Dynamical symmetries in relativistic kinetic theory

This paper is part of a program investigating symmetries that are defined at a physical or observational level rather than purely geometrically. Here we generalize previous work on dynamical matter symmetries of relativistic gases. If the matter symmetry vector is surface-forming with the dynamical Liouville vector, then Einstein's equations reduce it to a Killing symmetry of the metric. We show that this conclusion is unaltered if the gas particles are subject to a nongravitational force (including the electromagnetic force on charged particles) or if the gravitational field obeys higher-order field equations. In the Brans-Dicke theory, the matter symmetry reduces to a homothetic symmetry of the metric. This is also the case for a generalized conformal symmetry in Einstein's theory. We consider the problem of relaxing the surface-forming assumption in an attempt to determine whether there are dynamical symmetries that do not necessarily reduce to geometrical symmetries of the metric.     

A conformal mapping and isothermal perfect fluid model

Instead of the metric conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as minimally curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three-parameter family of perfect fluid solutions which are static and inhomogeneous. The density and pressure fall off in the curvature radial coordinates asR ^–2, for unbounded cosmological model with a barotropic equation of state. This is the characteristic of an isothermal fluid. We thus have an ansatz for an isothermal perfect fluid model. The solution can also represent bounded fluid spheres.     

Dynamics of Rotating Cylindrical Shells in General Relativity

Cylindrical spacetimes with rotation are studied using the Newmann–Penrose formulas. By studying null geodesic deviations, the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuity of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that 0 1, where is related to the mass per unit length of the shell. PACS numbers: 04.20Cv, 04.30.+x, 97.60.Sm, 97.60.Lf.     

Pseudoinstantons in Metric-Affine Field Theory

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible (pseudoinstanton). We suggest a metric-affine action which is a natural generalization of the Yang–Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.     

Stationary axisymmetric solutions in the vacuum Jordan-Brans-Dicke theory

It is shown that any stationary axisymmetric solution to the vacuum field equations of the Jordan-Brans-Dicke (JBD) theory may be obtained from a composition of any stationary axisymmetric vacuum Einstein spacetime with the Weyl class of metrics. Thus, generating solution techniques can be used to obtain any stationary axisymmetric JBD vacuum solution. In this manner, C. B. G. McIntosh's results concerning this topic are improved upon.     

Higher Spin Conserved Currents in Sp(2M) Symmetric Spacetime

An infinite set of higher spin conserved charges is found for the sp(2M) symmetric dynamical systems in M(M+ 1)/2-dimensional generalized spacetime _M. Since the dynamics in _M is equivalent to the conformal dynamics of infinite towers of fields in d-dimensional Minkowski spacetime with d = 3, 4, 6, 10, ... for M = 2, 4, 8, 16, ..., respectively, the constructed currents in _M generate infinite towers of (mostly new) higher spin conformal currents in Minkowski spacetime. The charges have a form of integrals of M-forms which are bilinear in the field variables and are closed as a consequence of the field equations. Conservation implies independence of a value of charge of a local variation of a M-dimensional integration surface _M analogous to Cauchy surface in the usual spacetime. The scalar conserved charge provides an invariant bilinear form on the space of solutions of the field equations that gives rise to a positive-definite norm on the space of quantum states.     

On the permanence of the null character of Maxwell fields

A critical review of known results about the permanence conditions for the null character of the solutions to the (vacuum) Maxwell equations, is presented. Concomitants of the electromagnetic field and the metric tensor are constructed, which give the principal directions of the field in covariant form. The known permanence conditions are generalized in order to includeall the (local) null fields; the above concomitants allow these conditions to be explicitly formulated in terms of the electromagnetic field.Supported in part by Conselleria de Cultura, Educació i Ciència de la Generalitat Valenciana.     

Lie groups and gravitation theory

A generalized space with torsion and curvature, defined by a fundamental group, is constructed by starting from the necessity of introducing standards of length and time in gravitation theory. The field variables coincide with the coefficients _i of the infinitesimal operator of the group. It is shown that the structural equations of the group depend on the transformation properties of the object to which they are applied. The simplest equations that the _i can satisfy are given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 40–47, December, 1970.     

Nonvacuum taub-type cosmological model

The Einstein universe is a simple model describing a static cosmological spacetime, having a constant radius and a constant curvature, and, as is well known, it does not describe our universe. We propose a model which is an extension of Einstein's. Our metric, havingR × S ³ topology, describes a nonisotropic homogeneous closed (finite) universe of Bianchi type IX. This metric is similar to that of Taub, but is simpler. Unlike the Taub solution (which is a cosmological extension of the NUT solution), however, the universe described by our metric contains matter. Like the Taub metric, our metric has two positive constants (, T). The gravitational red shift calculated from our metric is given. Similarly to the Schwarzschild metric, which has a singularity atr = 2m, this metric has the same kind of singularity att = 2. The maximal extension of the coordinates in our metric is fairly analogous to that of the Schwarzschild metric.     

