Late-time behavior of primordial gravitational waves in expanding universe

The results of two different approaches to the late-time behavior of primordial gravitational radiation are compared. Using asymptotic expansion in time of the tetrad components of the Riemann tensor one finds that initially chaotic behavior transfers itself into a radiative gravitational field of Petrov typeN ast. On the other hand, to accomplish the physical picture, we study the high-frequency behavior of the field variables in the same formalism. We show that the background spacetime is of general Petrov type I, and then calculate the tetrad components of the stress-energy tensor induced by the disappeared radiation with the help of Newman-Penrose equations.     

Broken Weyl Invariance and the Origin of Mass

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W₄, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields , representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction of the Weyl space W₄ to a pseudo-Riemannian space V₄ is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W₄ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with providing a potential for the Weyl vector fields . After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V₄. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on , coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields.     

Differential rotation of an electrically conducting fluid in general relativity

Starting from the Ricci identity for the 4-velocity vectoru _a ,a mathematical identity is derived, in terms of the kinematic quantities and the Riemann curvature tensor, for ,_a H ^a , the derivative along a magnetic field line of the magnitude of the vorticity of an electrically conducting fluid. Maxwell's field equations are not used in the derivation and the result is true for a fluid with finite, infinite, or even nonuniform, electric conductivity. Previous results [3–5] derived for an infinitely conducting fluid are obtained as special cases. By expressing the Riemann curvature tensor in terms of the Ricci and Weyl tensors, Einstein's field equations are introduced and the role played by the free gravitational field is examined. It is found that ,_aH^a does not depend on the electric part of the Weyl tensor and that, for an infinitely conducting fluid satisfying certain steady state conditions, ,_aH^a is independent of that part of the curvature determined locally through Einstein's field equations.     

Characterization of the Weyl solutions

For the Weyl solutions(z, x) of the Schrödinger and Dirac equations, asymptotics for |z| are obtained. This gives a possibility of selecting Weyl solutions by their behaviour when |z| . Some applications are given.     

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.     

Cancellations of infrared divergences in the two-dimensional non-linear σ models

In the two-dimensionalO(N) nonlinear models, the expectation value of anyO(N) invariant observable is shown to have an infrared finite weak coupling perturbative expansion, although it is computed in the wrong spontaneously broken symmetry phase. This result is proved by extracting all infrared divergences of any bare Feynman amplitude atD=2– dimension. The divergences cancel at any order only for invariant observables. The renormalization atD=2 preserves the infrared finiteness of the theory.     

Mass Generation by Weyl Symmetry Breaking

A massless electroweak theory for leptons is formulated in a Weyl space, W₄, yielding a Weyl invariant dynamics of a scalar field , chiral Dirac fermion fields _L and _R, and the gauge fields , A, Z, W, and W , allowing for conformal rescalings of the metric g and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, , representing the D(1) or dilatation gauge fields. The local group structure of this Weyl electroweak (WEW) theory is given by —or its universal coverging group for the fermions—with denoting the electroweak gauge group SU(2)_W × U(1)_Y. In order to investigate the appearance of nonzero masses in the theory the Weyl symmetry is explicitly broken by a term in the Lagrangean constructed with the curvature scalar R of the W₄ and a mass term for the scalar field. Thereby also the Z and W gauge fields as well as the charged fermion field (electron) acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D ² = 0, where is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory by breaking the D(1) gauge symmetry, is compared to the so-called spontaneous symmetry breaking in the standard electroweak theory, which is, actually, the choice of a particular (nonlinear ) gauge obtained by adopting an origin, , in the coset space representing , with being invariant under the electromagnetic, gauge group U(1)_e.m.. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl symmetry breaking, yielding a pseudo-Riemannian space, V₄, from a W₄ and a scalar field with a constant modulus . The quantity affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. The consequences of the broken WEW theory are worked out and the determination of the parameters of the theory is discussed.     

On the assumptions made in treating the gravitational wave problem by the high-frequency approximation

The assumptions that enter into the formulation of the gravitational wave problem in the limit of high frequency are discussed. It is shown that depending on the relation between the amplitude parameter and the frequency parameter the concept of back-reaction can have different physical interpretations. We also show that as a direct consequence of these assumptions high-frequency gravitational waves do not disperse in a vacuum. Finally, we make some conjectures about a coordinate-free characterization of high frequency for some cases and also about the use of the invariants of the Weyl tensor for the problem of finding a background space-time given a vacuum space-time containing high-frequency gravitational radiation.     

