Conformal Gravity Research Articles

Cubic interactions for massless and partially massless spin-1 and spin-2 fields

We perform a complete classification of the consistent two-derivative cubic couplings for a system containing an arbitrary number of massless spin-1, massless spin-2, and partially massless (PM) spin-2 fields in D-dimensional (anti-)de Sitter space. In addition to previously known results, we find a unique candidate mixing between spin-1 and PM spin-2 fields. We derive all the quadratic constraints on the structure constants of the theory, allowing for relative “wrong-sign” kinetic terms for any of the fields. In the particular case when the kinetic terms in each sector have no relative signs, we find that the unique consistent non-trivial theory is given by multiple independent copies of conformal gravity coupled to a Yang-Mills sector in D = 4. Our results strengthen the well-known no-go theorems on the absence of mutual interactions for massless and PM spin-2 fields.

Wave metrics in the Cotton and conformal Killing gravity theories

Wave metrics in the Cotton and conformal Killing gravity theories

Impossible symmetries and conformal gravity

We explore the physics of relativistic gapless phases defined by a mixed anomaly between two generalized conserved currents. The gapless modes can be understood as Goldstone modes arising from the nonlinear realization of (generically higher-form) symmetries arising from these currents. In some cases, the anomaly cannot be reproduced by any local and unitary theory, indicating that the corresponding symmetries are impossible, in the sense that they cannot appear in a Lorentzian physical system. We give a general construction and illustrate it with several examples. Most notably, we study conformal gravity from this perspective, describing the higher-form symmetries of the linear theory and showing how it can be understood in terms of anomalies. Along the way we clarify some aspects of electric-magnetic duality in linear conformal gravity. Published by the American Physical Society 2024

Running of the number of degrees of freedom in quantum conformal gravity

We study how the number of degrees of freedom in Weyl conformal gravity runs with the energy scale from the UV fixed point. To this end we employ two approaches. First, we utilize the Fradkin–Tseytlin prescription for the number of degrees of freedom and demonstrate that the one-loop result is highly dependent on the selected background. We then employ the counting methodology based on the a- and c-function, which are typically used to characterize the trace anomaly of conformal field theories in four dimensions. We compute these in the enhanced one-loop approximation and demonstrate that the degrees of freedom decrease monotonically from six degrees in the UV regime. This behavior is independent of the backgrounds considered. Further salient issues, such as the connection between the Fradkin–Tseytlin prescription and counting based on the a- and c-function, or the applicability of both methods in Einstein’s gravity, are also addressed.

Conformal Killing gravity: pp-waves; multiple matter sources

Abstract Recently Harada proposed a gravitational theory which is of third order in the derivatives of the metric tensor. This attracted some attention particularly as it predicts a late-time transition from cosmological deceleration to accelerated expansion without assuming the presence of dark energy or a non-zero cosmological constant. This theory has been dubbed conformal Killing gravity (CKG). The most general exact solutions of the Harada field equations are known for a number of important physical situations: homogeneous and isotropic cosmological models, static spherically symmetric vacuum and electrovac spacetimes. These are analogues of the well-known FRWL, Schwarzschild and Reissner–Nordström metrics of general relativity (GR). In this paper a subclass of pp-waves are studied and the most general exact solution obtained together with its specialization for plane waves. The generalization from GR to Harada’s theory is straightforward. The solutions have Petrov type N or 0 and the Ricci tensor is either zero or the Segré type is [(211)] with zero eigenvalue. For any metric in CKG it is shown that more than one possible matter source can generate the solution. If the metric admits one or more Killing vectors or Killing tensors, the ambiguity in the possible matter sources increases.

Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions

In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev’s holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) — i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields — as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.

Quasinormal modes of the Mannheim–Kazanas black holes

Abstract A spherically symmetric black hole solution in the conformal Weyl gravity was found by Mannheim and Kazanas in 1988. While the quasinormal modes (QNMs) of these black holes have been considered in a few works, here we complement these studies by considering in detail the regime of vanishing cosmological constant and negative values of the Weyl parameter for which we find quasinormal frequencies for scalar, Dirac and electromagnetic perturbations with the time-domain integration and WKB methods. In particular, we derive the compact and remarkably accurate analytic formula for the frequencies in the form of expansion in terms of the inverse multipole number. Comparison with the time-domain integration shows that the 6th order WKB method with the Padé approximants is quite accurate, unless the black hole is in the near extreme state.

Conformal Killing gravity in static spherically symmetric spacetimes

Conformal Killing gravity in static spherically symmetric spacetimes

Spherical symmetric solutions of conformal Killing gravity: black holes, wormholes, and sourceless cosmologies

Abstract The most general set of static and spherically symmetric solutions for conformal Killing gravity coupled to Maxwell fields is presented in closed form. These solutions, depending on six parameters, include non-asymptotically flat black holes or naked singularities, non-asymptotically flat traversable wormholes, and (possibly singularity-free) closed universes. We also consider the inverse problem, showing that the most general energy–momentum tensor generating a given static spherically symmetric metric depends on three parameters. Sourceless time-dependent isotropic solutions are also given. These solutions depending on the curvature, the cosmological constant, and a new integration constant α, present a rich variety, including singularity-free eternal cosmologies and universes evolving symmetrically from a big bang to a big crunch within a finite lapse of time.

