Conformal Yano–Killing tensor for the Kerr metric and conserved quantities

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by Rüdiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.

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Conserved quantities of spinning test particles in general relativity. II

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1983.0012 ◽

1983 ◽

Vol 385 (1788) ◽

pp. 229-239 ◽

Cited By ~ 14

Keyword(s):

General Relativity ◽

Equations Of Motion ◽

Conserved Quantities ◽

Killing Tensor ◽

Kerr Geometry ◽

Change Of Variables ◽

Test Particles ◽

Suitable Change

In this paper, which completes earlier work on conserved quantities of spinning test particles in general relativity (Rüdiger 1981 a ), quadratic conserved quantities are considered. It is shown that by a suitable change of variables the trivial conserved quantities, which result from a reducible Killing tensor, can essentially be separated from the non-trivial quantities. If the equations of motion are linearized in the spin, it is shown that nontrivial quantities of this type can be constructed for two classes of spacetimes including the Kerr geometry and the Friedman universe.

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

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000001119x ◽

1999 ◽

Vol 41 (2) ◽

pp. 248-259 ◽

Cited By ~ 1

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.

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Conserved Quantities for the General Linear Heat Equation

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v12i4.4418 ◽

2016 ◽

pp. 4437-4439

Author(s):

Adil Jhangeer ◽

Fahad Al-Mufadi

Keyword(s):

Evolution Equation ◽

Heat Equation ◽

Conservation Laws ◽

Exact Solutions ◽

Lagrangian Approach ◽

General Linear ◽

Conserved Quantities ◽

Linear Heat ◽

The Family ◽

Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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The Kerr solution

10.1093/oso/9780198786399.003.0048 ◽

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.

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Conservation laws

10.1093/oso/9780198786399.003.0007 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Dynamical System ◽

Angular Momentum ◽

Conservation Laws ◽

Total Energy ◽

Superposition Principle ◽

Momentum Conservation ◽

Conserved Quantities ◽

Angular Momentum Conservation ◽

Third Law ◽

The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

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On the accuracy of the binary-collision algorithm in particle-in-cell simulations of magnetically confined fusion plasmas

Journal of Plasma Physics ◽

10.1017/s0022377821000398 ◽

2021 ◽

Vol 87 (2) ◽

Author(s):

Timo P. Kiviniemi ◽

Eero Hirvijoki ◽

Antti J. Virtanen

Keyword(s):

Finite Difference ◽

Electric Fields ◽

Finite Size ◽

Binary Collision ◽

Conserved Quantities ◽

Fusion Plasmas ◽

Fusion Plasma ◽

Particle In Cell ◽

Magnetically Confined ◽

Scale Length

Ideally, binary-collision algorithms conserve kinetic momentum and energy. In practice, the finite size of collision cells and the finite difference in the particle locations affect the conservation properties. In the present work, we investigate numerically how the accuracy of these algorithms is affected when the size of collision cells is large compared with gradient scale length of the background plasma, a parameter essential in full- $f$ fusion plasma simulations. Additionally, we discuss implications for the conserved quantities in drift-kinetic formulations when fluctuating magnetic and electric fields are present: we suggest how the accuracy of the algorithms could potentially be improved with minor modifications.

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Mass and spin for classical strings in dS3

Journal of High Energy Physics ◽

10.1007/jhep05(2021)277 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Klaas Parmentier

Keyword(s):

Black Hole ◽

Evolution Equation ◽

Cosmic Strings ◽

Center Of Mass ◽

Open String ◽

Conserved Quantities ◽

Numerical Example ◽

All Solutions

Abstract We demonstrate that all rigidly rotating strings with center of mass at the origin of the dS3 static patch satisfy the Higuchi bound. This extends the observation of Noumi et al. for the open GKP-like string to all solutions of the Larsen-Sanchez class. We argue that strings violating the bound end up expanding towards the horizon and provide a numerical example. Adding point masses to the open string only increases the mass/spin ratio. For segmented strings, we write the conserved quantities, invariant under Gubser’s algebraic evolution equation, in terms of discrete lightcone coordinates describing kink collisions. Randomly generated strings are found to have a tendency to escape through the horizon that is mostly determined by their energy. For rapidly rotating segmented strings with mass/spin < 1, the kink collisions eventually become causally disconnected. Finally we consider the scenario of cosmic strings captured by a black hole in dS and find that horizon friction can make the strings longer.

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