We investigate the extension of isotropic interior solutions for static self-gravitating systems to include the effects of anisotropic spherically symmetric gravitational sources by means of the gravitational decoupling realised via the minimal geometric deformation approach. In particular, the matching conditions at the surface of the star with the outer Schwarzschild space-time are studied in great detail, and we describe how to generate, from a single physically acceptable isotropic solution, new families of anisotropic solutions whose physical acceptability is also inherited from their isotropic parent.

We investigate how a spherically symmetric fluid modifies the Schwarzschild vacuum solution when there is no exchange of energy-momentum between the fluid and the central source of the Schwarzschild metric. This system is described by means of the gravitational decoupling realised via the minimal geometric deformation approach, which allows us to prove that the fluid must be anisotropic. Several cases are then explicitly shown. recent direct observation of black holes through the detection of gravitational waves, which opens a new and promising era for gravitational physics [1,2].It is well known that general relativity predicts surprisingly simple solutions for black holes, characterised at most by three fundamental parameters, namely the mass M , angular momentum J and charge Q [3]. The original no-hair conjecture states that these solutions should not carry any other charges [4]. Therefore, as the observations of systems containing black holes improve, the degree of consistency of these observations with the predictions determined according to the general relativistic solutions (with parameters M , J and Q) will result in a direct test of the validity of general relativity in the strong field regime. There could in fact exist other charges associated with inner gauge symmetries (and fields), and it is now known that black holes could have (soft) quantum hair [5]. The existence of new fundamental fields, which leave an imprint on the structure of the black hole, thus leading to hairy black hole solutions, is precisely the scenario under study in this paper.Possible conditions for circumventing the no-go theorem have been investigated for a long time in different scenarios (see for some recent works and Refs. [16][17][18][19][20][21] for earlier works). In particular, a fundamental scalar field φ has been considered with great interest (see Ref.[22] and references therein). In this work, we will take a different and more general approach than most of the investigations carried out so far and, instead of considering specific fundamental fields to generate hair in black hole solutions, we shall just assume the presence of an additional completely generic source described by a conserved energy-momentum tensor θ µν . Of course, this θ µν could account for one or more fundamental fields, but the crucial property is that it gravitates but does not interact directly with the matter that sources the (hairless) black hole solutions we start from. This feature may seem fanciful, but can be fully justified, for instance, in the context of the dark matter. Achieving this level of generality in the classical scheme represented by general relativity is a non-trivial task, and the gravitational decoupling by Minimal Geometric Deformation (MGDdecoupling, henceforth) is precisely the method that was developed for this purpose in Ref. [23].The MGD approach was originally proposed [24,25] in the context of the brane-world [26, 27] and extended to investigate new black hole solutions in Refs. [28,29] (for some earlier works on the...

In the context of the Randall-Sundrum braneworld, the minimal geometric deformation approach (MGD) is used to generate a new physically acceptable interior solution to Einstein's field equations for a spherically symmetric compact distribution. This new solution is used to elucidate the role of exterior Weyl stresses from bulk gravitons on compact stellar distributions. We found strong evidences showing that the exterior dark radiation U + always increases both the pressure and the compactness of stellar structures, and that the exterior "dark pressure" P + always reduces them. 1 theory like never before, but also leaves GR as the only reliable gravitational theory to be used in the analysis of phenomena occurred in the strong field regime. Likewise, the ability of observing increasingly distant objects deep in the universe, and thus with a great gravitational red shift, leads inevitably to the conclusion that only using GR we can obtain an adequate analysis of these phenomena [3], [4]. Furthemore, with the recent results shown by PLANCK [5], which improve greatly the previous by WMAP [6], we can assure that the cosmological models based in GR enjoys a well-deserved and well-earned prestige.Despite the above facts, there are some fundamental questions associated to the gravitational interaction which GR cannot answer satisfatory. This can be broadly grouped in two fundamental issues, which most likely are closely related: 1) The inability of GR to explain satisfactorily the dark matter [7] and dark energy problem without the need of introducing some kind of unknown matter-energy to reconcile what predicts GR with the observed, namely, galactic rotation curves and accelerated expansion of the universe. 2) The impossibility, so far, to reconcile GR with the Standard Model of particle physics, or equivalently, the inability to quantize GR. This has strongly motivated the searching of a gravitational theory beyond GR that helps to explain satisfactorily part of the problems described above. If the new theory is a consistent quantum theory, this should lead to a generalization of GR at low energy, being likely this extension of GR at low energy which could accounts for the dark matter and dark energy problems. If the new theory is not a consistent quantum theory for gravity, this should also contain GR in a suitable limit, and somehow show greater tolerance to its quantum description.Extra-dimensional theories, which are mostly inspired by String/M-theory, are among the theories that lead to modifications to GR. One of these extra-dimensional theories is the Braneworld (BW) proposed by Randall and Sundrum (RS) [8] which has been largely studied and which explains, so pretty straightforward, one of the fundamental problems of Physics, i.e. the hierarchy problem (see also the ADD model [9] and [10]). This theory reduces the fundamental scale to the weak scale by considering extra-dimensional effects, thus explaining the weakness of gravity relative to the other forces. Because of this, its study and impact on GR is full...

Black holes with hair represented by generic fields surrounding the central source of the vacuum Schwarzschild metric are examined under the minimal set of requirements consisting of i) the existence of a well defined event horizon and ii) the strong or dominant energy condition for the hair outside the horizon. We develop our analysis by means of the gravitational decoupling approach. We find that trivial deformations of the seed Schwarzschild vacuum preserve the energy conditions and provide a new mechanism to evade the no-hair theorem based on a primary hair associated with the charge generating these transformations. Under the above conditions i) and ii), this charge consistently increases the entropy from the minimum value given by the Schwarzschild geometry. As a direct application, we find a non-trivial extension of the Reissner-Nordström black hole showing a surprisingly simple horizon. Finally, the non-linear electrodynamics generating this new solution is fully specified.

We consider a Hořava theory that has a consistent structure of constraints and propagates two physical degrees of freedom. The Lagrangian includes the terms of Blas, Pujolàs, and Sibiryakov. The theory can be obtained from the general Horava's formulation by setting λ = 1/3. This value of λ is protected in the quantum formulation of the theory by the presence of a constraint. The theory has two second-class constraints that are absent for other values of λ. They remove the extra scalar mode. There is no strong-coupling problem in this theory since there is no extra mode. We perform explicit computations on a model that put together a z = 1 term and the IR effective action. We also show that the lowest-order perturbative version of the IR effective theory has a dynamics identical to the one of linearized general relativity. Therefore, this theory is smoothly recovered at the deepest IR without discontinuities in the physical degrees of freedom.