On the analyticity of stationary vacuum solutions of Einstein's equation

1970 ◽  
Vol 68 (1) ◽  
pp. 199-201 ◽  
Author(s):  
H. Müller zum Hagen
Keyword(s):  
Tensor Field ◽  
Space Time ◽  
Analytic Structure ◽  
Stationary Vacuum ◽  
Einstein's Equation ◽  
Vacuum Solutions ◽  
Vacuum Space

AbstractIt is shown that a stationary C3 vacuum space-time has an analytic structure with respect to which the metric gab is an analytic tensor field.

On the analyticity of static vacuum solutions of Einstein's equations

1970 ◽  
Vol 67 (2) ◽  
pp. 415-421 ◽  
Author(s):  
H. Müller zum Hagen
Keyword(s):  
Coordinate System ◽  
Space Time ◽  
Coordinate Systems ◽  
Einstein’S Equations ◽  
Static Vacuum ◽  
Einstein's Equations ◽  
Vacuum Solutions ◽  
Vacuum Space

AbstractIt is shown that around each point of a C3static vacuum space-time there is a neighbourhood in which the metric is, in a certain suitable coordinate system, analytic. If two of these neighbourhoods overlap, then the transformation betweea these suitable coordinate systems is analytic (in the overlapping domain).

scholarly journals DUALITY INVARIANCE OF COSMOLOGICAL SOLUTIONS WITH TORSION

1999 ◽  
Vol 14 (31) ◽  
pp. 4953-4966 ◽  
Author(s):  
DEBASHIS GANGOPADHYAY ◽  
SOUMITRA SENGUPTA
Keyword(s):  
Physical Meaning ◽  
Tensor Field ◽  
Antisymmetric Tensor ◽  
Dilaton Field ◽  
Maximal Symmetry ◽  
Killing Vectors ◽  
Cosmological Solutions ◽  
Antisymmetric Tensor Field ◽  
Vacuum Solutions

We show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the O(d) ⊗ O(d) transformation on the vacuum solutions gives inequivalent solutions that are not maximally symmetric. We then show that the usual physical meaning of maximal symmetry can be made to remain unaltered even if torsion is present and illustrate this through two toy models by determining the torsion fields, the metric and Killing vectors. Finally we show that under the O(d) ⊗ O(d) transformation this generalized maximal symmetry can be preserved under certain conditions. This is interesting in the context of string related cosmological backgrounds.

A NEW APPROACH FOR SPACE-TIME-MATTER THEORY

2013 ◽  
Vol 10 (04) ◽  
pp. 1350004 ◽  
Author(s):  
AUREL BEJANCU
Keyword(s):  
Tensor Field ◽  
Differential Operators ◽  
Space Time ◽  
Horizontal Gradient ◽  
Tensor Fields ◽  
Tensor Calculus ◽  
New Approach ◽  
Horizontal Connection ◽  
Klein Theory ◽  
Kaluza Klein

This is the first paper in a series of three papers on a new approach for space-time-matter (STM) theory. The main purpose of this approach is to replace the Levi-Civita connection on the space-time from the classical Kaluza–Klein theory by what we call the Riemannian horizontal connection on the general Kaluza–Klein space. This is done by a development of a 4D tensor calculus whose geometrical objects live in a 5D space. The 4D tensor calculus and the Riemannian horizontal connection enable us to define in a 5D space some 4D differential operators: horizontal differential, horizontal gradient, horizontal divergence and horizontal Laplacian, which have a great role in the presentation of the STM theory in a covariant form. Finally, we introduce and study the horizontal electromagnetic tensor field, the horizontal Ricci tensor and the horizontal Einstein gravitational tensor field, which replace the well-known tensor fields from the classical Kaluza–Klein theory.

Einstein flow and cosmology

2015 ◽  
Vol 30 (18n19) ◽  
pp. 1530047 ◽  
Author(s):  
J. Kouneiher
Keyword(s):  
Space Time ◽  
Cosmological Models ◽  
Point Of View ◽  
Continuous Family ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Minkowski Metric ◽  
Global Topology ◽  
Einstein's Equation ◽  
Physical Case

The recent evolution of the observational technics and the development of new tools in cosmology and gravitation have a significant impact on the study of the cosmological models. In particular, the qualitative and numerical methods used in dynamical system and elsewhere, enable the resolution of some difficult problems and allow the analysis of different cosmological models even with a limited number of symmetries. On the other hand, following Einstein point of view the manifold [Formula: see text] and the metric should be built simultaneously when solving Einstein’s equation [Formula: see text]. From this point of view, the only kinematic condition imposed is that at each point of space–time, the tangent space is endowed with a metric (which is a Minkowski metric in the physical case of pseudo-Riemannian manifolds and an Euclidean one in the Riemannian analogous problem). Then the field [Formula: see text] describes the way these metrics depend on the point in a smooth way and the Einstein equation is the “dynamical” constraint on [Formula: see text]. So, we have to imagine an infinite continuous family of copies of the same Minkowski or Euclidean space and to find a way to sew together these infinitesimal pieces into a manifold, by respecting Einstein’s equation. Thus, Einstein field equations do not fix once and for all the global topology. [Formula: see text] Given this freedom in the topology of the space–time manifold, a question arises as to how free the choice of these topologies may be and how one may hope to determine them, which in turn is intimately related to the observational consequences of the space–time possessing nontrivial topologies. Therefore, in this paper we will use a different qualitative dynamical methods to determine the actual topology of the space–time.

scholarly journals Relation between Schrödinger’s Equation and Einstein’s Equation

2021 ◽  
Author(s):  
Aman Yadav
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Helmholtz Equation ◽  
Space Time ◽  
Wave Functions ◽  
Quantum Level ◽  
Schrödinger’S Equation ◽  
Einstein's Equation ◽  
The Relationship ◽  
Schrödinger's Equation

The relationship between Einstein's Field Equation and Schrodinger's Equation is examined in thiswork. I adjusted Schrodinger's Equation to offer the solution, and utilizing the wave equation, Icame up with two cases: In case 1, I discovered the structure and dimension of the equations in amanner similar to Einstein's Field Equation, and in case 2, the Helmholtz equation replaces themodified Schrodinger's equation. Finally, the findings suggested that wave functions may haverelevance beyond determining the position of a particle, and that they may be used to determinethe structure of space-time at the quantum level.

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