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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.

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.

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.

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

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.

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.

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

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

In this paper, we present and discuss the NPTOOLS package written for the computer algebra system Maple. The package contains the main routines “nulltran”, “inv”, “nullor”, “ornull” and “pptype”. The routine “nulltran” is an implementation of the 6-parameter Lorentz group in the Newman-Penrose (null tetrad) formalism, and “inv” checks for invariance of equations under subgroups of the Lorentz group. The routines “nullor” and “ornull” provide a translation between the Newman-Penrose quantities associated with the null tetrad and the orthonormal tetrad quantities associated with the Ellis-MacCallum formalism. Finally, “pptype” determines the Petrov-Plebanski type of the Ricci tensor.

In this paper we discuss the connection between the geometric and tetrad approaches in the quantum affine-metric gravity. The corresponding transition formulas are obtained at the one-loop level. As an example, the one-loop counterterms are calculated in the tetrad formalism in the theory with terms quadratic in the torsion field. This model possesses the extra local symmetries connected with transformation of the connection field. It is shown that the special gauge can be chosen so that the corresponding additional ghosts do not contribute to the one-loop divergent terms., Comment: LaTeX, 10 pages, to appear in Mod.Phys.Lett.A

It is well known that a complex four-component Dirac spinor defines an orthonormal tetrad on flat Minkowski spacetime. For example, a spinor solution to the Dirac equation determines the four-velocity and Pauli-Lubanski spin vector, comprising the timelike and first spacelike members of the particle’s tetrad, as well as two other spacelike members that are rarely discussed. Also, of particular note is the complex null tetrad formalism that provides a map from a two-component SL(2, C) spinor and its complex conjugate to a tetrad. The inverse problem is studied here. Given a tetrad, a complex Dirac spinor valued function of the tetrad is explicitly defined in such a way that certain sums of bilinear products of the components of this spinor exactly reproduce the tetrad. This spinor may be used in the Feynman propagator representation of the solution to the Dirac equation to generate a Dirac wave spinor with desired initial properties. The mappings studied here represent a new class of nonlinear mappings SO(...

In the context of a null tetrad formalism it is assumed that the two real vectors of the tetrad have neither expansion nor rotation and that they are nongeodesic and/or have shear. This assumption is rarely used in search of algebraically special spacetimes. General relations for the connection and for the Ricci and Weyl tensors are presented. An additional assumption that these vectors are principal leads to Petrov type D spacetimes, reducible in some cases to conformally flat ones. Twenty-two explicit metric forms, probably hitherto unknown, are found.

Starting from a constrained real $BF$-type action for general relativity that includes both the Immirzi parameter and the cosmological constant, we obtain the Ashtekar-Barbero variables used in the canonical approach to the quantization of the gravitational field. This is accomplished by explicitly solving the second-class constraints resulting from the Hamiltonian analysis of the considered action, and later imposing the time gauge. All throughout this work the tetrad formalism is left aside, obtaining the Ashtekar-Barbero variables entirely in terms of the $B$-fields that define the action., 12 pages, no figures

We present a new evolution system for Einstein's field equations which is based on tetrad fields and conformally compactified hyperboloidal spatial hypersurfaces which reach future null infinity. The boost freedom in the choice of the tetrad is fixed by requiring that its timelike leg be orthogonal to the foliation, which consists of constant mean curvature slices. The rotational freedom in the tetrad is fixed by the 3D Nester gauge. With these conditions, the field equations reduce naturally to a first-order constrained symmetric hyperbolic evolution system which is coupled to elliptic equations for the gauge variables. The conformally rescaled equations are given explicitly, and their regularity at future null infinity is discussed. Our formulation is potentially useful for high accuracy numerical modeling of gravitational radiation emitted by inspiraling and merging black hole binaries and other highly relativistic isolated systems., Corrected factor of 2 errors in Eqs. (A8) and (A9) and a few typos; final version

Starting with a brief description of dyons and gravito-dyons, combined field equations and the equation of motion for generalized electromagnetic and generalized gravito-Heavisidian fields are derived in a manifestly covariant way. A non-Abelian gauge theory of dyons and gravito-dyons is described in terms of a generalized Yang-Mills potential, field strengths, and generalized field equations each carrying electric and magnetic constituents. A null tetrad formulation of a generalized Yang-Mills potential and field strength of dyons is discussed in detail in terms of symmetric spinors and spin coefficients. Generalized Yang-Mills field equations of dyons are rewritten in terms of null tetrad notation, and dyon solutions of source-free Dirac equations are obtained.

