Geometrical formulation of relativistic mechanics

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler–Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad hoc formulation, and we briefly analyze the relativistic Liénard oscillator and the spacetime it implies.

Fixing redundant degrees of teleparallel equivalent of general relativity

It is well known that the field equation of teleparallel theory which is equivalent to general relativity completely agrees with the field equations of general relativity. However, teleparallel equivalent of general relativity has six redundant degrees of freedom which spoil the true physics. These extra degrees are related to the local Lorentz transformation. In this study, we give three different tetrad fields having cylindrical symmetry and depend only on the radial coordinate. One of these tetrads contains an arbitrary function, which is responsible to reproduce the other solutions, which come from local Lorentz transformation. We show by explicate calculations that this arbitrary function spoils the calculations of the conserved charges. We formulate a skew-symmetric tensor whose vanishing value puts a constraint on this arbitrary function. This constraint fixed the redundant degrees of freedom which characterize the teleparallel equivalent of general relativity.

Regularization off(T)Gravity Theories and Local Lorentz Transformation

We regularized the field equations off(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, and the second matrix represents a proper tetrad field which is a solution to the field equations off(T)gravitational theory (which are not invariant under LLT). This “general tetrad field” is then applied to the regularized field equations off(T). We show that the effect of the arbitrary function which is involved in the LLT invariably disappears.

Axially Symmetric-dS Solution in Teleparallelf(T)Gravity Theories

We apply a tetrad field with six unknown functions to Einstein field equations. Exact vacuum solution, which represents axially symmetric-dS spacetime, is derived. We multiply the tetrad field of the derived solution by a local Lorentz transformation which involves a generalization of the angleϕand get a new tetrad field. Using this tetrad, we get a differential equation from the scalar torsionT=TαμνSαμν. Solving this differential equation we obtain a solution to thef(T)gravity theories under certain conditions on the form off(T)and its first derivatives. Finally, we calculate the scalars of Riemann Christoffel tensor, Ricci tensor, Ricci scalar, torsion tensor, and its contraction to explain the singularities associated with this solution.

COMMENTS ON THE TETRAD (VIELBEINS)

We want to correct the misunderstandings on the tetrad (or vielbeins in general) appeared in many textbooks or review articles. The tetrad should be defined without any condition. eμa = ∂μXa with local Lorentz coordinates Xa is wrong in many senses: it gives the condition ∂μeνa = ∂νeμa, which leads us to the trivial result that the cyclic coefficients vanish identically and to the null Riemannian tensor. Also [Formula: see text] is not scalar under the local Lorentz transformation etc. We show how these deficits are remedied by the correct definition, eμa = DμZa with local (anti) de Sitter coordinates ZA.