Showing papers on "Singularity published in 1973"

Spin and torsion may avert gravitational singularities

Abstract: IT has long been recognized that isotropic models of the world are singular. The models, be they Newtonian or based on Einstein's equations of general relativity, lead to the prediction of at least one moment in the history of the Universe with an infinite density of matter. A similar singularity has been shown to occur as a result of spherically symmetric gravitational collapse of a cloud of dust1. For some time it was thought that the singularities might be a consequence of the high symmetry of the models.

Internal instability in a Reissner-Nordström black hole

Abstract: The question of the effect of asynumetries in gravitational collapse is investigated by considerations of test electromagnetic fields in an extended Reissner-Nordstrom background. It is found, with ths aid of computer calculations, that instabilities in the test field arise at the inner (Cauchy or anti-event) horizon, though not at the ouier (event) horizon. Thus it is reasonable to infer that in the full coupled Einstein-Maxwell theory the inner horizon will not survive as a non-singular bypersurface when asymmetric perturbations are present, but will instead become a space-time curvature singularity.

On the analysis of scattering and antenna problems using the singularity expansion technique

Abstract: In this paper, the possibility of analyzing scattering and antenna problems from a singularity expansion point of view is discussed. As an example of the method, a thin-wire scatterer is considered by first determining the locations of the exterior natural resonant frequencies and then constructing the time response of the current on the body, much in the same manner as in classical circuit theory. The numerical techniques used will be presented, and some advantages of the singularity expansion method over the other conventional ways of treating this problem will be mentioned.

Quantized matter fields and the avoidance of singularities in general relativity

Abstract: We consider a classical gravitational field minimally coupled to a quantized neutral scalar field possessing mass. We are especially concerned with the effects of particle creation and quantum coherence on the premises and conclusions of the singularity theorems, which imply the inevitability of singularities in classical general relativity. A closed Robertson-Walker geometry is used throughout. Nongravitational interactions are not considered. The source of the gravitational field in the Einstein equations is the expectation value of the energy-momentum tensor of the quantized scalar field. Lacking a general prescription for obtaining a finite operator from the divergent formal expression for the energy-momentum tensor, we confine our attention to situations in which plausible special methods are available. We show that quantum coherence effects in this semiclassical model can result in a violation of the energy conditions which enter into the singularity theorems. Then we exhibit numerical solutions of the coupled Einstein and scalar field equations in which a Friedmann-like collapse is stopped and converted to a Friedmann-like expansion. (In this calculation one mode of the quantum field was assumed dominant.) We conclude that quantum effects of the type considered here can sometimes lead to avoidance of the cosmological singularity, at least on the time scale of one Friedmann expansion.

Application of the renormalization group technique to the problem of phase transition in one-dimensional metallic systems. I. Invariant couplings, vertex, and one-particle Green's function

Abstract: A one-dimensional system of electrons interacting via a BCS-type interaction is investigated by renormalization group techniques in two successive approximations atT=0, keeping only a single energy variable ω. The first approximation is equivalent to the summation of leading logarithmic terms carried out by Bychkovet al., and correspondingly the vertex function displays a singularity at a finite value of ω. The second approximation accounts for the next leading logarithmic terms as well, and by this means the singularity is shown to be pushed down to ω=0. Due to important self-energy contributions, however, the invariant couplings behave differently and tend to a saturation value at ω=0.

Maximally slicing a black hole.

Abstract: Analytic and computer-derived solutions are presented of the problem of slicing the Schwarzschild geometry into asymptotically flat, asymptotically static, maximal spacelike hypersurfaces. The sequence of hypersurfaces advances forward in time in both halves (u greater than or equal to 0, u less than or equal to 0) of the Kruskal diagram, tending asymptotically to the hypersurface r = 3/2 M and avoiding the singularity at r = 0. Maximality is therefore a potentially useful condition to impose in obtaining computer solutions of Einstein's equations.

An improved method for the aerodynamic analysis of wing-body-tail configurations in subsonic and supersonic flow. Part 1: Theory and application

Abstract: A new method has been developed for calculating the pressure distribution and aerodynamic characteristics of wing-body-tail combinations in subsonic and supersonic potential flow. A computer program has been developed to perform the numerical calculations. The configuration surface is subdivided into a large number of panels, each of which contains an aerodynamic singularity distribution. A constant source distribution is used on the body panels, and a vortex distribution having a linear variation in the streamwise direction is used on the wing and tail panels. The normal components of velocity induced at specified control points by each singularity distribution are calculated and make up the coefficients of a system of linear equations relating the strengths of the singularities to the magnitude of the normal velocities. The singularity strengths which satisfy the boundary condition of tangential flow at the control points for a given Mach number and angle of attack are determined by solving this system of equations using an interactive procedure. Once the singularity strengths are known, the pressure coefficients are calculated, and the forces and moments acting on the configuration determined by numerical integration.

