Tetrad Field Research Articles

Renormalization of the Einstein–Cartan theory in first-order form

We examine the Einstein–Cartan (EC) theory in first-order form, which has a diffeomorphism as well as a local Lorentz invariance. We study the renormalizability of this theory in the framework of the Batalin–Vilkovisky formalism, which allows for a gauge invariant renormalization. Using the background field method, we discuss the gauge invariance of the background effective action and analyze the Ward identities which reflect the symmetries of the EC theory. As an application, we compute, in a general background gauge, the self-energy of the tetrad field at one-loop order.

Discrete Z4 Symmetry in Quantum Gravity

We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=−ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=−R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation.

Dark energy and dark matter as a kinematic-electromagnetic Abelian gauge effect

In this paper, we will discuss an alternative theory for the origin of dark matter and dark energy based on the new concept of tetrad gauge states of spacetime. The new tetrads already introduced new physics since it has been proved that local electromagnetic gauge transformations can boost the local tetrad fields in a four-dimensional curved Lorentz spacetime. It is within this context that we will present a different qualitative explanation for the so-called dark matter and dark energy and new possible topological solutions of a galactic nature for Einstein–Maxwell spacetimes in its relation to dark matter and new possible topological solutions of a cosmological nature for Einstein–Maxwell spacetimes in its relation to dark energy.

Chronology Protection in f(T) Gravity: The Case of Gott’s Pair of Moving Cosmic Strings

As a consequence of the spacetime structure, defined by the tetrad field instead of the metric tensor alone, f(T) gravity seems to harbor its own chronology protection agency. When Gott’s pair of moving cosmic strings is considered, it is shown that the requirement of having a global parallelization—i.e., a global smooth field of tetrads– drastically restricts the form of the tetrads on the junction surface between the two strings. The junction conditions on the tetrad field are satisfied only if the corresponding boosts needed to put the strings in motion are null on the matching surface. This seems to throw overboard Gott’s construction from the outset without the need for analyzing the divergence of the expectation value of the energy–momentum tensor on the Cauchy horizon, evading in this way bothersome quarrels concerning the choice of vacuum.

One-loop effective action in chiral Einstein–Cartan gravity

In chiral Einstein–Cartan gravity, a new gauge fixing procedure is implemented recently, leading to a very economical perturbation expansion of the action. Using this formulation and the relevant gauge fixing, we develop the ghost Lagrangian on an arbitrary Einstein background using the Becchi-Rouet-Stora-Tyutin (BRST) formalism. The novelty is the appearance of a new term quadratic in the tetrad field. We next compute the heat kernel coefficients and understand the divergences arising in the gravitational one-loop effective action. In our computation, the arising heat kernel coefficients depend only on the self-dual part of the Weyl curvature. We make a comparison between our results and what has been obtained for metric general relativity.

Anisotropic strange stars in extended [formula omitted] gravity with electromagnetic field

In the present article, we have studied the modified field equations for static, spherically symmetric spacetime in a more extended f(T,B,T) gravity in the framework of teleparallel geometries. In order to apply this gravity, we have explored anisotropic strange star configurations with the electromagnetic field. For this purpose, we have chosen off-diagonal tetrad fields to represent gravitational interaction. Also, a linear relationship between radial pressure and energy density through Bag constant C, defined as MIT Bag equation of state, has been employed to describe the strange quark matter distribution. A lot of tests have been done in the context of our solutions to confirm the stability and physical acceptability of our obtained stellar structure. In this connection, we have demonstrated a lot of graphs for two selected known compact stars VelaX−1 and SAXJ1808.4−3658, with respect to our calculations. Again, the significant role of T in our stellar configuration has been displayed by plotting all graphs for five selected values of β, the coefficient parameter of T in the functional form f(T,B,T). Finally, we have observed that stability criteria and physically well-behaved nature for thermodynamical parameters holds good to represent a strange star configuration in this model, corresponding to our considered constant parameters explicit expressions. Hence this proposed model becomes viable and acceptable to study strange star configuration in f(T,B,T) gravity.

