Hamiltonian Structure of Poincaré Gauge Theory and Separation of Non-Dynamical Variables in Exact Torsion Solutions

The canonical Hamiltonian of the Poincaré gauge theory of gravity is reanalyzed for generic Lagrangians. It is shown that the time components e₀^α and Γ₀^αβ of the tetrad and the linear connection fields of a Riemann-Cartan space-time U₄ constitute gauge degrees of freedom which remain non-dynamical during the time evolution of the system. Whereas the e₀^α are to be identified with the lapse and shift functions N^α known from the ADM formalism in Einstein's theory, the additional Lorentz degrces of freedom Γ₀^αβ are pertinent to Poincaré gauge models. These non-dynamical variables are instrumental in the derivation of exact torsion solutions obeying modified double duality conditions for the U₄-curvature. Thereby, in the case of spherical symmetry and for the charged Taub-NUT metric, we obtain the most general torsion configuration for a large class of quadratic Lagrangians. Previously found solutions are contained therein and can be recovered after fixing special “gauge”.     

Fermion Fields in Theories of Gravitation

After an introduction to the formalism used throughout the paper there follows a concise presentation of the theory of fermion fields in one-tetrad gravitational theories. That presentation gives a hint to the construction of a bi-tetrad theory, the two tetrad fields being denoted by h^A_k and h?^A_k. The tetrad field h^A_k. gives the Riemannian metric g_kl while the tetrad field h?h^A_k is orthonormalized with respect to the flat metric a_kl. Specializing h?^A_k in such a way that they have the form δ^A_k in the preferred coordinates of Minkowski space and using a matter Lagrangian which contains these h?^A_k only by the anholonomic components of the metric Christoffel symbols, we obtain a dynamical energy momentum tensor which is equal to the canonical one. Then we consider the relations of the bi-tetrad theory to other theories which are only covariant with respect to global Lorentz transformations from the beginning. As an example we formulate the main relations of the two-component neutrino theory.     

Initial-data formulation of tetrad gravity utilizing York's extrinsic time approach

An ability to analyze the geometrodynamic degrees of freedom and initial-data formulation is central to the canonical quantization of gravity. In the metric theory of gravity York provided the most powerful technique to analyze the dynamic degrees of freedom and to solve the initial-data problem. In this paper we extend York's analysis to tetrad gravity. Such an extension is necessary for the quantization of gravity when coupled to a half-integer-spin field. We present a comparative analysis of the geometric information carried by (1) a 3-metric of an initial hypersurface and (2) the spacelike triad of a time-gauged tetrad. We apply the tetrad initial-data formulation to Ashtekar's connection variables, and provide a comparison with other alternative choices of canonical tetrad variables.     

Canonical vielbein forms ofD=3 andD=4 gravity

We compare, in bothD=3 andD=4, two vielbein canonical formulations of gravity: that of Deser and Isham which starts with the metric replaced by vielbeins in the ADM action, and that starting directly from first order vielbein action, and compute the functional that generates the canonical transformation between these two formulations. InD=3, the generator thus exhibits the inherent simplicity of the Einstein action, starting from the DI form (which is as complicated as inD=4). InD=4, however, the same procedure, while leads to a somewhat different formulation from the existing ones, does not result in miraculous simplifications.     

Charged axially symmetric solution and energy in teleparallel theory equivalent to general relativity

An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity. The associated metric has the structure function G(ξ)=1-ξ²-2mAξ³-q²A²ξ⁴. The fourth order nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad whose metric has the structure function in a factorizable form (1-ξ²)(1+r₊Aξ)(1+r_-Aξ) with r_± as the horizons of Reissner–Nordström space-time. This new form has the advantage that its roots are now trivial to write down. Then, we study the singularities of this space-time. Using another coordinate transformation, we obtain a tetrad field. Its associated metric yields the Reissner–Nordström black hole. In calculating the energy content of this tetrad field using the gravitational energy-momentum, we find that the resulting form depends on the radial coordinate! Using the regularized expression of the gravitational energy-momentum in the teleparallel equivalent of general relativity we get a consistent value for the energy.     

Teleparallel equivalent theory of (1+1)-dimensional gravity

A theory of (1+1)-dimensional gravity is constructed on the basis of the teleparallel equivalent of general relativity.The fundamental field variables are the tetrad fields e i μ and the gravity is attributed to the torsion.A dilatonic spherically symmetric exact solution of the gravitational field equations characterized by two parameters M and Q is derived.The energy associated with this solution is calculated using the two-dimensional gravitational energy-momentum formula.     

Quantization of conformal gravity: Another approach to the renormalization of gravity

The non-renormalizability of quantum gravity poses a great problem to the construction of any unified field theory of all known interactions. Normally, we start with a unitary theory of gravity and investigate its renormalization properties. This is the first of a series of papers where we start with the opposite approach, beginning with a renormalizable theory and investigating its unitarity structure. In particular, we study non-perturbative approaches to the quantization of conformal gravity. Using ADM coordinates, we perform the canonical quantization of the Weyl action C_μναβ², which is renormalizable and is also local scale invariant. Although this theory is certainly not unitary in perturbation theory, we speculate that unitarity may be restored when we approach this theory non-perturbatively, by examining the possibility of different phase transitions.     

Static spherically symmetric solution of (<Emphasis Type="Italic">R</Emphasis> ± <Emphasis Type="Italic">μ</Emphasis><Superscript>4</Superscript>/<Emphasis Type="Italic">R</Emphasis>) gravity

The static spherically symmetric solution for R ± μ ⁴/R model of f(R) gravity is investigated. We obtain the metric for space-time in the solar system that reduces to the Schwarzschild metric, when μ tends to zero. For the obtained metric, the deviation from Einstein gravity is very small. This result is different from the other results have been obtained by equivalence between f(R) gravity and scalar tensor theory. Also it is shown that the vacuum solution in the solar system depends on the shape of matter distribution which differ from the Einstein’s gravity.     

