Commutator Structure of the Riemannian Connection


The well known commutators of Lie algebra in the Lorentz and Poincare groups of Minkowski spacetime are those of the boost, rotation and spacetime translation generators (the latter discovered by Wigner). These are infinitesimal generators. In a more general spacetime of any dimension, the connection must have a commutator symmetry made up of a more general Lie group generator. The same result holds for the Cartan spin connection. The general law is that any object is antisymmetric in mu and nu and any such object is generated by a commutator. One can therefore conclude that the internal structure of the Riemannian connection is the commutator of covariant derivatives as follows: [D sub mu, D sub nu] V sup rho = – gamma sup lambda sub mu nu D sub lambda V sup rho + ………….. The covariant derivatives are operators of a Lie group, called the Einstein group. For example, for each lambda, (or each a in the spin connection), the gamma mu nu or omega mu nu can be rotation generators. This has been discussed in an appendix of review paper 100.


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