In this work, we calculate the deflection angle of light in a spacetime that interpolates between regular black holes and traversable wormholes, depending on the free parameter of the metric. Afterwards, this angular deflection is substituted into the lens equations which allows us to obtain physically measurable results, such as the position of the relativistic images and the magnifications.

We consider a metric-affine extension to the gravitational sector of the Standard Model extension for the Lorentz-violating coefficients $u$ and $s^{\mu \nu }$. The general results, which are applied to a specific model called metric-affine bumblebee gravity, are obtained. A Schwarzschild-like solution, incorporating effects of the Lorentz symmetry breaking through the coefficient $X=\xi b^2$, is found. Furthermore, a complete study of the geodesic trajectories of particles is accomplished in this background, emphasizing the departure from general relativity. We also compute the advance of Mercury’s perihelion and the deflection of light within the context of the weak-field approximation, and we verify that there exist two new contributions ascribed to the Lorentz symmetry breaking. As a phenomenological application, we compare our theoretical results with observational data in order to estimate the coefficient $X$.

Within the framework of the spontaneous Lorentz symmetry breaking (LSB), we consider a metric-affine generalization of the gravitational sector of the Standard Model Extension (SME), including the Lorentz-violating (LV) coefficients u and s^μν . In this model, we derive the modified Einstein field equations in order to obtain a new axisymmetric vacuum spinning solution for a particular bumblebee's profile. Such a solution has the remarkable property of incorporating the effects of LSB through the LV dimensionless parameter X = ξb ², with ξ is the nonminimal coupling constant, and b ² = b^μb_μ , with b_μ is the vacuum expectation value of the bumblebee field; as the LSB is turned off, X = 0, we recover the well-established result, the Kerr solution, as expected. Afterwards, we calculate the geodesics, the radial acceleration and thermodynamic quantities for this new metric. We also estimate an upper bound for X by using astrophysical data of the advance of Mercury's perihelion.

We consider the space-time metric generated by a global monopole in an extension of general relativity (GR) of the form $f(\mathcal{R})=\mathcal{R}-\lambda {\mathcal{R}}^2$. The theory is formulated in the metric-affine (or Palatini) formalism, and exact analytical solutions are obtained. For $\lambda <0$, one finds that the solution has the same characteristics as the Schwarzschild black hole with a monopole charge in Einstein’s GR. For $\lambda >0$, instead, the metric is more closely related to the Reissner-Nordström metric with a monopole charge and, in addition, it possesses a wormhole-like structure that allows for the geodesic completeness of the space-time. Our solution recovers the expected limits when $\lambda =0$ and also at the asymptotic far limit. The angular deflection of light in this space-time in the weak field regime is also calculated.

We consider the metric-affine formulation of bumblebee gravity, derive the field equations, and show that the connection can be written as Levi-Civita of a disformally related metric in which the bumblebee field determines the disformal part. As a consequence, the bumblebee field gets coupled to all the other matter fields present in the theory, potentially leading to nontrivial phenomenological effects. To explore this issue we compute the post-Minkowskian, weak-field limit and study the resulting effective theory. In this scenario, we couple scalar and spinorial matter to the effective metric, and then we explore the physical properties of the VEV of the bumblebee field, focusing mainly on the dispersion relations and the stability of the resulting effective theory.