Weyl Geometry Research Articles

A note on Weyl gauge symmetry in gravity

Abstract A scale invariant theory of gravity, containing at most two derivatives, requires, in addition to the Riemannian metric, a scalar field and (or) a gauge field. The gauge field is usually used to construct the affine connection of Weyl geometry. In this note, we incorporate both the gauge field and the scalar field to build a generalised scale invariant Weyl affine connection. The Ricci tensor and the Ricci scalar made out of this generalised Weyl affine connection contain, naturally, kinetic terms for the scalar field and the gauge field. This provides a geometric interpretation for these terms. It is also shown that scale invariance in the presence of a cosmological constant and mass terms is not completely lost. It becomes a duality transformation relating various fields.

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Probing Weyl-type [formula omitted] gravity: Cosmological implications and constraints

In this paper, we investigate the cosmological implications and constraints of Weyl-type f(Q,T) gravity. This theory introduces a coupling between the non-metricity Q and the trace T of the energy–momentum tensor, using the principles of proper Weyl geometry. In this geometry, the scalar non-metricity Q, which characterizes the deviations from Riemannian geometry, is expressed in its standard Weyl form ∇μgαβ=−wμgαβ and is determined by a vector field wμ. To study the implications of this theory, we propose a deceleration parameter with a single unknown parameter χ, which we constrain by using the latest cosmological data. By solving the field equations derived from Weyl-type f(Q,T) gravity, we aim to understand the behavior of the energy conditions within this framework. In the present work, we consider two well-motivated forms of the function f(Q,T): (i) the linear model represented by f(Q,T)=αQ+β6κ2T, and (ii) the coupling model represented by f(Q,T)=γ6H02κ2QT, where α, β, and γ are free parameters. Here, κ2=116πG represents the gravitational coupling constant. In both of the models considered, the strong energy condition is violated, indicating consistency with the present accelerated expansion. However, the null, weak, and dominant energy conditions are satisfied in these models.

Quintessence in the Weyl-Gauss-Bonnet model

Quintessence models have been widely examined in the context of scalar-Gauss-Bonnet gravity, a subclass of Horndeski's theory, and were proposed as viable candidates for Dark Energy. However, the relatively recent observational constraints on the speed of gravitational waves c GW have resulted in many of those models being ruled out because they predict c GW ≠ c generally. While these were formulated in the metric formalism of gravity, we put forward a new quintessence model with the scalar-Gauss-Bonnet action but in Weyl geometry, where the connection is not metric compatible. We find the fixed points of the dynamical system under some assumptions and determine their stability via linear analysis. The past evolution of the Universe can be reproduced correctly, but the late Universe constraints on c GW are grossly violated. Moreover, at these later stages tensor modes suffer from the gradient instabilities. We also consider the implications of imposing an additional constraint c GW = c, but this does not lead to evolution that is consistent with cosmological observations.

Cosmological observational constraints on the power law f(Q) type modified gravity theory

In modern cosmology, the curiosity of ultimately understanding the nature of the dark energy controlling the recent acceleration of the Universe motivates us to explore its properties by using some novel approaches. In this work, to explore the properties of dark energy we adopt the modified f(Q) gravity theory, where the non-metricity scalar Q, emerging from Weyl geometry, plays the dynamical role. For the function f(Q) we adopt the functional form f(Q)=Q+6γH02(Q/Q0)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(Q)=Q+ 6\\gamma \\,H_0^2(Q/Q_0)^n$$\\end{document}, where n,γ,H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n,\\, \\gamma ,\\, H_0$$\\end{document} and Q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_0$$\\end{document} are constants. Then, we test our constructed model against the various observational datasets, such as the Hubble, and the Pantheon+SHOES samples, and their combined sample, through the Markov Chain Monte Carlo (MCMC) statistical analysis. We also employ the parameter estimation technique to constrain the free parameters of the model. In addition, we use the constrained values of the model parameters to explore a few implications of the cosmological model. A detailed comparison of the predictions of our model with the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model is also performed. In particular, we discuss in detail some cosmographic parameters, like the deceleration, the jerk, and the snap parameters, as well as the behavior of the dark energy and matter energy densities to see the evolution of various energy/matter profiles. The Om diagnostics is also presented to test the dark energy nature of our model, as compared to the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM paradigm. Our findings show that the considered version of the non-metric f(Q) type modified gravity theory, despite some differences with respect to the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM paradigm, can still explain the current observational results on the cosmological parameters, and provide a convincing and consistent account for the accelerating expansion of the Universe.

