Teleparallel Gravity: An Introduction: 173 (Fundamental Theories of Physics)
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Note that in the more general case with the field equations are covariant form. Nevertheless, the theory is not local Lorentz invariant. In case of and constant scalar torsion, , GR is recovered and field equations are covariant and the theory is Lorentz invariant. The tetrad field that is stationary and has axial symmetry takes the form where , , are unknown functions of the radial coordinate, , and the azimuthal angle. Applying tetrad field 17 to the field equations of GR, , we get a system of lengthy partial nonlinear differential equations. The solution of these system has the form The metric associated with tetrad field 17 after using 18 has the following form.
Solution 18 is a solution to Einstein field equations; however the main task of the present study is to find an axially symmetric-dS solution for. To do such aim, we multiply tetrad field 17 with the following matrix: However, if becomes a general function, that is, depends on , and the calculations will be very complicated. We rewrite 16 as where is the effective energy momentum tensor.
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To find a vacuum solution within theories, it is sufficient to find the condition that makes the effective energy momentum tensor vanishing. The simplest condition that satisfies this aim is the vanishing of the scalar torsion,. The imposition of is just one simplest solution, and in principle there can exist many more; for example, when the torsion scalar is constant the effective energy momentum tensor will also be vanishing under some constraints on the form of and its first derivative. Using 6 and tetrad field 22 one can obtain. With the use of 3 , 4 , and 5 we obtain the torsion scalar as where.
The exact solution of 24 has the form Using 25 and 22 in 23 we get a solution to gravitational theory provided that. Now we are going to study the singularities of the derived solution. For this purpose we search the value at which makes and tend to zero or. This procedure may reproduce singularities corresponding to coordinate singularities.
Therefore to procedure correct singularities we are going to demonstrate the invariants. In orthodox general relativity or its modifications the invariants are the Ricci scalar, the Kretschmann scalar, or other invariants constructed from Riemann tensor and its contractions. In the teleparallel geometry of gravity we have two approaches of finding invariants.
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In the second approach one uses the solution to construct metric and then construct the Levi-Civita connection and finally calculate curvature invariants such as the Ricci and Kretschmann scalars. The comparison of the two approaches is able to show differences between curvature and torsion gravity. In teleparallel theories we mean by singularity of spacetime the singularity of the scalar concomitants of the torsion and curvature tensors.
The curvature invariants that arise from the metric solution 19 through the calculation of the Levi-Civita connection are [ 61 ] where , , and are functions of and. Here , , and are the irreducible representation of the torsion tensor: Observing the forms of Ricci and Kretschmann torsion and its irreducible representation scalars in 26 we deduce that in the Kerr-dS case there are no divergence points at either or.
In these theories, it is not easy to find exact solutions. We have rewritten the field equations of these theories in a simple form. This form enables us to show the extra terms that are responsible for the deviation from GR theory. In the nonvacuum case, those terms can be regarded as the effective energy momentum tensor and generally they depend on the scalar torsion and its derivatives. For vacuum solution one must show that the effective energy momentum tensors are vanishing. So if the torsion scalar is vanishing then it turns out that the extra terms are vanishing provided some constraints on the form of the zero function, that is, and its first derivative.
By using the Levi-Civita scalar curvature for the metric 2 one gets the result of In spite of the fact that the tetrad field contains 16 components 6 more than the metric field , TEGR is invariant under local Lorentz transformations of the tetrad due to the existence of the divergence term. This behavior is evident in 13 because is invariant under local Lorentz transformations. Instead gravity, like other theories of amended gravity, possesses extra degrees of freedom.
In fact the dynamical equations 7 are sensitive to local Lorentz transformations of the tetrad except for the case i. This implies that the dynamical equations of gravitational theories contain information not only about the evolution of the metric but also about some extra degrees of freedom exclusively associated with the tetrad that are not present in the undeformed theory [ 24 , 25 , 33 — 51 ]. For theories, the Lagrangian changes under a local Lorentz transformation as In this case the divergence term remains free inside the function damaging the principle of invariance under local Lorentz transformation.
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The loss of the local Lorentz invariance leads to a preferred global reference frame defined by the autoparallel curves of the manifold that consistently solve the dynamical equations. This means that 7 not only determines the metric but also chooses some other properties of the tetrad field. The tetrads connected by local Lorentz transformations lead to the same metric; however they are different with respect to the parallel framework.
Indexed in Science Citation Index Expanded. The role of nonmetricity in metric-affine theories of gravity - Vitagliano, Vincenzo Class. This is a first step in deriving a special solution within gravitational theories, which proves that any GR solution can be regarded as a solution in under some conditions [ 58 ]. We appreciate your feedback. The Physical Basis of Chemistry. Lecture Notes on Newtonian Mechanics. Main Results and Discussion gravitational theories are modifications of the TEGR that try to deal with the recent problems appearing in cosmology.
Because of this fundamental property of theories, when one is searching for solutions of a given symmetry it is quite difficult to do an ansatz for the tetrad field. For the metric case symmetry helps us to choose suitable coordinates to write the metric. However, this does not say much about the ansatz for the tetrad due to local Lorentz transformation [ 66 , 67 ].
Certainly, in the context of theories, the proper frame which parallelizes the spacetime for a given symmetry of the geometry must be independent of the function [ 68 ]. This work is focused on how to find the parallelization for axially symmetric solutions in theories. In particular, we want to know whether Kerr-dS geometry survives or not in gravity.
To answer this question we should find the correct ansatz to solve 7. This search is greatly facilitated by invoking the following argument concerning the survival of certain TEGR solutions [ 67 ]: An Introduction to Geometrical Physics. Fundamental Theories of Physics Book How to write a great review. The review must be at least 50 characters long. The title should be at least 4 characters long.
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August 10, Imprint: You can read this item using any of the following Kobo apps and devices: The book here proposed will be the first one dedicated exclusively to TG, and will include the foundations of the theory, as well as applications to specific problems to illustrate how the theory works.
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