scholarly journals Some metrics admitting nonpolynomial first integrals of the geodesic equation

2021 ◽  
pp. 136483
Author(s):  
Anton Galajinsky
Keyword(s):  
Geodesic Equation ◽  
First Integrals

Low valence Killing spinors in Gödel's Universe

2017 ◽  
Vol 26 (13) ◽  
pp. 1750147 ◽  
Author(s):  
S. A. Cook
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Geodesic Equation ◽  
First Integrals ◽  
Coordinate Systems ◽  
Killing Spinors ◽  
Killing Tensors ◽  
Generalized Symmetries ◽  
Related Notion

Killing tensors have been of interest historically primarily for generating first integrals for the geodesic equation and for their use in finding separable coordinate systems. A related notion, that of Killing spinors, has recently been shown to be important in the study of generalized symmetries of Maxwell's equations. In a given spacetime, the generalized symmetries depend on the existence of Killing spinors of the spacetime of certain valences. The existence of Killing spinors for the curved metric of Gödel's Universe is investigated. There are five (1,1) Killing spinors, 14 (2,2) and five (1,5) Killing spinors of the spacetime, in addition to the unique (0,2) and (0,4) Killing spinors which are exhibited here as well.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Angular Momentum ◽  
Conservation Laws ◽  
Equations Of Motion ◽  
Geodesic Equation ◽  
First Integrals ◽  
Spacetime Symmetries ◽  
Conserved Quantities ◽  
Killing Vectors ◽  
Invariance Properties ◽  
Matter Energy

This chapter studies how the ‘spacetime symmetries’ can generate first integrals of the equations of motion which simplify their solution and also make it possible to define conserved quantities, or ‘charges’, characterizing the system. As already mentioned in the introduction to matter energy–momentum tensors in Chapter 3, the concepts of energy, momentum, and angular momentum are related to the invariance properties of the solutions of the equations of motion under spacetime translations or rotations. The chapter explores these in greater detail. It first turns to isometries and Killing vectors. The chapter then examines the first integrals of the geodesic equation, and Noether charges.

scholarly journals The separation of the Hamilton-Jacobi equation for the Kerr metric

1999 ◽  
Vol 41 (2) ◽  
pp. 248-259 ◽  
Author(s):  
G. E. Prince ◽  
J. E. Aldridge ◽  
S. E. Godfrey ◽  
G. B. Byrnes
Keyword(s):  
Jacobi Equation ◽  
Geodesic Equation ◽  
First Integrals ◽  
Physical Significance ◽  
Symplectic Reduction ◽  
Kerr Metric ◽  
Killing Tensor ◽  
Tensor Quantity ◽  
Recent Theorem ◽  
Hamilton Jacobi Equation

AbstractWe discuss the separability of the Hamilton-Jacobi equation for the Kerr metric. We use a recent theorem which says that a completely integrable geodesic equation has a fully separable Hamilton-Jacobi equation if and only if the Lagrangian is a composite of the involutive first integrals. We also discuss the physical significance of Carter's fourth constant in terms of the symplectic reduction of the Schwarzschild metric via SO(3), showing that the Killing tensor quantity is the remnant of the square of angular momentum.

Method of first integrals and interface, surface waves

2010 ◽  
Vol 32 (2) ◽  
pp. 107-120
Author(s):  
Pham Chi Vinh ◽  
Trinh Thi Thanh Hue ◽  
Dinh Van Quang ◽  
Nguyen Thi Khanh Linh ◽  
Nguyen Thi Nam
Keyword(s):  
Rayleigh Waves ◽  
Displacement Vector ◽  
Attenuation Factor ◽  
Electrical Potential ◽  
Polarization Vector ◽  
First Integrals ◽  
Dispersion Equations ◽  
Interface Surface ◽  
Electric Induction ◽  
The One

The method of first integrals (MFI) based on the equation of motion for the displacement vector, or  based on the one for the traction vector was introduced  recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves,  the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI  to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii)  Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

scholarly journals Entanglement entropy for $$ \mathrm{T}\overline{\mathrm{T}} $$, $$ \mathrm{J}\overline{\mathrm{T}} $$, $$ \mathrm{T}\overline{\mathrm{J}} $$ deformed holographic CFT

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Soumangsu Chakraborty ◽  
Akikazu Hashimoto
Keyword(s):  
Parameter Space ◽  
Wilson Loop ◽  
Lorentz Invariance ◽  
Entanglement Entropy ◽  
Geodesic Equation ◽  
Leading Order ◽  
Theoretic Analysis ◽  
Expectation Value ◽  
Spatial Coordinates ◽  
Single Trace

Abstract We derive the geodesic equation for determining the Ryu-Takayanagi surface in AdS3 deformed by single trace $$ \mu T\overline{T} $$ μT T ¯ + $$ {\varepsilon}_{+}J\overline{T} $$ ε + J T ¯ + $$ {\varepsilon}_{-}T\overline{J} $$ ε − T J ¯ deformation for generic values of (μ, ε+, ε−) for which the background is free of singularities. For generic values of ε±, Lorentz invariance is broken, and the Ryu-Takayanagi surface embeds non-trivially in time as well as spatial coordinates. We solve the geodesic equation and characterize the UV and IR behavior of the entanglement entropy and the Casini-Huerta c-function. We comment on various features of these observables in the (μ, ε+, ε−) parameter space. We discuss the matching at leading order in small (μ, ε+, ε−) expansion of the entanglement entropy between the single trace deformed holographic system and a class of double trace deformed theories where a strictly field theoretic analysis is possible. We also comment on expectation value of a large rectangular Wilson loop-like observable.

Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

2018 ◽  
Vol 24 (2) ◽  
pp. 175-183
Author(s):  
Jean-Claude Ndogmo
Keyword(s):  
Explicit Form ◽  
Nonlinear Equations ◽  
Variational Principles ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Maximal Symmetry ◽  
General Order ◽  
Linear And Nonlinear ◽  
Variational Symmetries ◽  
Variational Symmetry

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

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