On integrability of the Szekeres system. I

Gierzkiewicz, Anna; Golda, Zdzislaw A.

doi:10.1080/14029251.2016.1237199

Cited by 20 publications

(11 citation statements)

References 11 publications

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“…where (3) R is the spatial curvature of the three-dimensional hypersurfaces. The latter system is known as the Szekeres-Szafron system and has been widely studied in the literature [5,[64][65][66].…”

Section: Dynamical Analysismentioning

confidence: 99%

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

2020

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We investigate the existence of inhomogeneous exact solutions in Weyl Integrable theory with a matter source. In particular, we consider the existence of a dust fluid source while for the underlying geometry we assume a line element which belongs to the family of silent universes. We solve explicitly the field equations and we find the Szekeres spacetimes in Weyl Integrable theory. We show that only the isotropic family can describe inhomogeneous solutions where the LTB spacetimes are included. A detailed analysis of the dynamics of the field equations is given where the past and future attractors are determined. It is interesting that the Kasner spacetimes can be seen as past attractors for the gravitation models, while the unique future attractor describes the Milne universe similar with the behaviour of the gravitational model in the case of General Relativity.

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Section: Dynamical Analysismentioning

confidence: 99%

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

2020

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“…In order to prove that there actually exist solutions, the integrability of the system should be established [49,50,51,52,53,54,55]. We first study the integrability of static spherically symmetric spacetimes in Einstein-aether gravity with a matter source of the form of a perfect fluid with EoS µ = µ 0 + (h − 1) p, h > 1.…”

Section: Integrability and Modified Tolman-oppenheimer-volkoff Formulmentioning

confidence: 99%

Static spherically symmetric Einstein-æther models II: Integrability and the modified Tolman–Oppenheimer–Volkoff approach

León¹,

Coley²,

Paliathanasis³

2020

Annals of Physics

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We investigate the existence of analytic solutions for the field equations in the Einsteinaether theory for a static spherically symmetric spacetime and provide a detailed dynamical system analysis of the field equations. In particular, we investigate if the gravitational field equations in the Einstein-aether model in the static spherically symmetric spacetime possesses the Painlevè property, so that an analytic explicit integration can be performed. We find that analytic solutions can be presented in terms of Laurent expansion only when the matter source consists of a perfect fluid with linear equation of state (EoS) µ = µ 0 + (h − 1) p, h > 1. In order to study the field equations we apply the Tolman-Oppenheimer-Volkoff (TOV) approach and other approaches. We find that the relativistic TOV equations are drastically modified in Einstein-aether theory, and we explore the physical implications of this modification. We study perfect fluid models with a scalar field with an exponential potential. We discuss all of the equilibrium points and discuss their physical properties. Contents

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“…Because of its importance in physical applications, the Szekeres system has been widely investigated in the literature. With the use of Darboux polynomials and Jacobi's last multiplier method, the integrability of the Szekeres system with zero cosmological constant terms was found in [18]. Specifically, three time-independent conservation laws where derived which were used to reduce the four-dimensional Szekeres system into a two-dimensional system.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics and evolution for the Szekeres system with nonzero cosmological constant term

Paliathanasis¹,

León²

2021

Preprint

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The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in R 4 . In this study, we investigate the behavior of trajectories in the presence of cosmological constant. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, h and I 0 which indicate the integrability of the Hamiltonian system. We solve the Hamilton-Jacobi equation, and we reduce the Szekeres system from R 4 to an equivalent system defined in R 2 . Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and I 0 < 2.08. Otherwise, trajectories reach infinity. For I 0 > 0 the origin of trajectories in R 2 is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in R 5 . We see that the attractor at the finite regime in R 5 is related with the de Sitter universe for a positive cosmological constant.

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On integrability of the Szekeres system. I

Cited by 20 publications

References 11 publications

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

Static spherically symmetric Einstein-æther models II: Integrability and the modified Tolman–Oppenheimer–Volkoff approach

Global dynamics and evolution for the Szekeres system with nonzero cosmological constant term

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