Reference frames and black hole thermodynamics

In the context of the absolute parallelism formulation of General Relativity, and because of the fact that the scalar curvature can be written in purely torsional terms, it was known for a long time that a surface term based solely on the torsion tensor appears in the action. It was subsequently suggested that this term might play the role of the Gibbons-Hawking-York boundary term which, in turn, is associated to the free energy in the path integral approach, and then, to the black hole entropy by standard thermodynamic arguments. We show that the identification of the two boundary terms is rather incomplete, and that it strongly depends on the choice of the tetrad (frame) field used to reproduce a given metric. By considering variations of the tetrad field not necessarily subjected to Dirichlet-like conditions on the boundary surface, we find a class of frames adapted to the Schwarzschild spacetime in which the Gibbons-Hawking-York/torsion link is actually established, and conducing to the right black hole entropy without the need of any background subtraction. Remarkably, these frames are also responsible for the correct value of the gravitational energy as computed from the teleparallel energy-momentum pseudo-current.

Tetrad Fields, Reference Frames, and the Gravitational Energy–Momentum in the Teleparallel Equivalent of General Relativity

AbstractThe concept and definitions of the energy–momentum and angular momentum of the gravitational field in the teleparallel equivalent of general relativity (TEGR) are reviewed. The importance of these definitions is justified by three major reasons. First, the TEGR is a well established and widely accepted formulation of the gravitational field, whose basic field strength is the torsion tensor of the Weitzenböck connection. Second, in the phase space of the TEGR there exists an algebra of the Poincaré group. Not only the definitions of the gravitational energy–momentum and 4‐angular momentum satisfy this algebra, but also the first class constraints related to these definitions satisfy the algebra. And third, numerous applications of these definitions lead to physically consistent results. These definitions follow from a well established Hamiltonian formulation, and rely on the idea of localization of the gravitational energy. In this review, the concept of localizability of the gravitational energy is revisited, in light of results obtained in recent years. The behavior of free particles is studied in the space–time of plane fronted gravitational waves (pp‐waves). Free particles are here understood as particles that are not subject to external forces other than the gravitational acceleration due to pp‐waves. Since these particles acquire or loose kinetic energy locally, the transfer of energy from or to the gravitational field must also be localized. This theoretical result is considered an important and definite argument in favor of the localization of the gravitational energy–momentum, and by extension, of the gravitational 4‐angular momentum.

The effect of modified hybrid and logarithmic teleparallel gravity on the interior solutions of compact stars

This study aims to investigate spherically symmetric anisotropic solutions that describe compact stellar objects in the modified Rastall teleparallel (MRT) theory of gravity. In order to achieve this goal, we utilize the Karmarkar condition to evaluate the spherically symmetric components of the line element. We explore the field equations by selecting appropriate off-diagonal tetrad fields for two different scenarios. In the first scenario, we use a hybrid form of f(T)=βemTTn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T)=\\beta e^{m T} T^n$$\\end{document} and a linear equation of state (EoS) pr=ξρ+ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_r=\\xi \\rho +\\phi $$\\end{document}, where 0<ξ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\xi < 1$$\\end{document}, to evaluate h(T). In the second scenario, we again use a hybrid form of f(T)=βemTTn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T)=\\beta e^{m T} T^n$$\\end{document} and a logarithmic form of h(T)=ψlog(ϕTχ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h(T)=\\psi \\log (\\phi T^{\\chi })$$\\end{document}. We aim to investigate the possible forms of gravity modifications by evaluating the function for different values of m and n, reducing the gravity forms to hybrid, power law form, and exponential form. Our findings reveal that the exponential-logarithmic case is unstable in our scenario. To the best of our knowledge, we are the first to attempt to explore compact star models in MRT gravity. After obtaining the field equations, we investigate different physical parameters that demonstrate the stability and physical acceptability of the stellar models. We utilize observational data, such as the mass and radius of the PSRJ1416-2230\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PSRJ\\;1416-2230$$\\end{document} model, to ensure the physical plausibility of our findings.

A discrete discontinuity between the two phases of gravity

When tetrad (metric) fields are not invertible, the standard canonical formulation ofgravity cannot be adopted as it is. Here we develop a Hamiltonian theory of gravity fornoninvertible tetrad. In contrast to Einstein gravity, this phase is found to exhibit threelocal degrees of freedom. This reflects a discrete discontinuity in the limit of a vanishingtetrad determinant. For the particular case of vanishing lapse, the Hamiltonian constraintdisappears from the classical theory upon fixing the torsional gauge-freedom. Any statefunctional invariant under the internal gauge rotations and spatial diffeomorphisms is a formal solution of the associated quantum theory. The formulation here provides a Hamiltonianbasis to analyze gravity theory around a physical singularity, which corresponds to a zero ofthe tetrad determinant in curved spacetime.

