Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Gravity and gauge: A new perspective

Realization of the Poincaré groupP ₁₀ as a subgroup ofGL(5,R) that maps a 4-dimensional affine set into itself has been shown to lead to a direct Yang-Mills gauging process. This paper discusses the differences between direct gauge theory forP ₁₀ and previously published works. These differences are fundamental, both physically and mathematically, and lead to marked departures from previous concepts and interpretations. The translation subgroup is correctly gauged; the metric structure and metric compatibility are derived from the gauging process rather than assumed; spin structures are automatically incorporated in a consistent manner; the local holonomy group is shown to be the component of the Lorentz group connected to the identity; the geometric analog of Yang-Mills minimal coupling precludes dependence of the free gauge field Lagranian on torsion; and the theory reduces exactly to general relativity when the momentumenergy complex is symmetric and all matter fields are spin-free. Gravitational effects on neutral test particles are shown to arise from the compensating 1-forms for local action of Lorentz boosts. The compensating 1-forms for local action of the translation subgroup may be interpreted as space-time dislocations, while the compensating 1-forms for the rotation subgroup can be viewed as space-time disclinations. Unfortunately, there are no clear physical meanings that can be ascribed to space-time dislocations or disclinations.     

Two kinds of rotation: An argument for torsion

There are now many theories of gravity with a torsion field as well as the usual metric field. One of the arguments for allowing torsion is based upon a gauge theory analogy. The purpose of this paper is to clarify exactly which symmetries are being gauged in this process. The principal observation is that special relativity is invariant under two different kinds of Lorentz transformations. The first type rotate the fields and move them from one point to another in space-time. The second type merely rotate the fields at each point without changing their location. To gauge both types of rotations requires a torsion field as well as a metric field.This essay received honorable mention from the Gravity Research Foundation for the year 1980 (Ed.).     

Reformulation of general relativity as a gauge theory

The starting point is a spinor affine space-time. At each point, two-component spinors and a basis in spinor space, called “spin frame,” are introduced. Spinor affine connections are assumed to exist, but their values need not be known. A metric tensor is not introduced. Global and local gauge transformations of spin frames are defined with GL(2) as the gauge group. Gauge potentials B_μ are introduced and corresponding fields F_μν are defined in analogy with the Yang-Mills case. Gravitational field equations are derived from an action principle. Incases of physical interest SL(2, C) is taken as the gauge group, instead of GL(2). In the special case of metric space-times the theory is identical with general relativity in the Newman-Penrose formalism. Linear combinations of B_μ are generalized spin coefficients, and linear combinations of F_μν are generalized Weyl and Ricci tensors and Ricci scalar. The present approach is compared with other formulations of gravitation as a gauge field.     

Topological invariants and the dynamics of an axial vector torsion field

A generalized theory of gravitation is discussed which is based on a Riemann-Cartan space-time,U ₄, with an axial vector torsion field. Besides Einstein's equations determining the metric of theU ₄, a system of nonlinear field equations is established coupling an axial vector source current to the axial vector torsion field. The properties of the solutions of these equations are discussed assuming a London-type condition relating the axial current and torsion field. To characterize the solutions use is made of the Euler and Pontrjagin forms and the associated quadratic curvature invariants for theU ₄ space-time. It is found that there exists for a Riemann-Cartan space-time a relation between the zeros of the axial vector torsion field and the singularities of the Pontrjagin invariant, which is analogous to the well-known Hopf relation between the zeros of vector fields and the Euler characteristic.     

A new class of exact solutions of the vacuum quadratic Poincaré gauge field theory

A class of algebraically special exact solutions of the vacuum quadratic Poincaré gauge field theory is presented. These solutions are of type III and type N and have a nonexpanding, shear-free and twist-free geodesic repeated principal null congruence. The metric is of Kundt's class, and the torsion components are solutions of certain differential equations. The solutions have been obtained using a generalised spin coefficient formalism.     

The mathematical basis of physical laws: Relativistic mechanics,quantum mechanics,the Lorentz force,Maxwell's equations and non-electromagnetic forces for spin zero particles

We show that defining the observed proper velocity and acceleration of a spin zero particle as the first and second derivatives of the classical expectation value for the space-time position vector, defined on a manifold carrying the Lorentz metric, with respect to a conditioning parameter , yields directly: a Lorentz and gauge invariant quantum mechanics, the Lorentz force, Maxwell's equations and a field equation for a non-electromagnetic potential. This also provides a new basis for gauge conditions in the field theory and shows that only the Lorentz gauge condition is admissible in electromagnetic theory.     