Complex relativity and real solutions. III. Real type-N solutions from complexN ⊗N ones

Einstein's vacuum field equations are integrated in complex relativity in a major subcase of the class whose Weyl tensor is of the type NN, i.e., when the left and right Weyl spinors and are each of typeN. The subcase is the complex equivalent of the real nontwisting case. Five separate families of solutions are found. Three of these are complexified versions of the two families of plane-fronted waves and the Robinson-Trautman real type-N metrics and two are complex solutions which do not have any real slices of Lorentz signature. Before the equations are integrated, the relevant general theory and equations are developed in a tetrad frame which is well suited to the discussion of these and a wider class of complex solutions and is called aleft quarter flat frame. The relationship between this frame and the coordinates used and some other frames and coordinates, including the complexified version of the frame often used for real type-N metrics, is discussed.     

The nonsymmetric Kaluza-Klein (Jordan-Thiry) theory in the electromagnetic case

We present the nonsymmetric Kaluza-Klein and Jordan-Thiry theories as interesting propositions of physics in higher dimensions. We consider the five-dimensional (electromagnetic) case. The work is devoted to a five-dimensional unification of the NGT (nonsymmetric theory of gravitation), electromagnetism, and scalar forces in a Jordan-Thiry manner. We find interference effects between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric. Our unification, called the nonsymmetric Jordan-Thiry theory, becomes the classical Jordan-Thiry theory if the skew-symmetric part of the metric is zero. It becomes the classical Kaluza-Klein theory if the scalar field=1 (Kaluza's Ansatz). We also deal with material sources in the nonsymmetric Kaluza-Klein theory for the electromagnetic case. We consider phenomenological sources with a nonzero fermion current, a nonzero electric current, and a nonzero spin density tensor. From the Palatini variational principle we find equations for the gravitational and electromagnetic fields. We also consider the geodetic equations in the theory and the equation of motion for charged test particles. We consider some numerical predictions of the nonsymmetric Kaluza-Klein theory with nonzero (and with zero) material sources. We prove that they do not contradict any experimental data for the solar system and on the surface of a neutron star. We deal also with spin sources in the nonsymmetric Kaluza-Klein theory. We find an exact, static, spherically symmetric solution in the nonsymmetric Kaluza-Klein theory in the electromagnetic case. This solution has the remarkable property of describing mass without mass and charge without charge. We examine its properties and a physical interpretation. We consider a linear version of the theory, finding the electromagnetic Lagrangian up to the second order of approximation with respect toh _v =g _v –n _v . We prove that in the zeroth and first orders of approximation there is no skewonoton interaction. We deal also with the Lagrangian for the scalar field (connected to the gravitational constant). We prove that in the zeroth and first orders of approximation the Lagrangian vanishes.     

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.     

Gravitational,electromagnetic, and nuclear theory

Through an examination of neutron lifetimes we shall propose a new theory of gravitational, electromagnetic, and nuclear fields. The fundamental assumption of our theory is that the motion of a particle in a combination of gravitational, electromagnetic, and nuclear fields is determined from a variational principle of the form _A ^B d=0. The form of the physical time is determined from an examination of classical equations of motion. The field equations are determined from an examination of Maxwell-Einstein theory and Yukawa theory. Utilizing the standard elementary model of the deuteron, the theory predicts that at a proton-neutron separationr 10^–13 cm the neutron lifetime is infinite and that nucleons have a repulsive core. These predictions have been experimentally verified.     

Periodic inverse problem for a new hierarchy of coupled evolution equations

We study the inverse problem with periodic boundary condition for a new class of integrable nonlinear evolution equations. The multiphase periodic solutions for the nonlinear fields (p, q, r) are expressed in terms of the Riemann theta function, which is obtained via the linearization of the flows of the set of auxiliary variables _j on a Riemann surface. An explicit case is evaluated to obtain the form of the algebraic curve on which the variables _j move.     

