On pseudoparticle solutions in Yang's theory of gravity

Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to pure gravitational fields. In afour-dimensional Riemannian manifold it is shown that thedouble self-dual solutions obey Einstein's vacuum equations with the cosmological term, whereas the doubleanti-self-dual configurations satisfy the Rainich conditions of Wheeler'sgeometrodynamics. Conformal methods reveal that the gravitational analog of the instanton or pseudoparticle solution of Yang-Mills theory was already known to Riemann.     

Progress in metric-affine gauge theories of gravity with local scale invariance

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection acquire the status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective low-energy theory.Based on a plenary talk given by one of us (EWM) at the 53rd annual meeting of the Deutsche Physikalische Gesellschaft in Bonn on March 14, 1989.Supported by the German-Israeli Foundation for Scientific Research and Development (GIF), Jerusalem and Munich.Supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, project He 528/12-1.Supported in part by DOE Grant DE-FG05-85-ER40200.     

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

Dynamical space and gravitation within a purely geometrical framework: A new approach to gravitation

We state a purely geometrical framework apparently implementing Machian ideas on inertia. Only coupling constants dimensionless in natural units have been introduced in the theory. In anynonvacuum cosmos the field equations describing the gravitational phenomena in cosmological units turn out to be identical to Einstein's equations, with the Einstein gravitational coupling expressed in terms of the parameters defining the cosmological structure. This dependence, however, is not detectable. Indeed, such equations do not need to incorporate the standard Machian requirements (apart from the requirement that they are not conceivable in the total absence of matter) in order to be Machian, since, just on the basis of Mach's principle, one cannot expect to be able to detect Machian effects in Nature by using a system of units based on gravitational phenomena. On the contrary, the equations describing the gravitational field in local atomic units are Machian in the standard sense and, in particular, they incorporate the ideas that the frame has to be fixeddirectly in connection with the observed distribution and motion of matter and that there does not exist any kind of space-time in the total absence of matter. Finally, to reconcile, at least in the weak-field approximation, Einstein's equations (considered as equations describing the gravitational phenomena in local atomic units) with Mach's principle and to be in agreement with cosmological observations, we suggest that our cosmos be identified with a superuniverse model in which the background structure is homogeneous (in space and in time) and isotropic, while our universe is represented by one of the local inhomogeneities of the background. Then we prove that in any region of our universe in which the gravitational field issufficiently weak and smooth the equations, describing the gravitational field in local atomic units, are expected to approximate Einstein's equations all the better, the more the dimensions of our universe are negligible with respect to the dimensions of the background and the background curvature is small. As regards the experimental predictions of the present approach, any prediction for experiments involving only purely gravitational measurements is identical to that of Einstein's theory and the above result also guaranteesa fortiori the agreement with the available experimental data, also asnonpurely gravitational experiments are concerned.This paper appeared as Istituto Matematico L. Tonelli, preprint 78–10 (April 1978) (unpublished).     

An extension of the nonsymmetric unified field theory

The field equations, in the new formulation of Einstein's unified field theory, are extended from the present vacuum form to the general case in which sources are present. In this generalization the contracted torsion tensor corresponds to the electromagnetic four-potential. By this correspondence, Einsteins-gauge transformation becomes identical to the ordinary electromagnetic gauge symmetry. The generalized Bianchi identities are found and used to discuss deviations from the Einstein-Lorentz equations of motion.     

Epicyclic Orbital Oscillations in Newton's and Einstein's Dynamics

We apply Feynman's principle, The same equations have the same solutions, to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at r = r _ms.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

Kerr-type solution in Brans-Dicke theory

The Kerr-type solution in the Brans-Dicke theory should contain three parameters: a massm, a rotational parametera ₀, and a coupling parameter It goes over to the Kerr solution in Einstein's theory of general relativity in the limit 8. Using these conditions, we construct a special solution from Bruckman's solutions which can be regarded as a Kerr-type solution in the Brans-Dicke theory.     

Canonical Path Integral Quantization of Einstein's Gravitational Field

The connection between the canonical and the path integral formulations of Einstein's gravitational field is discussed using the Hamilton Jacobi method. Unlike conventional methods, its shown that our path integral method leads to obtain the measure of integration with no -functions, no need to fix any gauge and so no ambiguous determinants will appear.     

The Phase of a Quantum Mechanical Particle in Curved Spacetime

We investigate the quantum mechanical wave equations for free particles of spin 0, 1/2, 1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is S/ = p dx /, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple manageable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has exactly the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in . We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in . For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.     

Moyal Deformations of Gravity via SU(∞) Gauge Theories, Branes and Topological Chern-Simons Matrix Models

Moyal noncommutative star-product deformations of higher-dimensional gravitational Einstein-Hilbert actions via lower-dimensional SU(), W gauge theories are constructed explicitly based on the holographic reduction principle. New reparametrization invariant p-brane actions and their Moyal star product deformations follows. It is conjectured that topological Chern-Simons brane actions associated with higher-dimensional knots have a one-to-one correspondence with topological Chern-Simons Matrix models in the large N limit. The corresponding large N limit of Topological BF Matrix models leads to Kalb-Ramond couplings of antisymmetric-tensor fields to p-branes. The former Chern-Simons branes display higher-spin W symmetries which are very relevant in the study of W Gravity, the Quantum Hall effect and its higher-dimensional generalizations. We conclude by arguing why this interplay between condensed matter models, higher-dimensional extensions of the Quantum Hall effect, Chern-Simons Matrix models and Gravity needs to be investigated further within the framework of W Gauge theories.     