SPHERICALLY SYMMETRIC AND STATIC FIELDS IN LINEARIZED GAUGE THEORIES OF GRAVITATION

In this paper Newtonian limit in the Poincare gauge field theory of gravitation is investigated. In spherically symmetric and static cases interior and exterior solutions of the linearized field equations with gravitational sourtion are obtained by maens of Green's function for the five Lagrangians with out ghosts and tachyons. In cases of four Lagrangians,the space-time metrics outside gravitational source are the usual Schwarzschild one of the first-older, while in the case of the fifth hagrangian the space-time metric differs from the Schwarzschild one. Under both,Newtonian and-weak gravitational field approximations,the motion of a test particle without span should therefore be different from Newton's second law. As a result of the exchanged particles of spin o⁺ the deviation from Newton's second law is a Yukawa term which is attractive. A distance-dependent gravitational "constant" G(r) can be defined according to the new result. The difference between G(r) and Newton's gravitational constant G_∞ is due to a nonzero component of torsion tensor, the effect of which can be tested by measuring G(r).     

RIEMANNIAN STRUCTURE ON GAUGE GROUP AND METRIC COMPATIBLE YANG-MILLS THEORY

A gauge theory with gauge potentials that are compatible with right invariant metric of the gauge group is presented. It is shown that in the metric compatible torsion free gauge theory, gauge potentials can acquire the mass, without introducing the tliggs field. A plane-wave exact solution in vacuum is obtained.     

A spherically symmetric vacuum solution of the quadratic Poincaré gauge field theory of gravitation with newtonian and confinement potentials

We derive a stationary spherically symmetric vacuum solution in the framework of the Poincaré gauge field theory with a recently proposed quadratic lagrangian. We find a metric of the Schwarzschild-de Sitter type, both torsion and curvature are non vanishing, with torsion proportional to the mass and curvature proportional to the strong coupling constant κ. The metric exhibits two pieces, a newtonian potential describing the gravitational behavior of macroscopic matter, and a confining potential ～κr² presumably related to the strong-interaction properties of hadrons. To our knowledge this is a new feature of a classical solution of a Yang-Mills type gauge theory.     

Adjoint QCD1 + 1 in light-cone gauge, quantized at equal time

SU (2) gauge theory coupled to massless fermions in the adjoint representation is quantized in light-cone gauge by imposing the equal-time canonical algebra. The theory is defined on a space-time cylinder with “twisted” boundary conditions, periodic for one color component (the diagonal 3-component) and antiperiodic for the other two. The focus of the study is on the non-trivial vacuum structure and the fermion condensate. It is shown that the indefinite-metric quantization of free gauge bosons is not compatible with the residual gauge symmetry of the interacting theory. A suitable quantization of the unphysical modes of the gauge field is necessary in order to guarantee the consistency of the subsidiary condition and allow the quantum representation of the residual gauge symmetry of the classical Lagrangian: the 3-color component of the gauge field must be quantized in a space with an indefinite metric while the other two components require a positive-definite metric. The contribution of the latter to the free Hamiltonian becomes highly pathological in this representation, but a larger portion of the interacting Hamiltonian can be diagonalized, thus allowing perturbative calculations to be performed. The vacuum is evaluated through second order in perturbation theory and this result is used for an approximate determination of the fermion condensate.     

Stability ofR + T 2 theories of gravitation

In a framework with metric and spin affine connection as independent field variables, we show that the total energy of a genericR + T ²theory of gravitation is positive definite for an asymptotically flat space-time. This suggests that a more thorough treatment of the perturbative quantization of quadratic theories of gravity (including curvature and torsion squared terms) does not yield violation of unitarity.     

Torsion-induced spin precession

We investigate the motion of a spinning test particle in a spatially-flat FRW-type space-time in the framework of the Einstein–Cartan theory. The space-time has a torsion arising from a spinning fluid filling the space-time. We show that, for spinning particles with non-zero transverse spin components, the torsion induces a precession of the particle spin around the direction of the spin of the fluid. We also show that a charged spinning particle moving in a torsion-less spatially-flat FRW space-time in the presence of a uniform magnetic field undergoes a precession of a different character.     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian has strict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory. Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar field minimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian for scalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressed by gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.     

Spaces with flat symmetry in poincaré gauge theory of gravity

A Taub space is considered in the Poincare gauge theory of gravity. It is shown that the torsion tensor has four nonvanishing components, which can be split into two independent pairs S₀₁ ⁰, S₀₁ ¹, and S₂₃ ⁰, S₂₃ ¹. The analysis of the gravitational field equations leads to the conclusion that in this case only a flat space-time with torsion is possible, and that its metric coefficients and the components of the torsion tensor are described by a wave equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 92–98, April, 1990.     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian hasstrict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory.Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar fieldminimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian forscalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressedby gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.