Green's functions for off-shell electromagnetism and spacelike correlations

The requirement of gauge invariance for the Schwinger-DeWitt equations, interpreted as a manifestly covariant quantum theory for the evolution of a system in spacetime, implies the existence of a five-dimensional pre-Maxwell field on the manifold of spacetime and proper time . The Maxwell theory is contained in this theory; integration of the field equations over restores the Maxwell equations with the usual interpretation of the sources. Following Schwinger's techniques, we study the Green's functions for the five-dimensional hyperbolic field equations for both signatures ± [corresponding to O(4, 1) or O(3, 2) symmetry of the field equations] of the proper time derivative. The classification of the Green's functions follows that of the four-dimensional theory for massive fields, for which the mass squared may be positive or negative, respectively. The Green's functions for the five-dimensional field are then given by the Fourier transform over the mass parameter. We derive the Green's functions corresponding to the principal part _P and the homogeneous function ₁ ; all of the Green's functions can be expressed in terms of these, as for the usual field equations with definite mass. In the O(3, 2) case, the principal part function has support for x²², corresponding to spacelike propagation, as well as along the light cone x²=0 (for =0). There can be no transmission ofinformation in spacelike directions, with this propagator, since the Maxwell field, obtained by integration over , does not contain this component of the support. Measurements are characterized by such an integration. The spacelike field therefore can dynamically establish spacelike correlations.     

Shell sources of stationary axisymmetric gravitational fields

A number of solutions for material shell sources of stationary axisymmetric gravitational fields are presented. Explicit solutions are found for shells lying on equipotential hypersurfaces (g _tt = const) and generating static monopole fields in prolate and oblate spheroidal coordinates (Zipoy-Voorhees fields). Numerical solutions are found for shells lying on hypersurfaces of constantg /g and generating Kerr- and Tomimatsu-Sato ( = 2) fields. The shells have minimum areas allowed by the energy conditions of Hawking and Ellis.     

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.     

On Spherically Symmetric Minimally Coupled Brane Worlds

For a static, spherically symmetric brane in the framework of the RS2 concept, we study the conditions under which the 4D tensor E _v, arising from the 5D Weyl tensor, vanishes on the brane. Gravity on the brane is then decoupled from the bulk geometry, it is the so-called minimally coupled brane world (MCBW). Assuming E _v =0 in the whole bulk, we try to solve the 5D Einstein equations G _AB + _5gAB =0 and obtain an overdetermined set of equations for functions of the radial coordinate. Some special solutions are found, among which are the well-known black string solution with the Schwarzschild metric on the brane and its generalizations with Schwarzschild–(A)dS on-brane metrics. It is concluded that a MCBW can be embedded, in general, in a bulk where E_v is not identically zero but only vanishes on the brane. We also present some previous results on the general properties of scalar fields on the brane and give an example of a wormhole supported by a scalar field in a MCBW.     

Soliton concept in general relativity

Soliton physics has made considerable progress in solving nonlinear problems. This paper is meant to relate the soliten concept to the stationary axisymmetric vacuum fields in general relativity. We present a functional transformation which, working as a nonlinear creation operator, generates gravitational fields of isolated sources. When applied to flat space-time (gravitational vacuum) this operation leads to a nonlinear superposition of an arbitrary number of Kerr particles. This superposition also includes the Tomimatsu-Sato fields. The functional transformations form an infinite-parameter group which contains the Kinnersley-Geroch group as a subgroup.This essay received the second award from the Gravity Research Foundation for the year 1980.-Ed.     

Gravitational field strength and generalized Komar integral

We define a gravitational field strength in theories of the Einstein-Cartan type admitting a Killing vector. This field strength is a second rank, antisymmetric, divergence-free tensor, whose (Komar) integral over a closed 2-surface gives a physically meaningful quantity. We find conditions on the Lagrange density of the theory which ensure the existence of such a tensor, and show that they are satisfied forN=2-supergravity and for a special case of the bosonic sector ofN=4-supergravity. We discuss a possible application of the generalized Komar integral in the theory of stationary black holes. We also consider the field strength problem in Kaluza-Klein theory, where the application to black holes is particularly interesting.     

Einstein's relativity and quantum physics

There are reasons to reject the idea that a field in empty space is a real physical entity. The nonexistence of the electromagnetic field and the gravitational field as physical entities leads to far-reaching consequences. The basic equations sufficient to constructclassical electrodynamics (the Maxwell equations and the Lorentz force equation) are obtained by combining quantum considerations with two premises: (a) there exists a subatomic particle, theemon, each concrete emon having a specific electric property described by aspacelike four-vector; (b) every concrete charged particle possesses a specific electric property described by atimelike four-vector. Some other points of interest are also discussed, in particular, ones related to Einstein's gravitational field as well as the action-at-a-distance versus local-action issue. Einstein's second postulate of special relativity is also shown to need some revision of principle.