The main goal of this paper is to get in a straightforward form the field equations in metric f (R) gravity, using elementary variational principles and adding a boundary term in the action, instead of the usual treatment in an equivalent scalar-tensor approach. We start with a brief review of the Einstein-Hilbert action, together with the Gibbons-York-Hawking boundary term, which is mentioned in some literature, but is generally missing. Next we present in detail the field equations in metric f (R) gravity, including the discussion about boundaries, and we compare with the Gibbons-York-Hawking term in General Relativity. We notice that this boundary term is necessary in order to have a well defined extremal action principle under metric variation.

The relationship between the emission of coronal lines (CLs) and nuclear activity in 36 Type 1 and 18 Type 2 active galactic nuclei (AGNs) is analyzed, for the first time, based on near-infrared (0.8-2.4 μm) spectra. The eight CLs studied, of Si, S, Fe, Al, and Ca elements and corresponding to ionization potentials (IPs) in the range 125-450 eV, are detected (3σ ) . We found that the non-detection is largely associated with either loss of spatial resolution or increasing object distance: CLs are essentially nuclear and easily lose contrast in the continuum stellar light for nearby sources or get diluted by the strong AGN continuum as the redshift increases. Yet, there are AGNs where the lack of coronal emission, i.e., lines with IP 100 eV, may be genuine. The absence of these lines reflects a non-standard AGN ionizing continuum, namely, a very hard spectrum lacking photons below a few Kev. The analysis of the line profiles points out a trend of increasing FWHM with increasing IPs up to energies around 300 eV, where a maximum in the FWHM is reached. For higher IP lines, the FWHM remains nearly constant or decreases with increasing IPs. We ascribe this effect to an increasing density environment as we approach the innermost regions of these AGNs, where densities above the critical density of the CLs with IPs larger than 300 eV are reached. This sets a strict range limit for the density in the boundary region between the narrow and the broad region of 10 8 -10 9 cm −3 . A relationship between the luminosity of the CLs and that of the soft and hard X-ray emission and the soft X-ray photon index is observed: the coronal emission becomes stronger with both increasing X-ray emission (soft and hard) and steeper X-ray photon index, i.e., softer X-ray spectra. Thus, photoionization appears as the dominant excitation mechanism. These trends hold when considering Type 1 sources only; they get weaker or vanish when including Type 2 sources, very likely because the X-ray emission measured in the latter is not the intrinsic ionizing continuum.

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We find a new rotating black hole in three-dimensional anti-de Sitter space using an anisotropic perfect fluid inspired by the noncommutative black hole. We deduce the thermodynamical quantities of this black hole and compare them with those of a rotating BTZ solution.

In this paper we study the Geodesic Deviation Equation (GDE) in metric f (R) gravity. We start giving a brief introduction of the GDE in General Relativity in the case of the standard cosmology. Next we generalize the GDE for metric f (R) gravity using again the FLRW metric. A generalization of the Mattig relation is also obtained. Finally we give and equivalent expression to the Dyer-Roeder equation in General Relativity in the context of f (R) gravity.

The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute the energy momentum tensor to second order in perturbation theory in terms of gauge invariant quantities. Two posible scenarios are discussed, first we consider a FLRW background without magnetic field and we study the perturbation theory introducing the magnetic field as a perturbation. The second scenario, we consider a magnetized FLRW and build up the perturbation theory from this background. We compare the cosmological dynamo like equation in both scenarios. PACS numbers: 98.80.-k, 95.30.Qd.

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We present the Geodesic Deviation Equation (GDE) for the Friedmann-Robertson Walker(FRW) universe and we compare it with the equation for Bianchi type I model. We justify consider this cosmological model due to the recent importance the Bianchi Models have as alternative models in cosmology. The main property of these models, solutions of Einstein Field Equations (EFE) is that they are homogeneous as the FRW model but they are not isotropic. We can see this because they have a non-null Weyl tensor, which is zero for FRW model. We study some consequences of this Weyl tensor in the GDE.