In this paper, we analyze the modification of integrable models in the κ-deformed space-time. We show that two dimensional isotropic oscillator problem, Kepler problem and MICZ-Kepler problem in κ-deformed space-time admit integrals of motion as in the commutative space. We also show that the duality equivalence between κ-deformed Kepler problem and κ-deformed two-dimensional isotropic oscillator explicitly, by deriving Bohlin-Sundman transformation which maps these two systems. These results are valid to all orders the the deformation parameter.
In this paper, we analyze the Hawking radiation of a κ-deformed-Schwarzschild black hole and obtain the deformed Hawking temperature. For this, we first derive deformed metric for the κ-spacetime, which in the generic case, is not a symmetric tensor and also has a momentum dependence. We show that the Schwarzschild metric obtained in the κ-deformed spacetime has a dependence on energy. We use the fact that the deformed metric is conformally flat in the 1 + 1 dimensions to solve the κ-deformed Klein-Gordon equation in the background of the Schwarzschild metric. The method of Bogoliubov coefficients is then used to calculate the thermal spectrum of κ-deformed-Schwarzschild black hole and show that the Hawking temperature is modified by the noncommutativity of the κ-spacetime.
We derive bounds on the deformation parameter of the κ-spacetime by analyzing the effect of noncommutativity on astrophysical model. We study compact stars, taken to be degenerate Fermi gas, in non-commutative spacetime. Using tools of statistical mechanics, we derive the degeneracy pressure of the compact star in κ-spacetime and from the hydrostatic equilibrium conditions we obtain a bound on the deformation parameter. We independently derive this bound using generalized uncertainty principle, which is a characteristic feature of quantum gravity approaches, strengthening the bound obtained.
We study presence of an additional symmetry of a generic central potential in the κ-space-time. An explicit construction of Fradkin and Bacry, Ruegg, Souriau (FBRS) for a central potential is carried out and the piece-wise conserved nature of the vector is established. We also extend the study to Kepler systems with a drag term, particularly Gorringe-Leach equation is generalized to the κ-deformed space. The possibility of mapping Gorringe-Leach equation to an equation with out drag term is exploited in associating a similar conserved vector to system with a drag term. An extension of duality between two class of central potential is introduced in the κ-deformed space and is used to investigate the duality existing between two class of Gorringe-Leach equations. All the results obtained can be retraced to the correct commutative limit as we let a → 0. 02.30.Ik, 45.20.D, 02.40.Gh. PACS:
In this paper we regularize the Kepler problem on κ-spacetime in several different ways. First, we perform a Moser-type regularization and then we proceed for the Ligon-Schaaf regularization to our problem. In particular, generalizing Heckman-de Laat (J. Symplectic Geom. 10, (2012), 463-473) in the noncommutative context we show that the Ligon-Schaaf regularization map following from an adaptation of the Moser regularization can be generalized to the Kepler problem on κ-spacetime.MSC primary 53D20, 37J15, 70H05, 70H33
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