scholarly journals Generalized derivations and general relativity

2013 ◽  
Vol 91 (10) ◽  
pp. 757-763
Author(s):  
Michael Heller ◽  
Tomasz Miller ◽  
Leszek Pysiak ◽  
Wiesław Sasin
Keyword(s):  
General Relativity ◽  
Extra Dimension ◽  
Linear Mapping ◽  
Einstein Equations ◽  
Leibniz Rule ◽  
Generalized Derivations ◽  
Field Equations ◽  
Time Dimension ◽  
Smooth Functions ◽  
Kaluza Klein

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra [Formula: see text]. Such a derivation, introduced by Brešar in 1991, is given by a linear mapping [Formula: see text] such that there exists a usual derivation, d, of [Formula: see text] satisfying the generalized Leibniz rule u(ab) = u(a)b + ad(b) for all [Formula: see text]. The generalized geometry “is tested” in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study generalized general relativity. We define the Einstein–Hilbert action and deduce from it Einstein’s field equations. We show that for a special class of metrics containing, besides the usual metric components, only one nonzero term, the action reduces to the O’Hanlon action that is the Brans–Dicke action with potential and with the parameter ω equal to zero. We also show that the generalized Einstein equations (with zero energy–stress tensor) are equivalent to those of the Kaluza–Klein theory satisfying a “modified cylinder condition” and having a noncompact extra dimension. This opens a possibility to consider Kaluza–Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space–time dimension but appears because of the generalization of the derivation concept.

GENERALIZED BELINSKY–RUFFINI GEOMETRY

1991 ◽  
Vol 06 (24) ◽  
pp. 2189-2195
Author(s):  
AMIR LEVINSON ◽  
AHARON DAVIDSON
Keyword(s):  
Electric Charge ◽  
Magnetic Moment ◽  
Extra Dimension ◽  
Space Time ◽  
Einstein Equations ◽  
Axially Symmetric ◽  
Relativistic Limit ◽  
Kaluza Klein

Stationary, axially symmetric solutions of Einstein equations in a free 5-dimensional Kaluza–Klein space-time are derived. The electric charge and magnetic moment are generated by a fictitious boost involving the extra dimension. The associated gyromagnetic factor tends to unity at the ultra-relativistic limit. The solution derived interpolates between the Kerr and the Belinsky–Ruffini solutions.

ROLE OF EXTRA DIMENSION IN ACCELERATED EXPANSION

2008 ◽  
Vol 23 (06) ◽  
pp. 909-917 ◽  
Author(s):  
K. D. PUROHIT ◽  
YOGESH BHATT
Keyword(s):  
Cosmological Model ◽  
Extra Dimension ◽  
Accelerated Expansion ◽  
Field Equations ◽  
Four Dimensions ◽  
Scale Factors ◽  
Expansion Of The Universe ◽  
The Universe ◽  
Kaluza Klein

A five-dimensional FRW-type Kaluza–Klein cosmological model is taken to study the role of extra dimension in the expansion of the universe. Relation between scale factors corresponding to conventional four dimensions and the extra dimension has been established. Field equations are solved in order to find out the effect of pressure corresponding to these scale factors. Conditions for accelerated expansion are derived.

PHYSICAL PROPERTIES OF MATTER DERIVED FROM GEOMETRY OF KALUZA-KLEIN THEORY

1993 ◽  
Vol 02 (02) ◽  
pp. 163-170 ◽  
Author(s):  
P.S. WESSON ◽  
J. PONCE DE LEON ◽  
P. LIM ◽  
H. LIU
Keyword(s):  
General Relativity ◽  
Physical Properties ◽  
Field Equations ◽  
Klein Theory ◽  
Kaluza Klein

We ask if it is possible to geometrize properties of matter such as the density and pressure in terms of a Kaluza-Klein extension of general relativity. We find that this is possible for at least three important classes of problems, where acceptable 4D properties of matter are recovered as the extra parts of a 5D geometry with “empty” field equations. This suggests to us that matter may be purely geometrical in origin.

