Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

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.

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Separation of variables for the Klein-Gordon equation in special Stackel spacetimes

Classical and Quantum Gravity ◽

10.1088/0264-9381/7/1/008 ◽

1990 ◽

Vol 7 (1) ◽

pp. 19-25 ◽

Cited By ~ 4

Author(s):

V G Bagrov ◽

V V Obukhov

Keyword(s):

Gordon Equation ◽

Separation Of Variables ◽

Klein Gordon ◽

Klein Gordon Equation

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Exact solutions of the 2D Dunkl–Klein–Gordon equation: The Coulomb potential and the Klein–Gordon oscillator

Modern Physics Letters A ◽

10.1142/s0217732321501716 ◽

2021 ◽

pp. 2150171

Author(s):

R. D. Mota ◽

D. Ojeda-Guillén ◽

M. Salazar-Ramírez ◽

V. D. Granados

Keyword(s):

Angular Momentum ◽

Energy Spectrum ◽

Exact Solutions ◽

Coulomb Potential ◽

Gordon Equation ◽

Separation Of Variables ◽

Point Of View ◽

Algebraic Point ◽

Klein Gordon ◽

Klein Gordon Equation

We introduce the Dunkl–Klein–Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein–Gordon (KG) equation. We show that the generalization with Dunkl derivative of the z-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein–Gordon oscillator analytically and from an su(1, 1) algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.

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Separation of variables in the Klein ? Gordon equation. I

Soviet Physics Journal ◽

10.1007/bf00889957 ◽

1973 ◽

Vol 16 (11) ◽

pp. 1533-1538 ◽

Cited By ~ 3

Author(s):

V. G. Bagrov ◽

A. G. Meshkov ◽

V. N. Shapovalov ◽

A. V. Shapovalov

Keyword(s):

Gordon Equation ◽

Separation Of Variables ◽

Klein Gordon ◽

Klein Gordon Equation

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PHANTOM FIELD FROM CONFORMAL INVARIANCE

Modern Physics Letters A ◽

10.1142/s0217732307021093 ◽

2007 ◽

Vol 22 (04) ◽

pp. 307-316

Author(s):

MOKHTAR HASSAÏNE

Keyword(s):

Scalar Field ◽

Cosmological Constant ◽

Gordon Equation ◽

Complex Scalar ◽

Conformally Invariant ◽

Phantom Scalar ◽

Klein Gordon ◽

Complex Scalar Field ◽

Phantom Scalar Field ◽

Klein Gordon Equation

We establish a correspondence between a conformally invariant complex scalar field action (with a conformal self-interaction potential) and the action of a phantom scalar field minimally coupled to gravity (with a cosmological constant). In this correspondence, the module of the complex scalar field is used to relate conformally the metrics of both systems while its phase is identified with the phantom scalar field. At the level of the equations, the correspondence allows to map solution of the conformally nonlinear Klein–Gordon equation with vanishing energy–momentum tensor to solution of a phantom scalar field minimally coupled to gravity with cosmological constant satisfying a massless Klein–Gordon equation. The converse is also valid with the advantage that it offers more possibilities owing to the freedom of rewriting a metric as the conformal transformation of another metric. In three dimensions, the coupling of this matter action to conformal gravity is put in equivalence with topologically massive gravity with a cosmological constant and with a phantom source. Finally, we provide some examples of this correspondence.

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