The rotating body problem

1964 ◽  
Vol 60 (2) ◽  
pp. 279-285 ◽  
Author(s):  
H. Levy ◽  
W. J. Robinson
Keyword(s):  
Mass Term ◽  
Approximate Solutions ◽  
Multipole Expansion ◽  
Riemann Tensor ◽  
Stationary System ◽  
Vacuum Field ◽  
Static Case ◽  
Canonical Coordinates ◽  
Field Equations ◽  
Canonical Coordinate

AbstractIt is shown that for an axisymmetric stationary system there exists, in vacuo, a canonical coordinate system, analogous to Weyl's canonical coordinates for the static case. Some approximate solutions of the vacuum field equations are then obtained in these canonical coordinates.In the second part of this paper, the vacuum field equations for an isolated axisymmetric stationary system are set up and approximate solutions obtained by means of a multipole expansion. The physical components of the Riemann tensor are then examined and it is found that the ‘mass term’ predominates, the ‘rotation term’ being of the next order of magnitude.

A spatially homogeneous gravitational field

1966 ◽  
Vol 62 (1) ◽  
pp. 87-89 ◽  
Author(s):  
V. Joseph
Keyword(s):  
Gravitational Field ◽  
Canonical Form ◽  
Riemann Tensor ◽  
Vacuum Field ◽  
Type I ◽  
Field Equations ◽  
Parameter Group ◽  
Spatially Homogeneous ◽  
Homogeneous Gravitational Field ◽  
Type V

AbstractA solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.

scholarly journals Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theories

Universe ◽  
2019 ◽  
Vol 5 (6) ◽  
pp. 142 ◽  
Author(s):  
Laur Järv ◽  
Manuel Hohmann ◽  
Martin Krššák ◽  
Christian Pfeifer
Keyword(s):  
General Relativity ◽  
Coordinate Transformation ◽  
Teleparallel Gravity ◽  
Analytic Solutions ◽  
Vacuum Field ◽  
Spin Connection ◽  
Flat Connection ◽  
Canonical Coordinates ◽  
Field Equations ◽  
Levi Civita Connection

Teleparallel geometry utilizes Weitzenböck connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi–Civita connection. In extended teleparallel theories, for instance in f ( T ) or scalar-torsion gravity, the connection must obey its antisymmetric field equations. Thus far, only a few analytic solutions were known. In this note, we solve the f ( T , ϕ ) gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformation, we present the solution also in Boyer–Lindquist coordinates, often used to study rotating solutions in general relativity. The result hints for the existence of another branch of rotating solutions besides the Kerr family in extended teleparallel gravities.

DESITTER GAUGE THEORY OF GRAVITATION

2003 ◽  
Vol 14 (01) ◽  
pp. 41-48 ◽  
Author(s):  
G. ZET ◽  
V. MANTA ◽  
S. BABETI
Keyword(s):  
Gauge Theory ◽  
Riemann Tensor ◽  
Space Time ◽  
Point Of View ◽  
Field Equations ◽  
Minkowski Space Time ◽  
Strength Tensor ◽  
Group A ◽  
Theory Of Gravitation ◽  
Tensor Formula

A deSitter gauge theory of gravitation over a spherical symmetric Minkowski space–time is developed. The "passive" point of view is adapted, i.e., the space–time coordinates are not affected by group transformations; only the fields change under the action of the symmetry group. A particular ansatz for the gauge fields is chosen and the components of the strength tensor are computed. An analytical solution of Schwarzschild–deSitter type is obtained in the case of null torsion. It is concluded that the deSitter group can be considered as a "passive" gauge symmetry for gravitation. Because of their complexity, all the calculations, inclusive of the integration of the field equations, are performed using an analytical program conceived in GRTensorII for MapleV. The program allows one to compute (without using a metric) the strength tensor [Formula: see text], Riemann tensor [Formula: see text], Ricci tensor [Formula: see text], curvature scalar [Formula: see text], field equations, and the integration of these equations.

Next steps

2019 ◽  
pp. 109-116
Author(s):  
Steven Carlip
Keyword(s):  
Approximate Solutions ◽  
Cosmic Censorship ◽  
Field Equations ◽  
Singularity Theorems ◽  
Open Questions ◽  
Tensor Models ◽  
General Relativity And Gravitation ◽  
Active Research ◽  
Experimental Gravity

This final chapter consists of a brief discussion of where the reader can go from here: active research topics in general relativity and gravitation, open questions, and ideas for further study. Topics include exact and approximate solutions of the field equations, including numerical methods and perturbation theory; problems in mathematical relativity, including global geometric methods, singularity theorems, cosmic censorship, and asymptotic conditions; alternative models such as scalar-tensor models; approaches to quantum gravity; and experimental gravity. These topics are not discussed in any depth; rather, the chapter is meant as a “teaser” to encourage readers to look further.

