We study the evolution of time-like congruences in the vacuum solutions of Weyl conformal theory of gravity. Using the Raycaudhuri equation, we show that for positive values of the coeffcient of the linear term in the solution and in the absence of the cosmological constant, the incoming rays converge. The evolution of the congruence for negative values is investigated for different values of the parameters. The behavior of the congruence under conformal transformations is also studied.
We discuss locally Weyl (scale) covariant generalisation of quadratic curvature gravity theory in three dimensions using Riemann-Cartan-Weyl space-times. We show that this procedure of Weyl gauging yields a consistent generalisation for a particular class of quadratic curvature gravity theories which includes the New Massive Gravity theory.
A crucial property of Weyl gravity is its conformal invariance. It is shown how this gauge symmetry is exactly reflected by the two constraints in the Hamiltonian framework. Since the spatial 3-metric is one of the configuration variables, the phase space of Weyl gravity can be extended to include internal gauge freedom by triad formalism. Moreover, by canonical transformations, we obtain two new Hamiltonian formulations of Weyl gravity with an SU(2) connection as one of its configuration variables. The connection-dynamical formalisms lay the foundation to quantize Weyl gravity nonperturbative…
We consider scalar-tensor theories of gravity defined in Weyl integrable space-time and show that in the ADM formalism Weyl transformations corresponding to change of frames induce canonical transformations between different representations of the phase space. In this context, we discuss the physical equivalence of two distinct Weyl frames at the classical level.
Weyl gravity in the presence of a non-minimal matter-curvature coupling is presented. Some properties arising from the non-metricity give rise to a second order theory whose vacuum is compatible with a cosmological constant, under conditions on the vector field.
Within vacuum Weyl gravity, we obtain a solution by which, using different choices of the conformal factor, we derive metrics describing (i)~a bounce of the universe; (ii)~toroidal and spherical wormholes; and (iii)~a change in metric signature. It is demonstrated that singularities occurring in these systems are "masked". We give a simple explanation of the possibility of masking the singularities within Weyl gravity. It is shown that in the first and third cases the three-dimensional metrics form Ricci flows. The question of the possible applicability of conformal Weyl gravity as some phenom…
We investigate the classical stability of Schwarzschild-AdS black hole in a massive theory of the Einstein-Weyl gravity. It turns out that the linearized Einstein tensor perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional AdS black string, in contrast to the stable Schwarzschild-AdS black hole in the Einstein-Weyl gravity. We confirm that the instability of the black hole in the Einstein-Weyl gravity arises from the massiveness but not a feature of fourth-order derivative theory giving ghost states.
The usual interpretation of Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the …
Penrose suggested that issues with the origin of the second law of thermodynamics and the remarkable homogeneous and isotropic nature of the universe on large scales can be resolved with a concept of an entropy for gravitational fields captured by the Weyl curvature tensor. This has led to various proposals for an entropy density for gravity constructed from the Weyl tensor. Most work investigating such entropy densities have been restricted to general relativity and little work has been done to them in modified theories of gravity. To remedy this, we investigate the simplest proposal for an e…
We consider a non-minimally coupled curvature-matter gravity model (NMC) with a Weyl connection, a theory referred to as non-minimally coupled Weyl connection gravity (NMCWCG). The Weyl connection is an affine connection that is not compatible with the metric, and involves a vector field. Assuming a vacuum expectation value for the vector field and a matter Lagrangian that only contains the contributions of the vacuum energy, we show that the model admits solutions in the space-form with a reference curvature that can be fine-tuned to much smaller values than the contribution of the matter fie…
We construct charged asymptotically flat black hole solutions in Einstein-Maxwell-Weyl(EMW) gravity. These solutions can be interpreted as generalizations of two different groups: Schwarzschild black hole (SBH) and non-Schwarzschild black hole (NSBH) solutions. In addition, we discuss the thermodynamic properties of two groups of numerical solutions in detail, and show that they obey the first law of thermodynamics.
In this paper we obtain the linear Weyl gravity in de Sitter background. Its linear form in five-dimensional ambient space notation has also been obtained. From the group theoretical point of view, we show that this conformal invariant fourth order theory in its linear form does not transform as an unitary irreducible representation of the de Sitter group.
We discuss locally Weyl (scale) covariant generalisations of gravitational theories using Riemann-Cartan-Weyl space-times in arbitrary dimensions. We demonstrate the procedure of Weyl gauging on two examples in particular: General relativity, and topologically massive gravity in three dimensions.
