In this work, we obtain uncharged\charged Kiselev-like black holes as a new class of black hole solutions surrounded by perfect fluid in the context of Rastall theory. Then, we study the specific cases of the uncharged\charged black holes surrounded by regular matter like dust and radiation, or exotic matter like quintessence, cosmological constant and phantom fields. By comparing the Kiselevlike black hole solutions in Rastall theory with the Kiselev black hole solutions in GR, we find an effective perfect fluid behaviour for the black hole's surrounding field. It is shown that the corresponding effective perfect fluid has interesting characteristic features depending on the different ranges of the parameters in Rastall theory. For instance, Kiselev-like black holes surrounded by regular matter in Rastall theory may be considered as Kiselev black holes surrounded by exotic matter in GR, or Kiselev-like black holes surrounded by exotic matter in Rastall theory may be considered as Kiselev black holes surrounded by regular matter in GR.

A generalized version for the Rastall theory is proposed showing the agreement with the cosmic accelerating expansion. In this regard, a coupling between geometry and the pressureless matter fields is derived which may play the role of dark energy, responsible for the current accelerating expansion phase. Moreover, our study also shows that the radiation field may not be coupled to the geometry in a non-minimal way which represents that the ordinary energymomentum conservation law is respected by the radiation source. It is also shown that the primary inflationary era may be justified by the ability of the geometry to couple to the energy-momentum source in an empty flat FRW universe. In fact, this ability is independent of the existence of the energymomentum source and may compel the empty flat FRW universe to expand exponentially. Finally, we consider a flat FRW universe field by a spatially homogeneous scalar field evolving in potential V(φ), and study the results of applying the slow-roll approximation to the system which may lead to an inflationary phase for the universe expansion.

We profit by a recent paper of Visser claiming that Rastall gravity is equivalent to Einstein gravity to compare the two gravitational theories in a general way. Our conclusions are different from Visser's ones. We indeed argue that these two theories are not equivalent. In fact, Rastall theory of gravity is an "open" theory when compared to Einstein general theory of relativity. Thus, it is ready to accept the challenges of observational cosmology and quantum gravity.The framework of extended theories of gravity [1][2][3][4][5] is today considered an useful and popular approach to attempt to solve the important problems of the standard model of cosmology like Dark Energy [6,7] and Dark Matter [8,9]. A key point is that all of the potential alternatives to Einstein's general theory of relativity (GTR) must be viable. This implies that alternative theories must be in agreement with the Einstein's equivalence principle, which has today a strong, unchallengeable empirical evidence [2]. The fundamental consequence is that the alternatives to Einstein gravity must be metric theories [2]. Another important point is that such alternatives must pass the solar system tests. Hence, deviations from the standard GTR must be weak [1,2].Within the framework of extended gravity, one finds the theory proposed by Rastall in 1972 [10], which recently obtained a renewed interest in the literature [11][12][13][14]. This interest is due to some good behavior of the Rastall theory. In fact, on one hand it seems in good agreement with observational data on the Universe age and on the Hubble parameter [15]. On the other hand it may, in principle, provide an alternative description for the matter dominated era with respect to the GTR [16]. Observational data from the helium nucleosynthesis seem also in agreement with Rastall gravity [17]. Based on these observational evidences, there have been some recently studies of the various cosmic eras in the framework of Rastall gravity [18][19][20][21][22]. Other interesting a e-mail: cordac.galilei@gmail.com issues are that Rastall gravity seems to do not suffer from the entropy and age problems of standard cosmology [23] and is consistent with the gravitational lensing phenomena [24,25]. Further interesting works on Rastall gravity are given by [26][27][28][29][30] and references within.Differently from Einstein gravity, Rastall gravity considers a non-divergence-free energy-momentum . A different, but similar theory is the so called Curvature-matter theory of gravity [31][32][33][34][35]. In this theory the matter and geometry are coupled to each other in a non-minimal way [31][32][33][34][35]. Then, the standard energy-momentum conservation law does not work in this case too [31][32][33][34][35].In a recent paper [36], Visser claimed that Rastall gravity is equivalent to Einstein gravity. In this letter, we argue that these two theories are not equivalent instead. Let us see the key points of our interpretation.Rastall did not define a new energy-momentum tensor (EMT) in his origina...

The Reissner-Nordström black hole solution in a generic cosmological constant background in the the context of Rastall gravity is obtained. It is shown that the cosmological constant arises naturally from the consistency of the non-vacuum field equations of the Rastall theory for a spherical symmetric spacetime, rather than its ad-hoc introduction in the usual Einstein and Einstein-Maxwell field equations. The usual Reissner-Nordström, Schwarzschild and Schwarzschild-(anti)de Sitter black hole solutions in the framework of this theory are also addressed as the special independent subclasses of the obtained general solution.

In the present work, we study the general surrounded Vaidya solution by the various cosmological fields and its nature describing the possibility of the formation of naked singularities or black holes. Motivated by the fact that real astrophysical black holes as non-stationary and nonisolated objects are living in non-empty backgrounds, we focus on the black hole subclasses of this general solution describing a dynamical evaporating-accreting black holes in the dynamical cosmological backgrounds of dust, radiation, quintessence, cosmological constant-like and phantom fields, the so called "surrounded Vaidya black hole". Then, we analyze the timelike geodesics associated with the obtained surrounded black holes and we find that some new correction terms arise relative to the case of Schwarzschild black hole. Also, we address some of the subclasses of the obtained surrounded black hole solution for both dynamical and stationary limits. Moreover, we classify the obtained solutions according to their behaviors under imposing the positive energy condition and discuss how this condition imposes some severe and important restrictions on the black hole and its background field dynamics.

There are a number of reasons to study wormholes with generic cosmological constant Λ. Recent observations indicate that present accelerating expansion of the universe demands Λ > 0. On the other hand, some extended theories of gravitation such as supergravity and superstring theories posses vacuum states with Λ < 0. Even within the framework of general relativity, a negative cosmological constant permits black holes with horizons topologically different from the usual spherical ones. These solutions are convertible to wormhole solutions by adding some exotic matter. In this paper, the asymptotically flat wormhole solutions in a generic cosmological constant background are studied. By constructing a specific class of shape functions, mass function, energy density and pressure profiles which support such a geometry are obtained. It is shown that for having such a geometry, the wormhole throat r0, the cosmological constant Λ and the equation of state parameter ω should satisfy two specific conditions. The possibility of setting different values for the parameters of the model helps us to find exact solutions for the metric functions, mass functions and energymomentum profiles. At last, the volume integral quantifier, which provides useful information about the total amount of energy condition violating matter is discussed briefly.

The Bousso's D-bound entropy for the various possible black hole solutions on a 4-dimensional brane is checked. It is found that the D-bound entropy here is apparently different from that of obtained for the 4-dimensional black hole solutions. This difference is interpreted as the extra loss of information, associated to the extra dimension, when an extra-dimensional black hole is moved outward the observer's cosmological horizon. Also, it is discussed that N -bound entropy is hold for the possible solutions here. Finally, by adopting the recent Bohr-like approach to black hole quantum physics for the excited black holes, the obtained results are written also in terms of the black hole excited states. *

In this paper, we will analyze the energy dependent deformation of massive gravity using the formalism of massive gravity's rainbow. So, we will use the Vainshtein mechanism and the dRGT mechanism for the energy dependent massive gravity, and thus analyze a ghost free theory of massive gravity's rainbow. We study the energy dependence of a time-dependent geometry, by analyzing the radiating Vaidya solution in this theory of massive gravity's rainbow. The energy dependent deformation of this Vaidya metric will be performed using suitable rainbow functions. *