Approximate solutions of the general relativity field equations with a scalar meson field

1963 ◽  
Vol 59 (4) ◽  
pp. 739-741 ◽  
Author(s):  
J. Hyde
Keyword(s):  
General Relativity ◽  
Scalar Meson ◽  
Empty Space ◽  
Approximate Solutions ◽  
Spherically Symmetric ◽  
Coordinate Transformations ◽  
Field Equations ◽  
Spherically Symmetric Solution ◽  
Image Position ◽  
Scalar Meson Field

It was shown by Birkhoff ((1), p. 253) that every spherically symmetric solution of the field equations of general relativity for empty space,may be reduced, by suitable coordinate transformations, to the static Schwarzschild form:where m is a constant. This result is known as Birkhoff's theorem and excludes the possibility of spherically symmetric gravitational radiation. Different proofs of the theorem have been given by Eiesland(2), Tolman(3), and Bonnor ((4), p. 167).

DIMENSIONAL REDUCTION AND THE LANCZOS TENSOR

1989 ◽  
Vol 04 (28) ◽  
pp. 2739-2746 ◽  
Author(s):  
M.D. ROBERTS
Keyword(s):  
Dimensional Reduction ◽  
Weyl Tensor ◽  
Weak Field ◽  
Vacuum Field ◽  
Linear Wave ◽  
Field Equations ◽  
Weak Field Limit ◽  
Magnetic Theory ◽  
Electro Magnetic ◽  
The One

The Lanczos tensor Hαβγ is a potential for the Weyl tensor. Given the symmetries of these tensors it would be expected that the identification Hαβ5=Fαβ would give a reduction of the five dimensional vacuum field equations into equations related to the Einstein Maxwell equation, it is shown that this does not happen; furthermore it is shown that there is no dimensional reduction scheme involving the Lanczos tensor which agrees with the one devised by Kaluza and Klein in the weak field limit. The covariant derivative of the Weyl tensor can be expressed as a type of non-linear wave equation in the Lanczos tensor, the literature contains two incorrect expressions for this equation, here the correct expression is given for the first time. The expression for the Lanczos tensor in the case of weak fields is generalized. Some remarks are made on other approaches to include electro-magnetic theory into the theory of the Lanczos tensor.

Exact solutions of the field equations: twist-free pure radiation fields

1962 ◽  
Vol 270 (1342) ◽  
pp. 328-334 ◽  
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Radiation Field ◽  
Relativity Theory ◽  
Vacuum Field ◽  
Field Equations ◽  
General Relativity Theory ◽  
Pure Radiation ◽  
Radiation Fields ◽  
Null Congruence

This note is intended to give a rough survey of the results obtained in the study of twist-free pure radiation fields in general relativity theory. Here we are using the following Definition. A space-time ( V 4 of signature +2) is called a pure radiation field if it contains a distortion-free geodetic null congruence (a so-called ray congruence ), and if it satisfies certain field equations which we will specify below (e.g. Einstein’s vacuum-field equations). A (null) congruence is called twist-free if it is hypersurface-orthogonal (or ‘normal’). The results listed below were obtained by introducing special (‘canonical’) co-ordinates adapted to the ray congruence. Detailed proofs were given by Robinson & Trautman (1962) and by Jordan, Kundt & Ehlers (1961) (see also Kundt 1961). For the sake of completeness we include in our survey the subclass of expanding fields, and make use of some formulae first obtained by Robinson & Trautman.

Application of Spencer’s Ideal Soil Model to Granular Materials Flow

1976 ◽  
Vol 43 (1) ◽  
pp. 49-53 ◽  
Author(s):  
H. L. Morrison ◽  
O. Richmond
Keyword(s):  
Fluid Flow ◽  
Granular Materials ◽  
Plane Deformation ◽  
Yield Condition ◽  
Classical Equation ◽  
Maximum Flow ◽  
Approximate Solutions ◽  
Field Equations ◽  
Full Field ◽  
Materials Flow

In 1964, Spencer proposed a model for plane deformation of soils based upon the concept that the strain at any point may be considered as the resultant of simple shears on the two surface elements where Coulomb’s yield condition is met. Gravitation and acceleration terms were neglected in his full field equations. These terms are included in the present treatment, however, since they play an essential role in granular materials flow problems. It is shown that the field equations remain hyperbolic, and the characteristic equations are derived. In addition, a streamline equation, similar to Bernoulli’s classical equation for fluid flow, is derived and used together with the continuity equation to obtain one-dimensional approximate solutions to some typical hopper and chute problems. A solution is obtained for the nonsteady flow from a wedge-shaped hopper when the gate is suddenly opened, including an equation for the minimum slope necessary to prevent arching. Another solution is obtained for the profile of a hopper which has constant wall pressure. Still another solution is obtained for the relationship between the height of a chute and its exit velocity, and it suggests that the maximum trajectory usually is obtained with a horizontal exit since an upward-sloping exit requires a velocity jump at the minimum point in the chute, similar to a hydraulic jump in fluid flow. All of these solutions are ideal in the sense that they include no wall resistance to the flow, and therefore represent maximum flow rates.

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