In the realm of astrophysics, black holes exist within nonvacuum cosmological backgrounds, making it crucial to investigate how these backgrounds influence the properties of black holes. In this work, we first introduce a novel static spherically-symmetric exact solution of Einstein field equations representing a surrounded hairy black hole. This solution represents a generalization of the hairy Schwarzschild solution recently derived using the extended gravitational decoupling method. Then, we discuss how the new induced modification terms attributed to the primary hairs and various background fields affect the geodesic motion in comparison to the conventional Schwarzschild case. Although these modifications may appear insignificant in most cases, we identify specific conditions where they can be comparable to the Schwarzschild case for some particular background fields.

The stability of Einstein static universe against homogeneous scalar perturbations in the context of braneworld scenario is investigated. The stability regions are obtained in terms of the constant geometric linear equation of state parameter ωextr=pextr/ρextr and are studied for each evolutionary era of the universe. The results are discussed for the case of closed, open or flat universe in each era under the obtained restricting conditions. We also briefly investigate the stability against vector and tensor perturbations. Contrary to the classical general relativity, it is found that a stable Einstein static universe may exist in a braneworld theory of gravity against scalar, vector and tensor perturbations for some suitable values and ranges of the cosmological parameters

To understand the nature of a black hole shadow in dynamical spacetimes, we construct an analytical model of a dynamical photon sphere in the context of the Bonnor–Vaidya spacetime. Comparing the resulting photon sphere radius with the one in Vaidya spacetime, we find that the charge always decreases the radius of the photon sphere. We also prove that a naked singularity in Bonnor–Vaidya spacetime, unlike the static Reissner–Nordstrom naked singularity, may cast a shadow, and as a result, it cannot be distinguished from a black hole through its shadow.

We prove that for the Friedmann–Lemaitre–Robertson–Walker metric, the field equations of any generic gravity theory in arbitrary dimensions are of the perfect fluid type. The cases of general Lovelock and $${\mathcal {F}}(R, {\mathcal {G}})$$F(R,G) theories are given as examples.

In the present work, we generalize our previous work (Heydarzade and Darabi in arXiv:1710.04485, 2018on the surrounded Vaidya solution by cosmological fields to the case of Bonnor–Vaidya charged...