Letter: The Real Scalar Field in Schwarzschild-de Sitter Spacetime

Abstract

In this paper, the real scalar field equation in Schwarzschild-de Sitter spacetime is solved numerically with high precision. A method called 'polynomial' approximation is introduced to derive the relation between the tortoise coordinate x and the radius r. This method is different from the 'tangent' approximation [1] and leads to more accurate result. The Nariai black hole is then discussed in details. We find that the wave function is harmonic only near the horizons as I. Brevik and B. Simonsen [1] found. Howerver the wave function is not harmonic in the region of the potential peak, with amplitude increasing instead. Furthermore, we also find that, when cosmological constant decreases, the potential peak increases, and the maximum wave amplitude increases.

Modification to the van der waals equation of state

“…In Ref. (5), a method called polynomial approximation is introduced to get the relation of two variables. Here we adopt this method to get the relation of ( ρ is the reduced density of the gas and * 2 ρ is the reduced density of the liquid.…”

Thermodynamics of multi-horizon spacetimes

“…The line element for the Schwarzschild-de Sitter spacetime in the spherically symmetric coordinate is given by [37,[43][44][45][46][47],…”

The real solution to scalar field equation in 5D black string space

“…To the parameter Λ, we can always find an appropriate approximate method from the above. For a widely separated horizons model, we adopt the same value Λ = 10 −3 as appeared in [8] [9]. Then putting Λ into the Eq.…”

“…Since then, many people have used various methods and techniques to research black hole through the particles radiating from it, such as the simple Klein-Gordon particles and Dirac particles (for some early works, see Damour and Ruffini [3] and Chandrasekhar [4] respectively). Recently, as for the former scalar field, Higuchi et al [5] and Grispino et al [6] gave the scalar field solution outside a Schwarzschild black hole, Brady et al [7], Brevik et al [8] and Tian et al [9] studied the Schwarzschild-de Sitter case and Guo et al [10] made further studies in the Reissner-Nordström-de Sitter case and so on.…”