scholarly journals GRAVITATIONAL FIELD OF A ROTATING GRAVITATIONAL DYON

2002 ◽  
Vol 17 (15n17) ◽  
pp. 1091-1096 ◽  
Author(s):  
N. DADHICH ◽  
Z. YA. TURAKULOV
Keyword(s):  
Gravitational Field ◽  
Gordon Equation ◽  
Equations Of Motion ◽  
Polar Angle ◽  
Vacuum Equation ◽  
Axially Symmetric ◽  
Einstein Vacuum Equation ◽  
Einstein Vacuum ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation

We have obtained the general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charges. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion.

SEPARATING STATIONARY AXIALLY-SYMMETRIC ASYMPTOTICALLY FLAT METRICS

1989 ◽  
Vol 04 (12) ◽  
pp. 2953-2958 ◽  
Author(s):  
Z. YA. TURAKULOV
Keyword(s):  
Gordon Equation ◽  
Separation Of Variables ◽  
Vacuum Solution ◽  
Complete Separation ◽  
Kerr Metric ◽  
Axially Symmetric ◽  
Spherical System ◽  
Asymptotically Flat ◽  
Klein Gordon ◽  
Klein Gordon Equation

Stationary axially-symmetric asymptotically flat metrics allowing the complete separation of variables in the Klein-Gordon equation are considered. It is shown that if such metrics coincide at infinity with the metric of spherical system of coordinates, the variables for them in the Einstein equation are completely separable and the only vacuum solution is the Kerr metric.

INFLATION IN KALUZA-KLEIN COSMOLOGY I: TRANSFORMATION TO FOURTH-ORDER GRAVITY

1990 ◽  
Vol 05 (24) ◽  
pp. 4661-4669 ◽  
Author(s):  
HANS-JÜRGEN SCHMIDT
Keyword(s):  
Fourth Order ◽  
Arbitrary Dimension ◽  
Internal Space ◽  
Vacuum Equation ◽  
Scale Invariant ◽  
Einstein Vacuum Equation ◽  
Higher Dimensional ◽  
Einstein Vacuum ◽  
Klein Model ◽  
Kaluza Klein

The higher-dimensional Einstein vacuum equation with Λ-term is shown to be conformally equivalent to the four-dimensional field equation of scale-invariant fourth-order gravity. This holds for a general warped product between space-time and internal space of arbitrary dimension m which turns out to be an Einstein space. (The limit m → ∞ makes sense!) Thus, the results concerning the attractor property of the power-law inflationary solution derived for fourth-order gravity hold for the Kaluza-Klein model, too.

scholarly journals C4 Space-Time.. A Window to New Physics?

2020 ◽  
Author(s):  
Dimitris Mastoridis ◽  
Konstantinos Kalogirou
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Gordon Equation ◽  
Real Space ◽  
Geodesic Equation ◽  
Space Time ◽  
Quantum Theories ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation ◽  
Klein Gordon Equation