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

Inequivalence of first- and second-order formulations in D=2 gravity models

The usual equivalence between the Palalini and metric (or affinity and vielbein) formulations of Einstein theory fails in two spacetime dimensions for its Kaluza-Klein reduced (as well as for its standard) version. Among the differences is the necessary vanishing of the cosmological constant in the first-order forms. The purely affine Eddington formulation of Einstein theory also fails here.The present results were reported in the Proceedings of the Markov Memorial Quantum Gravity Seminar.This work was supported bt the NSF under grant #PHY-9315811.     

On the new symmetries of Maxwell equations

The known symmetries of the Maxwell equations are summarized. Then new symmetries of these equations found by the authors are reviewed. These symmetries are generated by infinitesimal integrodifferential operators of the eight-dimensional Lie algebra. Their physical meaning is not clear.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Bergmannian relativity and bracket spaces

We explore an avenue of higher-dimensional spacetimes based on generalized spinors which transform under the special linear groups and result in spacetime dimensions which are squares of integers. The Bergmannian chronometrics are not Riemannian, but Finslerian in the higher dimensions. The general concept of bracket space is introduced in order to show a variety of routes to hyperspace. The field equations found generalize Einstein's by replacing a factor of two by the spinorial dimension. A mass term is introduced in the action, which results in a hyper-stress-energy-momentum tensor. The chronometric is not required to be covariantly constant under the hyper-Palatini variations: there is torsion. Spherical symmetry in this spacetime is explored, an appropriate set of coordinates is introduced, and the invariant for nine-dimensional spherical symmetry is given.     

On the existence of interior solutions in general relativity

The existence problem of matter sources for given stationary axisymmetric solutions of Einstein's vacuum field equations is investigated. The existence of sources of a differentially or rigidly rotating perfect fluid can be proved at least in the neighborhood of boundary surfaces if these can be chosen suitably (theorem). In particular, there exist such half-local perfect fluid sources for the Kerr metric. Hence the existence of a global regular Kerr-interior solution cannot be excluded by local considerations in an arbitrarily small neighborhood of a possible boundary.Work supported in part by the Deutsche Forschungsgemeinschaft.     

A variational principle giving gravitational “superpotentials,” the affine connection,Riemann tensor,and Einstein field equations

A first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensorR _k in terms of the affine connection and metric; the definition of the affine connection in terms of the metric; the Einstein field equations; and the definition of a set of gravitational superpotentials closely connected with the Komar conservation laws [7]. Substitution of the definition of the affine connection into this Lagrangian results in a second-order Lagrangian, from which follow the definition of the fully covariant Riemann tensor in terms of the metric, the Einstein equations, and the definition of the gravitational superpotentials.Dedicated to Achille Papapetrou on the occasion of his retirement.     

4D Homogeneous Isotropic Cosmological Models Generated by the 5D Vacuum

4-Dimensional homogeneous isotropic cosmological models obtained from solutions of vacuum 5-dimensional Einstein equations are considered. It is assumed, that the G₅₅-component of the 5D metric simulates matter in the comoving frame of reference. The observable 4D metric is defined up to conformal transformations of the metric of the 4D section g, with a conformal factor as a function of the component G₅₅. It is demonstrated, that the form of this function determines the matter equation of state. Possible equations of state are analyzed separately for flat, open and closed models.     

On purely radiative space-times

The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein's vacuum field equations possessing a smooth structure at past null infinity which forms the future null cone at past timelike infinity with complete generators. The pure radiation problem is analysed where free initial data for Einstein's field equations are prescribed on the null cone at past time-like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.Work supported by a Heisenberg-fellowship of the Deutsche Forschungsgemeinschaft     

Time-energy uncertainty and relativistic canonical commutation relations in quantum spacetime

It is shown that the time operatorQ ⁰ appearing in the realization of the RCCR's [Q,P^v]=–jhg^v, on Minkowski quantum spacetime is a self adjoint operator on Hilbert space of square integrable functions over _m =×v _m , where is a timelike hyperplane. This result leads to time-energy uncertainty relations that match their space-momentum counterparts. The operators Q appearing in Born's metric operator in quantum spacetime emerge as internal spacetime operators for exciton states, and the condition that the metric operator should possess a ground exciton state assumes the significance of achieving minimal spacetime4-momentum uncertainty in fundamental standards for spacetime measurements.Supported in part by NSERC research grant No. A5206.     

A Crucial Ingredient of Inflation

Nonminimal coupling of the inflaton field to the Ricci curvature of spacetime is generally unavoidable, and the paradigm of inflation should be generalized by including the corresponding term R²/2 in the Lagrangian of the inflationary theory. This paper reports on the status of the programme of generalizing inflation. First, the problem of finding the correct value (or set of values) of the coupling constant is analyzed; the result has important consequences for the success or failure of inflationary scenarios. Then, the slow-roll approximation to generalized inflation is studied. Both the unperturbed inflating universe models and scalar/tensor perturbations are discussed, and open problems are pointed out.     

A plane-symmetric solution of Einstein's equations with perfect fluid and an equation of statep=γϱ

A new class of plane-symmetric solutions of Einstein's equations with perfect fluid source and an equation of statep= (=const.) is presented. It contains the static vacuum solution, a special Kasner solution and the flat Friedmann-Robertson-Walker spacetime as subclasses. The only class for which the matter distribution is truly inhomogeneous (class D in the sequel) represents matter concentrated around a planar orbit of the symmetry group in an expanding universe.