In this paper, we study rotating black holes in symmergent gravity, and use deviations from the Kerr black hole to constrain the parameters of the symmergent gravity. Symmergent gravity induces the gravitational constantGand quadratic curvature coefficient $$c_{\textrm{O}}$$cO from the flat spacetime matter loops. In the limit in which all fields are degenerate in mass, the vacuum energy $$V_{\textrm{O}}$$VO can be wholly expressed in terms ofGand $$c_{\textrm{O}}$$cO . We parametrize deviation from this degenerate limit by a parameter $${\hat{\alpha }}$$α^ such that the black hole spacetime is dS for $${\hat{\alpha }} < 1$$α^<1 and AdS for $${\hat{\alpha }} > 1$$α^>1 . In constraining the symmergent parameters $$c_{\textrm{O}}$$cO and $${\hat{\alpha }}$$α^ , we utilize the EHT observations on the M87* and Sgr. A* black holes. We investigate first the modifications in the photon sphere and shadow size, and find significant deviations in the photonsphere radius and the shadow radius with respect to the Kerr solution. We also find that the geodesics of time-like particles are more sensitive to symmergent gravity effects than the null geodesics. Finally, we analyze the weak field limit of the deflection angle, where we use the Gauss-Bonnet theorem for taking into account the finite distance of the source and the receiver to the lensing object. Remarkably, the distance of the receiver (or source) from the lensing object greatly influences the deflection angle. Moreover, $$c_{\textrm{O}}$$cO needs be negative for a consistent solution. In our analysis, the rotating black hole acts as a particle accelerator and possesses the sensitivity to probe the symmergent gravity.

In this paper, we report on exact charged black hole solutions in symmergent gravity with Maxwell field. Symmergent gravity induces the gravitational constant G, quadratic curvature coefficient c O , and the vacuum energy V O from the flat spacetime matter loops. In the limit in which all fields are degenerate in mass, the vacuum energy V O can be expressed in terms of G and c O . We parametrize deviation from this limit by a parameter α ˆ such that the black hole spacetime is de Sitter (dS) for α ˆ < 1 and anti-de Sitter (AdS) for α ˆ > 1 . In our analysis, we study horizon formation, shadow cast and gravitational lensing as functions of the black hole charge, and find that there is an upper bound on the charge. At relatively low values of charge, applicable to astronomical black holes, we determine constraints on c O and α ˆ using the Event Horizon Telescope (EHT) data from Sgr. A* and M87*. We apply these constraints to reveal how the shadow radius behaves as the observer distance r O varies. It is revealed that black hole charge directly influences the shadow silhouette, but the symmergent parameters have a tenuous effect. We also explored the weak field regime by using the Gauss–Bonnet theorem to study the weak deflection angle caused by the M87* black hole. We have found that impact parameters comparable to the actual distance D = 16.8 Mpc show the potential detectability of such an angle through advanced astronomical telescopes. Overall, our results provide new insights into the behavior of charged black holes in the context of symmergent gravity and offer a new way to test these theories against observational data.

Extended metric-Palatini gravity, quadratic in the antisymmetric part of the affine curvature, is known to lead to the general relativity plus a geometric Proca field. The geometric Proca, equivalent of the non-metricity vector in the torsion-free affine connection, qualifies to be a distinctive signature of the affine curvature. In the present work, we explore how shadow and photon motion near black holes can be used to probe the geometric Proca field. To this end, we derive static spherically symmetric field equations of this Einstein-geometric Proca theory, and show that it admits black hole solutions in asymptotically AdS background. We perform a detailed study of the optical properties and shadow of this black hole and contrast them with the observational data by considering black hole environments with and without plasma. As a useful astrophysical application, we discuss constraints on the Proca field parameters using the observed angular size of the shadow of supermassive black holes M87 $$^*$$∗ and Sgr A $$^*$$∗ in both vacuum and plasma cases. Overall, we find that the geometric Proca can be probed via the black hole observations.