Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

Rajaratnam, Krishan; McLenaghan, R. G.; Valero, Carlos

doi:10.3842/sigma.2016.117

Cited by 13 publications

(79 citation statements)

References 85 publications

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“…As in [11], we will write the transformation equations in terms of transcendental functions, when possible. In many cases these will be the Jacobi elliptic functions, snpu; aq, cnpu; aq, dnpu; aq, etc.…”

Section: Classification Of Separable Webs In H 3 and Dsmentioning

confidence: 99%

“…We then get nine separable webs, corresponding to the nine possible webs we may lift from dS 2 . For a list of the nine webs and their adapted coordinates on dS 2 , see [11].…”

Section: Restriction To Hmentioning

confidence: 99%

“…The approach used is this paper is based on the theory of concircular tensors and warped products developed by Rajaratnam [12], Rajaratnam and McLenaghan [13,14] and Rajaratnam, McLenaghan and Valero [11], which is applicable to pseudo-Riemannian spaces of constant curvature. This theory is derived from Eisenhart's [6] characterization of orthogonal separability by means of valence-two Killing tensors which have simple eigenvalues and orthogonally integrable eigendirections, called characteristic Killing tensors.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, this paper relies heavily on some elementary results regarding the classification of self-adjoint linear operators on E n ν . A review of the relevant material may be found in [11]; for convenience, in appendix A we have summarized the main results for the case of E n 1 , which is all that shall be needed in this paper (of course in E n all self-adjoint operators are orthogonally diagonalizable). Any reader not familiar with this subject should look at the appendix before proceeding.…”

Section: Introductionmentioning

confidence: 99%

“…We then get four separable webs, corresponding to the four possible webs we may lift from E 2 . These webs may be found in any standard reference; see [11] for example.…”

mentioning

confidence: 99%

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Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Valero

McLenaghan

2019

Journal of Mathematical Physics

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We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for Hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

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Section: Classification Of Separable Webs In H 3 and Dsmentioning

confidence: 99%

“…We then get nine separable webs, corresponding to the nine possible webs we may lift from dS 2 . For a list of the nine webs and their adapted coordinates on dS 2 , see [11].…”

Section: Restriction To Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We then get four separable webs, corresponding to the four possible webs we may lift from E 2 . These webs may be found in any standard reference; see [11] for example.…”

mentioning

confidence: 99%

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Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Valero

McLenaghan

2019

Journal of Mathematical Physics

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Discrete relativistic positioning systems

et al. 2020

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We discuss the design for a discrete, immediate, simple relativistic positioning system (rPS) which is potentially able of self-positioning (up to isometries) and operating without calibration or ground control assistance. The design is discussed in dimension two on spacetime (i.e. one spatial dimension plus one time dimension), in Minkowski and Schwarzschild solutions, as well as in dimension three (i.e. two spatial dimensions plus one time dimension) in Minkowski.The system works without calibration, clock synchronizations, or a priori knowledge about the motion of clocks, it is able to self-diagnose hypotheses break down (for example, if one clock temporarily becomes not-freely falling, or the gravitational field changes) and it is automatically back and operational when the assumed conditions are restored.In the Schwarzschild case, we show that the system can also best fit the gravitational mass of the source of the gravitational field and stress that no weak field assumptions are made anywhere. In particular, the rPS we propose can work in a region close to the horizon since it does not use approximations or PPN expansions. More generally, the rPS can be adapted as detectors for the gravitational field and we shall briefly discuss their role in testing different theoretical settings for gravity. In fact, rPS is a natural candidate for a canonical method to extract observables out of a gravitational theory, an activity also known as designing experiments to test gravity.

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Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation

Rajaratnam

McLenaghan

2014

Journal of Mathematical Physics

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We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature. C 2014 AIP Publishing LLC. [http://dx.

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Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

Cited by 13 publications

References 85 publications

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Discrete relativistic positioning systems

Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation

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