Weyl geometry

Abstract

We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under dilatations, but not the full conformal group.

27 pp Minor notational changes, additional example and conclusion added

References in corpus (2)

• Aspects of Quadratic Gravity
• The roots of scalar-tensor theory: an approximate history

Cited by in corpus (37)

• Comparing Equivalent Gravities: common features and differences
• Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces in Weyl-type $f(Q,T)$ gravity
• Cosmological Inflation in f(Q, T) Gravity
• Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential
• Weyl gauge symmetry and its spontaneous breaking in Standard Model and inflation
• Gauging scale symmetry and inflation: Weyl versus Palatini gravity
• Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation
• Standard Model in Weyl conformal geometry
• Power-law cosmology in Weyl-type $f(Q,T)$ gravity
• Weyl gauge theories of gravity do not predict a second clock effect
• Einstein manifolds with torsion and nonmetricity
• Nonmetricity theories and aspects of gauge symmetry
• Schrödinger connection with selfdual nonmetricity vector in 2+1 dimensions
• The Non-Relativistic Geometric Trinity of Gravity
• Note on the absence of the second clock effect in Weyl gauge theories of gravity
• An action principle for the Einstein-Weyl equations
• Schrödinger Connections: From Mathematical Foundations Towards Yano-Schrödinger Cosmology
• Conformal Metric-Affine Gravities
• Action Principle for Scale Invariance and Applications (Part I)
• Perturbation Spectra of Warm Inflation in $f(Q,T)$ Gravity
• Phenomenological signatures of gauge invariant theories of gravity with vectorial nonmetricity
• General relativity as a biconformal gauge theory
• Modified gravity from Weyl connection and the $f(R,\mathcal{A})$ extension
• Extended Geometric Trinity of Gravity
• Projective and amplified symmetries in metric-affine theories
• Axiomatic de Sitter quantum Yang-Mills theory with color confinement and mass gap
• The Scale Invariant Vacuum Paradigm: Main Results and Current Progress Review (Part II)
• Elucidating the Dark Energy and Dark Matter Phenomena Within the Scale-Invariant Vacuum (SIV) Paradigm
• Logarithmic and Strong Coupling Models in Weyl-Type $f(Q,T)$ Gravity
• Quantum de Sitter geometry
• SO(4; 2) and derivatively coupled dRGT massive gravity
• Probing Primordial Symmetry Breaking with Cosmic Microwave Background Anisotropy
• Yang-Mills sources in biconformal gravity
• Internal symmetry in Poincare gauge gravity
• Minisuperspace Cosmology in Extended Geometric Trinity of Gravity
• Geometric formulation of $k$-essence and late-time acceleration
• Inflation and primordial fluctuations in $f(Q,T)$ gravity