Lie and Noether symmetries of geodesic equations and collineations

Tsamparlis, Michael; Paliathanasis, Andronikos

doi:10.1007/s10714-010-1054-9

Cited by 97 publications

(116 citation statements)

References 17 publications

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“…Let us now apply the Noether Symmetry Approach [536] (see also [537]) to a general class of f (T ) gravity models where the corresponding Lagrangian of the field equations is given by (641). We start by performing the analysis for arbitrary space-times, and then we focus on static spherically-symmetric geometries.…”

Section: Spherically Symmetric Solutions By Noether Symmetry Approachmentioning

confidence: 99%

f(T) teleparallel gravity and cosmology

et al. 2016

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Over the past decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T ) gravity, where f (T ) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T ) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T ) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T ) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations, or alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T ) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T ) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to enlighten the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications. Contents

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Section: Spherically Symmetric Solutions By Noether Symmetry Approachmentioning

confidence: 99%

f(T) teleparallel gravity and cosmology

et al. 2016

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“…In particular, the scope of the current article is (a) to investigate which of the available f (R) models admit extra Lie and Noether point symmetries, and (b) for these models to solve the system of the resulting field equations and derive analytically (for the first time to our knowledge) the main cosmological functions (the scale factor, the Hubble expansion rate etc.). We would like to remind the reader that a fundamental approach to derive the Lie and Noether point symmetries for a given dynamical problem living in a Riemannian space has been published recently by Tsamparlis & Paliathanasis [36] (a similar analysis can be found in [49][50][51][52][53][54][55]). …”

Section: Introductionmentioning

confidence: 99%

Constraints and analytical solutions off(R)theories of gravity using Noether symmetries

2011

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We perform a detailed study of the modified gravity f (R) models in the light of the basic geometrical symmetries, namely Lie and Noether point symmetries, which serve to illustrate the phenomenological viability of the modified gravity paradigm as a serious alternative to the traditional scalar field approaches. In particular, we utilize a model-independent selection rule based on first integrals, due to Noether symmetries of the equations of motion, in order to identify the viability of f (R) models in the context of flat FLRW cosmologies. The Lie/Noether point symmetries are computed for six modified gravity models that include also a cold dark matter component. As it is expected, we confirm that all the proposed modified gravity models admit the trivial first integral namely energy conservation. We find that only the f (R) = (R b − 2Λ) c model, which generalizes the concordance Λ cosmology, accommodates extra Lie/Noether point symmetries. For this f (R) model the existence of non-trivial Noether (first) integrals can be used to determine the integrability of the model. Indeed within this context we solve the problem analytically and thus we provide for the first time the evolution of the main cosmological functions such as the scale factor of the universe and the Hubble expansion rate.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

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“…It turns out that general plane symmetric space-times can admit 4, 5, 6, 7, 8, 9, 11, or 17 Noether symmetries. The maximum number of Noether symmetries (i.e., 17) appears only for the Minkowski space-time and 10 of these symmetries correspond to the generators of isometries associated with this metric.…”

Section: Resultsmentioning

confidence: 99%

“…In the past few decades, a significant amount of relativistic literature has been devoted to the study of symmetries in general relativity [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In [7], the authors considered Killing vectors (KVs) of spherically symmetric static space-times and concluded that they admit either ten KVs (corresponding to de Sitter, Minkowski, and anti-de Sitter metrics), seven KVs (corresponding to Einstein and anti-Einstein metrics), six KVs (incorporating the Bertotti-Robinson and two other metrics), or only four KVs (the minimal set of isometries).…”

Section: Introductionmentioning

confidence: 99%

“…For example, they have been computed in terms of the special projective group and homotheties (including KVFs) of the metric in [17]. Tsamparlis derives in [23] the Lie and Noether condition for the equations of motion of a dynamical system in an -dimensional Riemannian space and expresses the symmetry generating vectors in terms of the homothetic and projective collineations of the space.…”

Section: Introductionmentioning

confidence: 99%

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Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times

Jamil

Ферозе

Vargas

2017

Advances in Mathematical Physics

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The aim of this paper is to give the geometrical/physical interpretation of the conserved quantities corresponding to each Noether symmetry of the geodetic Lagrangian of plane symmetric space-times. For this purpose, we present a complete list of plane symmetric nonstatic space-times along with the generators of all Noether symmetries of the geodetic Lagrangian. Additionally, the structure constants of the associated Lie algebras, the Riemann curvature tensors, and the energy-momentum tensors are obtained for each case. It is worth mentioning that the list contains all classes of solutions that have been obtained earlier during the classification of plane symmetric space-times by isometries and homotheties.

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Lie and Noether symmetries of geodesic equations and collineations

Cited by 97 publications

References 17 publications

f(T) teleparallel gravity and cosmology

f(T) teleparallel gravity and cosmology

Constraints and analytical solutions off(R)theories of gravity using Noether symmetries

Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times

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