Background Geometry in Gauge Gravitation Theory

Dirac fermion fields are responsible for spontaneous symmetry breaking in gauge gravitation theory because the spin structure associated with a tetrad field is not preserved under general covariant transformations. Two solutions of this problem can be suggested. (i) There exists the universal spin structure $S\to X$ such that any spin structure $S^h\to X$ associated with a tetrad field $h$ is a subbundle of the bundle $S\to X$. In this model, gravitational fields correspond to different tetrad (or metric) fields. (ii) A background tetrad field $h$ and the associated spin structure $S^h$ are fixed, while gravitational fields are identified with additional tensor fields q^\la{}_\m describing deviations \wt h^\la_a=q^\la{}_\m h^\m_a of $h$. One can think of \wt h as being effective tetrad fields. We show that there exist gauge transformations which keep the background tetrad field $h$ and act on the effective fields by the general covariant transformation law. We come to Logunov's Relativistic Theory of Gravity generalized to dynamic connections and fermion fields.Comment: 12 pages, LaTeX, no figure

Should there be a spin-rotation coupling for a Dirac particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We compute the energy operator in the inertial and the rotating frame, using three tetrad fields: one for each of two frameworks proposed to select the tetrad field so as to solve this non-uniqueness problem, and one proposed by Ryder. We find that Mashhoon's term is there if the tetrad rotates as does the reference frame --- but then it is also there in the energy operator for the inertial frame. In fact, the Dirac Hamiltonian operators in two reference frames in relative rotation, but corresponding to the same tetrad field, differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.Comment: 29 pages in standard 12pt. V2: Introduction reinforced. New Section 3 on the dependences of the Hamiltonian on the reference frame and on the tetrad field. New reference

Gauge aspect of tetrad field in gravity

In general relativity, an inertial frame can only be established in a small region of spacetime, and a locally inertial frame is mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform a local Lorentz transformation in an inertial frame, and there exists freedom to choose the locally inertial frame at each spacetime. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom, corresponding to the Lorentz gauge $\partial^\mu\mathcal{A}_\mu=0$ and Coulomb gauge $\partial^i\mathcal{A}_i=0$ in electrodynamics, the Lorentz gauge and Coulomb gauge for the tetrad field are proposed in the present work. Moreover, properties of the Lorentz gauge and Coulomb gauge for tetrad field are discussed, which show the similarities to those in electromagnetic field.Comment: 4 pages, no figure, comments are welcome

Regularization of f(T)f(T) gravity theories and local Lorentz transformation

We regularized the field equations of $f(T)$ gravity theories such that the effect of Local Lorentz Transformation (LLT), in the case of spherical symmetry, is removed. A "general tetrad field", with an arbitrary function of radial coordinate preserving spherical symmetry is provided. We split that tetrad field into two matrices; the first represents a LLT, which contains an arbitrary function, the second matrix represents a proper tetrad field which is a solution to the field equations of $f(T)$ gravitational theory, (which are not invariant under LLT). This "general tetrad field" is then applied to the regularized field equations of $f(T)$. We show that the effect of the arbitrary function which is involved in the LLT invariably disappears.Comment: 12 page

Tetrads in Geometrodynamics

A new tetrad is introduced within the framework of geometrodynamics for non-null electromagnetic fields. This tetrad diagonalizes the electromagnetic stress-energy tensor and allows for maximum simplification of the expression of the electromagnetic field. The Einstein-Maxwell equations will also be simplified