The General Static Spherically Symmetric Solution of the "Weak" Unified Field Equations

1962 ◽  
Vol 14 ◽  
pp. 568-576 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Ricci Tensor ◽  
Spherically Symmetric ◽  
Dimensional Manifold ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Spherically Symmetric Solution ◽  
Geometric Objects ◽  
Symmetric Connection ◽  
Image Position

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where(1)Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:(2a)(2b)In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notationfor the symmetric and skew-symmetric parts of geometric objects Q is employed.

The general static spherically symmetric solution in Einstein’s unified field theory

1952 ◽  
Vol 210 (1102) ◽  
pp. 427-434 ◽  
Keyword(s):  
Magnetic Field ◽  
Electric Field ◽  
Spherically Symmetric ◽  
Unified Field Theory ◽  
Singular Surfaces ◽  
Field Equations ◽  
Unified Field ◽  
Spherically Symmetric Solution ◽  
Spherically Symmetric Solutions ◽  
Static Spherically Symmetric Solutions

The static spherically symmetric solutions of Einstein’s unified field equations previously given refer to an electric field alone or to a magnetic field alone. The general solutions in the case where both types of field exist together are now derived. After appropriate boundary conditions have been applied, the solutions may be interpreted to represent a magnetic field arising from a point pole, and an electric field arising from a dispersed charge distribution, but tending asymptotically to that of a point charge. The solutions have an infinity of singular surfaces, contain no arbitrary constant corresponding to the mass of the system, and in them the charge distributions contain both positive and negative electricity at different places. It appears that the only static spherically symmetric solutions likely to have any physical significance are certain of those referring to an electric field alone.

Static spherically symmetric solutions in Einstein’s unified field theory

1951 ◽  
Vol 209 (1098) ◽  
pp. 353-368 ◽  
Keyword(s):  
Charge Density ◽  
Point Charge ◽  
Spherically Symmetric ◽  
Constant Sign ◽  
Unified Field Theory ◽  
Magnetic Pole ◽  
Field Equations ◽  
Unified Field ◽  
Spherically Symmetric Solution ◽  
Magnetic Case

A new static spherically symmetric solution of Einstein’s unified field equations is derived. Certain boundary conditions are applied to this solution and to those already known, and the nature of the resulting fields is investigated. The only solution in the magnetic case corresponds to a magnetic pole without mass. In the electric case all the solutions correspond to continuous charge distributions, and the fields tend asymptotically to that of a point charge in classical theory. Several of the solutions are singular at an infinity of values of r , the radial co-ordinate, and in these the charge density is not of constant sign; but there are two solutions which have no singularities for finite values of r greater than 2 m (where m is a constant associated with the mass), and in which the charge density has constant sign throughout the field.

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).

The static spherically symmetric solutions in a unified field theory

1957 ◽  
Vol 53 (2) ◽  
pp. 489-493 ◽  
Author(s):  
John Moffat
Keyword(s):  
Charged Particle ◽  
Electric Energy ◽  
Gravitational Theory ◽  
Spherically Symmetric ◽  
Unified Field Theory ◽  
Field Equations ◽  
Schwarzschild Solution ◽  
Fundamental Properties ◽  
Unified Field ◽  
Conditions At Infinity

ABSTRACTA brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.

scholarly journals TELEPARALLEL LAGRANGE GEOMETRY AND A UNIFIED FIELD THEORY: LINEARIZATION OF THE FIELD EQUATIONS

2013 ◽  
Vol 10 (03) ◽  
pp. 1250092 ◽  
Author(s):  
M. I. WANAS ◽  
NABIL L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Field Theory ◽  
Approximate Solution ◽  
Order Of Approximation ◽  
Unified Field Theory ◽  
Field Equations ◽  
First Order ◽  
Unified Field ◽  
Vertical Field ◽  
Geometric Objects ◽  
Linearized Theory

This paper is a natural continuation of our previous paper: "Teleparallel Lagrange geometry and a unified field theory, Class. Quantum Grav.27 (2010) 045005 (29 pp)". In this paper, we apply a linearization scheme on the field equations obtained in the above-mentioned paper. Three important results under the linearization assumption are accomplished. First, the vertical fundamental geometric objects of the EAP-space lose their dependence on the positional argument x. Secondly, our linearized theory in the Cartan-type case coincides with the GFT in the first-order of approximation. Finally, an approximate solution of the vertical field equations is obtained.

scholarly journals VARIATIONAL FORMULATION OF EISENHART'S UNIFIED THEORY

2009 ◽  
Vol 24 (20n21) ◽  
pp. 3975-3984
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
Variational Formulation ◽  
Ricci Tensor ◽  
Maxwell Equations ◽  
Affine Connection ◽  
Unified Theory ◽  
Field Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Electromagnetic Field Tensor

Eisenhart's classical unified field theory is based on a non-Riemannian affine connection related to the covariant derivative of the electromagnetic field tensor. The sourceless field equations of this theory arise from vanishing of the torsion trace and the symmetrized Ricci tensor. We formulate Eisenhart's theory from the metric-affine variational principle. In this formulation, a Lagrange multiplier constraining the torsion becomes the source for the Maxwell equations.

scholarly journals Charged throats in the Hořava–Lifshitz theory

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
Alvaro Restuccia ◽  
Francisco Tello-Ortiz
Keyword(s):  
Coupling Parameter ◽  
Static Solution ◽  
Space Time ◽  
Coupling Constants ◽  
Spherically Symmetric ◽  
Dimensional Manifold ◽  
Field Equations ◽  
Flat Side ◽  
Minkowski Space Time ◽  
Spherically Symmetric Solution

AbstractA spherically symmetric solution of the field equations of the Hořava–Lifshitz gravity–gauge vector interaction theory is obtained and analyzed. It describes a charged throat. The solution exists provided a restriction on the relation between the mass and charge is satisfied. The restriction reduces to the Reissner–Nordström one in the limit in which the coupling constants tend to the relativistic values of General Relativity. We introduce the correct charts to describe the solution across the entire manifold, including the throat connecting an asymptotic Minkowski space-time with a singular 3+1 dimensional manifold. The solution external to the throat on the asymptotically flat side tends to the Reissner–Nordström space-time at the limit when the coupling parameter, associated with the term in the low energy Hamiltonian that manifestly breaks the relativistic symmetry, tends to zero. Also, when the electric charge is taken to be zero the solution becomes the spherically symmetric and static solution of the Hořava–Lifshitz gravity.

scholarly journals An Interpretation of the Field Tensor in the Unified Field Theory

1954 ◽  
Vol 7 (1) ◽  
pp. 1 ◽  
Author(s):  
NW Taylor
Keyword(s):  
Field Theory ◽  
Current Density ◽  
Free Space ◽  
Spherically Symmetric ◽  
Symmetric Part ◽  
Field Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Electric And Magnetic Fields

It is assumed that the skew symmetric part of the field tensor gik is a complex, self-dual tensor. This permits the whole set of field equations for free space to be derived directly from the theory without the introduction of an electric current density tensor. However, with this assumption it appears impossible for spherically symmetric electric and magnetic fields to exist in free space.

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