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.

Asymptotically flat spaces in asymptotically null-spherical coordinates

1970 ◽  
Vol 320 (1542) ◽  
pp. 349-368 ◽  
Keyword(s):  
Flat Space ◽  
Riemann Tensor ◽  
Polar Coordinates ◽  
Field Equations ◽  
Simple Criterion ◽  
Tetrad Formalism ◽  
Metric Tensor ◽  
Asymptotically Flat ◽  
Null Tetrad ◽  
Gravitational Fields

The behaviour of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism. For this, a system of coordinates u , r , θ and ɸ is used, such that at spatial infinity u = const, is a null hypersurface and r , θ and ɸ reduce to the usual spherical polar coordinates. A set of four vectors (a tetrad) is also chosen with the only restriction that they are everywhere null. The metric tensor and the four vectors are expanded in inverse powers of r ; the rotation coefficients and the tetrad components of the Riemann tensor are then calculated in a similar expansion; and the first two terms in the expansion beyond their values for a flat space are retained. The field equations in these approximations are derived explicitly and their effect on the expansion of the tetrad components of the Riemann tensor is studied; and the total energy and linear momentum are examined. In this paper three main results are derived: (i) the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad; (ii) the generalized expression for the news function of the field; (iii) a simple criterion for recognizing certain classes of non-radiating fields.

scholarly journals Post-Newtonian limit: second-order Jefimenko equations

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
David Pérez Carlos ◽  
Augusto Espinoza ◽  
Andrew Chubykalo
Keyword(s):  
Linear Equations ◽  
Space Time ◽  
Second Order ◽  
Field Equations ◽  
Gravitational Fields ◽  
Mass Distributions ◽  
Minkowski Space Time ◽  
Theory Of Gravitation ◽  
Gravity Probe ◽  
Gravitational Equations

Abstract The purpose of this paper is to get second-order gravitational equations, a correction made to Jefimenko’s linear gravitational equations. These linear equations were first proposed by Oliver Heaviside in [1], making an analogy between the laws of electromagnetism and gravitation. To achieve our goal, we will use perturbation methods on Einstein field equations. It should be emphasized that the resulting system of equations can also be derived from Logunov’s non-linear gravitational equations, but with different physical interpretation, for while in the former gravitation is considered as a deformation of space-time as we can see in [2–5], in the latter gravitation is considered as a physical tensor field in the Minkowski space-time (as in [6–8]). In Jefimenko’s theory of gravitation, exposed in [9, 10], there are two kinds of gravitational fields, the ordinary gravitational field, due to the presence of masses, at rest, or in motion and other field called Heaviside field due to and acts only on moving masses. The Heaviside field is known in general relativity as Lense-Thirring effect or gravitomagnetism (The Heaviside field is the gravitational analogous of the magnetic field in the electromagnetic theory, its existence was proved employing the Gravity Probe B launched by NASA (See, for example, [11, 12]). It is a type of gravitational induction), interpreted as a distortion of space-time due to the motion of mass distributions, (see, for example [13, 14]). Here, we will present our second-order Jefimenko equations for gravitation and its solutions.

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 SPIN-1 GRAVITATIONAL WAVES: THEORETICAL AND EXPERIMENTAL ASPECTS

2006 ◽  
Vol 03 (03) ◽  
pp. 451-469 ◽  
Author(s):  
F. CANFORA ◽  
L. PARISI ◽  
G. VILASI
Keyword(s):  
Physical Properties ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Lie Algebra ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Gravitational Fields ◽  
Killing Fields

Exact solutions of Einstein field equations invariant for a non-Abelian bidimensional Lie algebra of Killing fields are described. Physical properties of these gravitational fields are studied, their wave character is checked by making use of covariant criteria and the observable effects of such waves are outlined. The possibility of detection of these waves with modern detectors, spherical resonant antennas in particular, is sketched.

scholarly journals Axially Symmetric, Asymptotically Flat Vacuum Metric with a Naked Singularity and Closed Timelike Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Debojit Sarma ◽  
Faizuddin Ahmed ◽  
Mahadev Patgiri
Keyword(s):  
Naked Singularity ◽  
Empty Space ◽  
Space Time ◽  
Axially Symmetric ◽  
Field Equations ◽  
Closed Timelike Curves ◽  
Asymptotically Flat ◽  
Type D ◽  
Initial Hypersurface ◽  
Petrov Classification

We present an axially symmetric, asymptotically flat empty space solution of the Einstein field equations containing a naked singularity. The space-time is regular everywhere except on the symmetry axis where it possesses a true curvature singularity. The space-time is of type D in the Petrov classification scheme and is locally isometric to the metrics of case IV in the Kinnersley classification of type D vacuum metrics. Additionally, the space-time also shows the evolution of closed timelike curves (CTCs) from an initial hypersurface free from CTCs.

