Schwarzschild-de Sitter black hole in canonical quantization

Abstract

We solve Wheeler-De Witt (WDW) metric probability wave equation on the apparent horizon hypersurface of the Schwarzschild de Sitter (SdS) black hole. To do so we choose radial dependent mass function $M(r)$ for its internal regions in the presence of a dynamical massless quantum matter scalar field $\psi(r)$ and calculate canonical supper hamiltonian constraint on $t-$constant hypersurface near the horizon $r=M(r)$. In this case $M(r)$ become geometrical degrees of freedom while $\psi(r)$ is matter degrees of freedom of the apparent horizon. However our solution is obtained versus the quantum harmonic oscillator which defined against the well known hermit polynomials. In the latter case we obtain quantized mass of the SdS quantum black hole as $\sqrt{\Lambda}M(n)=\big(\frac{2n+1}{12\sqrt{2}}\big)^{\frac{1}{3}}$ in which $\Lambda$ is the cosmological constant and $n=0,1,2,\cdots $ are quantum numbers of the hermit polynomials. This shows that a quantized SdS in its ground state has a nonzero value for the mass $M(0)=0.38914/\sqrt{\Lambda}$. Thus one can infer that the latter result satisfies Penrose hypotheses for cosmic censorship where a causal singularity may be covered by a horizon surface.