Discontinuities in spherically symmetric gravitational fields and shells of radiation

1958 ◽  
Vol 248 (1254) ◽  
pp. 404-414 ◽  
Keyword(s):  
Boundary Conditions ◽  
Spherical Shell ◽  
Previous Treatment ◽  
Empty Space ◽  
Point Of View ◽  
Spherically Symmetric ◽  
Junction Conditions ◽  
Metric Tensor ◽  
Gravitational Fields

Boundary conditions at a 3-space of discontinuity ∑ are considered from the point of view of Lichnerowicz. The validity of the O’Brien—Synge junction conditions is established for co-ordinates derivable from Lichnerowicz’s ‘admissible co-ordinates’ by a transformation which is uniformly differentiable across ∑. The co-ordinates r , θ , ϕ , t , used by Schwarzschild and most later authors when dealing with spherically symmetric fields, are shown to be of this type. In Schwarzschild’s co-ordinates, the components of the metric tensor can always be made continuous across Ʃ, and simple relations are derived connecting the jumps in their first derivatives. A spherical shell of radiation expanding in empty space is examined in the light of the above ideas, and difficulties encountered by Raychaudhuri in a previous treatment of this problem are cleared up. A particular model is then discussed in some detail.

Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time

1962 ◽  
Vol 270 (1340) ◽  
pp. 103-126 ◽  
Keyword(s):  
Boundary Conditions ◽  
Gravitational Waves ◽  
Superposition Principle ◽  
Flat Space ◽  
Space Time ◽  
Axially Symmetric ◽  
Field Equations ◽  
Spatial Infinity ◽  
Metric Tensor ◽  
Gravitational Fields

Gravitational fields containing bounded sources and gravitational radiation are examined by analyzing their properties at spatial infinity. A convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced. Then suitable boundary conditions are set. The group of co-ordinate transformations that preserves the boundary conditions is analyzed. Different possible gravitational fields are characterized intrinsically by a combination of (i) characteristic initial data, and (ii) Dirichlet data at spatial infinity. To determine a particular solution one must specify four functions of three variables and three functions of two variables; these functions are not subject to constraints. A method for integrating the field equations is given; the asymptotic behaviour of the metric and Riemann tensors for large spatial distances is analyzed in detail; the dynamical variables of the radiation modes are exhibited; and a superposition principle for the radiation modes of the gravitational field is suggested. Among the results are: (i) the group of allowed co-ordinate transformations contains the inhomogeneous orthochronous Lorentz group as a subgroup; (ii) each of the five leading terms in an asymptotic expansion of the Riemann tensor has the algebraic structure previously predicted from analyzing the Petrov classification; (iii) gravitational waves appear to carry mass away from the interior; (iv) time-dependent periodic solutions of the field equations which obey the stated boundary conditions do not exist. It was found that the general fields studied in the present work are in many ways very similar to the axially symmetric fields recently studied by Bondi, van der Burg & Metzner.

Spherically Symmetric Gravitational Fields

1965 ◽  
Vol 6 (1) ◽  
pp. 1-5 ◽  
Author(s):  
P. G. Bergmann ◽  
M. Cahen ◽  
A. B. Komar
Keyword(s):  
Spherically Symmetric ◽  
Gravitational Fields

Automatic Generation of Anisotropic Patterns of Geometric Features for Industrial Design

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Diego Andrade ◽  
Ved Vyas ◽  
Kenji Shimada
Keyword(s):  
Boundary Conditions ◽  
Computer Aided Design ◽  
Industrial Design ◽  
Automatic Generation ◽  
Free Form ◽  
Geometric Features ◽  
Target Domain ◽  
Metric Tensor ◽  
Target Region ◽  
Free Form Surfaces

While modern computer aided design (CAD) systems currently offer tools for generating simple patterns, such as uniformly spaced rectangular or radial patterns, these tools are limited in several ways: (1) They cannot be applied to free-form geometries used in industrial design, (2) patterning of these features happens within a single working plane and is not applicable to highly curved surfaces, and (3) created features lack anisotropy and spatial variations, such as changes in the size and orientation of geometric features within a given region. In this paper, we introduce a novel approach for creating anisotropic patterns of geometric features on free-form surfaces. Complex patterns are generated automatically, such that they conform to the boundary of any specified target region. Furthermore, user input of a small number of geometric features (called “seed features”) of desired size and orientation in preferred locations could be specified within the target domain. These geometric seed features are then transformed into tensors and used as boundary conditions to generate a Riemannian metric tensor field. A form of Laplace's heat equation is used to produce the field over the target domain, subject to specified boundary conditions. The field represents the anisotropic pattern of geometric features. This procedure is implemented as an add-on for a commercial CAD package to add geometric features to a target region of a three-dimensional model using two set operations: union and subtraction. This method facilitates the creation of a complex pattern of hundreds of geometric features in less than 5 min. All the features are accessible from the CAD system, and if required, they are manipulable individually by the user.

scholarly journals GRIBOV PROBLEM FOR GAUGE THEORIES: A PEDAGOGICAL INTRODUCTION

2004 ◽  
Vol 01 (04) ◽  
pp. 423-441 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
DIEGO N. PELLICCIA ◽  
FRANCESCO ZACCARIA
Keyword(s):  
Cross Sections ◽  
Gauge Theories ◽  
Functional Integral ◽  
Point Of View ◽  
Spherically Symmetric ◽  
Abelian Gauge ◽  
Gauge Transformations ◽  
Recent Developments ◽  
Gribov Ambiguity ◽  
Gribov Copies

The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a nonlinear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.

