Invariant prolongation of the Killing tensor equation

We study dynamics of a test charged particle moving in a weakly charged extreme Kerr throat. Dynamical equations of the particle motion are solved in quadratures. We show explicitly that the Killing tensor of the Kerr spacetime becomes reducible in the extreme Kerr throat geometry. Special types of motion of particles and light are discussed.

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Complete set of quasi-conserved quantities for spinning particles around Kerr

SciPost Physics ◽

10.21468/scipostphys.12.1.012 ◽

2022 ◽

Vol 12 (1) ◽

Author(s):

Geoffrey Compère ◽

Adrien Druart

Keyword(s):

Linear Order ◽

Conserved Quantities ◽

Killing Tensor ◽

Kerr Geometry ◽

Spinning Particles ◽

Kerr Spacetime ◽

Killing Tensors ◽

Mixed Symmetry ◽

Constants Of Motion ◽

Complete Set

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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A Tensor Equation of Elliptic Type

Canadian Journal of Mathematics ◽

10.4153/cjm-1953-059-4 ◽

1953 ◽

Vol 5 ◽

pp. 524-535 ◽

Cited By ~ 1

Author(s):

G. F. D. Duff

Keyword(s):

Riemannian Manifold ◽

Differential Equations ◽

Tensor Field ◽

Differential Operators ◽

Elliptic Type ◽

Tensor Equation ◽

Tensor Theory ◽

Elliptic Differential Equations ◽

Invariant Differential ◽

Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

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Harmonic and Killing tensor fields in Riemannian spaces with boundary.

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/01040430 ◽

1958 ◽

Vol 10 (4) ◽

pp. 430-437 ◽

Cited By ~ 4

Author(s):

Kentaro YANO

Keyword(s):

Killing Tensor ◽

Tensor Fields ◽

Riemannian Spaces

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Iterative solution of quadratic tensor equations for mutual polarisation

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202110246 ◽

2002 ◽

Vol 32 (5) ◽

pp. 301-312 ◽

Cited By ~ 1

Author(s):

Wynand S. Verwoerd

Keyword(s):

Dipole Moment ◽

Linear Approximation ◽

Iterative Solution ◽

Second Order ◽

Local Fields ◽

Bulk Materials ◽

Tensor Equation ◽

Order Tensor ◽

Intrinsic Values ◽

Induced Dipole

To describe mutual polarisation in bulk materials containing high polarisability molecules, local fields beyond the linear approximation need to be included. A second order tensor equation is formulated, and it describes this in the case of crystalline or at least locally ordered materials such as an idealised polymer. It is shown that this equation is solved by a set of recursion equations that relate the induced dipole moment, linear polarisability, and first hyperpolarisability in the material to the intrinsic values of the same properties of isolated molecules. From these, macroscopic susceptibility tensors up to second order can be calculated for the material.

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