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Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.

By exploiting the null tetrad formalism of Jogia and Griffiths and the technique of differential forms on a non-Riemannian space-time, non-static conformally flat, Petrov-type D, spherically symmetric solutions of the Einstein–Cartan field equations; when Weyssenhoff fluid is the source of curvature and spin, are obtained. In general, the solution is expanding, accelerating, rotating and non-shearing. However, the dynamic solution is expanding and rotating with zero acceleration and shear; whereas the static solution is accelerating and rotating with ‘expansion free and shear free “congruences” ’.

We present a tetrad–gauge theory of gravity based on the local Lorentz group in a four-dimensional Riemann–Cartan space–time. Using the tetrad formalism allows avoiding problems connected with the noncompactness of the group and includes the possibility of choosing the local inertial reference frame arbitrarily at any point in the space–time. The initial quantities of the theory are the tetrad and gauge fields in terms of which we express the metric, connection, torsion, and curvature tensor. The gauge fields of the theory are coupled only to the gravitational field described by the tetrad fields. The equations in the theory can be solved both for a many-body system like the Solar System and in the general case of a static centrally symmetric field. The metric thus found coincides with the metric obtained in general relativity using the same approximations, but the interpretation of gravity is quite different. Here, the space–time torsion is responsible for gravity, and there is no curvature because the curvature tensor is a linear combination of the gauge field tensors, which are absent in the case of pure gravity. The gauge fields of the theory, which (together with the tetrad fields) define the structure of space–time, are not directly coupled to ordinary matter and can be interpreted as the fields describing dark energy and dark matter.

In the case of the cosmic microwave background, the observable information is contained in the phase space distribution of photons. The latter are particles with internal degrees of freedom, i.e. polarization, and the corresponding quantum states can be superposed, which leads to the consideration of matrix-valued distributions. In this chapter we derive in detail the corresponding general-relativistic Boltzmann equation, including also the case of fermions, since the latter interact with photons. Our equations are derived within the tetrad formalism, which allows one to embed exactly and straightforwardly the quantum field theory amplitudes describing microscopic collisions in the macroscopically curved geometry. As a concrete example, we provide the lowest-order collision term for a fluid of photons, electrons and protons that includes the polarization/spin information. Although our construction follows standard practice for the most part, our understanding of the involved assumptions and symmetries leads us to a different definition of the collision term than the one found in the literature, yielding in particular different results beyond the lowest-order particle interactions.

In this paper, we examine static cylindrical symmetric fluid in unimodular $f(R)$ gravity. Unimodularity for cylindrical symmetric spacetime geometry is ensured with a coordinate transformation by using tetrad formalism. Field equation solutions for given matter and the spacetime are obtained. Resource of cosmic matter is discussed by using Equation of State (EoS) and energy conditions. We investigate weak and strong energy conditions in unimodular $f(R)$ gravity context with the help of Raychaudhuri equations. Obtained results show that fluids which violate the energy conditions cause the proposed unimodular and cylindrical symmetric wormholes, but this kind of wormholes do not satisfy traversability conditions. Furthermore, the obtained solutions indicate that $\omega $ coefficient in EoS must have the condition $\omega \neq -1$ for physically meaningful fluids. It is discussed whether the result is compatible with the unimodular gravitation theories.

The standard description of cosmological observables is incomplete, because it does not take into account the correct angular parametrization of the sky, i.e. the one determined by the observer frame. The corresponding corrections must be taken into account for reliable results at non-linear orders. This can be accomplished by introducing an orthonormal basis, or "tetrad", at the observer point, representing the frame with respect to which observations are performed. In this work we consider the tetrad formulation of General Relativity, thus associating tetrads to sources as well, and develop a new formalism for describing cosmological observables associated with localized sources. It is based on a manifold which we call the "observer space-time", whose coordinates are the proper time, redshift and angles an observer uses to parametrize measurements, and on which the rest of the observables are defined. This manifold does not have to be diffeomorphic to the true space-time and allows us to resolve caustics in the latter, in contrast to similar coordinate-based formalisms. As a concrete example, we work out the definitions and equations for the angular diameter distance, weak lensing and number count observables. As for the observables associated to the CMB, they lie inside the phase space distribution of the photon fluid, so we also revisit the construction of general-relativistic matrix kinetic theory from the tetrad formalism viewpoint. Here too the latter appears as the natural approach for relating the macroscopic dynamics to the microscopic quantum field theory, and therefore for constructing the matrix Boltzmann equations, without any approximation on the gravitational side. (... more in the manuscript), 111 pages

The constraint algebra is derived in the second order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts and the Hassan–Rosen transform (characterized as ‘a complicated redefinition of the shift variable’ according to the authors) appears here not as an ansatz but as fixing of a Lagrange multiplier. A comparison of this approach with the other ones is provided.

