A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

A NEW EXTENSION OF GENERAL RELATIVISTIC THEORY

1994 ◽  
Vol 03 (02) ◽  
pp. 393-419 ◽  
Author(s):  
MASATOSHI YAZAKI
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Relativistic Theory ◽  
Maxwell's Equations ◽  
Geometrical Structure ◽  
Finsler Geometry ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
General Relativistic ◽  
Einstein’S General Relativity

The possibility of a new extension of the general relativistc theory will be considered using Finsler geometry. The extension of Einstein’s general relativity can be expected to regard gravitational, electroweak, and strong interactive fields as geometrical structure of a spacetime based on Finsler geometry. Indeed, it will be shown that this theory can include the general theory of relativity under a certain special condition. In addition, Maxwell’s equations will be expressed using new metric representations of the electromagnetic vector and its tensor. Moreover, it will be suggested that this theory may include metric representations of weak and strong interactive fields.

scholarly journals GENERAL RELATIVITY AND WEYL FRAMES

2011 ◽  
Vol 03 ◽  
pp. 27-35
Author(s):  
C. ROMERO ◽  
J. B. FONSECA-NETO ◽  
M. L. PUCHEU
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Riemannian Geometry ◽  
Vacuum Solution ◽  
Gravitational Theory ◽  
Mathematical Formalism ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Weyl Geometry ◽  
Starting Point

We present the general theory of relativity in the language of a non-Riemannian geometry, namely, Weyl geometry. We show that the new mathematical formalism may lead to different pictures of the same gravitational phenomena, by making use of the concept of Weyl frames. We show that, in this formalism, it is possible to construct a scalar-tensor gravitational theory that is invariant with respect to the so-called Weyl tranformations and reduces to general relativity in a particular frame, the Riemann frame. In this approach the Weyl geometry plays a fundamental role since it appears as the natural geometrical setting of the theory when viewed in an arbitrary frame. Our starting point is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in more familiar terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke theory of gravity. We illustrate this point with an example in which a known Brans-Dicke vacuum solution may appear when reinterpreted in a particular Weyl frame.

scholarly journals GENERAL RELATIVITY AND WEYL FRAMES

2011 ◽  
Vol 26 (22) ◽  
pp. 3721-3729 ◽  
Author(s):  
C. ROMERO ◽  
J. B. FONSECA-NETO ◽  
M. L. PUCHEU
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Riemannian Geometry ◽  
Vacuum Solution ◽  
Gravitational Theory ◽  
Mathematical Formalism ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Weyl Geometry ◽  
Starting Point

We present the general theory of relativity in the language of a non-Riemannian geometry, namely, Weyl geometry. We show that the new mathematical formalism may lead to different pictures of the same gravitational phenomena, by making use of the concept of Weyl frames. We show that, in this formalism, it is possible to construct a scalar-tensor gravitational theory that is invariant with respect to the so-called Weyl tranformations and reduces to general relativity in a particular frame, the Riemann frame. In this approach the Weyl geometry plays a fundamental role since it appears as the natural geometrical setting of the theory when viewed in an arbitrary frame. Our starting point is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in more familiar terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke theory of gravity. We illustrate this point with an example in which a known Brans-Dicke vacuum solution may appear when reinterpreted in a particular Weyl frame.

The general theory of relativity is correct!

1988 ◽  
Vol 155 (7) ◽  
pp. 517-527 ◽  
Author(s):  
Ya.B. Zel'dovich ◽  
Leonid P. Grishchuk
Keyword(s):  
General Theory ◽  
Theory Of Relativity ◽  
General Theory Of Relativity

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

scholarly journals On the 2PN Pericentre Precession in the General Theory of Relativity and the Recently Discovered Fast-Orbiting S-Stars in Sgr A*

Universe ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 37
Author(s):  
Lorenzo Iorio
Keyword(s):  
General Theory ◽  
Present Author ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Orbital Elements ◽  
Analytical Expression ◽  
Potential Interest ◽  
Short Period ◽  
The Future ◽  
Sgr A

Recently, the secular pericentre precession was analytically computed to the second post-Newtonian (2PN) order by the present author with the Gauss equations in terms of the osculating Keplerian orbital elements in order to obtain closer contact with the observations in astronomical and astrophysical scenarios of potential interest. A discrepancy in previous results from other authors was found. Moreover, some of such findings by the same authors were deemed as mutually inconsistent. In this paper, it is demonstrated that, in fact, some calculation errors plagued the most recent calculations by the present author. They are explicitly disclosed and corrected. As a result, all of the examined approaches mutually agree, yielding the same analytical expression for the total 2PN pericentre precession once the appropriate conversions from the adopted parameterisations are made. It is also shown that, in the future, it may become measurable, at least in principle, for some of the recently discovered short-period S-stars in Sgr A*, such as S62 and S4714.

The representation of matter in the general theory of relativity

1973 ◽  
Vol 17 (1) ◽  
pp. 122-128 ◽  
Author(s):  
V. A. Wynne ◽  
G. H. Derrick
Keyword(s):  
General Theory ◽  
Theory Of Relativity ◽  
General Theory Of Relativity
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