R J Petti 2006 Class. Quantum Grav. 23 737 doi:10.1088/0264-9381/23/3/012
R J Petti
Show affiliationsHow to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle and (3) using the affine connection and the zero section to define an 'associated solder form', whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however, it violates translational equivariance/gauge symmetry. This is the natural geometric framework for Einstein–Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry.
04.20.Gz Spacetime topology, causal structure, spinor structure
04.20.Fy Canonical formalism, Lagrangians, and variational principles
04.50.-h Higher-dimensional gravity and other theories of gravity
83C75 Space-time singularities, cosmic censorship, etc.
53A15 Affine differential geometry
83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)
Issue 3 (7 February 2006)
Received 24 October 2005
Published 12 January 2006
R J Petti 2006 Class. Quantum Grav. 23 737
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