Higher-dimensional charged black holes in f(R)-gravity's rainbow surrounded by cloud of strings: Exact solution, shadow, and effective potential barrier

Abstract

We construct a new class of spherically symmetric black hole solutions in f(R)-gravity's rainbow1 framework which is surrounded by string cloud configuration. In the thermodynamic analysis, we show that the black hole's temperature grows up for such small values of b parameter and also big values of it, graphically. In addition to the mass and charge of the black holes, we show that the presence of the rainbow function gr(ε) and cloud's strength b can also affect the size of the shadow. For the black hole solution, it is shown that the shadow size increases with the parameter b, but decreases with the parameter gr(ε). In addition, we have shown that the energy emission rate decreases with increase in gr(ε) and decrease in b parameter. We have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. The most important result derived from this study is that the height of this potential increases with decrease in gr(ε) parameter and increases in the coupling constant λ and the b parameter. We also investigate the impact of these parameters on the other thermodynamical quantities like temperature and heat capacity. There is a minimum on that between for one special value of b. From the temperature diagram versus the b parameter (T−b) and analogy with T−r+ diagrams, we can say that the b parameter behaves as the event horizon radius r+. Regarding the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we investigate the critical behavior of our black hole solutions. In d=5, the P-V diagrams are more complex than that of the standard Van der Waals. These diagrams behave like the Born–Infeld-AdS P-V diagrams. When d=6, the P-V diagrams are also more complex than that of the standard Van der Waals and we decompose them into two parts. One part behaves like the standard Van der Waals system. The other part is similar to the Schwarzschild-AdS black hole P-V diagrams where the isotherms turn and enter a region with negative pressures. The null geodesic equations are computed in d=5 spacetime dimensions by using the concept of symmetries and Hamilton-Jacobi equation and Carter separable method. With the null geodesics in hand, we then evaluate the celestial coordinates (x, y) and the radius Rs of the black hole shadow and represent it graphically.