scholarly journals The spacetime between Einstein and Kaluza–Klein

2020 ◽  
Vol 35 (36) ◽  
pp. 2030020
Author(s):  
Chris Vuille
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Differential Operators ◽  
Mathematical Structure ◽  
Field Equations ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Klein Theory ◽  
Klein Gordon ◽  
Kaluza Klein

In this paper I introduce tensor multinomials, an algebra that is dense in the space of nonlinear smooth differential operators, and use a subalgebra to create an extension of Einstein’s theory of general relativity. In a mathematical sense this extension falls between Einstein’s original theory of general relativity in four dimensions and the Kaluza–Klein theory in five dimensions. The theory has elements in common with both the original Kaluza–Klein and Brans–Dicke, but emphasizes a new and different underlying mathematical structure. Despite there being only four physical dimensions, the use of tensor multinomials naturally leads to expanded operators that can incorporate other fields. The equivalent Ricci tensor of this geometry is robust and yields vacuum general relativity and electromagnetism, as well as a Klein–Gordon-like quantum scalar field. The formalism permits a time-dependent cosmological function, which is the source for the scalar field. I develop and discuss several candidate Lagrangians. Trial solutions of the most natural field equations include a singularity-free dark energy dust cosmology.

scholarly journals NON-ABELIAN GAUGE SYMMETRIES INDUCED BY THE UNOBSERVABILITY OF EXTRA DIMENSIONS IN A KALUZA–KLEIN APPROACH

2006 ◽  
Vol 21 (03) ◽  
pp. 265-274 ◽  
Author(s):  
FRANCESCO CIANFRANI ◽  
GIOVANNI MONTANI
Keyword(s):  
Extra Dimensions ◽  
Gauge Coupling ◽  
Einstein Equations ◽  
Field Equations ◽  
Abelian Gauge ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Gauge Symmetries ◽  
The Right ◽  
Kaluza Klein

In this work we deal with the extension of the Kaluza–Klein approach to a non-Abelian gauge theory; we show how we need to consider the link between the n-dimensional model and a four-dimensional observer physics, in order to reproduce field equations and gauge transformations in the four-dimensional picture. More precisely, in field equations any dependence on extra coordinates is canceled out by an integration, as consequence of the unobservability of extra dimensions. Thus, by virtue of this extra dimension unobservability, we are able to recast the multidimensional Einstein equations into the four-dimensional Einstein–Yang–Mills ones, as well as all the right gauge transformations of fields are induced. The same analysis is performed for the Dirac equation describing the dynamics of the matter fields and, again, the gauge coupling with Yang–Mills fields are inferred from the multidimensional free fields theory, together with the proper spinors transformations.

scholarly journals Is gravity actually the curvature of spacetime?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944021
Author(s):  
Sebastian Bahamonde ◽  
Mir Faizal
Keyword(s):  
General Relativity ◽  
Casimir Effect ◽  
Boundary Effect ◽  
Experimental Tests ◽  
Classical Field ◽  
Einstein Equations ◽  
Field Equations ◽  
Tests Of General Relativity ◽  
Boundary Terms ◽  
Theories Of Gravity

The Einstein equations, apart from being the classical field equations of General Relativity, are also the classical field equations of two other theories of gravity. As the experimental tests of General Relativity are done using the Einstein equations, we do not really know if gravity is the curvature of a torsionless spacetime or torsion of a curvatureless spacetime or if it occurs due to the nonmetricity of a curvatureless and torsionless spacetime. However, as the classical actions of all these theories differ from each other by the boundary terms, and the Casimir effect is a boundary effect, we propose that a novel gravitational Casimir effect between superconductors can be used to test which of these theories actually describe gravity.

scholarly journals THE COSMOLOGICAL CONSTANT PROBLEM AND KALUZA–KLEIN THEORY

2001 ◽  
Vol 10 (06) ◽  
pp. 905-912 ◽  
Author(s):  
PAUL S. WESSON ◽  
HONGYA LIU
Keyword(s):  
Field Theory ◽  
General Relativity ◽  
Energy Density ◽  
Cosmological Constant ◽  
Field Equations ◽  
Cosmological Constant Problem ◽  
Klein Theory ◽  
Kaluza Klein

