Scalar Field at the Phase Transition Point of RNdS Space

The real scalar field equation between the outer black hole horizon and the cosmological horizon is solved in the Reissner-Nordstroem-de Sitter space when it is at the phase transition point. We use an accurate approximation,the polynomial approximation, to approximate the tortoise coordinate x(r) for obtaining the inverse function r=r(x) and then for solving the wave equation. The case where the two horizons are very close to each other is discussed in detail. It is found that the wave function is characteristically similar to the harmonic in the whole range with x as the independent coordinate, while the waves pile up near the horizons with r as the independent coordinate. Furthermore, we find that the height of the potential increases as the cosmological constant A decreases.