Anisotropic Viscous Cosmology with Variable G and Λ

Einstein's equations with variable gravitational and cosmological constants are considered in the presence of bulk viscosity for a Bianchi type I model in a way which conserves the energy momentum tensor. Several solutions are found, one of which corresponds to the earlier solution found by Tarkeshwar Singh et al. for the isotropic case. Our approach is compared with that of Arbab.     

Massive gravity as a quantum gauge theory

We present a new point of view on the quantization of the massive gravitational field, namely we use exclusively the quantum framework of the second quantization. The Hilbert space of the many-gravitons system is a Fock space F⁺ (H_graviton) where the one-particle Hilbert space H_graviton carries the direct sum of two unitary irreducible representations of the Poincaré group corresponding to two particles of mass m > 0 and spins 2 and 0, respectively. This Hilbert space is canonically isomorphic to a space of the type Ker(Q)/Im(Q) where Q is a gauge charge defined in an extension of the Hilbert space H_graviton generated by the gravitational field h and some ghosts fields u, (which are vector Fermi fields) and v (which is a vector Bose field).Then we study the self interaction of massive gravity in the causal framework. We obtain a solution which goes smoothly to the zero-mass solution of linear quantum gravity up to a term depending on the bosonic ghost field. This solution depends on two real constants as it should be; these constants are related to the gravitational constant and the cosmological constant. In the second order of the perturbation theory we do not need a Higgs field, in sharp contrast to Yang-Mills theory.     

On gravitational shock waves

The discontinuity planes of the Riemann curvature tensorR _klm ⁱ in the Einsteinian vacuumR _kl =0 are isotropic hypersurfaces. These surfaces are to be conceived as being constructed of lightlike geodesics, which form, in the eikonal approximation, gravitational radiation. The discontinuity planes themselves describe the wave fronts of disturbances of the metricg _ik , propagating with the velocity of light. By successively applying continuity conditions for the derivatives of theg _ik that follow from Einstein's equations, we obtain the universal expression of gravitational wave fields in space-time strips (or representations) of arbitrarily selected Einstein spaces.     

A natural framework for the minimal supersymmetric gauge theories

We show that the sequence of Jordan algebras M _inf3 ^sup1 , M _inf3 ^sup2 , M _inf3 ^sup4 , and M _inf3 ^sup8 , whose elements are in the 3×3 Hermitean matrices over , , , and O, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.     

General equations of motion for test particles in space-time with torsion

The momentum and spin equations of motion for test particles possessing different spins in space-time with torsion are derived from the most general functional form of _M . The same kinds of equations in general relativity and in Kibble's gauge theory of gravitation are special cases of our equations.     

On theories of gravitation with higher-order field equations

Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourth-order field equations which in empty space admit the Schwarzschild metric as a solution. These theories, Like Einstein's, follow from a variational principle and thus imply differential identities. If, as in Einstein's theory, the sources are taken to be proportional to the energy-momentum tensorT , these identities imply the vanishing of the covariant divergence ofT ^v. It is shown here that in the presence of extended sources, Weyl's and Eddington's theories (as well as all other higher-order metric theories derivable from an action principle) contradict Newton's law of gravitation in the nonrelativistic limit. To entail this law would require a modification of the source term of the field equations which in general is not compatible withT ^v _;v alternatively, one could require only asymptotic agreement with Newton's law, which is compatible with supplementary higher-order terms in Einstein's equations, but which requires the introduction of universal constants of the dimensions of length. None of the generalizations of Einstein's equations considered here admits Birkhoff's theorem.Dedicated to Achille Papapetrou on the occasion of his retirement.     

Plane-Symmetric Solitons of Spinor and Scalar Fields

We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms S= and P=i⁵; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant = _,^,, that becomes linear at 0. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.     

Primary Matter Creation in a Weyl–Dirac Cosmological Model

In the framework of an integrable Weyl–Dirac (W–D) theory a cosmological model is proposed. It describes a universe that began its expansion from a primary pre-Planckian geometric entity containing no matter. During the pre-Planckian period, from R ₀ =5.58×10 ^–36 cm to R_I=5.58×10 ^–34 cm, this embryonic universe has undergone a very rapid expansion and cosmic matter was created by geometry. At R_I the universe was already filled with matter having the Planckian density _P and being in the state of prematter (P=–), while the Weylian geometric elements were insignificant. This state is the Planckian egg that has served as the initial state of the singularity-free cosmological model ⁽¹⁾ considered in the framework of Einstein's general theory of relativity. The W–D character of the geometry and the cosmological constant are significant in the pre-Planckian period during the matter creation. In the dust-dominated period a relic of the W–D geometry causes a global dark matter effect. In between the pre-Planckian and dust period one has Einstein's framework and is negligible.