Spherically symmetric electrovac spacetimes in conformal Killing gravity

Recently Harada proposed a third-order gravitational theory for the derivatives of the metric. Mantica and Molinari showed that Harada’s theory may be recast into the form of Einstein’s field equations (EFEs) with an additional source term which is a second-order conformal Killing tensor. Accordingly they named the theory conformal Killing gravity. However, they overlooked the fact that all solutions of the new theory (except those satisfying EFEs) admit a non-trivial second-order Killing tensor. Harada derived an analog of the Schwarzschild solution. Recently Tarciso et al obtained a generalization of Harada’s vacuum solution analogous to the Reissner–Nordström solution. However, like Harada they assumed a restricted form for a static spherically symmetric metric. In this study the most general spherically symmetric static vacuum solution of Harada’s theory and its generalization with a Maxwell electromagnetic field as source were obtained. The validity of Birkhoff’s theorem for static spherically symmetric electrovac fields in conformal Killing gravity is investigated.

Unification of conformal gravity and internal interactions

Based on the observation that the dimension of the tangent space is not necessarily equal to the dimension of the corresponding curved manifold and on the known fact that gravitational theories can be formulated in a gauge theoretic way, we discuss how to describe all known interactions in a unified manner. This is achieved by enlarging the tangent group of the four-dimensional manifold to SO(2, 16), which permits the inclusion of both gauge groups, the one that describes gravity as a gauge theory as well as the SO(10) describing the internal interactions. Moreover it permits the use of both Weyl and Majorana conditions imposed on the fermions, as to avoid the duplication of fermion multiplets of SO(10) appearing in previous attempts. The gravity theory discussed in the present work is the Conformal Gravity which, after a spontaneous symmetry breaking, can lead either to Weyl Gravity or to the usual Einstein Gravity.

Black bounces in conformal Killing gravity

In this work, we analyse black bounce solutions in the recently proposed “Conformal Killing gravity” (CKG), by coupling the theory to nonlinear electrodynamics (NLED) and scalar fields. The original motivation of the theory was essentially to fulfill specific criteria that are absent in existing gravitational theories, namely, to obtain the cosmological constant as an integration constant, derive the energy–momentum conservation law as a consequence of the gravitational field equations, rather than assuming it, and not necessarily considering conformally flat metrics as vacuum solutions. In this work, we extend the static and spherically symmetric solutions obtained in the literature, and explore the possibility of black bounces in CKG, coupled to NLED and scalar fields. We find novel NLED Lagrangian densities and scalar potentials, and extend the class of black bounce solutions found in the literature. Furthermore, within black bounce geometries, we find generalizations of the Bardeen-type and Simpson–Visser geometries and explore the regularity conditions of the solutions.

Holographic study on Einstein ring for a charged black hole in conformal gravity

Under the help of AdS/CFT correspondence, the Einstein ring of a charged black hole in conformal gravity has been studied. As an alternative to standard Einstein’s theory, the conformal gravity is both renormalizable and unitary. And such conformal theory can naturally solve some troublesome problems in physics, such as the quantum gravity problem, the dark matter problem and so on. Considering the conformal gravity has many advantages, therefore in this paper, we study Einstein ring and the effect of gravity-related parameters c0, temperature T and chemical potential u on such Einstein ring in the conformal gravity through the holographic method. Explicitly, imposing an oscillating Gauss source on one side of the AdS boundary which propagates in the bulk, the response functions are derived on the other side of the boundary. Given the proposed wave optics system, the Einstein rings are observed as expected. All these results reveal that when the observer locates at the north pole, there always exists the Einstein ring and surrounding concentric stripes. And such ring becomes into a luminosity-deformed ring or light spot while the observer departs away from the north pole. We also investigate the effect of temperature T, chemical potential u and gravity-related parameters c0 on the ring radius. We find the ring radius increases with the decrease of the temperature, increase of the chemical potential, and increase of the gravity-related parameters respectively. And more, in conformal gravity, because the charge does not appear in the metric but is given by the constraint relation, the chemical potential affects the radius of the ring differently compared with that in Einstein’s gravity. To check the results in the framework of holography, we investigate the ingoing angle of photon at the photon ring via geometric optics and find it is consistent with the angle of Einstein ring obtained via holography.

Constructing massive particles surfaces in static spacetimes

Massive particle surface is a generalization of the photon surface, both of which translate the dynamical properties of geodesics orbiting the gravitating object onto the geometric description. The procedure for constructing the massive particle surfaces in static space-times is described in detail and the equivalence of the main results with the results of the geodesic approach is demonstrated for the following examples: Schwarzschild, Reissner–Nordström, dilatonic black holes, conformal gravity, and Culetu model.

