Tangent space symmetries in general relativity and teleparallelism

Abstract

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. “Parallel maps” are used to relate tangent space at different points. “Parallelisms” across a finite region of the manifold may be built up from them. These are used to define Weitzenböck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.