Regularity condition on the anisotropy induced by gravitational decoupling in the framework of MGD
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Regular Article - Theoretical Physics
Regularity condition on the anisotropy induced by gravitational decoupling in the framework of MGD G. Abellán1, V. A. Torres-Sánchez2 , E. Fuenmayor1, E. Contreras3,a 1
Grupo de Campos y Partículas, Facultad de Ciencias, Universidad Central de Venezuela, AP 47270, Caracas 1041, Venezuela School of Physical Sciences and Nanotechnology, Yachay Tech University, 100119 Urcuquí, Ecuador 3 Departamento de Física, Colegio de Ciencias e Ingeniería, Universidad San Francisco de Quito, Quito, Ecuador
2
Received: 28 January 2020 / Accepted: 14 February 2020 © The Author(s) 2020
Abstract We use gravitational decoupling to establish a connection between the minimal geometric deformation approach and the standard method for obtaining anisotropic fluid solutions. Motivated by the relations that appear in the framework of minimal geometric deformation, we give an anisotropy factor that allows us to solve the quasi–Einstein equations associated to the decoupling sector. We illustrate this by building an anisotropic extension of the well known Tolman IV solution, providing in this way an exact and physically acceptable solution that represents the behavior of compact objects. We show that, in this way, it is not necessary to use the usual mimic constraint conditions. Our solution is free from physical and geometrical singularities, as expected. We have presented the main physical characteristics of our solution both analytically and graphically and verified the viability of the solution obtained by studying the usual criteria of physical acceptability.
1 Introduction
pr = −(ρ + pr )
In 1916, Karl Schwarzschild obtained the first interior solution of the Einstein field equations [1]. This solutions describe a self-gravitating object sustained by a perfect and incompressible fluid which is embedded in a static and spherically symmetric vacuum space-time. Following the strategy of Schwarzschild, other interior solutions can be constructed providing suitable equations of state to close the system. However, in some cases the system obtained can not be analytically integrated and numerical models are required. Besides proposing an equation of state to relate thermodynamical quantities, we can use geometrical constraints on the functions. Indeed, following this program Tolman obtained a family of eight isotropic solutions [2]. a e-mail:
For many years isotropic solutions were considered as well posed models to study stellar interiors. However, as was shown by Delgaty and Lake [3], very few of this solutions can be considered physically acceptable (for a list of physical conditions of interior solutions see, for example, [4]). Nevertheless, even when acceptable solutions can be found the perfect fluid model is evidently not valid when local anisotropy of pressure is assumed. Regardingly, anisotropic models have been considered as very reasonable for describing the matter distribution under a variety of circumstances [5–21]. Now, as it is well known, assumption of local anisotropy in the fluid
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