On the problem of the initial state in the isotropic scalar-tensor cosmology of Brans-Dicke

Abstract: In connection with the problem of the initial singularity in the framework of the Brans-Dicke scalar-tensor isotropic cosmology, the dynamics of the early stages of the expansion is studied on the basis of the general analytical solutions for flat models withP=ne (0≤n≤1). The sourceless scalar field, which plays a role of effective source of the geometry, entirely changes the character of the initial expansion, whereas in the absense of it the dynamics of the models almost does not differ from GR-case. If the connecting parameter ω<−6 then the sourceless scalar field removes the initial singularity for any equations of state, and provides the regular transition form the compression to the expansion through the non-stationary state with $$\dot \varphi e 0$$ . If ω is positive, the singularity of vacuum nature necessarily exists; the sourceless scalar-field being prevalent over the material sources (at least for P≤1/3e) near the singularity. For P>1/3e parallel with the vacuum singularity the singular initial state with dominant role of gravitating matter is possible. In the course of expansion, the influence of gravitating matter and curvature-which as nonessential at the early stage in comparison with sourceless ϕ-field — becomes dominant, and all models approach the ‘Machian’ ones which differs slightly from corresponding GR-models. If ω is negative (ω<−6) the sourceless ϕ-field slows down the initial expansion and increases essentially the real age of the Universe; it can also influence the primordial nuclear synthesis by diminishing in particular the content of He4. It is shown that a knowledge of the signĠ gives us the information about the nature of the initial Universe state.

Theory of elastic constants of heterogeneous media

Abstract: The theory of elastic equilibrium of heterogeneous, macrohomogeneous bodies is cast in the form of a linear integral equation with the following properties: (1) The equation contains a constant elastic tensor which can be chosen freely to optimize any approximation method. (2) The singularity of the kernel is such that the integral must be interpreted as the principal value, thereby providing a clear separation between structure‐dependent and structure‐independent contributions. (3) The mean elastic tensor equals the above‐mentioned constant tensor plus the space average of the dependent variable. (4) A term proportional to this average appears separately in the equation. (5) The integral equation facilitates the rederivation of several well‐known approximations and opens new possibilities, of which some are discussed, in particular, a self‐consistent imbedding method and an approximation based on correlation functions.

The singularity at the crest of a finite amplitude progressive Stokes wave

Abstract: Expansions have been given in the past for steady Stokes waves at or near a largest wave with a 120° corner. It is shown here that the solution is more complicated than has been assumed: that the corner is not a regular singular point, and that waves of less than maximum amplitude have singularities of a different order.

Application of the renormalization group technique to the problem of phase transition in one-dimensional metallic systems. II. Response functions and the ground-state problem

Abstract: The results of the preceding paper for the invariant coupling are used to calculate some response functions in a one-dimensional metallic system. Three generalized susceptibilities, characterizing the possible superconducting or antiferromagnetic behavior of the system and the appearance of density waves, are calculated by means of the Lie equations of the renormalization group. Due to the nonsingular behavior of the invariant couplings, the response functions can diverge at ω=0 only, and this singularity is of power-law type. Depending on the sign and relative value of the bare coupling constants, the model system tends to superconducting or antiferromagnetic order at T=0. In certain cases the period of the system is doubled.

A Minisuperspace Example: The Gowdy T 3 Cosmology

Abstract: The Gowdy ${T}^{3}$ inhomogeneous cosmology is an exact vacuum solution of Einstein's equations representing gravitational waves propagating in an expanding closed universe. As shown by Berger, it provides an example showing graviton pair creation as a quantum effect near the initial singularity, together with the reaction this induces in the cosmological expansion. The truncated quantization given here by superspace methods develops the "single residual constraint" quantization method proposed by Moncrief. The quantum wave equation is soluble by separation of variables, and shows that graviton number is a concept unsuited to the description of the initial (velocity-dominated) singularity. Thus a good description of the pair creation occurring in even this soluble model has not yet been developed.

2-SPEED, a single-gimbal control moment gyro attitude control system.