In this paper, the cosmological "constant" and the Hubble parameter are considered in the Weyl theory of gravity, by taking them as functions of $r$ and $t$, respectively. Based on this theory and in the linear approximation, we obtain the values of $H_0$ and $Λ_0$ which are in good agreement with the known values of the parameters for the current state of the universe.
In this work, we study the issue of the past and future cosmological singularities in the context of the fourth-order conformal Weyl gravity. In particular, we investigate the emergent universe scenario proposed by Ellis {\it et al}, and find the stability conditions of the corresponding Einstein static state using the fixed point approach. We show that depending on the values of the parameters of the conformal Weyl gravity theory, there are possibilities for having initially stable emergent states for an FRW universe with both the positive and negative spatial curvatures. This represents that…
In this work we find analytical solutions to the null geodesics around a black hole in the conformal Weyl gravity. Exact expressions for the horizons are found, and they depend on the cosmological constant and the coupling constants of the conformal Weyl gravity. Then, we study the radial motion from the point of view of the proper and coordinate frames, and compare it with that found in spacetimes of general relativity. The angular motion is also examined qualitatively by means of an effective potential; quantitatively, the equation of motion is solved in terms of $\wp$-Weierstrass elliptic f…
We present locally scale (Weyl) covariant generalisation of Minimal Massive Gravity theory using the language of exterior differential forms on Riemann-Cartan-Weyl space-times. The theory is expressed by a locally scale invariant action and locally scale covariant field equations are found by a first order variational formalism.
Recently the vector inflation has been proposed as the alternative to inflationary models based on scalar bosons and quintessence scalar fields. In the vector inflationary model, the vector field non-minimally couples to gravity. We should, however, inquire if there exists a relevant fundamental theory which supports the inflationary scenario. We investigate the possibility that Weyl's gauge gravity theory could be such a fundamental theory. That is the reason why the Weyl's gauge invariant vector and scalar fields are naturally introduced. After rescaling the Weyl's gauge invariant Lagrangian…
In this work we find analytical solutions to the null geodesics in the hyperbolic topological black hole spacetime of conformal Weyl gravity in an invariant $2$-plane given by the orbits of an azimuthal Killing vector. Exact expressions for the (non-compact) horizons are found, which depend on the cosmological constant and the coupling constants of conformal Weyl gravity. The angular motion is examined qualitatively by means of an effective potential; quantitatively, the equation of motion is solved in terms of the $\wp$-Weierstraß elliptic function. Thus, we find the deflection angle for phot…
We present the exact exterior solution for a static and neutral cylindrically symmetric source in locally conformal invariant Weyl gravity. As a special case the general relativity analogue still can be attained, however only as a sub-family of solutions. Our solution contains a linear term that would thus result in a potential that grows linearly over large distances. This may have implications for exotic astrophysical structures as well as matter fields on the extremely small scale.
We investigate the cosmological evolution for the physical parameters in Weyl integrable gravity in a Friedmann--Lema\^ıtre--Robertson--Walker universe with zero spatially curvature. For the matter component, we assume that it is an ideal gas, and of the Chaplygin gas. From the Weyl integrable gravity a scalar field is introduced by a geometric approach which provides an interaction with the matter component. We calculate the stationary points for the field equations and we study their stability properties. Furthermore, we solve the inverse problem for the case of an ideal gas and prove that t…
We study the Hamiltonian formalism of the non-minimally coupled Weyl connection gravity (NMCWCG) in order to check whether Ostrogradsky instabilities are present. The Hamiltonian of the NMCWCG theories is obtained by foliating space-time into a real line (representing time) and \mbox{3-dimensional} space-like hypersurfaces, and by considering the spatial metric and the extrinsic curvature of the hypersurfaces as the canonical coordinates of the theory. Given the fact that the theory we study contains an additional dynamical vector field compared to the usual NMC models, which do not have Ostro…
We calculate the deflection angle of light from a distant source by a galaxy cluster in Weyl's conformal gravity. The general method of calculation is first applied to calculate the deflection angle in Schwarzschild-de Sitter (Kottler) spacetime. The deflection angle calculated in Kottler spacetime includes the contribution of the cosmological constant, which quantitatively agrees with one work and disagrees with many works in the literature. We then calculate the deflection angle in Mannheim-Kazanas spacetime in two conformally related coordinate systems and find that the result includes cont…
We reconsider the status of the so-called second clock effect in Weyl gauge theories of gravity, which are invariant both under local Poincaré transformations and local changes of scale. In particular, we revisit and extend our previous demonstration that the second clock effect does not occur in such theories, in order both to clarify our argument and confirm our original findings in response to recent counterclaims in the literature.