For pt.IV see ibid., vol.9, p.1711, (1992). The authors set up and solve the general (no symmetries assumed) first-order equations for dust (p=0) relative to an FLRW background in observational coordinates using the fluid-ray (FR) tetrad formalism, with data on the past light cone C-(p0) consisting of galaxy redshifts, observer area distances, galaxy number counts and null shear measures. Cosmological proper motions are not needed in the data set. A detailed framework for constructing solutions from any such data set is given, although no concrete solutions are presented. The character of these solutions will depend heavily on the data themselves.

Bondi's approach to the construction of a coordinate system is used with a different choice of gauge, in accordance with which the radial coordinate r is an affine parameter, to cast the metric tensor into a form suitable for use with the Newman-Penrose null tetrad formalism. The choice of tetrad has the result that the equations and all the functions that appear in them are real-valued. A group classification of the Sachs equations in this gauge leads to a unique expression for the first of the five independent elements $\Psi_0$ of the Weyl spinor, and to the corresponding exact solutions for two of the metric functions on an initial null hypersurface. A proof is presented that the result for $\Psi_0$ constitutes the appropriate characteristic initial value function for all physically realistic axisymmetric, non-rotating vacuum spacetimes. Integration of the field equations on the axis of symmetry when an equatorial symmetry plane is also present, and on the equatorial plane itself when time rates of change can be neglected, shows that these data produce results consistent with Newton's laws and with the Schwarzschild solution in the appropriate limits. The solution on the axis of symmetry indicates that the Weyl curvature increases without limit between black holes as their separation decreases., Comment: 12 pages. Published in Class. Quantum Grav. 15, 2509-2522

The Hamiltonian form of the Hilbert action in the first order tetrad formalism is examined. We perform a non-linear field redefinition of the canonical variables isolating the part of the spin connection which is canonically conjugate to the tetrad. The geometrical meaning of the constraints written in these new variables is examined., 12 pages, Latex. Minor presentation changes and some references added. Version to appear in Classical and Quantum Gravity

The usual method of solution of problems in General Relativity was to use a local coordinate basis to suit a given problem. However, it has been found useful to employ non-coordinate basis techniques in some problems. This is the tetrad formalism which consists of setting up four linearly independent vector fields called a tetrad basis at each point of a region of spacetime. The relevant equations are written in this basis and if the tetrad is aligned according to the symmetries of the spacetime an easier set of equations may ensue which could be amenable to solution.

Ashtekar's polynomial constraints for a new canonical gravity are examined in the first-order tetrad formalism. The action is given by the sum of Einstein's action and a complex total divergence term. We find the canonical form of the Hamiltonian explicitly. In the Hamiltonian, the coefficients of Gauss-law constraints are not arbitrary functions but must satisfy nonpolynomial relations. They guarantee the reality condition of canonical variables. In quantum theory an additional measure factor must be required in the inner product of state space.

The Ernst metric is analyzed in terms of the null tetrad formalism of Newman and Penrose. Some applications of the formalism are also given.

The sets of equations that form the basis for the tetrad formalism approach in general relativity contain considerable redundancy. Papapetrou has determined this redundancy explicitly in the form of three sets of identities and employed these in investigations of the Newman-Penrose tetrad formalism. In this paper Papapetrou's work is reviewed and some of his results that do not seem to be well known are emphasized, along with some general implications. The main new result that is established concerns the Geroch-Held-Penrose formulation of the tetrad formalism. When the sets of equations that are usually used in this formulation are considered in the light of Papapetrou's identities, it is found that certain formal simplifications can be made and that the Geroch-Held-Penrose formulation can be presented more concisely. It is emphasized that the results in this paper apply in the most general case only. Any special cases (e.g., simplified tetrad and/or Riemann tensor) need to be considered separately.

We give a modification of the Palatini Lagrangian for the free gravitational field that yields the vanishing of the torsion as a result of the field equations and requires only the assumption of the symmetry of the metric. We transcribe this Lagrangian into the tetrad formalism and show how the tetrad form of the Einstein field equations follows from it. Some remarks on possible generalization to a theory with nonvanishing torsion in the presence of matter conclude the paper.