In previous chapters (i.e. 4-6), it is observed that the tetrad formalisms provide an efficient way for finding the Lanczos potential in some general situations. Using those prescriptions, the Lanczos potential for some well-known solutions of Einstein and Einstein-Maxwell equations has been obtained here. Some other methods, apart from tetrad formalism, for finding the Lanczos potential have also been discussed.

The Newman–Penrose formalism (also known as spin-coefficient formalism) is a tetrad formalism with special choice of the basis vector. The beauty of this formalism, when it was first proposed by Newman and Penrose in 1962 (Newman and Penrose in J Math Phys 3:566–578, 1962 [13]), was precisely in their choice of a null basis. A detailed study of this formalism has been made here.

When we deal with the quantities that rescale under spin and boost transformations, we then have another example of tetrad formalism. This is known as Groch-Held-Penrose (GHP) formalism and the details of this formalism are given here.

The Lanczos tensor and its importance in general relativity have been discussed. The basic equations describing the gravitational field, in terms of Lanczos tensor, have been expressed in the language of NP and GHP formalisms along with other related equations.

The gravitational perturbation of the Morris-Thorne wormhole has been derived by using the Newman-Penrose formalism. We apply Teukolsky equation to the wormhole spacetime, compute the perturbed Weyl scalars, $\Psi_{4}^{(1)}$ and obtain its master equation, decomposed in spin weighted spherical harmonics with spin weight $-2$. For simplicity, we consider the perturbation provoked by a single gaussian pulse of pressureless dust matter., Comment: 20 pages, 6 figures

By applying a recent method --based on a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor-- to the simple spherical static case, we found that the only static solution with homogeneous energy density is the Schwarzschild solution and that there are no spherically symmetric dynamic solutions consistent with the homogeneous energy density assumption. Finally, a circular equivalence is shown among the most frequent conditions considered in the spherical symmetric case: homogeneous density, isotropy in pressures, conformally flatness and shear-free conditions. We demonstrate that, due to the regularity conditions at the center of the matter distribution, the imposition of two conditions necessarily leads to the static case., Comment: 17 pages

We present the consistent theory of a free massive spin-2 field with 5 degrees of freedom propagating in spacetimes with an arbitrary geometry. We obtain this theory via linearizing the equations of the ghost-free massive gravity expressed in the tetrad formalism. The theory is parameterized by a non-symmetric rank-2 tensor whose 16 components fulfill 11 constraints implied by the equations. When restricted to Einstein spaces, the theory reproduces the standard description of massive gravitons. In generic spacetimes, the theory does not show the massless limit and always propagates five degrees of freedom, even for the vanishing mass parameter. We illustrate these features by an explicit calculation for a homogeneous and isotropic cosmological background. It turns out that the spin-2 particles are always stable if they are sufficiently massive, hence they may be a part of the Dark Mater., 36 pages

We use the tetrad formalism to calculate a matrix square root occurring in the de Rham-Gabadadze-Tolley potential. In the minimal case, we obtain the constraints and their algebra. We show that the number of gravitational degrees of freedom corresponds to the massless and massive gravity fields. The Boulware-Deser ghost is eliminated because of two second-class constraints.

The possibility of testing gravity theories with the help of gravitational wave detections has become an interesting arena of recent research. In this paper, we follow this direction by investigating the quasinormal modes (QNMs) of the axial perturbations for charged black holes in the Palatini-type theories of gravity, specifically ($i$) the Palatini $f(R)$ gravity coupled with Born-Infeld nonlinear electrodynamics and ($ii$) the Eddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwell electromagnetic fields. The coupled master equations describing perturbations of charged black holes in these theories are obtained with the tetrad formalism. By using the Wentzel-Kramers- Brillouin (WKB) method up to 6th order, we calculate the QNM frequencies of the EiBI charged black holes, the Einstein-Born-Infeld black holes, and the Born-Infeld charged black holes within the Palatini $R+\alpha R^2$ gravity. The QNM spectra of these black holes would deviate from those of the Reissner-Nordstr\"om black hole. In addition, we study the QNMs in the eikonal limit and find that for the axial perturbations of the EiBI charged black holes, the link between the eikonal QNMs and the unstable null circular orbit around the black hole is violated., Comment: 18 pages, 5 figures

In this thesis, both the geometrical and action principle approach to Einstein’s field equations are developed, providing an intuitive fundamental path also with a more formal development given by an extremal principle. After this solid introduction, derivations of the metric of two distinct theoretical models of black holes are presented. Firstly, the Reissner-Nordstrom model is studied and its metric is obtained by solving the differential ¨ field equations. Secondly, the Kerr-Newman model is approached by the Newman-Jannis algorithm that provides an easy and straightaway procedure to obtain its metric and energymomentum tensor just by identifying a seed metric and applying a change of variables. Finally, the solutions are studied, horizons and regions of interest of both black holes are commented. Universidad de Sevilla. Doble Grado en Física y Matemáticas