We explore the possibility to form a physical theory in $C^4$. We argue that the expansion of our usual 4-d real space-time to a 4-d complex space-time, can serve us to describe geometrically electromagnetism and nuclear fields and unify it with gravity, in a different way that Kaluza-Klein theories do. Specifically, the electromagnetic field $A_\mu$, is included in the free geodesic equation of $C^4$. By embedding our usual 4-d real space-time in the symplectic 8-d real space-time (symplectic $R^8$ is algebraically isomorphic to $C^4$), we derive the usual geodesic equation of a charged particle in gravitational field, plus new information which is interpreted. Afterwards, we formulate and explore the extended special relativity and extended general relativity an $C^4$ or$R^8$. After embedding our usual 4-d space-time in $R^8$, two new phenomena rise naturally, that are interpreted as "dark matter" and "dark energy". A new cosmological model is presented, while the geometrical terms associated with "dark matter" and "dark energy" are investigated. Similarities, patterns and differences between "dark matter", "dark energy", ordinary matter and radiation are presented, where "dark energy" is a dynamic entity and "dark matter" reveal itself as a "mediator" betwen ordinary matter and "dark energy". Moreover, "dark matter" is deeply connected with "dark energy". Furthermore, the extended Hamilton-Jacobi equation of the extended space-time, is transformed naturally as an extended Klein-Gordon equation, in order to get in contact with quantum theories. By solving the Klein-Gordon equation analytically, we derive an eigenvalue for Higg's boson mass value at 125,173945 $Gev/c^{2}$. The extended Klein-Gordon equation, also connects Higg's boson (or vacuum) with Cosmology, due to the existence of our second "time" T (cosmological time), which serve us to connect quantum theories with Cosmology. Afterwards, in the general case, we explore the symmetries of the curved Hamilton-Jacobi equation locally, in order to investigate the consequences of a $C^4$ space-time in Standard Model. An extension to Standard Model is revealed, especially in the sector of strong nuclear field. The Stiefel manifold $SU(4)/SU(2)$ seems capable not only to describe the strong nuclear field but give us,as well, enough room to explore in the future, the possibility to explain quark confinment. Our extension, flavors firstly the unification of nuclear fields and afterwards the unification of nuclear fields with electromagnetic field. The desired grand unification, is achieved locally, through the symmetry group $GL(4,C)\simeq SO(4,4)\cap U(4)$ and we present a potential mechanism to reduce the existing particle numbers to just six. Afterwards,23 present the extended Dirac equation in $C^4$ space-time (Majorana-Weyl representation) plus a preliminary attempt to introduce a pure geometric structure for fermions. Finally, we consider a new geometric structure through n-linear forms in order to give geometric explanation for quantisation

scholarly journals Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2227
Author(s):  
Andrey Bochkarev ◽  
Aleksandr Zemlyanukhin ◽  
Vladimir Erofeev ◽  
Aleksandr Ratushny
Keyword(s):  
Elastic Medium ◽  
Gordon Equation ◽  
Modulation Instability ◽  
Solvable Models ◽  
Axially Symmetric ◽  
Wave Dynamics ◽  
Periodic Traveling Waves ◽  
Bending Waves ◽  
Klein Gordon ◽  
Nonlinear Elastic

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

Exact treatment of the relativistic double ring-shaped Kratzer potential using the quantum Hamilton–Jacobi formalism

2015 ◽  
Vol 30 (16) ◽  
pp. 1550082 ◽  
Author(s):  
A. Gharbi ◽  
S. Touloum ◽  
A. Bouda
Keyword(s):  
Gordon Equation ◽  
Relativistic Quantum ◽  
Separable Potential ◽  
Relativistic Energy ◽  
Double Ring ◽  
Vector Potentials ◽  
Jacobi Formalism ◽  
Klein Gordon ◽  
Equal Scalar ◽  
Hamilton Jacobi Equation

We study the Klein–Gordon equation with noncentral and separable potential under the condition of equal scalar and vector potentials and we obtain the corresponding relativistic quantum Hamilton–Jacobi equation. The application of the quantum Hamilton–Jacobi formalism to the double ring-shaped Kratzer potential leads to its relativistic energy spectrum as well as the corresponding eigenfunctions.

Pilot-Wave Theory for a Relativistic Spin Zero Particle

2020 ◽  
Author(s):  
Dan N. Vollick
Keyword(s):  
Gordon Equation ◽  
Wave Theory ◽  
Particle Trajectories ◽  
Superluminal Motion ◽  
Pilot Wave ◽  
Nonlinear Generalization ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation ◽  
Klein Gordon Equation ◽  
Pilot Wave Theory

In the usual approach to the pilot-wave theory for a spin zero particle one starts with the Klein-Gordon equation, which is the relativistic generalization of the Schrodinger equation. This approach encounters several difficulties including superluminal motion and particle trajectories that move backwards in time. In this paper I start with the relativistic classical Hamilton-Jacobi equation and introduce the quantum potential in a way that avoids the above mentioned difficulties. Particle trajectories are timelike or null and are future pointing. The wave equation satisfied by the field is a nonlinear generalization of the Klein-Gordon equation.

scholarly journals Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 473
Author(s):  
Joshua Baines ◽  
Thomas Berry ◽  
Alex Simpson ◽  
Matt Visser
Keyword(s):  
Jacobi Equation ◽  
Gordon Equation ◽  
Rotation Parameter ◽  
Slow Rotation ◽  
Square Root ◽  
Killing Tensor ◽  
Constant Of Motion ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation ◽  
Klein Gordon Equation

Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

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