General Relativity and Weyl Geometry

Abstract

We show that the general theory of relativity can be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar-tensor gravitational theory, is invariant with respect to Weyl tranformations. A kew point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke gravitational theory. In this scenario, the gravitational field is not described by the metric tensor only, but by a combination of both the metric and a geometrical scalar field. We illustrate this point by, firstly, discussing the Newtonian limit in an arbitrary frame, and, secondly, by examining how distinct geometrical and physical pictures of the same phenomena may arise in different frames. To give an example, we discuss the gravitational spectral shift as viewed in a general Weyl frame. We further explore the analogy of general relativity with scalar-tensor theories and show how a known Brans-Dicke vacuum solution may appear as a solution of general relativity theory when reinterpreted in a particular Weyl frame. Finally, we show that the so-called WIST gravity theories are mathematically equivalent to Brans-Dicke theory when viewed in a particular frame.

LATEX, 22 pages

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Cited by in corpus (64)

• Invariant quantities in the scalar-tensor theories of gravitation
• Selected topics in scalar-tensor theories and beyond
• Unimodular Gravity vs General Relativity: A status report
• From Brans-Dicke gravity to a geometrical scalar-tensor theory
• Weyl geometry
• Gravity localization and mass hierarchy in scalar-tensor branes
• Weyl Connections and their Role in Holography
• Transformation properties and general relativity regime in scalar-tensor theories
• Some remarks about non-minimally coupled scalar field models
• Einstein manifolds with torsion and nonmetricity
• The Weyl-Cartan Gauss-Bonnet gravity
• Local conformal symmetry in non-Riemannian geometry and the origin of physical scales
• Probing Geometric Proca in Metric-Palatini Gravity with Black Hole Shadow and Photon Motion
• A note on the problem of proper time in Weyl space-time
• Space-time singularities in Weyl manifolds
• Quadratic curvature theories formulated as Covariant Canonical Gauge theories of Gravity
• Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory
• Cosmological models in Weyl geometrical scalar-tensor theory
• Schrödinger connection with selfdual nonmetricity vector in 2+1 dimensions
• The Non-Relativistic Geometric Trinity of Gravity
• Conformally invariant proper time with general non-metricity
• Induced geometry from disformal transformation
• Underdetermination in Classic and Modern Tests of General Relativity
• Present accelerated expansion of the universe from new Weyl-Integrable gravity approach
• A metric-affine version of the Horndeski theory
• Weyl relativity: A novel approach to Weyl's ideas
• The Maxwell-Chern-Simons gravity and its cosmological implications
• An action principle for the Einstein-Weyl equations
• Acceleration in Weyl integrable spacetime
• Gauge invariant fluctuations of the metric during inflation from new scalar-tensor Weyl-Integrable gravity model
• Black holes in massive conformal gravity
• Weyl geometric effects on the propagation of light in gravitational fields
• Topological Features of the Quantum Vacuum
• A correspondence between $1^{st}$ and $2^{nd}$ order formalism by a metricity constraint
• A generalized Weyl structure with arbitrary non-metricity
• Dark Matter Production in Weyl $R^2$ Inflation
• Gauging the spacetime metric -- looking back and forth a century later
• The Chameleon Effect in the Jordan Frame of the Brans--Dicke Theory
• Black hole stability in Jordan and Einstein frames
• Inflation in Weyl Scaling Invariant Gravity with $R^3$ Extensions
• Einstein-aether theory in Weyl integrable geometry
• A Noncompact Weyl-Einstein-Yang-Mills Model: A Semiclassical Quantum Gravity
• A conformal gauge theory of solids: insights into a class of electromechanical and magnetomechanical phenomena
• (2+1)-Dimensional Gravity in Weyl Integrable Spacetime
• Modified gravity from Weyl connection and the $f(R,\mathcal{A})$ extension
• Projective and amplified symmetries in metric-affine theories
• On the Cauchy Problem for Weyl-Geometric Scalar-Tensor Theories of Gravity
• Integrability and Cosmological Solutions in Einstein-aether-Weyl theory
• Tolman-Ehrenfest-Klein Law in non-Riemannian geometries
• Inhomogeneous spacetimes in Weyl integrable geometry with matter source
• Extended Cosmologies
• New exact and analytic solutions in Weyl Integrable cosmology from Noether symmetry analysis
• Refinements of the Weyl pure geometrical thick branes from information-entropic measure
• An exact five dimensional Weyl-Geometry Gauss-Bonnet Black Hole
• Pure Geometric Branes and Mass Hierarchy
• Primordial Gravitational Waves in Generalized Palatini Gravity
• Spinning Particles around Einstein-Geometric Proca AdS Compact Objects
• On the physical interpretation of non-metricity in Brans-Dicke gravity
• D-dimensional black holes in extended Gauss-Bonnet gravity
• Probing Primordial Symmetry Breaking with Cosmic Microwave Background Anisotropy
• From weak interaction to gravity
• Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach
• $f(R)$ gravity with broken Weyl gauge symmetry, cosmological backreaction, and its effects on CMB anisotropy
• Behavior of vacuum and naked singularity under smooth gauge function in Lyra geometry