Physically viable strange quark star models in modified teleparallel gravity

The aim of this paper is to develop the isotropic and anisotropic quark stars configurations in the context of [Formula: see text] gravity in the static spherically symmetric background. To explore the combined effects of torsion scalar [Formula: see text] and the trace of energy–momentum tensor (EMT) [Formula: see text] on relativistic astrophysics, we use diagonal as well as non-diagonal tetrad fields. By considering the conformal Killing vectors along with the MIT bag model, the interior solutions of the field equations corresponding to the linear [Formula: see text] model (in which [Formula: see text] are the constants and [Formula: see text] indicates the cosmological constant) are calculated. The feasibility of the obtained solutions is confirmed by implementing several physical tests. The model parameters are constrained subject to the existence and stability of the quark star models. We formulate the energy constraints, stability equations, mass function, compactness and redshift factor, and present the graphical analysis of all physical quantities. It is found that the derived solutions for both diagonal and non-diagonal tetrad exhibit well-behaved profiles in the framework of modified teleparallel gravity.

Second-order teleparallel gauge theory

In this work, we analyze second-order extensions of teleparallel theories of gravity as gauge theories for the translation group. We use Utiyama’s approach to gauge theories and show that it is possible to include second-order derivative terms in the Lagrangian of the gauge potential and preserve gauge invariance. Besides the usual field strength, a new object has to be introduced in order to preserve both gauge and diffeomorphism invariances. From this new object, we obtain a set of fourteen independent invariants which leads to equations that are linear in the fourth derivative of the tetrad field. We analyze a particular example with one of these invariants and evaluate the weak field limit, showing that the effective gravitational potential is a combination of Newton and Yukawa potentials.

Quintessence anisotropic stellar models in quadratic and Born–Infeld modified teleparallel Rastall gravity

This study aims to discuss anisotropic solutions that are spherically symmetric in the quintessence field, which describe compact stellar objects in the modified Rastall teleparallel theory of gravity. To achieve this goal, the Krori and Barua arrangement for spherically symmetric components of the line element is incorporated. We explore the field equations by selecting appropriate off-diagonal tetrad fields. Born–Infeld function of torsion f(T)=βλT+1−1 and power law form h(T) = δTn are used. The Born–Infeld gravity was the first modified teleparallel gravity to discuss inflation. We use the linear equation of state pr = ξρ to separate the quintessence density. After obtaining the field equations, we investigate different physical parameters that demonstrate the stability and physical acceptability of the stellar models. We use observational data, such as the mass and radius of the compact star candidates PSRJ 1416-2230, Cen X-3, & 4U 1820-30, to ensure the physical plausibility of our findings.

On torsion axial vector and gravitational energy in Lewis–Papapetrou space-time in the theory of teleparallel gravity

The teleparallel version of Lewis–Papapetrou space-time is investigated. For this space-time, the true discussion of the geometric and physical properties is given. We show that the value of the space-like torsion-axial vector does not depend on the choice of tetrad field. Consequently, the spin procession of the Dirac particle and the corresponding Hamiltonian have physical meaning. The energy and momentum densities for such space-time are calculated. We show that when choosing two sets of tetrad fields, the dependence and independence of the aforementioned quantities on the tetrad field depend on the Lorentz factor, which links the two sets of tetrad fields.

A structural analysis of self-gravitating anisotropic stars via equation of state in modified teleparallel gravity

This work aims to explore spherically symmetric anisotropic solutions describing compact stellar objects in the extended teleparallel theory of gravity with matter coupling. For this purpose, we use the Adler Finch Skea geometry for both the off-diagonal and diagonal tetrads. We explore the complete matching of the tetrad fields for two different choices of the function f(T,T), i.e., linear and nonlinear models. We obtain complete solutions to the stellar models by taking the strange quark matter distribution after defining the MIT bag model equation of state, pr=13(ρ−4BG). By using the proper choice of the tetrad forms, we incorporate the generic model f(T,T)=αTn(r)+βT(r), which can be treated as linear and nonlinear models depending upon the value of parameter n. After obtaining the field equations, we investigate different physical parameters showing the stability and physical acceptability of the stellar models by using the observational data like mass and radius of PSRJ1416−2230, 4U1608−52, CenX−3, EXO1785−248, and SMCX−1 models.

