Constructing conformally invariant equations by using Weyl geometry

Abstract

We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined by the metric tensor and the Weyl vector $W$, it becomes equivalent to a Riemann space when $W$ is gradient. ii) Any homogeneous differential equation written in a Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to a Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrates the efficiency of the present method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk-Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover this equation reduces to the Drew-Gegenberg and Deser-Nepomechie equations in respectively Minkowski and de Sitter spaces.

13 pages, no figures

References in corpus (9)

• Bosonic conformal higher--spin fields of any symmetry
• On Conformal, SL(4,R) and Sp(8,R) Symmetries of 4d Massless Fields
• Tractors, Mass and Weyl Invariance
• Conformally covariant quantization of Maxwell field in de Sitter space
• Conformal invariant powers of the Laplacian, Fefferman-Graham ambient metric and Ricci gauging
• Conformal use of retarded Green's functions for the Maxwell field in de Sitter space
• Conformal invariance: from Weyl to SO(2,d)
• The Eastwood-Singer gauge in Einstein spaces
• SO(2,4)-covariant quantization of the Maxwell field in a conformally flat space

Cited by in corpus (8)

• Black Holes in Bi-scalar Extensions of Horndeski Theories
• Partially Massless Graviton on Beyond Einstein Spacetimes
• Conformally invariant wave equation for a symmetric second rank tensor ("spin-2") in d-dimensional curved background
• Scale and Conformal Invariance on (A)dS
• Refinements of the Weyl pure geometrical thick branes from information-entropic measure
• Conformally Invariant Spinorial Equations in Six Dimensions
• Local(ish) gravity theories in conformal superspace
• A spherically symmetric gravitational solution of nearly conformally flat metric~measure~space