scholarly journals A relativistic theory of the field II: Hamilton's principle and Bianchi's identities

2021 ◽  
Vol 24 (3) ◽  
pp. 12-24
Author(s):  
Mississippi Valenzuela
Keyword(s):  
General Relativity ◽  
Variational Principle ◽  
Long Range ◽  
Relativistic Theory ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Current Formulation ◽  
Long Range Interactions ◽  
Gravitational Equations

As gravitation and electromagnetism are closely analogous long-range interactions, and the current formulation of gravitation is given in terms of geometry. Thence emerges a relativistic theory of the field by generalization of the general relativity. The derivation presented shows how naturally we can extend general relativity theory to a non-symmetric field, and that the field-equations are really the generalizations of the gravitational equations. With curvature tensor and the variational principle, we will deduce the field equations and Bianchi's identities. In consecuense, the field equations will find from Bianchi's identities.

scholarly journals A relativistic theory of the field

2021 ◽  
Vol 24 (2) ◽  
pp. 72-79
Author(s):  
Mississippi Valenzuela
Keyword(s):  
General Relativity ◽  
Long Range ◽  
Relativistic Theory ◽  
Field Equations ◽  
Current Formulation ◽  
Long Range Interactions

As gravitation and electromagnetism are closely analogous long-range interactions, and the current formulation of gravitation is given in terms of geometry, we expect the latter also to appear through the geometry. This unification has however, remained an unfulfilled goal. Thence emerges a relativistic theory of the asymmetric field by generalization of the general relativity. It will demonstrate in a new way that the field-equations chosen for the non-symmetric fields are really the natural ones.

Radiation and Gravitational Equations of Motion

1951 ◽  
Vol 3 ◽  
pp. 195-207 ◽  
Author(s):  
L. Infeld ◽  
A. E. Scheidegger
Keyword(s):  
General Relativity ◽  
Equations Of Motion ◽  
Classical Field ◽  
Relativity Theory ◽  
Field Theories ◽  
Field Equations ◽  
General Relativity Theory ◽  
Principle Of Superposition ◽  
Classical Field Theories ◽  
Gravitational Equations

Among the classical field theories, general relativity theory occupies a somewhat peculiar place. Unlike those of most other field theories, the field equations in relativity theory are non-linear. This implies that many facts, well known in linear theories, have no analogues in general relativity theory, and conversely. The equations of motion of the sources of the gravitational field are contained in the field equations, a fact which does not apply for the motion of an electron in the electromagnetic field. Conversely, it is difficult to define the notion of a wave (familiar in electrodynamics) in relativity theory; for, the linear principle of superposition is crucial for the existence of waves (at least in the sense that the notion of a wave is normally used).

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.

scholarly journals On Hilbert's world-function

1927 ◽  
Vol 113 (765) ◽  
pp. 496-511 ◽  
Keyword(s):  
Dynamical System ◽  
General Relativity ◽  
Minimum Principle ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Least Action ◽  
Principle Of Least Action ◽  
The Universe ◽  
World Function

The well-known theorem that the motion of any conservative dynamical system can be determined from the “Principle of Least Action” or “Hamilton’s Principle” was carried over into General Relativity-Theory in 1915 by Hilbert, who showed that the field-equations of gravitation can be deduced very simply from a minimum-principle. Hilbert generalised his ideas into the assertion that all physical happenings (gravitational electrical, etc.) in the universe are determined by a scalar “world-function” H, being, in fact, such as to annul the variation of the integral ∫∫∫∫H√(−g)dx 0 dx 1 dx 2 dx 3 where ( x 0 , x 1 , x 2 , x 3 ) are the generalised co-ordinates which specify place and time, and g is (in the usual notation of the relativity-theory) the determinant of the gravitational potentials g v q , which specify the metric by means of the equation dx 2 = ∑ p, q g vq dx v dx q . In Hilbert’s work, the variation of the above integral was supposed to be due to small changes in the g vq 's and in the electromagnetic potentials, regarded as functions of x 0 , x 1 , x 2 , x 3 .

Non-linear Lagrangians and Palatine's device

1960 ◽  
Vol 56 (4) ◽  
pp. 396-400 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
General Relativity ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Invariant Action ◽  
Action Integral ◽  
Independent Variation ◽  
Non Linear

ABSTRACTField equations in general relativity theory have sometimes been generated by subjecting, in an invariant action integral, the components of linear connexion and the components of a covariant tensor of valence 2 to independent variation. The conceptual objections to this process, and some of the manifold formal difficulties inherent in it, are discussed in some detail. At the same time certain results obtained elsewhere are strengthened and in part corrected.

scholarly journals Reciprocal Static Solutions of the Equations of the Gravitational Field

1956 ◽  
Vol 9 (1) ◽  
pp. 13 ◽  
Author(s):  
HA Buchdahl
Keyword(s):  
General Relativity ◽  
Equation Of State ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Remarkable Property ◽  
Reciprocal Transformation ◽  
Total Energy Density ◽  
Static Solutions ◽  
Line Elements

This paper deals with reciprocal static line-elements, previously defined by the author, the condition that their Ricci tensors vanish being no longer imposed. If the indices i, k run from 1 to n with the exception of the fixed index a (the line-elements being static with respect to xa) a certain quantity appears with the remarkable property that in a reciprocal transformation is invariant, whilst merely changes sign. is closely related to the Hamiltonian derivative of the Gaussian curvature, so that the general results obtained may be applied to the field equations of General Relativity Theory, with n=a=4. is then the total energy density; and formally every static distribution of matter has a " reciprocal distribution" associated with it. In particular, the equation of state of a distribution of fluid reciprocal to a distribution of fluid possessing a given equation of state may be obtained directly from the latter, i.e. without the solution of the field equations being known.

scholarly journals A scale-invariant, Machian theory of gravitation and electrodynamics unified

2018 ◽  
Vol 15 (10) ◽  
pp. 1850178 ◽  
Author(s):  
Ram Gopal Vishwakarma
Keyword(s):  
Long Range ◽  
Geometric Theory ◽  
Riemann Tensor ◽  
Field Equations ◽  
Scale Invariant ◽  
Current Formulation ◽  
Long Range Interactions ◽  
Electromagnetic Field Tensor ◽  
Respective Field ◽  
Theory Of Gravitation

As gravitation and electromagnetism are closely analogous long-range interactions, and the current formulation of gravitation is given in terms of geometry, we expect the latter also to appear through the geometry. This unification has however, remained an unfulfilled goal. The goal is achieved here in a new theory, which results from the principles of equivalence and Mach supplemented with a novel insight that the field tensors in a geometric theory of gravitation and electromagnetism must be traceless, since these long-range interactions are mediated by virtual exchange of massless particles whose mass is expected to be related to the trace of the field tensors. Hence, the Riemann tensor, like the analogous electromagnetic field tensor, must be traceless. Thence emerges a scale-invariant, Machian theory of gravitation and electrodynamics unified, wherein the vanishing of the Ricci tensor appears as a boundary condition. While the field equations of the theory are given by the vanishing divergence of the respective field tensors and their duals, the matter and charge emerge from the spacetime. A quantitative formulation thereof, embodied in “energy-momentum super tensors”, follows from the respective Bianchi identities for the two fields. The resulting theory is valid at all scales and explains the observations without invoking the non-baryonic dark matter, dark energy or inflation. Moreover, it answers the questions that the general relativity-based standard paradigm could not address.

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