Translational Spacetime Symmetries in Gravitational Theories
arXiv:1804.06730 · doi:10.1088/0264-9381/23/3/012
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.
16 pages, 1 table, 1 figure. Published in Classical and Quantum Gravity, 23 (2006) p 737-751. This version makes two changes to the version published in Class. Quantum Grav. (1) It corrects equation (10) in section 2.7 by including a term that was omitted. (2) It has a a table of contents at the end that was not included in Class. Quantum Grav