Time, Topology, and the Twin Paradox

2011 ◽  
Author(s):  
Jean‐Pierre Luminet
Keyword(s):  
High Speed ◽  
Reference Frames ◽  
Thought Experiment ◽  
Flat Space ◽  
Space Time ◽  
Twin Paradox ◽  
Theory Of Relativity ◽  
Apparent Paradox ◽  
Curved Space ◽  
Gravitational Fields

This chapter notes that the twin paradox is the best-known thought experiment associated with Einstein's theory of relativity. An astronaut who makes a journey into space in a high-speed rocket will return home to find he has aged less than his twin who stayed on Earth. This result appears puzzling, as the homebody twin can be considered to have done the travelling with respect to the traveller. Hence, it is called a “paradox”. In fact, there is no contradiction, and the apparent paradox has a simple resolution in special relativity with infinite flat space. In general relativity (dealing with gravitational fields and curved space-time), or in a compact space such as the hypersphere or a multiply connected finite space, the paradox is more complicated, but its resolution provides new insights about the structure of space–time and the limitations of the equivalence between inertial reference frames.

scholarly journals Axially Symmetric Null Dust Space-Time, Naked Singularity, and Cosmic Time Machine

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Faizuddin Ahmed
Keyword(s):  
Gravitational Lensing ◽  
Cosmic Time ◽  
Naked Singularity ◽  
Space Time ◽  
Axially Symmetric ◽  
Time Machine ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Closed Orbits ◽  
Group A

We present a gravitational collapse null dust solution of the Einstein field equations. The space-time is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. The space-time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine. The radial geodesics near the singularity and the gravitational lensing (GL) will be discussed. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It was demonstrated that this solution depends on the local gravitational field consisting of two components with amplitudes Ψ2 and Ψ4.

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.

scholarly journals SPIN-3/2 FERMIONS IN TWISTOR FORMALISM

2001 ◽  
Vol 16 (32) ◽  
pp. 2103-2113 ◽  
Author(s):  
MITSUO J. HAYASHI
Keyword(s):  
Twistor Space ◽  
Integrability Condition ◽  
Local Existence ◽  
Flat Space ◽  
Space Time ◽  
Necessary Condition ◽  
Field Equations ◽  
Weyl Spinor ◽  
Cartan Theory ◽  
The Right

Consistency conditions for the local existence of massless spin-3/2 fields have been explored to find the facts that the field equations for massless helicity-3/2 particles are consistent if the space–time is Ricci-flat, and that in Minkowski space–time the space of conserved charges for the fields is its twistor space itself. After considering the twistorial methods to study such massless helicity-3/2 fields, we show in flat space–time that the charges of spin-3/2 fields, defined topologically by the first Chern number of their spin-lowered self-dual Maxwell fields, are given by their twistor space, and in curved space–time that the (anti-)self-duality of the space–time is the necessary condition. Since in N=1 supergravity torsions are the essential ingredients, we generalize our space–time to that with torsion (Einstein–Cartan theory), and investigate the consistency of existence of spin-3/2 fields in this theory. A simple solution to this consistency problem is found: The space–time has to be conformally (anti-)self-dual, left-(or right-) torsion-free. The integrability condition on α-surface shows that the (anti-)self-dual Weyl spinor can be described only by the covariant derivative of the right-(left-)handed torsion.

EXACT SOLUTIONS FOR SELF-DUAL SU(2) GAUGE THEORY WITH AXIAL SYMMETRY

2001 ◽  
Vol 16 (11) ◽  
pp. 685-692 ◽  
Author(s):  
G. ZET ◽  
V. MANTA ◽  
C. BANDAC
Keyword(s):  
Gauge Theory ◽  
Axial Symmetry ◽  
Dimensional Space ◽  
Space Time ◽  
Field Equations ◽  
Metric Tensor ◽  
Gauge Invariant ◽  
Local Gauge ◽  
Dimensional Space Time ◽  
Strength Tensor

A model of SU(2) gauge theory is constructed in terms of local gauge-invariant variables defined over a four-dimensional space–time endowed with axial symmetry. A metric tensor gμν is defined starting with the components [Formula: see text] of the strength tensor and its dual [Formula: see text]. The components gμν are interpreted as new local gauge-invariant variables. Imposing the condition that the new metric coincides with the initial metric we obtain the field equations for the considered ansatz. We obtain the same field equations using the condition of self-duality. It is concluded that the self-dual variables are compatible with the axial symmetry of the space–time. A family of analytical solutions of the gauge field equations is also obtained. The solutions have the confining properties. All the calculations are performed using the GRTensorII computer algebra package, running on the MapleV platform.

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