We consider rigidly rotating states in thermal equilibrium on static spherically symmetric spacetimes. Using the Maxwell-Juttner equilibrium distribution function, onstructed as a solution of the relativistic Boltzmann equation, the equilibrium particle flow four-vector, stress-energy tensor and the transport coefficients in the Marle model are computed. Their properties are discussed in view of the topology of the speed-of-light surface induced by the rotation for two classes of spacetimes: maximally symmetric (Minkowski, de Sitter and anti-de Sitter) and charged (Reissner-Nordstrom) black-hole spacetimes. To facilitate our analysis, we employ a non-holonomic comoving tetrad field, obtained unambiguously by applying a Lorentz boost on a fixed background tetrad., 18 pages, 10 figures

We examine the stability of Einstein–Maxwell perfect fluid configurations with a privileged radial direction by means of a $$1+1+2$$ -tetrad formalism. We use this formalism to cast in a quasilinear symmetric hyperbolic form the equations describing the evolution of the system. This hyperbolic reduction is used to discuss the stability of linear perturbations in some special cases. By restricting the analysis to isotropic fluid configurations, we assume a constant electrical conductivity coefficient for the fluid. As a result of this analysis we provide a complete classification and characterization of various stable and unstable configurations. We find, in particular, that in many cases the stability conditions are strongly determined by the constitutive equations and the electric conductivity. A threshold for the emergence of the instability appears in both contracting and expanding systems.

The Riemann–Silberstein–Majorana–Oppenheimer approach to the Maxwell electrodynamics in the presence of electrical sources, arbitrary media and curved space-time is investigated within the matrix formalism and tetrad method. Symmetries of the matrix Maxwell equation under transformations of the local gauge complex rotation group SO(3,C) is demonstrated explicitly. Equivalence of the approach to general covariant Proca technique and spinor formalism is shown.

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics extended to the case of arbitrary pseudo-Riemannian space-time applied to solve the Maxwell equations are solved exactly on the background of simplest static cosmological models, spaces of constant curvature of Riemann and Lobachevsky parameterized by spherical coordinates. Separation of variables is realized in the basis of Schrodinger–Pauli type, description of angular dependence in electromagnetic complex 3-vectors is given in terms of Wigner D-functions. In the case of compact Riemann model a discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. In the case of hyperbolic Lobachevsky model no discrete spectrum for frequencies of electromagnetic modes arises.

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors _as such_ cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is (nearly) Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag(-1,1,1,1)$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and _values_ of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the (conformal part of the) metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O(1,3)$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors (related to fermions, with half-integral spin) and tensors and the like (related to bosons, with integral spin) is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited.

The exact solution of Einstein-Cartan field equations for static, conformally flat spherically symmetric space-time is derived and it is proved to be Petrov-type D.

Newman-Penrose (NP)formalism [5, 13], in general relativity is a tetrad formalism in which various geometrical quantities are projected on a chosen null tetrad basis. Spin coefficients in NP formalism replace connection coefficients in geometry. Applications of spin coefficients are discussed by many authors, in diverse fields of general relativity and cosmology, like Ahsan [1], Ahsan et al [2], Hasmani and Ahsan [8], Hasmani and Katkar [10] and others. The computations involved are very lengthy and complicated. Motivated by such a large number of applications and looking to the complexity involved, we have exploited some features of computer algebra system Mathematica to carry out algebraic computation of spin coefficients in Newman-Penrose formalism for given metric and null tetrad. In this paper the required Mathematica program is also included.

A new classical theory of gravitation within the framework of general relativity is presented. It is based on a matrix formulation of four-dimensional Riemann-spaces and uses no artificial fields or adjustable parameters. The geometrical stress-energy tensor is derived from a matrix-trace Lagrangian, which is not equivalent to the curvature scalar R. To enable a direct comparison with the Einstein-theory a tetrad formalism is utilized, which shows similarities to teleparallel gravitation theories, but uses complex tetrads. Matrix theory might solve a 27-year-old, fundamental problem of those theories (sec. 4.1). For the standard test cases (PPN scheme, Schwarzschild-solution) no differences to the Einstein-theory are found. However, the matrix theory exhibits novel, interesting vacuum solutions., Comment: 24 pages

