Κ-deformed Spacetime Research Articles

Neutron Star in Quantized Space-Time

We construct and analyze a model of a neutron star in a κ-deformed space-time. This is conducted by first deriving the κ-deformed generalization of the Einstein tensor, starting from the non-commutative generalization of the metric tensor. By generalizing the energy-momentum tensor to the non-commutative space-time and exploiting the κ-deformed dispersion relation, we then set up Einstein’s field equations in the κ-deformed space-time. As we adopt a realization of the non-commutative coordinates in terms of the commutative coordinates and their derivatives, our model is constructed in terms of commutative variables. Using this, we derive the κ-deformed generalization of the Tolman–Oppenheimer–Volkoff equation. Now, by treating the interior of the star as a perfect fluid as in the commutative space-time, we investigate the modification of the neutron star’s mass due to the non-commutativity of space-time, valid up to first order in the deformation parameter. We show that the non-commutativity of space-time enhances the mass limit of the neutron star. We show that the radius and maximum mass of the neutron star depend on the deformation parameter. Further, our study shows that the mass increases as the radius increases for fixed values of the deformation parameter. We show that maximum mass and radius increase as the deformation parameter increases. We find that the mass varies from 0.26M⊙ to 3.68M⊙ as the radius changes from 8.45 km to 18.66 km. Using the recent observational limits on the upper bound of the mass of a neutron star, we find the deformation parameter to be |a|∼10−44 m. We also show that the compactness and surface redshift of the neutron star increase with its mass.

Newtonian Cosmology and Evolution of κ-Deformed Universe

Considering space-time to be non-commutative, we study the evolution of the universe employing the approach of Newtonian cosmology. Generalizing the conservation of energy and the first law of thermodynamics to κ-deformed space-time, we derive the modified Friedmann equations, valid up to the first order, in the deformation parameter. Analyzing these deformed equations, we derive the time evolution of the scale factor in cases of radiation-dominated, matter-dominated, and vacuum (energy)-dominated universes. We show that the rate of change of the scale factor in all three situations is modified by the non-commutativity of space-time, and this rate depends on the sign of the deformation parameter, indicating a possible explanation for the observed Hubble tension. We undertake this investigation for two different realizations of non-commutative space-time coordinates. In both cases, we also argue for the existence of bounce in the evolution of the universe.

Time-space noncommutativity and Casimir effect

We show that the Casimir force and energy are modified in the κ-deformed space-time. This is shown by solving the Green's function corresponding to κ-deformed scalar field equation in presence of two parallel plates, modelled by δ-function potentials. Exploiting the relation between Energy-Momentum tensor and Green's function, we calculate corrections to Casimir force, valid up to second order in the deformation parameter. The Casimir force is shown to get corrections which scale as L−4 and L−6 and both these types of corrections produce attractive forces. Using the measured value of Casimir force, we show that the deformation parameter should be below 10−23 m.

Conformal quantum mechanics and holography in noncommutative space–time

We analyze the effects of noncommutativity in conformal quantum mechanics (CQM) using the κ-deformed space–time as a prototype. Up to the first order in the deformation parameter, the symmetry structure of the CQM algebra is preserved but the coupling in a canonical model of the CQM gets deformed. We show that the boundary conditions that ensure a unitary time evolution in the noncommutative CQM can break the scale invariance, leading to a quantum mechanical scaling anomaly. We calculate the scaling dimensions of the two and three point functions in the noncommutative CQM which are shown to be deformed. The AdS2/CFT1 duality for the CQM suggests that the corresponding correlation functions in the holographic duals are modified. In addition, the Breitenlohner–Freedman bound also picks up a noncommutative correction. The strongly attractive regime of a canonical model of the CQM exhibit quantum instability. We show that the noncommutativity softens this singular behaviour and its implications for the corresponding holographic duals are discussed.

MICZ-Kepler systems in noncommutative space and duality of force laws

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 in the deformation parameter.

The covariant and on-shell statistics in κ-deformed space-time

It has been a long-standing issue to construct the statistics of identical particles in κ-deformed space-time. In this letter, we investigate different ideas on this problem. Following the ideas of Young and Zegers, we obtain the covariant and on shell kappa two-particle state in 1+1 D in a simpler way. Finally, a procedure to get such state in higher dimension is proposed.

κ-deformed spacetime from twist

We twist the Hopf algebra of igl(n,R) to obtain the κ-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The κ-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional κ-Minkowski spacetime, but its Hopf algebra structure is different from the well-known κ-deformed Poincaré algebra in that it has larger symmetry algebra than the κ-Minkowski case. There are some physical models which consider this symmetry [R. Percacci, Phys. Lett. B 144 (1984) 37; R. Percacci, Geometry of Nonlinear Field Theories, World Scientific, Singapore, 1986; L. Smolin, Nucl. Phys. B 132 (1978) 138; R. Floreanini, R. Percacci, Class. Quantum Grav. 7 (1990) 975]. Incidentally, we obtain the canonical (θ-deformed) non-commutative spacetime from canonically twisted igl(n,R) Hopf algebra.

A symmetry invariant integral on κ-deformed spacetime

In this note we present an approach using both constructive and Hopf algebraic methods to contribute to the not yet fully satisfactory definition of an integral on κ-deformed spacetime. The integral presented here is based on the inner product of differential forms and it is shown that this integral is explicitly invariant under the deformed symmetry structure.