Spherical symmetry and mass-energy in higher dimensions

The components of the Einstein tensor and other relations are given for a spherically symmetric metric in null coordinates in higher dimensions. These relations are particularly relevant to the study of gravitational collapse of a perfect fluid with heat flow but without viscosity, where the exterior space cannot be considered as vacuum and matching to Schwarzschild space-time is not suitable. The analysis generalizes to higher dimensions work of Cahill and McVittie in 4D space-time. Using the expression for the mass function, it is observed that pressure vanishes at the boundary of the distribution for a perfect fluid in the higher-dimensional case also, but the same is not true when heat flow is considered.     

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general tilting case.     

Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem

In a nongeometrical interpretation of gravity,the metric g(x) = + (x)is interpreted as an effective metric, whereas(x) is interpreted as afundamental gravitational field, propagated in spacetime which isactually flat. Some advantages and disadvantages of suchan interpretation are discussed. The main advantage isa natural resolution of the flatness problem.     

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.     

Probability Backflow for a Dirac Particle

The phenomenon of probability backflow, previously quantified for a free nonrelativistic particle, is considered for a free particle obeying Dirac's equation. It is shown that probability backflow can occur in the opposite direction to the momentum; that is to say, there exist positive-energy states in which the particle certainly has a positive momentum in a given direction, but for which the component of the probability flux vector in that direction is negative. It is shown that the maximum possible amount of probability that can flow backwards, over a given time interval of duration T, depends on the dimensionless parameter = (4/mc²T)^1/2, where m is the mass of the particle and c is the speed of light. At = 0, the nonrelativistic value of approximately 0.039 for this maximum is recovered. Numerical studies suggest that the maximum decreases monotonically as increases from 0, and show that it depends on the size of m, , and T, unlike the nonrelativistic case.     

Cylindrically symmetric Zel'dovich fluid distributions in general theory of relativity

The problem of charged perfect fluid distribution is investigated when the space-time is described by the Einstein-Rosen metric. It is shown that with assumed cylindrical symmetry the cosmological constant vanishes, the electromagnetic field becomes source-free, and the perfect fluid reduces to Zel'dovich fluid withp=. Sets of exact solutions for this case have been obtained and the corresponding solutions for Brans-Dicke-Maxwell fields have been derived. For these solutions the Einstein-Rosen metric, however, goes over to three-parameter Marder metric in Brans-Dicke theory.     

The metric in a static cylindrical elastic medium and in an empty rotating frame as solutions of Einstein's field equations

Using the Weyl-type canonical coordinates, an integration of Einstein's field equations in the cylindrosymmetric case considered by Kurunolu is reexamined. It is made clear that the resulting metric is not describing the spacetime in a rotating frame, but in astatic cylindrical elastic medium. The conclusion of Kurunolu that for an observer on a rotating disk there is no way of escape from a curved spacetime is therefore not valid. The metric in an empty rotating frame is found as a solution of Einstein's field equations, and is not orthogonal. It is shown that the corresponding orthogonal solution represents spacetime in an inertial frame expressed in cylindrical coordinates. Introducing a noncoordinate basis, the metric in a rotating frame is given the static form of Kurunolu's solution. The essential role played by the nonvanishing structure coefficients in this case is made clear.     

Experimental investigation of infrared fluorescence of molecular gases under the action of CO2-laser radiation

In the paper, the experimental investigation is carried out of the IR fluorescence of organic substances of different classes in the range 600–3000 cm^–1 under the action of the continuous radiation of a CO₂ laser with a power of 40 W/cm². The procedure is given for obtaining the fluorescence spectra. For all the substances investigated, the magnitudes of the effective absorption cross section were determined experimentally at the generation frequency of the laser. Fluorescence is observed only in the case when _v 0; if _v 0, fluorescence is absent. It is shown that the fluorescence intensity depends significantly on the magnitude of the absorption cross section at the laser frequency. A comparison of the fluorescence and absorption spectra shows that only vibrations allowed in the IR region appear in the emission; in a number of cases, a sharp increase of the fluorescence intensity of certain overtones and compound vibrations is observed: when the frequencies are multiples of the laser emission.Translated from Izvestiya Uchebnykh Zavedenii, Fizika, No. 3, pp. 50–53, March Vysshikh 1979.     

Singular Shell Embedded into a Cosmological Model

We generalize Israel's formalism to cover singular shells embedded in a non-vacuum Universe. That is, we deduce the relativistic equation of motion for a thin shell embedded in a Schwarzschild/Friedmann-Lemaître-Robertson-Walker spacetime. Also, we review the embedding of a Schwarzschild mass into a cosmological model using curvature coordinates and give solutions with (Sch/FLRW) and without the embedded mass (FLRW).     