We constructed the conformally invariant model for scalar particle creation induced by strong gravitational fields. Starting from the "usual" hydrodynamical description of the particle motion written in the Eulerian coordinates we substituted the particle number conservation law (which enters the formalism) by "the particle creation law", proportional to the square of the Weyl tensor (following the famous result by Ya.B.Zel`dovich and A. A.Starobinsky). Then, demanding the conformal invariance of the whole dynamical system, we have got both the (Weyl)-conformal gravity and the Einstein--Hilber…
We study the case in which the five dimensional theory is the transformed Weyl$-$Yang$-$Kaluza$-$Klein gravity. The dimensionally reduced equations of motion are derived by considering an alternative form of the main equation of the theory in the coordinate basis. The conformal transformation rules are applied to the invariants. We also discuss the possible specific cases and the new Lorentz force density term, in detail.
In this paper, we use the Gauss Bonnet theorem to obtain the deflection angle by the photons coupled to Weyl tensor in a Schwarzschild black hole and Schwarzschild-like black hole in bumblebee gravity in the weak limit approximation. To do so, we first calculate the corresponding optical metrics, and then we find the Gaussian curvature to use in Gauss-Bonnet theorem, which is first done by Gibbons and Werner. Hence, in the leading order terms we show the deflection angle, that is affected by the coupling between the photon and Weyl tensor, and there is a deviation from the deflecting angle as …
I state and prove, in the context of a space having only the metrical and affine structure imposed by the geometrized version of Newtonian gravitational theory, a theorem analagous to that of Weyl's for a Lorentz manifold. The theorem says that a projective structure and a suitably defined compatible conformal structure jointly suffice for fixing the metric structure of a Newtonian spacetime model up to constant factors, and for fixing its affine structure as well. The theorem allows one to give a natural, physically compelling interpretation of the spatiotemporal geometry of a geometrized New…
Metric-affine geometry provides a non-trivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the space-time (with non-vanishing torsion and non-metricity). In this paper we study the generic form of action in this formalism, and then construct the Weyl-invariant version of this theory. It is shown that in Weitzenbock space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally the related field equations are obtained in the general case.
In this paper we consider quasi-normal modes of test scalar and electromagnetic fields in the background of a non-Schwarzschild black hole in the Einstein-Weyl gravity. We find the dependence of the modes on the new parameter $p$, which is related to the coupling constant $α$, for different values of the angular parameter $l$ and show that when $l$ is increasing, the frequencies tend to linear dependence on $p$. We have also examined the case of a massive scalar field for which we have shown that the arbitrarily long lived modes (called quasi-resonances) exist in the spectrum.
We calculate the characteristics of the "black hole shadow" for a rotating, neutral black hole in fourth-order conformal Weyl gravity. It is shown that the morphology is not significantly affected by the underlying framework, except for very large masses. Conformal gravity black hole shadows would also significantly differ from their general relativistic counterparts if the values of the main conformal gravity parameters, $γ$ and $κ$, were increased by several orders of magnitude. Such increased values for $γ$ and $κ$ are currently ruled out by gravitational phenomenology. Therefore, it is unl…
We investigate gauge invariant scalar fluctuations of the metric during inflation in a non-perturbative formalism in the framework of a recently introduced scalar-tensor theory of gravity formulated on a Weyl-Integrable geometry. We found that the Weyl scalar field can play the role of the inflaton field in this theory. As an application we study the case of a power law inflation. In this case the quasi-scale invariance of the spectrum for scalar fluctuations of the metric is achieved for determined values of the $ω$ parameter of the scalar-tensor theory. In our formalism the physical inflaton…
In this paper, we analyse the well-posedness of the initial value formulation for particular kinds of geometric scalar-tensor theories of gravity, which are based on a Weyl integrable space-time. We will show that, within a frame-invariant interpretation for the theory, the Cauchy problem in vacuum is well-posed. We will analyse the global in space problem, and, furthermore, we will show that geometric uniqueness holds for the solutions. We make contact with Brans-Dicke theory, and by analysing the similarities with such models, we highlight how some of our results can be translated to this we…
We consider modified Weyl gravity where a Dirac spinor field is nonminimally coupled to gravity. It is assumed that such modified gravity is some approximation for the description of quantum gravitational effects related to the gravitating spinor field. It is shown that such a theory contains solutions for a class of metrics which are conformally equivalent to the Hopf metric on the Hopf fibration. For this case, we obtain a full discrete spectrum of the solutions and show that they can be related to the Hopf invariant on the Hopf fibration. The expression for the spin operator in the Hopf coo…
We construct a Weyl transverse diffeomorphism invariant theory of symmetric teleparallel gravity by employing the Weyl compensator formalism. The low-energy dynamics has a single spin two gravition without a scalar degree of freedom. By construction, it is equivalent to the unimodular gravity (as well as the Einstein gravity) at the non-linear level.