We present technical results which extend previous work and show that the cosmological constant of general relativity is an artefact of the reduction to 4D of 5D Kaluza–Klein theory (or 10D superstrings and 11D supergravity). We argue that the distinction between matter and vacuum is artificial in the context of ND field theory. The concept of a cosmological "constant" (which measures the energy density of the vacuum in 4D) should be replaced by that of a series of variable fields whose sum is determined by a solution of ND field equations in a well-defined manner.

scholarly journals CAUSAL ANOMALIES IN KALUZA–KLEIN GRAVITY THEORIES

1998 ◽  
Vol 13 (18) ◽  
pp. 3181-3191 ◽  
Author(s):  
M. J. REBOUÇAS ◽  
A. F. F. TEIXEIRA
Keyword(s):  
General Relativity ◽  
Perfect Fluid ◽  
Momentum Tensor ◽  
Causality Principle ◽  
Field Equations ◽  
Perfect Fluid Solution ◽  
Dust Solution ◽  
Gravity Theories ◽  
Induced Matter ◽  
Kaluza Klein

Causal anomalies in two Kaluza–Klein gravity theories are examined, particularly as to whether these theories permit solutions in which the causality principle is violated. It is found that similarly to general relativity the field equations of the space–time–mass Kaluza–Klein (STM-KK) gravity theory do not exclude violation of causality of Gödel type, whereas the induced matter Kaluza–Klein (IM-KK) gravity rules out noncausal Gödel-type models. The induced matter version of general relativity is shown to be an efficient therapy for causal anomalies that occurs in a wide class of noncausal geometries. Perfect fluid and dust Gödel-type solutions of the STM-KK field equations are studied. It is shown that every Gödel-type perfect fluid solution is isometric to the unique dust solution of the STM-KK field equations. The question as to whether 5D Gödel-type noncausal geometries induce any physically acceptable 4D energy–momentum tensor is also addressed.

scholarly journals General relativity as a two-dimensional CFT

2015 ◽  
Vol 24 (12) ◽  
pp. 1544024
Author(s):  
Tim Adamo
Keyword(s):  
General Relativity ◽  
Perturbative Expansion ◽  
Einstein Equations ◽  
Tree Level ◽  
Two Dimensional ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Conformal Field ◽  
Full Nonlinearity ◽  
Hilbert Action

The tree-level scattering amplitudes of general relativity (GR) encode the full nonlinearity of the Einstein field equations. Yet remarkably compact expressions for these amplitudes have been found which seem unrelated to a perturbative expansion of the Einstein–Hilbert action. This suggests an entirely different description of GR which makes this on-shell simplicity manifest. Taking our cue from the tree-level amplitudes, we discuss how such a description can be found. The result is a formulation of GR in terms of a solvable two-dimensional conformal field theory (CFT), with the Einstein equations emerging as quantum consistency conditions.

General Relativity

2019 ◽  
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
High Energy Physics ◽  
Hamiltonian Formulation ◽  
Experimental Tests ◽  
High Energy ◽  
Gravitational Energy ◽  
Field Equations ◽  
Physics First ◽  
Condensed Matter Theory ◽  
Introductory Textbooks

This work is a short textbook on general relativity and gravitation, aimed at readers with a broad range of interests in physics, from cosmology to gravitational radiation to high energy physics to condensed matter theory. It is an introductory text, but it has also been written as a jumping-off point for readers who plan to study more specialized topics. As a textbook, it is designed to be usable in a one-quarter course (about 25 hours of instruction), and should be suitable for both graduate students and advanced undergraduates. The pedagogical approach is “physics first”: readers move very quickly to the calculation of observational predictions, and only return to the mathematical foundations after the physics is established. The book is mathematically correct—even nonspecialists need to know some differential geometry to be able to read papers—but informal. In addition to the “standard” topics covered by most introductory textbooks, it contains short introductions to more advanced topics: for instance, why field equations are second order, how to treat gravitational energy, what is required for a Hamiltonian formulation of general relativity. A concluding chapter discusses directions for further study, from mathematical relativity to experimental tests to quantum gravity.

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