe nucleosynthesis is treated differently but within the same conformal-fluid structure. There the modified background expansion gives $$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$7 with a coefficient about $$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$8 times the standard ACDM radiation-dominated coefficient, so the MCG universe expands more slowly at the same temperature (Faria, 2023). In the minimal particle-content case this leads to a baryon-to-photon ratio at nucleosynthesis

$$S = \int d^{4}x \sqrt{-g}\left[ \varphi^{2}R + 6\,\partial^{\mu}\varphi\partial_{\mu}\varphi -\frac{1}{2\alpha^2}C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu} \right] + \int d^4x\,\mathcal{L}_m,$$9

which is deemed inconsistent with observed light-element abundances (Faria, 2023). The proposed remedy is the existence of right-handed sterile neutrinos, which increase the effective relativistic degrees of freedom from $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$0 to $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$1 before $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$2 annihilation and from $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$3 to $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$4 afterward, yielding

$C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$5

and making the theory consistent with observed primordial abundances except for lithium (Faria, 2023).

The thermodynamic viability of the MCG cosmological solution has also been tested through the generalized second law. Using the scale factor

$C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$6

the total entropy production rate is derived as

$C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$7

so that $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$8 for the flat and open cases but not for the closed case (Faria, 22 May 2025). Because the flat case is said there to be inconsistent with observational $C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}$9 and $W_{\mu\nu}$0, the open MCG universe is identified as the physically viable solution, and in that case $W_{\mu\nu}$1 as well (Faria, 22 May 2025).

The perturbative structure-formation analysis extends this background cosmology. Starting from the conservation of the conformal perfect-fluid energy-momentum tensor, the perturbed continuity and Euler equations are

$W_{\mu\nu}$2

and in the Newtonian limit the growth equation becomes

$W_{\mu\nu}$3

(Faria et al., 12 Aug 2025). For large scales, dropping the gradient term gives the formally standard growth equation, but with the distinct MCG background $W_{\mu\nu}$4. The resulting growing modes are

$W_{\mu\nu}$5

at early times, implying $W_{\mu\nu}$6, and

$W_{\mu\nu}$7

at late times, implying $W_{\mu\nu}$8 (Faria et al., 12 Aug 2025). The conclusion drawn is that MCG enhances cosmic structure formation at high redshift and suppresses it at low redshift relative to $W_{\mu\nu}$9CDM (Faria et al., 12 Aug 2025).

7. Related theories, correspondences, and terminological boundaries

Several nearby theories are frequently discussed together with MCG but are not identical to it. The most prominent is new massive conformal gravity (NMCG), obtained by adding an Einstein-Hilbert term $\varphi$00 to the MCG action, thereby breaking conformal invariance (Myung, 2014). In that framework the linearized Ricci tensor, rather than the metric perturbation, is used to describe the massive spin-2 field, and the theory is said to propagate five massive graviton polarizations plus one conformal scalar, for a total of six degrees of freedom (Myung, 2014). Minkowski spacetime is stable when $\varphi$01, whereas small Schwarzschild black holes are unstable against $\varphi$02-mode massive graviton perturbations (Myung, 2014).

The later NMCG black-hole analysis pushes this distinction further. Because the Einstein-Hilbert term enforces $\varphi$03, the linearized Ricci scalar satisfies $\varphi$04, and the Schwarzschild perturbation problem reduces to a Lichnerowicz-Ricci tensor equation

$\varphi$05

with instability threshold

$\varphi$06

(Myung et al., 2019). That instability is interpreted as signaling a bifurcation to a non-BBMB black hole with both Ricci-tensor hair and primary conformal scalar hair (Myung et al., 2019). The paper explicitly presents this mechanism as available in NMCG but obstructed in conformally invariant MCG because conformal symmetry prevents the same reduction (Myung et al., 2019).

Other papers use “conformal massive gravity” in an even broader sense. One study begins from Lorentz-breaking massive gravity and uses a conformal rescaling to construct static and rotating regular black holes characterized by $\varphi$07, $\varphi$08, $\varphi$09, $\varphi$10, and a conformal scale $\varphi$11; photon shadows depend on $\varphi$12, $\varphi$13, and $\varphi$14, while the deflection of relativistic massive particles is strongly affected by $\varphi$15 (1911.07520). This is a related conformal extension of massive gravity rather than the standard dilaton-plus-Weyl MCG action.

A different kind of relation is established at the level of exact black-hole solutions. In a comparison between four-dimensional conformal gravity and Vegh/dRGT-type massive gravity, the conformal-gravity metric function

$\varphi$16

is made identical to the massive-gravity black-hole metric by the parameter identifications

$\varphi$17

(Panah et al., 2019). That paper does not study an action explicitly named MCG, but it is relevant to the MCG literature because it shows a solution-level correspondence between conformal-gravity black holes and a massive-gravity branch (Panah et al., 2019).

These neighboring constructions matter because the MCG literature is not terminologically uniform. Some papers focus on the strictly conformally invariant dilaton–Weyl theory; others on non-conformally invariant descendants such as NMCG; and others on black-hole geometries in broader conformal extensions of massive gravity. A plausible implication is that many disputes in the literature concern not only dynamics and phenomenology, but also which precise conformal massive-gravity model is being analyzed.