We investigate the perihelion shift of planetary motion in conformal Weyl gravity using the metric of the static, spherically symmetric solution discovered by Mannheim \& Kazanas (1989). To this end we employ a procedure similar to that used by Weinberg for the Schwarzschild solution, which has also been used recently to study the solar system effects of the cosmological constant $Λ$. We show that besides the general relativistic terms obtained earlier from the Schwarzschild - de Sitter solution, the expression for the perihelion shift includes a negative contribution which arises from the lin…
In this paper, we try to consider the stability conditions of a modified gravity coupled by Weyl tensor. In this way, we indicate the suitable conditions for a successful bounce while the equation of state (EoS) parameter crosses the phantom divider for our new corrected modified gravity. In the spatially flat Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) universe, according to the ordinary version of the holographic dark energy (HDE) model, describing accelerated expansion of the universe, will be considered. Our model's stability conditions and its general properties of attractors for scalar …
We investigate (2+1)-dimensional gravity in a Weyl integrable spacetime (WIST). We show that, unlike general relativity, this scalar-tensor theory has a Newtonian limit for any dimension and that in three dimensions the congruence of world lines of particles of a pressureless fluid has a non-vanishing geodesic deviation. We present and discuss a class of static vacuum solutions generated by a circularly symmetric matter distribution that for certain values of the parameter w corresponds to a space-time with a naked singularity at the center of the matter distribution. We interpret all these re…
The conformal gravity fit to observed galactic rotation curves requires γ>0. On the other hand, conventional method for light deflection by galaxies gives a negative contribution to Schwarzschild value for γ>0, which is contrary to observation. Thus, it is very important that the contribution to bending should in principle be positive, no matter how small its magnitude is. Here we show that the Rindler-Ishak method gives a positive contribution to Schwarzschild deflection for γ>0, as desired. We also obtain the exact local coupling term derived earlier by Sereno. These results indicate that co…
We study the Weyl vector fields which can play an important role in quantum gravity. The metric obtains its dynamical content after dynamical symmetry breaking in the phase of the effective Einstein gravity which is induced by quantum Weyl corrections. In low energy regime with scalar field there is a relation between the Weyl vector fields and the torsion fields. If this condition is given to Weyl vector fields and torsions, then the Lagrangian becomes like Maxwell type.
We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We s…
Thermodynamics of the Weyl Geometric Gravity Black Holes
Chulalongkorn University（朱拉隆功大学）；National Institute of Physics and Nuclear Engineering (IFIN-HH)（国家物理与核工程研究所（IFIN-HH））；Babes-Bolyai University（巴贝什-玻尧伊大学）
It is shown that in a quadratic gravity based on Weyl's conformal geometry, not only the Einstein-Hilbert action emerges but also a Weyl gauge field becomes massive in the Weyl gauge condition, $\tilde R = k$, for a Weyl gauge symmetry within the framework of the BRST formalism. We also consider a more general gravitational theory with a scalar field in the Weyl geometry and see that the Einstein-Hilbert action can be induced from spontaneous symmetry breakdown of the Weyl gauge symmetry. Thus, it turns out that Weyl's conformal gravity is quantum mechanically equivalent to Einstein's general …
Weyl invariant gravity has been investigated as the fundamental theory of the vector inflation. Accordingly, we consider a Weyl invariant extension of Dirac-Born-Infeld type gravity. We find that an appropriate choice of the metric removes the scalar degree of freedom which is at the first sight required by the local scale invariance of the action, and then a vector field acquires mass. Then nonminimal couplings of the vector field and curvatures are induced. We find that the Dirac-Born-Infeld type gravity is a suitable theory to the vector inflation scenario.
Weyl's conformal theory of gravity is an extension of Einstein's theory of general relativity which associates metrics with 1-forms . In the case of locally integrable (closed non-exact) 1-forms the spacetime manifolds are no more simply connected. The Weil connections yield curvature tensors which satisfy the basic properties of Riemann curvature tensors. The Ricci tensors are symmetric, conformally invariant, and the Einstein tensors computed with the Weyl connections implicate a cosmological term replacing the cosmological constant by a function of spacetime, and a shear stress tensor. A to…
We construct Horava-Lifshitz gravities that are invariant under anisotropic Weyl scaling. This construction is based on an extension of the group of symmetries of healthy extended Horava-Lifshitz gravity and RFDiff invariant Horava-Lifshitz gravity. We find their Hamiltonian formulation and determine their constraint structure.
