Invariances and Conservation Laws Based on Some FRW Universes

Camcı, Uğur; Jamal, Sameerah; Kara, A. H.

doi:10.1007/s10773-013-1948-x

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…Since then the group classification problem has been studied intensely for fundamental equations arising from models in engineering and physics, e.g. the symmetry classification of the geodesic equations of Riemannian spaces [3,4], the symmetry classification of the two and three dimensional Newtonian systems [5,6,7,8] and many others [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Furthermore, a symmetry analysis of wave equation in a power-law Bianchi III spacetime spacetime can be found in [26] and a symmetry analysis of the wave equation on static spherically symmetric spacetimes, with higher symmetries, was recently carried out in [27] In [28], it was proved that for a linear, in the derivatives, second order partial differential equation (PDE) the Lie point symmetries are related with the conformal algebra of the geometry defined by the PDE.…”

Section: Introductionmentioning

confidence: 99%

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

Paliathanasis

Tsamparlis

Mustafa

2015

Int. J. Geom. Methods Mod. Phys.

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In this work we perform the symmetry classification of the Klein Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also and Noether symmetries for the Klein Gordon equation. We use these resutls in order to determine all the potentials in which the Klein Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.Received (Day Month Year) Revised (Day Month Year)

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Section: Introductionmentioning

confidence: 99%

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

Paliathanasis

Tsamparlis

Mustafa

2015

Int. J. Geom. Methods Mod. Phys.

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“…Here we note that it is only found one Noether symmetry in Reference [19] which is X 2 given in (24), and the remaining ones are not appeared in this reference. It follows from E L = 0 that a = 0, that is, a(t) = a 0 = constant, which is the Minkowski spacetime recovered in vacuum and so I 2 = I 3 = 0, I 4 = 4 f 0 a 3 0 and I 5 = 8 f 0 a 3/2 0 by (28).…”

Section: Vacuum Casementioning

confidence: 73%

“…Firstly, one can consider a strict Noether symmetry approach [18][19][20][21] which yields £ X L = 0, where £ X is the Lie derivative operator along X. On the other side, one could use the classical Noether symmetry approach with a functional term [22][23][24][25] which is a generalization of the strict Noether symmetry approach in the sense that the Noether symmetry equation includes a divergence of a functional boundary term. The classical Noether symmetry approach was originally established by Emmy Noether [26] and it gives a connection between a Noether symmetry and the existence of a first integral expressed in a simple form.…”

Section: Introductionmentioning

confidence: 99%

F(R,G) Cosmology through Noether Symmetry Approach

Camcı¹

2018

Symmetry

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The F ( R , G ) theory of gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant, is studied in the context of existence the Noether symmetries. The Noether symmetries of the point-like Lagrangian of F ( R , G ) gravity for the spatially flat Friedmann-Lemaitre-Robertson-Walker cosmological model is investigated. With the help of several explicit forms of the F ( R , G ) function it is shown how the construction of a cosmological solution is carried out via the classical Noether symmetry approach that includes a functional boundary term. After choosing the form of the F ( R , G ) function such as the case ( i ) : F ( R , G ) = f 0 R n + g 0 G m and the case ( i i ) : F ( R , G ) = f 0 R n G m , where n and m are real numbers, we explicitly compute the Noether symmetries in the vacuum and the non-vacuum cases if symmetries exist. The first integrals for the obtained Noether symmetries allow to find out exact solutions for the cosmological scale factor in the cases (i) and (ii). We find several new specific cosmological scale factors in the presence of the first integrals. It is shown that the existence of the Noether symmetries with a functional boundary term is a criterion to select some suitable forms of F ( R , G ) . In the non-vacuum case, we also obtain some extra Noether symmetries admitting the equation of state parameters w ≡ p / ρ such as w = − 1 , − 2 / 3 , 0 , 1 etc.

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“…First, one can take a strict Noether symmetry approach [4][5][6], which results in £ X L = 0, where £ X is the Lie derivative operator along X. On the other hand, one can employ the Noether symmetry approach with a gauge term [7][8][9][10][11][12][13], a generalization of the strict Noether symmetry approach where the Noether symmetry equation includes the gauge term. Noether symmetries with a gauge term are equally valuable in addressing a variety of problems in physics and applied mathematics.…”

Section: Introductionmentioning

confidence: 99%

Noether Symmetry Analysis of the Klein–Gordon and Wave Equations in Bianchi I Spacetime

Camci

2024

Symmetry

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We investigate the Noether symmetries of the Klein–Gordon Lagrangian for Bianchi I spacetime. This is accomplished using a set of new Noether symmetry relations for the Klein–Gordon Lagrangian of Bianchi I spacetime, which reduces to the wave equation in a special case. A detailed Noether symmetry analysis of the Klein–Gordon and the wave equations for Bianchi I spacetime is presented, and the corresponding conservation laws are derived.

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Invariances and Conservation Laws Based on Some FRW Universes

Cited by 11 publications

References 15 publications

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

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

F(R,G) Cosmology through Noether Symmetry Approach

Noether Symmetry Analysis of the Klein–Gordon and Wave Equations in Bianchi I Spacetime

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