We obtain a class of regular black hole solutions in four-dimensional $f(R)$ gravity, $R$ being the curvature scalar, coupled to a nonlinear electromagnetic source. The metric formalism is used and static spherically symmetric spacetimes are assumed. The resulting $f(R)$ and nonlinear electrodynamics functions are characterized by a one-parameter family of solutions which are generalizations of known regular black holes in general relativity coupled to nonlinear electrodynamics. The related regular black holes of general relativity are recovered when the free parameter vanishes, in which case one has $f(R)\propto R$. We analyze the regularity of the solutions and also show that there are particular solutions that violate only the strong energy condition.

In this paper, we determine regular black hole solutions using a very general f(R) theory, coupled to a non-linear electromagnetic field given by a Lagrang

In this paper, we investigate the gravitational lensing effect for the Schwarzschild-like black hole space-time in the background of a Kalb-Ramond (KR) field proposed by Yang et al. [ Phys. Rev. D 108, 124004 (2023)]. The solution is characterized by a single extra parameter $l$, which is associated to the Lorentz symmetry breaking induced by the KR field. First, we calculate the exact deflection angle of massive and massless particles for finite distances using elliptic integrals. Then we study this effect in the weak and strong field regimes, discussing the correction of the KR parameter on the coefficients of the expansions in both limits. We also find that increasing $l$ decreases the deflection angle. Furthermore, we use the available data from the Sagittarius $A^{\star }$ object, which is believed to be a supermassive black hole at the center of our Galaxy, to calculate relevant observables, such as the image position, luminosity, and delay time. The values found could be potentially measured in the weak field regime, though for strong fields one would have to wait for the next generation of interferometers.

We reconstruct the ΛCDM model for f ( T ,  ) theory, where T is the torsion scalar and  the trace of the energy-momentum tensor. The result shows that the action of ΛCDM is a combination of a linear term, a constant ( − 2 Λ ) and a nonlinear term given by the product − T F g [ ( T 1 / 3 / 16 π G ) ( 16 π G  + T + 8 Λ ) ] , with F _g being a generic function. We show that to maintain conservation of the energy-momentum tensor, we should impose that F g [ y ] must be linear on the trace  . This reconstruction decays in f ( T ) theory for F g ≡ Q , with Q a constant. Our reconstruction describes the cosmological eras to the present time. The model present stability within the geometric and matter perturbations for the choice F g = y , where y = ( T 1 / 3 / 16 π G ) ( 16 π G  + T + 8 Λ ) , except for the geometric part in the de Sitter model. We impose the first and second laws of thermodynamics to ΛCDM and find the condition where they are satisfied, that is, T A , G eff > 0 , however where this is not possible in the cases that we choose, this leads to a breakdown of positive entropy and Misner–Sharp energy.