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arXiv · 1804.06730

Translational Spacetime Symmetries in Gravitational Theories

R. J. Petti
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Abstract

How 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.

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R. J. Petti. 2018-04-16. Translational Spacetime Symmetries in Gravitational Theories. https://doi.org/10.1088/0264-9381%2F23%2F3%2F012

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