Weyl conformal geometry vs Weyl anomaly

Weyl conformal geometry is a gauge theory of scale invariance that naturally brings together the Standard Model (SM) and Einstein gravity. The SM embedding in this geometry is possible without new degrees of freedom beyond SM and Weyl geometry, while Einstein gravity is generated by the broken phase of this symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl gauge boson becomes massive and decouples, as discussed in the past [1–3]. However, Weyl anomaly could break explicitly this gauge symmetry, hence we study it in Weyl geometry. We first note that in Weyl geometry metricity can be restored with respect to a new differential operator ( hat{nabla} ) that also enforces simultaneously a Weyl-covariant formulation. This leads to a metric-like Weyl gauge invariant formalism that enables one to do quantum calculations directly in Weyl geometry, rather than use a Riemannian (metric) geometry picture. The result is the Weyl-covariance in d dimensions of all geometric operators ( hat{R} , etc) and of their derivatives ( hat{nabla} μ hat{R} , etc), including the Euler-Gauss-Bonnet term. A natural (geometric) Weyl-invariant dimensional regularisation of quantum corrections exists and Weyl gauge symmetry is then maintained and manifest at the quantum level. This is related to a non-trivial current of this symmetry, the divergence of which cancels the trace of the energy-momentum tensor. The “usual” Weyl anomaly and Riemannian geometry are recovered in the (spontaneously) broken phase. The relation to holographic Weyl anomaly is discussed.

Projective transformations in metric-affine and Weylian geometries

We discuss generalizations of the notions of projective transformations acting on affine model of Riemann–Cartan and Riemann–Cartan–Weyl gravity which preserve the projective structure of the light-cones. We show how the invariance under some projective transformations can be used to recast a Riemann–Cartan–Weyl geometry either as a model in which the role of the Weyl gauge potential is played by the torsion vector, which we call torsion-gauging, or as a model with traditional Weyl (conformal) invariance.

Conformal invariance and phenomenology of particle creation: Weyl geometry vs. Riemannian geometry

Conformal invariance and phenomenology of particle creation: Weyl geometry vs. Riemannian geometry

Конформная инвариантность и феноменология рождения частиц: сравнение геометрии Вейля с римановой геометрией

На примере действия идеальной жидкости с переменным числом частиц изучено феноменологическое описание процессов рождения частиц на фоне сильных внешних полей, в частности гравитационного и скалярного полей. Эта модель рассматривается как для геометрии Вейля, так и для римановой геометрии. Новый инвариант, связанный с взаимодействием вектора Вейля с частицами, инкорпорирован в действие материи. Показана конформная инвариантность слагаемого в действии материи, ответственного за рождение частиц.

Weyl Geometry, Particle Production, and Induced Gravity

Weyl Geometry, Particle Production, and Induced Gravity

Weyl-ambient geometries

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal R+-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.

Compact stellar structures in Weyl geometric gravity

We consider the structure and physical properties of specific classes of neutron, quark, and Bose-Einstein condensate stars in the conformally invariant Weyl geometric gravity theory. The basic theory is derived from the simplest conformally invariant action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To keep the theory conformally invariant the trace condition is imposed on the matter energy-momentum tensor. The field equations are derived by varying the action with respect to the metric tensor, Weyl vector field and scalar field. By adopting a static spherically symmetric interior geometry, we obtain the field equations, describing the structure and properties of stellar objects in Weyl geometric gravity. The solutions of the field equations are obtained numerically, for different equations of state of the neutron and quark matter. More specifically, constant density stellar models, and models described by the stiff fluid, radiation fluid, quark bag model, and Bose-Einstein condensate equations of state are explicitly constructed numerically in both general relativity and Weyl geometric gravity, thus allowing an in depth comparison between the predictions of these two gravitational theories. As a general result it turns out that for all the considered equations of state, Weyl geometric gravity stars are more massive than their general relativistic counterparts. As a possible astrophysical application of the obtained results we suggest that the recently observed neutron stars, with masses in the range of $2{M}_{\ensuremath{\bigodot}}$ and $3{M}_{\ensuremath{\bigodot}}$, respectively, could be in fact conformally invariant Weyl geometric neutron or quark stars.

