Revisiting black holes surrounded by cloud and fluid of strings in general relativity

This paper revisits black hole solutions surrounded by clouds and fluids of strings within the framework of general relativity. We introduce a generalized equation of state for a fluid of strings with a variable parameter and derive a general solution to Einstein field equations for this system. We allow the parameter $\alpha$ in the equation of state for a fluid of strings ($\rho/p = \alpha$) to vary as a function of the radial coordinate. A particular solution with $M=0$ is explored, focusing on positive ranges of the equation of state parameter in analogy with the reduced Kiselev solution. Furthermore, we present a novel regular black hole solution that reduces to the Schwarzschild solution with a cloud of strings when the radial coordinate is much larger than a control parameter $r_0$. As an additional contribution of this work, we show that the energy-momentum tensor for the fluid of strings can be decomposed into contributions from three components: an isotropic perfect fluid, an electromagnetic field, and a scalar field minimally coupled. The paper examines the connection between the fluid of strings and Kiselev anisotropic fluid, revealing structural similarities through their energy-momentum tensors. The study also highlights how the obtained solutions influence the horizons and thermodynamic properties of black holes. The new solutions allow for the emergence of geometries with multiple horizons and non-trivial temperature behaviors. As an additional test of the general solution, it is shown that particular choices for the function $\alpha(r)$ reproduce well-known results in the literature. Finally, we study geodesic motion of particles, geodesic completeness and shadows in the geometry of the novel regular black hole.