scholarly journals WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

2011 ◽  
Vol 03 ◽  
pp. 87-97 ◽  
Author(s):  
F. P. POULIS ◽  
J. M. SALIM
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Riemannian Geometry ◽  
Axiomatic Approach ◽  
Space Time ◽  
Weyl Geometry ◽  
Test Particles ◽  
Variational Formalism ◽  
Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

GEOMETRY OF SPACETIME AND FINSLER GEOMETRY

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1678-1685 ◽  
Author(s):  
REZA TAVAKOL
Keyword(s):  
General Relativity ◽  
String Theory ◽  
Riemannian Geometry ◽  
Lorentz Invariance ◽  
Finsler Geometry ◽  
Spacetime Geometry ◽  
Test Particles ◽  
Light Rays ◽  
Recent Developments ◽  
Underlying Geometry

A central assumption in general relativity is that the underlying geometry of spacetime is pseudo-Riemannian. Given the recent attempts at generalizations of general relativity, motivated both by theoretical and observational considerations, an important question is whether the spacetime geometry can also be made more general and yet still remain compatible with observations? Here I briefly summarize some earlier results which demonstrate that there are special classes of Finsler geometry, which is a natural metrical generalization of the Riemannian geometry, that are strictly compatible with the observations regarding the motion of idealised test particles and light rays. I also briefly summarize some recent attempts at employing Finsler geometries motivated by more recent developments such as those in String theory, whereby Lorentz invariance is partially broken.

scholarly journals On the methods of extending Dirac's equation of the electron to general relativity

1942 ◽  
Vol 7 (1) ◽  
pp. 39-50
Author(s):  
D Martin
Keyword(s):  
General Relativity ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Space Time ◽  
Point Of View ◽  
Dirac Equations ◽  
Usual Form ◽  
Galilean System ◽  
Second Category ◽  
Working Out

1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.

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 ON THE DETECTION OF SCALAR FIELD INDUCED SPACE–TIME TORSION

2002 ◽  
Vol 17 (07) ◽  
pp. 421-428 ◽  
Author(s):  
T. DERELI ◽  
R. W. TUCKER
Keyword(s):  
Scalar Field ◽  
Alternative Hypothesis ◽  
Space Time ◽  
Current Time ◽  
Civita Connection ◽  
Test Particles ◽  
Perihelion Shift ◽  
Theories Of Gravity ◽  
Levi Civita Connection

We argue that the geodesic hypothesis based on autoparallels of the Levi–Cività connection may need refinement in the scalar–tensor theories of gravity. Based on a reformulation of the Brans–Dicke theory in terms of a connection with torsion determined dynamically in terms of the gradient of the Brans–Dicke scalar field, we compute the perihelion shift in the orbit of Mercury on the alternative hypothesis that its worldline is an autoparallel of a connection with torsion. If the Brans–Dicke scalar field couples significantly to matter and test particles move on such worldlines, the current time keeping methods based on the conventional geodesic hypothesis may need refinement.

scholarly journals Space–Time Physics in Background-Independent Theories of Quantum Gravity

Universe ◽  
2021 ◽  
Vol 7 (7) ◽  
pp. 251
Author(s):  
Martin Bojowald
Keyword(s):  
General Relativity ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Riemannian Geometry ◽  
Loop Quantum Gravity ◽  
Space Time ◽  
Time Analysis ◽  
Complete Space ◽  
Background Independence ◽  
Starting Point

Background independence is often emphasized as an important property of a quantum theory of gravity that takes seriously the geometrical nature of general relativity. In a background-independent formulation, quantum gravity should determine not only the dynamics of space–time but also its geometry, which may have equally important implications for claims of potential physical observations. One of the leading candidates for background-independent quantum gravity is loop quantum gravity. By combining and interpreting several recent results, it is shown here how the canonical nature of this theory makes it possible to perform a complete space–time analysis in various models that have been proposed in this setting. In spite of the background-independent starting point, all these models turned out to be non-geometrical and even inconsistent to varying degrees, unless strong modifications of Riemannian geometry are taken into account. This outcome leads to several implications for potential observations as well as lessons for other background-independent approaches.

scholarly journals Weyl geometry and gauge-invariant gravitation

2014 ◽  
Vol 23 (11) ◽  
pp. 1450091 ◽  
Author(s):  
F. P. Poulis ◽  
J. M. Salim
Keyword(s):  
General Relativity ◽  
Gauge Symmetry ◽  
Proper Time ◽  
Gauge Invariant ◽  
Weyl Geometry ◽  
Test Particles ◽  
Constant Solution ◽  
Intimate Relation ◽  
Theory Of Gravitation ◽  
Particles Trajectories

In this paper, we provide a gauge-invariant theory of gravitation in the context of Weyl Integrable Spacetimes. After making a brief review of the theory's postulates, we carefully define the observers' proper-time and point out its relation with spacetime description. As a consequence of this relation and the theory's gauge symmetry we recover all predictions of general relativity. This feature is made even clearer by a new exact solution we provide which reveals the importance of a well defined proper-time. The thermodynamical description of the source fields is given and we observe that each of the geometric fields have a certain physical significance, despite the gauge-invariance. This is shown by two examples, where one of them consists of a new cosmological constant solution. Our conclusions highlight the intimate relation among test particles trajectories, proper-time and spacetime description which can also be applied in any other situation, whether or not it recovers general relativity results and also in the absence of a gauge symmetry.

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.

scholarly journals On causality violation in Lyra Geometry

2018 ◽  
Vol 15 (08) ◽  
pp. 1850143
Author(s):  
W. D. R. Jesus ◽  
A. F. Santos
Keyword(s):  
General Relativity ◽  
Riemannian Geometry ◽  
Trivial Solution ◽  
Lyra Geometry ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Weyl Geometry ◽  
Causality Violation ◽  
Different Sources ◽  
Gödel Universe

In this paper, the causality issues are discussed in a non-Riemannian geometry, called Lyra geometry. It is a non-Riemannian geometry originated from Weyl geometry. In order to compare this geometry with the Riemannian geometry, the Einstein field equations are considered. It is verified that the Gödel and Gödel-type metric are consistent with this non-Riemannian geometry. A non-trivial solution for Gödel universe in the absence of matter sources is determined without analogue in general relativity. Different sources are considered and then different conditions for causal and non-causal solutions are discussed.

scholarly journals Null geodesics and red–blue shifts of photons emitted from geodesic particles around a noncommutative black hole space–time

2018 ◽  
Vol 33 (16) ◽  
pp. 1850098 ◽  
Author(s):  
Ravi Shankar Kuniyal ◽  
Rashmi Uniyal ◽  
Anindya Biswas ◽  
Hemwati Nandan ◽  
K. D. Purohit
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Noncommutative Geometry ◽  
Equatorial Plane ◽  
Mass Parameter ◽  
Space Time ◽  
Geodesic Motion ◽  
Schwarzschild Black Hole ◽  
Gravitational Fields ◽  
Test Particles

We investigate the geodesic motion of massless test particles in the background of a noncommutative geometry-inspired Schwarzschild black hole. The behavior of effective potential is analyzed in the equatorial plane and the possible motions of massless particles (i.e. photons) for different values of impact parameter are discussed accordingly. We have also calculated the frequency shift of photons in this space–time. Further, the mass parameter of a noncommutative inspired Schwarzschild black hole is computed in terms of the measurable redshift of photons emitted by massive particles moving along circular geodesics in equatorial plane. The strength of gravitational fields of noncommutative geometry-inspired Schwarzschild black hole and usual Schwarzschild black hole in General Relativity is also compared.

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