To those familiar with general relativity, it is highly reminiscent of the tetrad formalism although there are significant conceptual differences.
The equivalence principle is not assumed, but instead follows from the fact that the gauge covariant derivative is minimally coupled.
GTG was first proposed by Lasenby, Doran, and Gull in 1998[1] as a fulfillment of partial results presented in 1993.
A displacement by some arbitrary function f gives rise to the position-gauge field defined by the mapping on its adjoint, which is linear in its first argument and a is a constant vector.
Similarly, a rotation by some arbitrary rotor R gives rise to the rotation-gauge field We can define two different covariant directional derivatives or with the specification of a coordinate system where × denotes the commutator product.
For those more familiar with general relativity, it is possible to define a metric tensor from the position-gauge field in a manner similar to tetrads.
The Greek index μ is raised or lowered by multiplying and contracting with the spacetime's metric tensor.
The parenthetical Latin index (a) is a label for each of the four tetrads, which is raised and lowered as if it were multiplied and contracted with a separate Minkowski metric tensor.
For example, in the study of a point mass, the choice of a "Newtonian gauge" yields a solution similar to the Schwarzschild metric in Gullstrand–Painlevé coordinates.