Segregation of Components as a Result of High‐Power Laser Irradiation of Heterogeneous Condensed Media

The effects of periodic segregation of components in metastable (supercooled or supersaturated) binary alloys in the course of kinetic phase transformations as a result of laser irradiation of heterogeneous systems were studied analytically. Nonlinear processes of temporal and spatial selforganization of concentrationrelated structures were simulated using (i) a selfconsistent system of timedependent twodimensional equations for the distribution function for the sizes and spatial coordinates of the newphase particles and (ii) balance equations for the temperature and concentration of dissolved components; the latter equations account for nonlinearity of the particlesource function, sinks, for dependences of the phasetransition temperature on the surface curvature of particles and on the concentration of components, and for diffusive motion of particles in space. The domain of existence for the instabilities under consideration and the characteristics of the formed crystallizationrelated periodic structures are determined. It is established that nanoclusters formed during supersaturation of crystallizing material may play an important role in generation of selfoscillatory crystallization modes. Hydrodynamic aspects of liquidphase concentrationrelated stratification in heterogeneous systems based on immiscible components are considered.     

On symmetric solutions of the relativistic Vlasov-Poisson system

Spherically symmetric solutions to the Cauchy problem for the relativistic Vlasov-Poisson system are studied in three space dimensions. If the energy is positive definite (the plasma physics case), global classical solutions exist. In the case of indefinite energy, small radial solutions exist in the large, but large data solutions (those with negative energy) will blow-up in finite time.Research supported in part by NSF MCS 8319944     

The physical properties of linear and action-angle coordinates in classical and quantum mechanics

The quantum harmonic oscillator is described in terms of two basic sets of coordinates: linear coordinates x, p_x and angular coordinates eⁱ, P (action-angle variables). The angular coordinate eⁱ is assumed unitary, the conjugate momentum p is assumed Hermitian, and eⁱ and p are assumed to be a canonical pair. Two transformations are defined connecting the angular coordinates to the linear coordinates. It is found that x, p_x can be physical, i.e., Hermitian and canonical, only under constraints on the p eigenvalue spectrum. The conclusion is that eⁱ can be a unitary operator. A parallel analysis of the classical harmonic oscillator is done with equivalent results.     

Parametrized Field Theory

A theory is presented in which a field depends not only on spacetime coordinates x, but also on a Lorentz-invariant parameter . Such a theory is conceptually and technically simple and manifestly covariant at every step. The generator of evolution and the generator of spacetime translations and Lorentz transformations are obtained in a straightforward way. In the quantized theory the Heisenberg equation of motion is written in a covariant form and is equivalent to the field equation. The equal commutator between the field and its canonically conjugate momentum is just proportional to the spacetime function. Finally comparison with the conventional field theory is done, and it is found that the expectation value of the momentum operator in the on shell states is the same.     

Perfect Fluid Spacetimes with a Purely Magnetic Weyl Tensor

We present a solution of the gravitational fieldequations which is similar in form to that given byWainwright. Several cases are considered, in particularwe find a general algebraic perfect fluid solution with equation of state p = whose Weyl tensor is of the purely magnetic type within a finiteregion of the spacetime. It is shown, for an observerwith four-velocity, u_mag say, that themetric's Weyl tensor is purely magnetic within the finiteregion while it is purely electric, as read by anotherobserver with four-velocity u_ele, elsewhere.Another observer, independent of the observers whomeasure the Weyl tensor to be purely electric ormagnetic, interprets the perfect fluid to have anequation of state p = . The Petrov type of themetric, in this case, is I(M) by theArianrhod-McIntosh classification and therefore there exists noconformally related metric which is vacuum. The vacuumseed metrics are derived for the perfect fluidsolutions.     

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 Hamiltonian of Asymptotically Friedmann-Lemaître-Robertson-Walker Spacetimes

We obtain the correct Hamiltonian which describes the dynamics of classes of asymptotic open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, which includes Tolman geometries. We calculate the surface term that has to be added to the usual Hamiltonian of General Relativity in order to obtain an improved Hamiltonian with well defined functional derivatives. For asymptotic flat FLRW spaces, this surface term is zero, but for asymptotic negative curvature FLRW spaces it is not null in general. In the particular case of the Tolman geometries, they vanish. The surface term evaluated on a particular solution of Einstein's equations may be viewed as the energy of this solution with respect to the FLRW spacetime they approach asymptotically. Our results are obtained for a matter content described by a dust fluid, but they are valid for any perfect fluid, including the cosmological constant.     

SO(5, 1) dynamical symmetry for electron Zitterbewegung

Electron rest-frame internal canonical coordinates are reobtained by the free-particle Foldy-Wouthuysen transformation: Schrödinger microscopic momentum, Barut-Bracken microscopic coordinate, and the rest Hamiltonian, which describe Zitterbewegung in this frame.SO(4, 1) Snyder space-time invariant quantization is considered in order to construct a dynamical group for Zitterbewegung. The electron's internal structure appears associated with its secondorder self-energy process and governed by the 15-parameter dynamical groupSO(5, 1). This is a generalization of Barut-Bracken symmetry which describes Zitterbewegung as generated by an algebra of the rotation groupSO(5). This noncompact symmetrySO(5, 1) permits a natural interpretation for the operators of its algebra and introduces a generalization to higher-dimensional fermionic representations.     

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's exact set of fields for the vacuum equations.Dedicated to the memory of H. Rund     

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.     

Horizons and singularities in static,spherically symmetric spacetimes

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (static) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.