Revisiting the gravitational atom: Exact scalar cloud solution

In this work, we investigate perfectly bound states of the Klein–Gordon equation, well-known as the neutral scalar cloud, in the Dyonic Kerr-Sen black hole (DKSBH) spacetime. Because the system closely mimics the proton-electron structure in a hydrogen atom, the black hole bearing a boson cloud is referred as a gravitational atom. DKSBH itself is the most general 3+1 dimensional black hole solution of the Einstein–Maxwell-Dilaton–Axion theory, generalizes all Kerr-family solutions, including Kerr, Kerr–Newmann, Kerr-EMDA, and their respective dilatonic versions. Conventionally, the scalar cloud is investigated via approximation technique known as the AAM technique, which works in ultralight scalar field and low black hole’s spin regime. In this work, we aim to revisit the theory of the black hole’s scalar cloud, by firstly finding and investigating the exact solution to the massive Klein–Gordon equation in the black hole spacetime. We find that the scalar cloud exact solution is obtained in terms of the Confluent Heun function and its exact spectrum corresponds to the polynomial condition of the radial solution. Working with exact solution, we can examine the scalar cloud without being troubled by ultralight scalar mass or the slow black hole limit. Finally, we discover that the exact characteristic frequency of the black hole’s scalar cloud takes into account the effect comes from both the black hole’s event horizon and the Cauchy horizon.