Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

Paliathanasis, Andronikos; Tsamparlis, Michael; Mustafa, M. T.

doi:10.1142/s0219887815500334

Cited by 24 publications

(36 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From (26) it follows that T ,tt = 0, that is T (t) = T 0 + T 1 t, while from (26) we have that Z A = ψT 1 tY A or Z A = Y A and T 1 . Condition (27) provides an infinite number of trivial symmetries, which is a well-known result for the heat equation. Thus, from Theorem 1 follows:…”

Section: Theoremmentioning

confidence: 82%

Lie symmetries for systems of evolution equations

Paliathanasis

Tsamparlis

2018

Journal of Geometry and Physics

View full text Add to dashboard Cite

The Lie symmetries for a class of systems of evolution equations are studied. The evolution equations are defined in a bimetric space with two Riemannian metrics corresponding to the space of the independent and dependent variables of the differential equations. The exact relation of the Lie symmetries with the collineations of the bimetric space is determined.Lie symmetries is a powerful method for the determination of solutions in the theory of differential equations. A Lie symmetry is important as it provides invariants which can be used to write a new differential equation with less degree of freedom. Furthermore, a solution of such an equation is also a solution of the original differential equation [1,2]. The reduction process is the main application of Lie symmetries, however, it is not a univocal approach. Symmetries can be used for the determination of conservation currents [3], for the classification of differential equations [4][5][6][7][8][9][10] and for the reconnaissance of some well-known systems [11][12][13][14][15].In the recent literature, it has been shown that there is a close relation between the Lie symmetries of a second order differential equation and the geometry of the space where motion occurs. For example, the conservation of energy and angular momentum in Newtonian Physics is a result of the Lie point symmetries, generated by the Killing vectors of translations and rotations respectively. The general result for a holonomic autonomous dynamical system moving in a Riemannian space is that the Lie point symmetries of the equations of motion *

show abstract

Section: Theoremmentioning

confidence: 82%

Lie symmetries for systems of evolution equations

Paliathanasis

Tsamparlis

2018

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…It was shown that the Lie and Noether symmetries of that system are related with the special projective algebra of the underlying space. This geometric approach has been extended to the determination of the Lie and Noether point symmetries of some families of partial differential equations [15,16].…”

Section: Introductionmentioning

confidence: 99%

Variational contact symmetries of constrained Lagrangians

Terzis

Dimakis

Christodoulakis

et al. 2016

Journal of Geometry and Physics

View full text Add to dashboard Cite

The investigation of contact symmetries of re-parametrization invariant Lagrangians of finite degrees of freedom and quadratic in the velocities is presented. The main concern of the paper is those symmetry generators which depend linearly in the velocities. A natural extension of the symmetry generator along the lapse function N (t), with the appropriate extension of the dependence inṄ (t) of the gauge function, is assumed; this action yields new results. The central finding is that the integrals of motion are either linear or quadratic in velocities and are generated, respectively by the conformal Killing vector fields and the conformal Killing tensors of the configuration space metric deduced from the kinetic part of the Lagrangian (with appropriate conformal factors). The freedom of re-parametrization allows one to appropriately scale N (t), so that the potential becomes constant; in this case the integrals of motion can be constructed from the Killing fields and Killing tensors of the scaled metric. A rather interesting result is the non-necessity of the gauge function in Noether's theorem due to the presence of the Hamiltonian constraint.

show abstract

“…In order to determine the Lie and the Noether symmetries of the WDW equation we will follow the results of [39,40] which relate the point symmetries of the WDW equation with the conformal algebra of the space which defines the Laplace operator. This means that we can separate the problem in two steps: (a) we will study the conformal algebra of the minisuperspace and (b) we will determine the unknown potential.…”

Section: Lie Symmetries Of the Wdw Equationmentioning

confidence: 99%

“…In [39,40], it has been shown how to extract the conservation laws and compute the Noether integrals for a given classical Lagrangian from the Lie symmetries of the WDW equation. In this section, for each Lie group of transformations in terms of which the WDW Eq.…”

Section: Conservation Laws and Analytical Solutions Of The Field Equamentioning

confidence: 99%

Closed-form solutions of the Wheeler–DeWitt equation in a scalar-vector field cosmological model by Lie symmetries

Paliathanasis

Vakili

2015

Gen Relativ Gravit

View full text Add to dashboard Cite

We apply as selection rule to determine the unknown functions of a cosmological model the existence of Lie point symmetries for the Wheeler-DeWitt equation of quantum gravity. Our cosmological setting consists of a flat Friedmann-RobertsonWalker metric having the scale factor a(t), a scalar field with potential function V (φ) minimally coupled to gravity and a vector field of its kinetic energy is coupled with the scalar field by a coupling function f (φ). Then, the Lie symmetries of this dynamical system are investigated by utilizing the behavior of the corresponding minisuperspace under the infinitesimal generator of the desired symmetries. It is shown that by applying the Lie symmetry condition the form of the coupling function and also the scalar field potential function may be explicitly determined so that we are able to solve the Wheeler-DeWitt equation. Finally, we show how we can use the Lie symmetries in order to construct conservation laws and exact solutions for the field equations.

show abstract

Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

Cited by 24 publications

References 39 publications

Lie symmetries for systems of evolution equations

Lie symmetries for systems of evolution equations

Variational contact symmetries of constrained Lagrangians

Closed-form solutions of the Wheeler–DeWitt equation in a scalar-vector field cosmological model by Lie symmetries

Contact Info

Product

Resources

About