A definition of perturbations of space-times in general relativity is proposed. The definition leads in a natural way to a concept of gauge invariance, and to an extension of a lemma of Sachs (1964). Coupled equations governing linearized perturbations of certain tetrad components of scalar, electromagnetic, and gravitational fields are derived by the use of Geroch, Held & Penrose’s (1973) version of the tetrad formalism of Newman & Penrose (1962). It is shown that these perturbations are gauge invariant if and only if the unperturbed space-time is vacuum of algebraic type {22} or, equivalently, if and only if the perturbation equations decouple. Finally the maximal subclass of type {22} space-times for which the decoupled perturbation equations can be solved by separation of variables is found. This class comprises all the nonaccelerating type {22} space-times, including that of Kerr, thus elucidating earlier results of Bardeen & Press (1972) and Teukolsky (1973).

We adapt the Newman-Penrose tetrad formalism to a systematic and efficient evaluation of Feynman integrals in the light-cone gauge. Our procedure exploits the traditional assumptions of locality and power counting as well as the useful tensor technique. It is shown that this program yields the same results as the much lengthier Feynman parameter method. There are no double poles to one-loop order.

The tetrad formalism is used to construct the four-dimensional covariant kinematics for an elastic medium. The deformation tensors which are generalizations of the Almansi, Green, and Cauchy tensors are determined. Kinematic equations are derived to relate the stress-propagation tensor to the rate of change of the deformation tensor.

The behaviour of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism. For this, a system of coordinates u, r, H and 0 is used, such that at spatial infinity u = const. is a null hypersurface and r, 0 and 5 reduce to the usual spherical polar coordinates. A set of four vectors (a tetrad) is also chosen with the only restriction that they are everywhere null. The metric tensor and the four vectors are expanded in inverse powers of r; the rotation coefficients and the tetrad components of the Riemann tensor are then calculated in a similar expansion; and the first two terms in the expansion beyond their values for a flat space are retained. The field equations in these approximations are derived explicitly and their effect on the expansion of the tetrad components of the Riemann tensor is studied; and the total energy and linear momentum are examined. In this paper three main results are derived: (i) the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad; (ii) the generalized expression for the news function of the field; (iii) a simple criterion for recognizing certain classes of non-radiating fields. 1. INTRODIUCTION During the last decade a great deal of work has been done on asymptotically flat spaces in general relativity. Bondi, van der Burg & Metzner (i962) investigated the case of a gravitational field with an axis and a plane of symmetry while Sachs (i962) investigated the general case of a field with no symmetries. Among their results, two important ones were the mass-loss formula and the so-called peeling theorem. Using a slightly different approach, Newman & Penrose (I962) and Newman & Unti (I962) derived the peeling theorem from different assumptions and analysed in more detail the field at infinity. And finally, Newman & Penrose (I965) discovered ten absolute constants of the field with physical meanings still unknown. All the foregoing investigations were carried out in a system of coordinates

The integrability conditions of conformal motions are written in the null tetrad formalism of Newman and Penrose. The maximum order of the group of conformal motions admitted by nonflat empty space‐times of given Petrov type is shown to be at most one greater than the maximum order of the group of Killing motions. The symmetries of those empty space‐times which possess hypersurface orthogonal geodesic rays with nonvanishing divergence are determined. Among these space‐times is one of type III which admits a group of Killing motions of order three. This provides a counter example to a result of Petrov which states that the maximum order of the group of Killing motions for such space‐times is two.

In the low-energy limit, superstring theory goes over into supergravity, and it is well known that in supergravity the basic entity is not the metric but the Cartan variables: the field of tetrads e/sub ..mu..//sup a/ and the Lorentz connection omega/sub b..mu..//sup a/. Therefore, it appears to us that the field theories of closed superstrings and the heterotic string must contain the tetrad string field E/sup a/ and the string connection Omega/sub b//sup a/, their mode expansions leading to the appearance of e/sub ..mu..//sup a/ and omega/sub b..mu..//sup a/, respectively. In this paper, the authors discuss their possible transformation properties and a mechanism for constructing a gauge-invariant theory

In the frame work of non-linear generalizations of TREDER's tetrad theory of gravitation considered in part I. a pure bimetric gravitation theory results for the LAGRANGian Ω(1)F with ω2 = 1. The discussion of the post-NEWTONian approximation given in I. has demonstrated that must be: ω2 = −1 − 2ω1. - However, a LAGRANGian with ω1 = − ω2 = −1 is identical with GUPTA's post-NEWTONian approximation for EINSTEIN's general relativistic LAGRANGian. Therefore, for ω1 = − ω2 = − 1 the EINSTEIN effects are resulting evidently and the question discussed in I. the tetrad formalism becomes non-important.

The tetrad formalism is applied to certain cosmological models and it is found that the vector 0. has the time component associated with Hubble's expansion.