Dark matter as a Weyl geometric effect

We investigate the possibility that the observed behavior of test particles outside galaxies, which is usually explained by assuming the existence of dark matter, is the result of the dynamical evolution of particles in a Weyl type geometry, and its associated conformally invariant Weyl geometric quadratic gravity. As a first step in our investigations we write down the simplest possible conformally invariant gravitational action, constructed in Weyl geometry, and containing the Weyl scalar, and the strength of the Weyl vector only. By introducing an auxiliary scalar field, the theoretical model can be reformulated in the Riemann geometry as scalar-vector-tensor theory, containing a scalar field, and the Weyl vector, respectively. The field equations of the theory are derived in the metric formalism, in the absence of matter. A specific static, spherically symmetric model, in which the Weyl vector has only a radial component, is considered. In this case, an exact analytic solution of the gravitational field equations can be obtained. The behavior of the galactic rotation curves is also considered in detail, and it is shown that an effective geometric mass term, with an associated density profile, can also be introduced. Three particular cases, corresponding to some specific functional forms of the Weyl vector, are also investigated. A comparison of the model with a selected sample of galactic rotation curves is also performed when an explicit breaking of conformal invariance is introduced, which allows the fix of the numerical values of the free parameters of the model. Our results show that Weyl geometric models can be considered as a viable theoretical alternative to the dark matter paradigm.

Non-metric geometry as the origin of mass in gauge theories of scale invariance

We discuss gauge theories of scale invariance beyond the Standard Model (SM) and Einstein gravity. A consequence of gauging this symmetry is that their underlying 4D geometry is non-metric (nabla _mu g_{alpha beta }!not =!0). Examples of such theories are Weyl’s original quadratic gravity theory and its Palatini version. These theories have spontaneous breaking of the gauged scale symmetry to Einstein gravity. All mass scales have a geometric origin: the Planck scale (M_p), cosmological constant (Lambda ) and the mass of the Weyl gauge boson (omega _mu ) of scale symmetry are proportional to a scalar field vev that has an origin in the (geometric) {tilde{R}}^2 term in the action. With omega _mu of non-metric geometry origin, the SM Higgs field also has a similar origin, generated by Weyl boson fusion in the early Universe. This appears as a microscopic realisation of “matter creation from geometry” discussed in the thermodynamics of open systems applied to cosmology. Unlike in local scale invariant theories (with no omega _mu present) with an underlying pseudo-Riemannian geometry, in our case: (1) there are no ghosts and no additional fields beyond the SM and underlying Weyl or Palatini geometry, (2) the cosmological constant is positive and is small because gravity is weak, (3) the Weyl or Palatini connection shares the Weyl (gauge) symmetry of the action, and: (4) there exists a non-trivial, conserved Weyl current of this symmetry. An intuitive picture of non-metricity and its relation to mass generation is also provided from a solid state physics perspective where it is common and is associated with point defects (metric anomalies) of the crystalline structure.

Weyl covariance, second clock effect and proper time in theories of symmetric teleparallel gravity

Just after Weyl’s paper (Weyl in Gravitation und Elektrizität, Sitzungsber. Preuss. Akad., Berlin, 1918) Einstein claimed that a gravity model written in a spacetime geometry with non-metricity suffers from a phenomenon, the so-called second clock effect. We give a new prescription of parallel transport of a vector tangent to a curve which is invariant under both of local general coordinate and Weyl transformations in order to remove that effect. Thus since the length of tangent vector does not change during parallel transport along a closed curve in spacetimes with non-metricity, a second clock effect does not appear in general, not only for the integrable Weyl spacetime. We have specially motivated the problem from the point of view of symmetric teleparallel (or Minkowski–Weyl) geometry. We also conclude that if nature respects Lorentz symmetry and Weyl symmetry, then the simplest geometry in which one can develop consistently alternative gravity models is the symmetric teleparallel geometry; Q_{mu nu }ne 0, ; T^mu =0, ; R^mu {}_nu =0. Accordingly we discuss the proper time, the orbit equation of a spinless test body and the Lagrangian for symmetric teleparallel gravity.