The transport coefficients induced by the Anderson-Witting approximation of the collision term in the relativistic Boltzmann equation are derived for close to equilibrium flows in general relativity. Using the tetrad formalism, it is shown that the expression for these coefficients is the same as that obtained on flat space-time, in agreement with the generalized equivalence principle., Comment: 6 pages, 1 figure, proceedings of TIM 15-16 conference (26-28 May 2016, Timisoara, Romania)

We revisit the action principle for general relativity motivated by the path integral approach to quantum gravity. We consider a spacetime region whose boundary has piecewise $C^2$ components, each of which can be spacelike, timelike or null and consider metric variations in which only the pullback of the metric to the boundary is held fixed. Allowing all such metric variations we present a unified treatment of the spacelike, timelike and null boundary components using Cartan's tetrad formalism. Apart from its computational simplicity, this formalism gives us a simple way of identifying corner terms. We also discuss "creases" which occur when the boundary is the event horizon of a black hole. Our treatment is geometric and intrinsic and we present our results both in the computationally simpler tetrad formalism as well as the more familiar metric formalism. We recover known results from a simpler and more general point of view and find some new ones., Comment: Latex, 22 pages, 5 figures. Minor revisions and corrections and some clarifications in the text

In this paper, we perform a parallel analysis to the model proposed in [25]. By considering the central co-tetrad (instead of the central metric), we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian $\kappa$-deformations provides metrics, which lead either to cosmological constant or spatial curvature type solutions of non-vacuum Einstein equations. Among vacuum solutions, we find pp-wave type., Comment: 12 pages, published version, title modified

We deduce the explicit form of a phase-locked (PL) tetrad, adapted to a small spinning particle (a test gyroscope) following an arbitrary geodesic orbit in the Schwarzschild geometry of a gravitational source. We subsequently obtain the analytic expression of the gyroscopic precessional velocity Ω in a non-circular orbit, by means of the Fermi rotation coefficients related to the tetrad’s transport law. As an application, we compute the orbital shift in the spin vector, considering a spinning test particle (the gyroscope) in a slightly non-circular orbital motion in the weak gravitational field limit. We compare our results with those of other previous works.

We consider an application of the tetrad formalism introduced by Cardall et al. [Phys. Rev. D 88 (2013) 023011] to the problem of a rigidly rotating relativistic gas in thermal equilibrium and discuss the possible applications of this formalism to relativistic lattice Boltzmann simulations. We present in detail the transformation to the comoving frame, the choice of tetrad, as well as the explicit calculation and analysis of the components of the equilibrium particle flow four-vector and of the equilibrium stress-energy tensor., 20 pages, 1 figure, accepted for publication in Annals of West University of Timisoara - Physics

The present status of the shear-free fluid conjecture in general relativity is discussed: a review is given of recent partial proofs, a new and complete proof is given for the case of a linear equation of state, including a non-zero cosmological constant, and a number of useful results are presented which might help in proving the conjecture for a general equation of state., 28 pages, minor corrections

This review is devoted to problems of defining the reference frames in the tetrad formalism of General Relativity. Tetrads are the expansion coefficients of components of an orthogonal basis over the differentials of a coordinate space. The Hamiltonian cosmological perturbation theory is presented in terms of these invariant differential forms. This theory does not contain the double counting of the spatial metric determinant in contrast to the conventional Lifshits-Bardeen perturbation theory. We explicitly write out the Lorentz transformations of the orthogonal-basis components from the cosmic microwave background (CMB) reference frame to the laboratory frame, moving with a constant velocity relative to the CMB frame. Possible observational consequences of the Hamiltonian cosmological perturbation theory are discussed, in particular, the quantum anomaly of geometric interval and the shift of the origin, as a source of the CMB anisotropy, in the course of the universe evolution.

A fundamental aspect of many plasma-related astrophysical problems is the kinetic description of magnetized rel- ativistic plasmas in intense gravitational fields, such as in accretion disks around compact gravitating bodies. The goal of this paper is to formulate a gyrokinetic description for a Vlasov-Maxwell plasma within the framework of general relativity. A closed set of relativistic gyrokinetic equations, consisting of the collisionless gyrokinetic equation and corresponding expres- sions for the four-current density, is derived for an arbitrary four-dimensional coordinate system. General relativity effects are taken into account via the tetrad formalism. The guiding-center dynamics of charged particles and the gyrokinetic transforma- tion are obtained accurate to the second order of the ratio of the Larmor radius to the nonuniformity scale length. The wave terms with arbitrary wavelength (kρL ∼ 1) are described in the second-order (nonlinear) approximation with respect to the amplitude of the wave. The same approximations are used in the derivation of the gyrophase-averaged Maxwell equations. The derivation is based on the perturbative Lagrangian approach with a fully relativistic, four-dimensional covariant formulation. Its results improve on existing limitations of the gyrokinetic theory.