A discretization of Holst’s action for general relativity

We present a simplicial model for gravity written in terms of a discretized Lorentz connection and a discretized tetrad field. The continuum limit of its action is Holst’s action for general relativity. With the intention of using it to construct spin foam modes for quantum gravity, we write two other equivalent models written in terms of a discretized and constrained B field. The differences between our model and existing models are most likely inessential in the sense that a quantization would lead to equivalent quantum theories in the Wilsonian continuum limit. Nevertheless, we mention two features leading to possible advantages: Curvature degrees of freedom are described at the level of each 4-simplex. Our model offers a picture of bulk geometry leading to actions for matter couplings that split as a sum over 4-simplices.

Lorentz Gauge and Coulomb Gauge for Tetrad Field of Gravity

In general relativity, an inertial frame can only be established in a small region of spacetime, and local inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations in local inertial frames, and there exists freedom to choose the local inertial frame at each spacetime point. 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 ∂μAμ=0 and the Coulomb gauge ∂iAi=0 in electrodynamics, the Lorentz gauge and Coulomb gauge for tetrad fields are proposed in the present work. Moreover, properties of the Lorentz gauge and the Coulomb gauge for tetrad fields are discussed to show their similarities to those in electromagnetic fields.

A pilot study on canonical gravity with mass dimension one fermions

The mass dimension one (MDO) fermionic field is built on a complete set of dual-helicity eigenspinors of the charge conjugation operator, which obeys the statistic of Fermi-Dirac. These spinors are a potential candidates for the description of dark matter. The redefinition of the dual structure of this object holds a local adjacent theory and completely satisfies the Lorentz invariance. In this work, we investigate important aspects of the interaction of this fermion with gravity in the light of a canonical formulation in ADM formalism. We construct an action via tetrad fields using a manifold on a family of spacelike surfaces Σt that carries the MDO matter field, and additionally we propose a condition for the action to have a term associated with dark energy. We found the Hamiltonian and diffeomorphism constraints at the classical level for gravitational dynamics with the immersion of this material content in space-time, which leads us to the interpretation of the contribution of dark matter energy density, parallel to the lapse function of foliation and its directional flux of energy density in the hypersurface of manifold.

Unification of elementary forces in gauge SL(2N,C) theories

We argue that the gauge SL(2N,C) theories may point to a possible way where the known elementary forces, including gravity, could be consistently unified. Remarkably, while all related gauge fields are presented in the same adjoint multiplet of the SL(2N,C) symmetry group, the tensor field submultiplet providing gravity can be naturally suppressed in the weak-field approach developed for accompanying tetrad fields. As a result, the whole theory turns out to effectively possess the local SL(2,C)×SU(N) symmetry so as to naturally lead to the SL(2,C) gauge gravity, on the one hand, and the SU(N) grand unified theory, on the other. Since all states involved in the SL(2N,C) theories are additionally classified according to their spin values, many possible SU(N) GUTs – including the conventional one-family SU(5) theory – appear not to be relevant for the standard 1/2 spin quarks and leptons. Meanwhile, the SU(8) grand unification for all three families of composite quarks and leptons that stems from the SL(16,C) theory seems to be of special interest that is studied in some detail.

Geodesic equation in non-commutative gauge theory of gravity* *Supported by PRFU Research Project (B00L02UN050120190001), Univ. Batna 1, Algeria

In this study, we construct a non-commutative gauge theory of the modified structure of the gravitational field using the Seiberg-Witten map and the general tetrad fields of Schwarzschild space-time to show that the non-commutative geometry removes the singularity at the origin of the black hole, thus obtaining a non-singular Schwarzschild black hole. The geodetic structure of this black hole presents new types of motion next to the event horizon within stable orbits that are not allowed by the ordinary Schwarzschild spacetime. The noncommutative periastron advance of the Mercury orbit is obtained, and with the available experimental data, we find a parameter of non-commutativity on the order of . This result shows that the new fundamental length, , is on the order of .