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Register a new userUsing MGD gravitational decoupling to extend the isotropic solutions of Einstein equations to the anisotropical domain
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The aim of this work is to obtain new analitical solutions for Einstein equations in the anisotropical domain. This will be done via the minimal geometric deformation (MGD) approach, which is a simple and systematical method that allow us to decouple the Einstein equations. It requires a perfect fluid known solution that we will choose to be Finch-Skeas(FS) solution. Two different constraints were applied, and in each case we found an interval of values for the free parameters, where necesarly other physical solutions shall live.
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We implement the Gravitational Decoupling through the Minimal Geometric Deformation method and explore its effect on exterior solutions by imposing a regularity condition in the Tolman--Oppenheimer--Volkoff equation of the decoupling sector. We obtain that the decoupling function can be expressed formally in terms of an integral involving the $g_{tt}$ component of the metric of the seed solution. As a particular example, we implement the method by using the Schwarzschild exterior as a seed and we obtain that the asymptotic behavior of the extended geometry corresponds to a manifold with constant curvature.
The Einstein-Maxwell (E-M) equations in a curved spacetime that admits at least one Killing vector are derived, from a Lagrangian density adapted to symmetries. In this context, an auxiliary space of potentials is introduced, in which, the set of potentials associated to an original (seed) solution of the E-M equations are transformed to a new set, either by continuous transformations or by discrete transformations. In this article, continuous transformations are considered. Accordingly, originating from the so-called $gamma_A$-metric, other exact solutions to the E-M equations are recovered and discussed.
Exact solutions to the Einstein field equations may be generated from already existing ones (seed solutions), that admit at least one Killing vector. In this framework, a space of potentials is introduced. By the use of symmetries in this space, the set of potentials associated to a known solution are transformed into a new set, either by continuous transformations or by discrete transformations. In view of this method, and upon consideration of continuous transformations, we arrive at some exact, stationary axisymmetric solutions to the Einstein field equations in vacuum, that may be of geometrical or/and physical interest.
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 decoupler 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.
We extend the monumental result of Christodoulou-Klainerman on the global nonlinear stability of the Minkowski spacetime to the global nonlinear stability of a class of large dispersive spacetimes. More precisely, we show that any regular future causally geodesically complete, asymptotically flat solution to the Einstein-scalar field system which approaches the Minkowski spacetime sufficiently fast for large times is future globally nonlinearly stable. Combining our main theorem with results of Luk-Oh, Luk-Oh-Yang and Kilgore, we prove that a class of large data spherically symmetric dispersive solutions to the Einstein-scalar field system are globally nonlinearly stable with respect to small non-spherically symmetric perturbations. This in particular gives the first construction of an open set of large asymptotically flat initial data for which the solutions to the Einstein-scalar field system are future causally geodesically complete.
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