Slowly Rotating Black Holes in 4D Gauss-Bonnet Gravity

Weak gravitational lensing and shadow cast by rotating black holes in axionic Chern–Simons theory

“…The black hole shadow, a dark area that occupies the center of the bright accretion disk, is an impressive visual characteristic of a black hole. The shadow is a consequence of the black hole's intense gravitational field, which deflects and confines light, ultimately creating a photon sphere [75,76,134,138,139]. The size and shape of the shadow of a black hole depend on its mass, rotation and proximity to Earth, presenting an exceptional opportunity to scrutinize the features of black holes.…”

“…Our objective is to find the angle at which the photon hits the observer's plane perpendicular to the r-direction. To do so, we need the tangential vector at that specific point in space, which can be expressed as [138,139] U = − ṙ e r + r obs θ e θ + r obs sin θ 0 φ e φ ,…”

“…Further, we assume that r obs θ2 + sin 2 θ 0 φ2 1 in these expressions because we are considering the limit where r obs approaches infinity. By using the geodesic equations, we can relate these angles to the angular momentum, or alternatively, can express the angular momentum of the geodesic in terms of these angles, given by [138,139] j = r obs sin δ , L z = r obs sin θ 0 cos α sin δ.…”

Thermal geometries and the Joule–Thomson expansion of modified charged and slowly rotating black holes

“…Here, we explore and study the general quantities of thermodynamic geometry for the considered BH. Weinhold's and Ruppeiner's formalisms are used in order to justify the phase transition for the charged BH in 4D Gauss-Bonnet gravity [59][60][61][62][63]. Furthermore, based on the Hessian matrix of the BH mass, we introduce thermodynamic geometric methods and give its scalar curvatures (Ruppeiner and Weinhold).…”

“…In this development, we are interested in developing Schwarzschild-like solutions, where h(r) = f(r). By setting x = cos(θ), our line element can be written as [61]…”