On the Gauge Aspects of Gravity

Abstract

We give a short outline, in Sec.\ 2, of the historical development of the gauge idea as applied to internal ($U(1),SU(2),\dots$) and external ($R^4,SO(1,3),\dots$) symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec.\ 3, experimental results with neutron interferometers in the gravitational field of the earth, as inter- preted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge- theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs.\ 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group $R^4 \semidirect GL(4,R)$ or its Poincaré subgroup $R^4\semidirect SO(1,3)$. We briefly report on these results in Sec.\ 6 (metric-affine geometry) and in Sec.\ 7 (metric-affine field equations (\ref{zeroth}, \ref{first}, \ref{second})). Finally, in Sec.\ 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed.

51 pages LATEX, uses epsf.tex, uuencoded with 4 eps-figures, fifth figure available from the authors. Based on an invited Erice lecture in May 1995

References in corpus (7)

• Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance
• Normal frames for non-Riemannian connections
• A Unified Conformal Model for Fundamental Interactions without Dynamical Higgs Field
• Towards complete integrability of two dimensional Poincaré gauge gravity
• The renormalization group flow of the dilaton potential
• Yang--Mills Configurations from 3D Riemann--Cartan Geometry
• The Use of Computer Algebra in Maxwell's Theory

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• Cosmological signature change in Cartan Gravity with dynamical symmetry breaking
• Kaluza-Klein Reduction of a Quadratic Curvature Model
• Irreducible decompositions in metric-affine gravity models
• Teleparallel Space-Time with Defects yields Geometrization of Electrodynamics with quantized Charges
• Gravity, Cartan geometry, and idealized waywisers
• Conservation Laws on Riemann-Cartan, Lorentzian and Teleparallel Spacetimes
• An introduction to the physics of Cartan gravity
• Non-commutative Weitzenboeck geometry, gerbe modules, and WZW branes
• A New Representation of the Field Equations of Quadratic Metric-Affine Gravity
• On the equation of motion for test particles in an ambient gravitational field as the Wong equation for a generalized gauge theory
• Bosonic (p - 1)-forms in Einstein-Cartan theory of gravity
• Novel Analysis of Spinor Interactions and non-Riemannian Geometry