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by McIntosh and Hickman [1]. Tetrads of real metrics can be formally complexified by complex coordinate transformations and tetrad rotations and their properties investigated from the viewpoint of complex relativity. First, complex bivectors are examined and classified, partly by using the fundamental quadric surface of a metric in projective complex 3-space P³-an elegant but not well-known method of investigating the null structure of a metric. A generalization of the Mariot-Robinson theorem from real relativity is then given and related to various canonical forms of complex bivectors. The second part of the paper discusses four classes of complex metrics. Real metrics of the first class are ones with a null congruence whose wave surfaces have equal curvature. The second class, a subcase of the first one, is the main one; it contains integrable double Kerr-Schild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typs-D vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) Kerr-Schild form.     

Existence of black hole solutions for the Einstein-Yang/Mills equations

This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge groupSU(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/r ², asr tends to infinity.Research supported in part by the NSF, Contract No. DMS-89-05205Research supported in part by the DE, Contract No. De-FG 02-88 EF 25065     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.     

A new class of interior solutions for a stress-energy tensor of Segré characteristic [111, 1]

The interior solutions of (the tetrad versions of) Einstein's field equations withT _AB having Segré characteristic [111, 1] (which has all four eigenvalues distinct), are investigated. For this purpose amixed method, which combines Synge'sg method andT method, is introduced. Some of the tetrad equations are solved for the metric functions while the remaining equations are used to define the corresponding components ofT _AB . As necessary conditions of the consistency of the mixed method the conservation equationsT ^AB _B =0 are explicitly verified. Several simplifications and analysis of some differential inequalities show the existence of a new class of solutions which, in addition to having Segré characteristic [111, 1], also satisfy the strong energy conditions of Hawking and Ellis.This Author is a member of the Theoretical Sciences Institute, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada.     

Canonical structure of tetrad bimetric gravity

We perform the complete canonical analysis of the tetrad formulation of bimetric gravity and confirm that it is ghost-free describing the seven degrees of freedom of a massless and a massive gravitons. In particular, we find explicit expressions for secondary constraints, one of which is responsible for removing the ghost, whereas the other ensures the equivalence with the metric formulation. Both of them have a remarkably simple form and, being combined with conditions on Lagrange multipliers, can be written in a covariant way.     

Covariantized Nœther identities and conservation laws for perturbations in metric theories of gravity

A construction of conservation laws and conserved quantities for perturbations in arbitrary metric theories of gravity is developed. In an arbitrary field theory, with the use of incorporating an auxiliary metric into the initial Lagrangian covariantized N?ther identities are carried out. Identically conserved currents with corresponding superpotentials are united into a family. Such a generalized formalism of the covariantized identities gives a natural basis for constructing conserved quantities for perturbations. A new family of conserved currents and correspondent superpotentials for perturbations on arbitrary curved backgrounds in metric theories is suggested. The conserved quantities are both of pure canonical N?ther and of Belinfante corrected types. To test the results each of the superpotentials of the family is applied to calculate the mass of the Schwarzschild-anti-de Sitter black hole in the Einstein–Gauss–Bonnet gravity. Using all the superpotentials of the family gives the standard accepted mass.     

Geodesic deviation equation in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

In this paper we study the Geodesic Deviation Equation (GDE) in metric f (R) gravity. We start giving a brief introduction of the GDE in General Relativity in the case of the standard cosmology. Next we generalize the GDE for metric f (R) gravity using again the FLRW metric. A generalization of the Mattig relation is also obtained. Finally we give and equivalent expression to the Dyer-Roeder equation in General Relativity in the context of f (R) gravity.     

Induced classical gravity

We review the induced-gravity approach according to which the Einstein gravity is a long-wavelength effect induced by underlying fundamental quantum fields due to the dynamical-scale symmetry breaking. It is shown that no ambiguities arise in the definition of the induced Newton and cosmological constants if one works with the path integral for fundamental fields in the low-scale region. The main accent is on a specification of the path integral which enables us to utilize the unitarity condition and thereby avoid ambiguities. Induced Einstein equations appear from the symmetry condition that the path integral of fundamental fields for a slowly varying metric is invariant under the local GL(4, R)-transformations of a tetrad, which contain the local Euclidean Lorentz, O(4)-rotations as a subgroup. The relationship to induced quantum gravity is briefly outlined.     

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L ²-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric. During the preparation of this work, the first two authors were partially supported by NSF grant DMS-0504367.     

Scale Transformation,Modified Gravity,and Brans-Dicke Theory

A model of Einstein-Hilbert action subject to the scale transformation is studied. By introducing a dilaton field as a means of scale transformation a new action is obtained whose Einstein field equations are consistent with traceless matter with non-vanishing modified terms together with dynamical cosmological and gravitational coupling terms. The obtained modified Einstein equations are neither those in f(R) metric formalism nor the ones in f(ℛ) Palatini formalism, whereas the modified source terms are formally equivalent to those of f(R)=\frac12R²f({\mathcal{R}})=\frac{1}{2}{\mathcal{R}}^{2} gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with w = -\frac32\omega=-\frac{3}{2} is explicitly shown, provided the dilaton field is condensated to its vacuum state.     

ON THE ADM EQUATIONS FOR GENERAL RELATIVITY

The Arnowitt–Deser–Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in . An example of this construction is given in the case of a closed Friedmann–Lemaitre–Robertson–Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.