scholarly journals EDGE CURRENTS AND VERTEX OPERATORS FOR CHERN-SIMONS GRAVITY

1993 ◽  
Vol 08 (04) ◽  
pp. 653-682 ◽  
Author(s):  
G. BIMONTE ◽  
K.S. GUPTA ◽  
A. STERN
Keyword(s):  
Boundary Conditions ◽  
Vertex Operator ◽  
Vacuum State ◽  
Space Time ◽  
Moody Algebra ◽  
Functional Integrals ◽  
Gravitational Fields ◽  
Dimensional Gravity ◽  
Discrete Mass ◽  
Chern Simons

We apply elementary canonical methods for the quantization of 2+1 dimensional gravity, where the dynamics is given by E. Witten’s ISO(2, 1) Chern-Simons action. As in a previous work, our approach does not involve choice of gauge or clever manipulations of functional integrals. Instead, we just require the Gauss law constraint for gravity to be first class and also to be everywhere differentiable. When the spatial slice is a disc, the gravitational fields can either be unconstrained or constrained at the boundary of the disc. The unconstrained fields correspond to edge currents which carry a representation of the ISO(2, 1) Kac-Moody algebra. Unitary representations for such an algebra have been found using the method of induced representations. In the case of constrained fields, we can classify all possible boundary conditions. For several different boundary conditions, the field content of the theory reduces precisely to that of 1+1 dimensional gravity theories. We extend the above formalism to include sources. The sources take into account self-interactions. This is done by punching holes in the disc, and erecting an ISO(2, 1) Kac–Moody algebra on the boundary of each hole. If the hole is originally sourceless, a source can be created via the action of a vertex operator V. We give an explicit expression for V. We shall show that when acting on the vacuum state, it creates particles with a discrete mass spectrum. The lowest mass particle induces a cylindrical space-time geometry, while higher mass particles give an n fold covering of the cylinder. The vertex operator therefore creates cylindrical space-time geometries from the vacuum.

scholarly journals Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Mikhail Z. Iofa
Keyword(s):  
Spherical Shell ◽  
Coordinate Transformation ◽  
Time Dependent ◽  
Thin Shells ◽  
Junction Conditions ◽  
Coordinate Systems ◽  
Normal Coordinate ◽  
Normal Coordinates

Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates. We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications to the Israel junction conditions are discussed.

scholarly journals PARAMETRIC INSTABILITY IN SCALAR GRAVITATIONAL FIELDS

2012 ◽  
Vol 27 (24) ◽  
pp. 1230023 ◽  
Author(s):  
TREVOR B. DAVIES ◽  
CHARLES H.-T. WANG ◽  
ROBERT BINGHAM ◽  
J. TITO MENDONÇA
Keyword(s):  
Gravitational Field ◽  
Field Effect ◽  
Strong Field ◽  
Parametric Instability ◽  
Spherically Symmetric ◽  
Scalar Tensor Theory ◽  
Subsequent Work ◽  
Dynamical Mechanism ◽  
Gravitational Fields ◽  
Tensor Theory

We present a brief review on a new dynamical mechanism for a strong field effect in scalar–tensor theory. Starting with a summary of the essential features of the theory and subsequent work by several authors, we analytically investigate the parametric excitation of a scalar gravitational field in a spherically symmetric radially pulsating neutron star.

scholarly journals On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem ◽  
Mieczysław Cichoń
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Point Of View ◽  
Integral Boundary ◽  
Special Cases

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

GRAVITATIONAL BAG EFFECTS ON KALUZA KLEIN AND STRING EXCITATIONS

1989 ◽  
Vol 04 (19) ◽  
pp. 5119-5131 ◽  
Author(s):  
E. I. GUENDELMAN
Keyword(s):  
Extra Dimensions ◽  
Blow Up ◽  
Point Of View ◽  
Spherically Symmetric ◽  
Massive String ◽  
Free Objects ◽  
Higher Dimensional ◽  
Central Regions ◽  
The Masses ◽  
Kaluza Klein

Gravitational Bags are spherically symmetric solutions of higher-dimensional Kaluza Klein (K – K) theories, where the compact dimensions become very large near the center of the geometry, although they are small elsewhere. The K – K excitations therefore become very light when located near the center of this geometry and this appears to affect drastically the naive tower of the masses spectrum of K – K theories. In the context of string theories, string excitations can be enclosed by Gravitational Bags, making them not only lighter, but also localized, as observed by somebody, that does not probe the central regions. Strings, however, can still have divergent sizes, as quantum mechanics seems to demand, since the extra dimensions blow up at the center of the geometry. From a projected 4-D point of view, very massive string bits may lie inside their Schwarzschild radii, as pointed out by Casher, Gravitational Bags however are horizon free objects, so no conflict with macroscopic causality arises if the string excitations are enclosed by Gravitational Bags.

scholarly journals The theory of spherically symmetric thin shells in conformal gravity

2018 ◽  
Vol 27 (06) ◽  
pp. 1841012 ◽  
Author(s):  
Victor Berezin ◽  
Vyacheslav Dokuchaev ◽  
Yury Eroshenko
Keyword(s):  
General Relativity ◽  
Weyl Tensor ◽  
Delta Function ◽  
Einstein Gravity ◽  
Riemann Tensor ◽  
Momentum Tensor ◽  
Thin Shells ◽  
Spherically Symmetric ◽  
Gravitational Fields ◽  
Gravitational Theories

The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy–momentum tensor of the thin shell is proportional to Diracs delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl–Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in detail the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless ([Formula: see text] massless) shell, it is shown that their dynamics cannot be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl–Einstein gravity, the trajectory of the same type of shell is completely restored even without knowledge of the outside solution.

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