Abstract
We propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1 + 1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical Hamiltonian H Δ. The holonomy correction in H Δ is implemented by the -scheme regularization with a Planckian area scale Δ (which often chosen as the minimal area gap in loop quantum gravity). The effective dynamics recovers the semiclassical Schwarzschild geometry at low curvature regime and resolves the black hole singularity with Planckian curvature, e.g. R μνρσ R μνρσ ∼ 1/Δ2. Our model predicts that the evolution of the black hole at late time reaches the charged Nariai geometry dS2 × S 2 with Planckian radii . The Nariai geometry is stable under linear perturbations but may be unstable by nonperturbative quantum effects. Our model suggests the existence of quantum tunneling of the Nariai geometry and a scenario of black-hole-to-white-hole transition.
Highlights
M Han and H Liu (Some figures may appear in colour only in the online journal) is expected that the effective dynamics should be valid for black hole, or at least be a crucial first step toward understanding full quantum effects of black hole
We propose a new model of the spherical symmetric LQG black hole and study the effective dynamics
As a reason of choosing the μ-scheme regularization, it has the nice properties that the improved effective dynamics from HΔ has infinitely many conserved charges corresponding to spatial diffeomorphisms, which play an interesting role in our discussion
Summary
The reduced phase space formulation couples gravity to clock fields at classical level. Where T, S j=1,2,3 are clock fields and defines time and space coordinates in the dust reference frame. Eaj(y)|T(y)≡t, Sj(y)≡σ j, where σ, t are physical space and time coordinates in the dust reference frame. Aaj(σ, t), Eaj(σ, t) depending only on values of dust fields are independent of gauge choices of coordinates y. They are proven to be invariant (on the constraint surface) under gauge transformations generated by diffeomorphism and Hamiltonian constraints [41,42,43]. H0 formally coincides with smearing the gravity Hamiltonian C with the unit lapse, while here C(σ) is in terms of Dirac observables Aaj(σ), Eaj(σ) and σ j=1,2,3 are dust coordinates on S. HΔ to be constructed in section 3 is expected to describe the quantum effective theory of H , such that the gravity-dust theory discussed is the low-energy effective theory
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