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.

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.

ON THE PHYSICAL INTERPRETATION OF ASYMPTOTICALLY FLAT GRAVITATIONAL FIELDS

2008 ◽  
Vol 17 (13n14) ◽  
pp. 2599-2606
Author(s):  
CARLOS KOZAMEH ◽  
EZRA T. NEWMAN ◽  
GILBERTO SILVA-ORTIGOZA
Keyword(s):  
Vector Fields ◽  
Center Of Mass ◽  
Flat Space ◽  
Einstein Equations ◽  
Position Vector ◽  
Approximate Symmetry ◽  
Asymptotically Flat ◽  
Gravitational Fields ◽  
Maxwell Fields ◽  
Physical Information

A problem in general relativity is how to extract physical information from solutions to the Einstein equations. Most often information is found from special conditions, e.g., special vector fields, symmetries or approximate symmetries. Our concern is with asymptotically flat space–times with approximate symmetry: the BMS group. For these spaces the Bondi four-momentum vector and its evolution, found at infinity, describes the total energy–momentum and the energy–momentum radiated. By generalizing the simple idea of the transformation of (electromagnetic) dipoles under a translation, we define (analogous to center of charge) the center of mass for asymptotically flat Einstein–Maxwell fields. This gives kinematical meaning to the Bondi four-momentum, i.e., the four-momentum and its evolution is described in terms of a center of mass position vector, its velocity and spin-vector. From dynamical arguments, a unique (for our approximation) total angular momentum and evolution equation in the form of a conservation law is found.

scholarly journals Four interactions in the sedenion curved spaces

2019 ◽  
Vol 16 (02) ◽  
pp. 1950019 ◽  
Author(s):  
Zi-Hua Weng
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Flat Space ◽  
Field Equations ◽  
Curved Space ◽  
Gravitational Fields ◽  
Curved Spaces ◽  
Spatial Parameters ◽  
Physical Quantities

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially, the field equations of the four quantum-fields in the quantum mechanics. Apparently, the paper and General Relativity both confirm and succeed to the Cartesian academic thought of ‘the space is the extension of substance’.

New conservation laws for zero rest-mass fields in asymptotically flat space-time

1968 ◽  
Vol 305 (1481) ◽  
pp. 175-204 ◽  
Keyword(s):  
Conservation Laws ◽  
Rest Mass ◽  
Power Series Expansion ◽  
Flat Space ◽  
Arbitrary Spin ◽  
Space Time ◽  
Maxwell Theory ◽  
Asymptotically Flat ◽  
Gravitational Fields ◽  
Higher Spin

Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s = 0,½,1,...) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff’s integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4 s + 2 real quantities, for spin s , giving a D ( s , 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of ‘tails’ to the outgoing waves (or back-scattered field),which destroys the stationary nature of the final field.

Symmetries and conservation laws of some asymptotically symmetric spacetimes of interest in gravitational waves

2019 ◽  
Vol 16 (10) ◽  
pp. 1950152 ◽  
Author(s):  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
B. B. I. Gadjagboui ◽  
Ghulam Shabbir
Keyword(s):  
Conservation Laws ◽  
Gravitational Waves ◽  
Flat Space ◽  
Einstein Equations ◽  
Necessary Condition ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Asymptotically Flat ◽  
Symmetric Spacetimes ◽  
Flat Spacetimes

In this paper, we discuss symmetries and the corresponding conservation laws of certain exact solutions of the Einstein field equations (EFEs) representing a Schwarzschild black hole and gravitational waves in asymptotically flat space times. Of particular interest are symmetries of asymptotically flat spacetimes because they admit a property that identifies them for the existence of gravitational waves there. In the light of this fact, we discuss symmetry algebras of a few recently published solutions of Einstein equations in asymptotically flat metrics. Given the fact that gravitational waves are of great interest in relativity, we focus in this paper on finding the type of symmetries they admit and their corresponding conservation laws. We also show how these symmetries are radically different from the other well-known symmetries and present necessary condition that distinguishes them.

On the static field in general relativity

1977 ◽  
Vol 81 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Jamal N. Islam
Keyword(s):  
Harmonic Function ◽  
Power Series Expansion ◽  
Static Solution ◽  
Flat Space ◽  
Static Field ◽  
Vacuum Field ◽  
Field Equations ◽  
Axisymmetric Solution ◽  
Asymptotically Flat ◽  
Coordinate Conditions

AbstractThe possibility is explored that the general static solution of Einstein's vacuum field equations is given in terms of a single harmonic function, i.e. solution of the flat space Laplace equation. It is shown that coordinate conditions can be chosen such that one of Einstein's equations reduces to Laplace's equation. To examine the extent to which the metric is determined by the resulting harmonic function ø, a power series expansion is considered in which in the first non-trivial order one gets a system of eight equations for six unknowns. This is reduced to a system of three equations for three unknowns. This is some indication that the coordinate conditions are consistent and shows the sense in which, to this order, the metric is determined by ø. This system of three equations is then solved explicitly for a particular choice of ø, thereby giving an approximate non-axisymmetric solution of Einstein's vacuum equations. This solution is asymptotically flat and could represent the approximate gravitational field of a non-axisymmetric bounded rigid body. The form of a class of time-dependent metric related to the general static metric is discussed.

Quadratic Langrangians and static gravitational fields

1973 ◽  
Vol 74 (1) ◽  
pp. 145-148 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Scalar Curvature ◽  
Conformally Flat ◽  
Field Equations ◽  
Asymptotically Flat ◽  
Gravitational Fields ◽  
Zero Scalar Curvature

AbstractIt is shown that the solutions of the field equations generated by the most general effective quadratic Lagrangian aR2 + bRklRkl which are static, singularity-free, of class C4 and asymptotically flat (of rank A3) are (i) conformally flat when 3a + b = 0, (ii) flat when 3a + b ≠ 0 (b ≠ 0), and (iii) have zero scalar curvature when b = 0.

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.

On a non-symmetric theory of the pure gravitational field

1958 ◽  
Vol 54 (1) ◽  
pp. 72-80 ◽  
Author(s):  
D. W. Sciama
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Symmetric Tensor ◽  
Momentum Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Unified Field

ABSTRACTIt is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

scholarly journals Cylindrically Symmetric, Asymptotically Flat, Petrov Type D Spacetime with a Naked Curvature Singularity and Matter Collapse

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Faizuddin Ahmed
Keyword(s):  
Cosmic Time ◽  
Petrov Type ◽  
Energy Conditions ◽  
Anisotropic Fluid ◽  
Curvature Singularity ◽  
Field Equations ◽  
Asymptotically Flat ◽  
Type D ◽  
Cylindrically Symmetric ◽  
Petrov Type D

We present a cylindrically symmetric, Petrov type D, nonexpanding, shear-free, and vorticity-free solution of Einstein’s field equations. The spacetime is asymptotically flat radially and regular everywhere except on the symmetry axis where it possesses a naked curvature singularity. The energy-momentum tensor of the spacetime is that for an anisotropic fluid which satisfies the different energy conditions. This spacetime is used to generate a rotating spacetime which admits closed timelike curves and may represent a Cosmic Time Machine.

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