Abstract: In 2-SPEED (Two Scissored Pair Ensemble, Explicit Distribution) four single-gimbal control moment gyros (SGCMGs) configured into two scissored pairs are combined with an explicit distribution of angular momentum between pairs to produce a system relatively insensitive to the singularity problems which have plagued other SGCMG concepts. In this system, the singularity surfaces in momentum space degenerate to discrete curves. Further, the system permits a smooth passage through these remaining singularities with, at worst, a temporary delay while momentum redistribution takes place. Finally, CMG-out operation is possible within the full volume of the reduced momentum envelope.

Can spin avert singularities

Abstract: TRAUTMAN has suggested1 that the introduction of intrinsic spin effects into general relativity through the Einstein-Cartan torsion theory may avert a gravitational singularity. In this note we suggest that the singularity is avoided only because of the high symmetry of the model used.

Stress singularities at concentrated loads

Abstract: The stress singularity created by a concentrated load applied at the boundary of a half-plane was studied by transforming it into an optical singularity by the optical method of caustics. The half-plane was considered to be elastic, isotropic and under generalized plane-stress conditions. According to the method of caustics, the light rays impinging normally at the thin plate are partly reflected from either the front or the rear faces of the plate. The reflected rays are deviated because of the important constraint of the plate at the vicinity of the applied load and the significant variation of the refractive index there. The deviated light rays, when projected on a reference screen, are concentrated along a singular curve which is, therefore, strongly illuminated and forms a caustic. It is shown that the shape and size of the caustic depends on the stress singularity at the point of application of the load. Thus, by measuring the dimensions of this singular curve, one can evaluate the state of stress at the singularity. The characteristic properties of the caustic created by such a singularity were studied in relation with the loading mode of the plate.

The space of Lorentz metrics

Abstract: The set of allC 2 Lorentz metrics on a non-compact four-manifold is given the Whitney fineC 2 topology. It is shown that this provides the correct framework within which to discuss the global properties of spacetime manifolds in general, and the singularity theorems in particular. The main result is a theorem showing that the Robertson-Walker big bang (global infinite density singularity in the finite past) is stable under sufficiently small, but otherwise arbitrary, finiteC 2 perturbations of the metric tensor.

Boundary-Layer Separation From Downstream Moving Boundaries

Abstract: The laminar boundary-layer equations for incompressible flow with a mild adverse pressure gradient were numerically solved for flows over downstream moving boundaries. It was demonstrated that the vanishing of skin friction in this case is not related to separation.2 Indeed the integration proceeds smoothly through a point of vanishing skin friction and further downstream a Goldstein-type singularity appears at a station where all the properties of separation according to the model of Moore, Rott, and Sears are present. It is also numerically demonstrated that the singular behavior is not uniform with n, the distance perpendicular to the wall, but it is initiated at a point away from the wall leaving below a region of nonsingular flow. The foregoing points provide numerical justification of the general theoretical models of unsteady boundary-layer separation suggested by Sears and Telionis.

Numerical investigation of unsteady boundary‐layer separation

Abstract: A numerical scheme of integration is devised to integrate the unsteady laminar boundary‐layer equations with partly reversed flows. The neighborhood of the vanishing skin friction which has been considered up to now as the location of separation, is found to be nonsingular while a typical Goldstein‐type of a traveling singularity is discovered farther downstream. This singularity is interpreted according to the theory of Sears and Telionis as unsteady boundary‐layer separation. The features of the flow in the neighborhood of the singularity are investigated.

The singularity problem in the dynamic input-output model†

Abstract: A transformation of the control variables is introduced which overcomes matrix singularity problems arising out of the transposition of the dynamic input-output model into the 9tato-sp formulation.

An Integral Equation Approach to the Semi-Infinite Strip Problem

Abstract: A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. The integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. The stress singularity at the strip corner is obtained from the singular integral equation, which is then solved numerically. Stresses along the rigid end are determined, and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.

Singularity and Matter Creation in Cosmological Models

Abstract: UNDER “reasonable assumptions” the singularity present in the classical Friedmann cosmological models and in the gravitational collapse of massive objects, is an unavoidable consequence of general relativity1,2 As a rule, singular solutions are unpalatable to theoreticians and suggest something wrong with the theory Nevertheless some cosmologists make a virtue of this drawback by arguing that the cosmological singularity describes an unusual event, the creation of the Universe and as such should not be judged by the usual yardstick Whatever the merit of this argument, it certainly cannot be applied to the case of collapsing massive objects

Equivalent currents for a ring discontinuity

Abstract: Based on the geometrical theory of diffraction, equivalent filamentary currents are derived for a linear element of a surface singularity for general incidence and polarization. The currents are used to obtain an analytical expression for the bistatic field diffracted by a ring singularity, valid within the caustic region as well as at wide angles.