Scattering state and bound state of scalar field in Schwarzschild spacetime: Exact solution

Abstract

The main aim of this paper is twofold. (1) Exact solutions of a scalar field in the Schwarzschild spacetime are presented. The exact wave functions of scattering states and bound-states are presented. Besides the exact solution, we also provide explicit approximate expressions for bound-state eigenvalues and scattering phase shifts. (2) By virtue of the exact solutions, we give a direct calculation for the discontinuous jump on the horizon for massive scalar fields, while in literature such a jump is obtained from an asymptotic solution by an analytic extension treatment.Corresponding to the region outside the horizon, i.e., r ∈ [2M, ∞), the range of the variable z in the confluent Heun equation is z ∈ [1, ∞). The confluent Heun equation (3.1) has two singular points, z = 1 and z → ∞, in the region z ∈ [1, ∞) [46]. These two singular points just correspond to the two singular points of the Schwarzschild spacetime, r = 2M and r → ∞.

Cylindrical Gravitational Wave: Source and Resonance

“…With the conservation law of the energy, we may define the energy of the cylindrical gravitational radiation in our framework. We can also consider the resonance between matter waves and gravitational waves based on our previous works on scattering [37][38][39].…”

Cylindrical gravitational waves: radiation and resonance

“…With the conservation law of the energy, we may define the energy of the cylindrical gravitational radiation in our framework. We can also consider that the matter wave resonates with the gravitational radiation based on the preceding work on scattering [29][30][31].…”

Stable orbit of Massive Klein Gordon Field near Schawarzschild black hole and its Quasi-Normal modes

“…The focus of my work revolves around addressing the unresolved issues mentioned in the previous paragraph. In the first section, I calculate the potential energy using the tortoise coordinate system [1,2,3,4,5,6] for the Klein-Gordon equation in the Schwarzschild black hole [6,7,8,9,10,11]. Furthermore, I present the minimum distance from the event horizon required for a stable orbit.…”