The scalar field in extreme Reissner-Nordstrom-de Sitter space

By generalizing the method of I. Brevik et al. the scalar field equation between the outer black hole horizon and the cosmological horizon in the extreme Reissner-Nordstrom-de Sitter (RNdS) geometry is solved. The field amplitude, as well as the potential, is shown graphically by introducing the "tangent" approximation, which is more exact than that used by I. Brevik et al., of the tortoise coordinate. There are two limiting cases of our special interest. The first one is when the cosmological horizon is very close to the outer horizon of the "black hole." The second one is when they are far apart. And the reflection and transmission coefficients are worked out in the two cases respectively.