The κ-Model under the Test of the SPARC Database

Our main goal here is to conduct a comparative analysis between the well-known MOND theory and a more recent model called the κ-model. An additional connection, between the κ-model and two other novel MOND-type theories, Newtonian Fractional-Dimension Gravity (NFDG) and Refracted Gravity (RG), is likewise presented. All these models are built to overtake the DM paradigm, or at least to strongly reduce the dark matter content. Whereas they rely on different formalisms, however, all four seem to suggest that the universal parameter, a0, appearing in MOND theory could intrinsically be correlated to either the sole baryonic mean mass density (RG and κ-model) and/or to the dimension of the object under consideration (NFDG and κ-model). We then confer to parameter a0 a more flexible status of multiscale parameter, as required to explain the dynamics together in galaxies and in galaxy clusters. Eventually, the conformal gravity theory (CFT) also seems to have some remote link with the κ-model, even though the first one is an extension of general relativity, and the second one is Newtonian in essence. The κ-model has been tested on a small sample of spiral galaxies and in galaxy clusters. Now, we test this model on a large sample of galaxies issued from the SPARC database.

Observational redshift from general spherically symmetric black holes

In this work, we obtain an expression for the total observational frequency shift of photons emitted by massive geodesic particles circularly orbiting a black hole in a general spherically symmetric background. Our general relations are presented in terms of the metric components and their derivatives that characterize the black hole parameters. As a concrete example of this general relativistic approach, a special case is studied by applying the formalism to a nonsingular black hole conformally related to the Schwarzchild solution that possesses a length scale parameter l and an integer parameter N in addition to the black hole mass. Besides, we express the nonsingular black hole mass in terms of the observational redshift/blueshift. Finally, we investigate the effects of the free parameters of the conformal gravity theory on the observational frequency shift and compare results with those of the standard Schwarzschild black hole.

Images of nonsingular nonrotating black holes in conformal gravity

The accretion disk around a black hole and its emissions play an essential role in theoretical analysis of the black hole image. In the literature, two analytical toy models of accretions are widely adopted: the spherical model and the thin disk model. They are different geometrically but both thin optically. We polish them for free-falling accretions around static spherical black holes. As an application, we investigate the images of a class of nonsingular black holes conformally related to the Schwarzschild black hole. These black holes are vacuum solutions of a family of conformal gravity theories. Results are compared with the Schwarzschild black hole of the same mass. Our results indicate that the conformal factor does not affect the shadow radius seen by distant observers, but it leaves an imprint on the intensity image of black hole.

Quantum conformal gravity

We present the manifestly covariant canonical operator formalism of a Weyl invariant (or equivalently, a locally scale invariant) gravity whose classical action consists of the well-known conformal gravity and Weyl invariant scalar-tensor gravity, on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism. It is shown that there exists a Poincaré-like IOSp(8|8) global symmetry as in Einstein’s general relativity, which should be contrasted to the case of only the Weyl invariant scalar-tensor gravity where we have a more extended Poincaré-like IOSp(10|10) global symmetry. This reduction of the global symmetry is attributed to the presence of the Stückelberg symmetry.

(Regular) Black holes in conformal Killing gravity coupled to nonlinear electrodynamics and scalar fields

In this work, we explore new solutions with static and spherical symmetry in 4D for black holes and regular black holes in the recently proposed conformal Killing gravity (CKG). This theory is of third order in the derivatives of the metric tensor and essentially satisfies three theoretical criteria for gravitational theories beyond general relativity (GR). The criteria essentially stipulate the following, that one should: (i) obtain the cosmological constant as an integration constant; (ii) derive the energy conservation law as a consequence of the field equations, rather than assuming it; (iii) and not necessarily consider conformally flat metrics as vacuum solutions. In fact, existing modified theories of gravity, including GR, do not simultaneously fulfil all of these three criteria. Here, we couple CKG to nonlinear electrodynamics (NLED) and scalar fields, and we explore solutions of black holes and regular black holes. More specifically, by solving the field equations of CKG, we find specific forms for the NLED Lagrangian, the scalar field and the field potential, and analyse the regularity of the solutions through the Kretschmann scalar. We find generalizations of the Schwarschild–Reissner-Nordström–AdS solutions, and consequently further extend the class of (regular) black hole solutions found in the literature.

Gravitational Light Bending in Weyl Gravity and Schwarzschild–de Sitter Spacetime

The topic of gravitational lensing in the Mannheim–Kazanas solution of Weyl conformal gravity and the Schwarzschild–de Sitter solution in general relativity has featured in numerous publications. These two solutions represent a spherical massive object (lens) embedded in a cosmological background. In both cases, the interest lies in the possible effect of the background non-asymptotically flat spacetime on the geometry of the local light curves, particularly the observed deflection angle of light near the massive object. The main discussion involves possible contributions to the bending angle formula from the cosmological constant Λ in the Schwarzschild–de Sitter solution and the linear term γr in the Mannheim–Kazanas metric. These effects from the background geometry, and whether they are significant enough to be important for gravitational lensing, seem to depend on the methodology used to calculate the bending angle. In this paper, we review these techniques and comment on some of the obtained results, particularly those cases that contain unphysical terms in the bending angle formula.