James Thomas Wheelerhttps://works.bepress.com/james_wheeler/Recent works by James Thomas Wheeleren-usCopyright (c) 2019 All rights reserved.Wed, 13 Jun 2018 07:00:00 +00003600Weyl Geometryhttps://works.bepress.com/james_wheeler/150/<p>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 local dilatations, but not the full conformal group.</p>
Wed, 13 Jun 2018 07:00:00 +0000https://works.bepress.com/james_wheeler/150/Selected articlesWeyl Geometryhttps://works.bepress.com/james_wheeler/149/<div class="line" id="line-35"><span style="color: rgb(51, 51, 51); font-family: Georgia, serif; font-size: 17px;">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 local dilatations, but not the full conformal group.</span></div>James Thomas WheelerWed, 13 Jun 2018 00:00:00 +0000https://works.bepress.com/james_wheeler/149/Selected articlesVariation of the biconformal actionhttps://works.bepress.com/james_wheeler/148/<div class="line" id="line-35">The biconformal action, expressed using differential forms, requires a number of techniques to vary. Here we describe in detail the variation of the biconformal action, presenting those techniques along with some useful nomenclature.</div><div class="line" id="line-45"><br></div>James Thomas WheelerMon, 22 Jan 2018 00:00:00 +0000https://works.bepress.com/james_wheeler/148/Gravitational gauge theory: biconformalWeyl geometryhttps://works.bepress.com/james_wheeler/147/<div class="line" id="line-35"><span style="font-family: SFRM0900; font-size: 9pt;">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. </span></div>James Thomas WheelerMon, 09 Jan 2017 00:00:00 +0000https://works.bepress.com/james_wheeler/147/Gravitational gauge theory: Gravity in Weyl geometryDynamical spacetime symmetryhttps://works.bepress.com/james_wheeler/125/<p>According to the Coleman-Mandula theorem, any gauge theory of gravity combined with an internal symmetry based on a Lie group must take the form of a direct product in order to be consistent with basic assumptions of quantum field theory. However, we show that an alternative gauging of a <em>simple</em> group can lead <em>dynamically</em> to a spacetime with compact internal symmetry. The biconformal gauging of the conformal symmetry of n-dimensional Euclidean space doubles the dimension to give a symplectic manifold. Examining one of the Lagrangian submanifolds in the flat case, we find that in addition to the expected SO(n) connection and curvature, the solder form necessarily becomes Lorentzian. General coordinate invariance gives rise to an SO(n−1,1) connection on the spacetime. The principal fiber bundle character of the original SO(n) guarantees that the two symmetries enter as a direct product, in agreement with the Coleman-Mandula theorem.</p>
Fri, 15 Apr 2016 07:00:00 +0000https://works.bepress.com/james_wheeler/125/Selected articlesLinearized Conformal gravityhttps://works.bepress.com/james_wheeler/118/<div class="line" id="line-5">We examine the linearization of Weyl conformal gravity, showing that the only solutions are also solutions to linearized general relativity.</div><div class="line" id="line-7"><br></div>James Thomas WheelerThu, 28 Jan 2016 00:00:00 +0000https://works.bepress.com/james_wheeler/118/Gravitational gauge theory: Weyl gravityTime in conformal General Relativityhttps://works.bepress.com/james_wheeler/119/<div class="line" id="line-5"><span style="font-family: Calibri; font-size: 18pt;"> Gravity theories based on the conformal group give general relativity augmented by local dilatational covariance. In one of these theories, biconformal gravity, the dimension of the original space doubles, giving a symplectic manifold with conformal general relativity on a lagrangian submanifold. </span></div><div class="line" id="line-7"><span style="font-family: Calibri; font-size: 18pt;"> The restriction of the Killing form to the biconformal space is non-degenerate and signature-changing. This opens several possibilities, from time arising only within solutions to a Euclidean gravity theory, to novel approaches to the AdS/CFT correspondence, or the natural emergence of an SO(n) Yang-Mills field on gravitating spacetime. We briefly discuss some of our investigations into these options.</span></div>James Thomas WheelerFri, 02 Oct 2015 00:00:00 +0000https://works.bepress.com/james_wheeler/119/Gravitational gauge theory: biconformalA solution in Weyl gravity with planar symmetryhttps://works.bepress.com/james_wheeler/117/<p>We solve the Bach equation for Weyl gravity for the case of a static metric with planar symmetry. The solution is not conformal to the solution to the corresponding Einstein equation.</p>
James Thomas WheelerSat, 23 May 2015 00:00:00 +0000https://works.bepress.com/james_wheeler/117/Gravitational gauge theory: Weyl gravityThe spacetime co-torsion in torsion-free biconformal spaceshttps://works.bepress.com/james_wheeler/116/<p>In preceding studies, [TR Gamma minus, TR Gamma plus] we showed that the solution for the connection of flat biconformal space also solves the curved space field equations for the torsion and co-torsion. We continued this investigation with an attempt to solve the full set of torsion and co-torsion field equations, with only the assumption of vanishing torsion and the known form of the metric. We successfully reduced the torsion equations to a single equation. Here, we reduce that equation to its essential degrees of freedom. We find that the spacetime co-torsion is entirely determined by the scale vector and certain contractions of the totally symmetric part of the difference of the symmetric connections.</p>
James Thomas WheelerMon, 09 Feb 2015 00:00:00 +0000https://works.bepress.com/james_wheeler/116/Technical ReportsVariation of the Weyl actionhttps://works.bepress.com/james_wheeler/115/<p>We show how to vary the fourth order Weyl gravity action to derive the Bach equation.</p>
James Thomas WheelerWed, 04 Feb 2015 00:00:00 +0000https://works.bepress.com/james_wheeler/115/Gravitational gauge theory: Weyl gravity