In this paper we present a Bardeen solution surrounded by a cloud of strings fluid. We show how this model has the same event horizon characteristic as the Bardeen solution, however, the parameter of the strings make the solution singular at the origin. We also analyze the solution from a thermodynamic point of view. We calculate the system's state function, enthalpy; the temperature; and the other potentials; as a function of entropy. By analyzing the thermodynamic coefficients, we show that the solution presents three distinct phases, two of which are stable and one unstable. These three phases could be best visualized using a plot of Gibbs free energy versus temperature. In the end, we calculate the critical exponents and find that they are the same as those found in Van der Walls theory.
In this paper we propose to use the Shapiro time delay as a tool to distinguish between different black hole solutions. We calculate the analytic Shapiro time using first order expansions for four solutions. These are Schwarzschild, Reissner–Nordström, Bardeen, and Ayón-Beato and García. We created a numerical experiment based on measurements in the solar system consisting of the outward and return paths of light traversing a black hole at the center. We obtained different delay times on the order of $$10^{-4}~\hbox {s}$$ 10 - 4 s and $$10^{-6}~\hbox {s}$$ 10 - 6 s for a stellar black hole; and variations on the order of hours for a supermassive hole. Considering that the accuracies currently achieved in solar system measurements are on the order of $$10^{-12}~\hbox {s}$$ 10 - 12 s , we believe that this mechanism could be used in the determination of black hole models in the near future.
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