Cosmological particle creation in Weyl geometry

We investigated the possibility of the homogeneous and isotropic cosmological solution in Weyl geometry, which differs from the Riemannian geometry by adding the so called Weyl vector. The Weyl gravity is obtained by constructing the gravitational Lagrangian both to be quadratic in curvatures and conformal invariant. It is found that such solution may exist provided there exists the direct interaction between the Weyl vector and the matter fields. Assuming the matter Lagrangian is that of the perfect fluid, we found how such an interaction can be implemented. Due to the existence of quadratic curvature terms and the direct interaction the perfect fluid particles may be created straight from the vacuum, and we found the expression for the rate of their production which appeared to be conformal invariant. In the case of creating the Universe ‘from nothing’ in the vacuum state, we investigated the problem, whether this vacuum may persist or not. It is shown that the vacuum may persist with respect to producing the non-dust matter (with positive pressure), but cannot resist to producing the dust particles. These particles, being non-interactive, may be considered as the candidates for dark matter.

BRST formalism of Weyl conformal gravity

We present the BRST formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge-fixing for the Weyl invariance we find that there is a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symmetry and account for how the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry. The corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.

Palatini formulation of the conformally invariant fleft( R,L_mright) gravity theory

We investigate the field equations of the conformally invariant models of gravity with curvature-matter coupling, constructed in Weyl geometry, using the Palatini formalism. We consider the case in which the Lagrangian is given by the sum of the square of the Weyl scalar, the strength of the field associated to the Weyl vector, and a conformally invariant geometry-matter coupling term, constructed from the matter Lagrangian and the Weyl scalar. After substituting the Weyl scalar in terms of its Riemannian counterpart, the quadratic action is defined in Riemann geometry and involves a nonminimal coupling between the Ricci scalar and the matter Lagrangian. For the sake of generality, a more general Lagrangian, in which the Weyl vector is nonminimally coupled with an arbitrary function of the Ricci scalar, is also considered. By varying the action independently with respect to the metric and the connection, the independent connection can be expressed as the Levi-Civita connection of an auxiliary Ricci scalar- and Weyl vector-dependent metric, which is related to the physical metric by means of a conformal transformation. The field equations are obtained in both the metric and the Palatini formulations. The cosmological implications of the Palatini field equations are investigated for three distinct models corresponding to different forms of the coupling functions. A comparison with the standard Lambda CDM model is also performed, and we find that the Palatini type cosmological models can give an acceptable description of the observations.

Coupling matter and curvature in Weyl geometry: conformally invariant fleft( R,L_mright) gravity

We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form L_m{tilde{R}}^2, where L_m is the ordinary matter Lagrangian, and {tilde{R}} is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing {tilde{R}}^2 with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant fleft( R,L_mright) type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar. We obtain the gravitational field equations of the theory, as well as the energy–momentum balance equations. The divergence of the matter energy–momentum tensor does not vanish, and an extra force, depending on the Weyl vector, and matter Lagrangian is generated. The thermodynamic interpretation of the theory is also discussed. The generalized Poisson equation is derived, and the Newtonian limit of the equations of motion is considered in detail. The perihelion precession of a planet in the presence of an extra force is also considered, and constraints on the magnitude of the Weyl vector in the Solar System are obtained from the observational data of Mercury. The cosmological implications of the theory are also considered for the case of a flat, homogeneous and isotropic Friedmann–Lemaitre–Robertson–Walker geometry, and it is shown that the model can give a good description of the observational data for the Hubble function up to a redshift of the order of zapprox 3.

The Constitution of Weyl’s Pure Infinitesimal World Geometry

The Constitution of Weyl’s Pure Infinitesimal World Geometry