Weak cosmic censorship conjecture in Kerr-(anti-)de Sitter black hole with scalar field

Bogeun Gwak

doi:10.1007/jhep09(2018)081

Abstract

We investigate the weak cosmic censorship conjecture in Kerr-(anti-)de Sitter black holes under the scattering of a scalar field. We test the conjecture in terms of whether the black hole can exceed the extremal condition with respect to its change caused by the energy and angular momentum fluxes of the scalar field. Without imposing the laws of thermodynamics, we prove that the conjecture is valid in all the initial states of the black hole (non-extremal, near-extremal, and extremal black holes). The validity in the case of the near-extremal black hole is different from the results of similar tests conducted by adding a particle because the fluxes represent the energy and angular momentum transferred to the black hole during the time interval not included in the tests involving the particle. Using the time interval, we show that the angular velocity of the black hole with the scalar field of a constant state takes a long time for saturation to the frequency of the scalar field.

Highlights

JHEP09(2018)081 singularity of the black hole located inside the horizon
We assume that the K(A)de Sitter (dS) black hole infinitesimally changes during the infinitesimal time interval because of the transferred energy and angular momentum from their fluxes of the scalar field when the scalar field is scattered at the outer horizon of the K(A)dS black hole
We focus on the fluxes at the outer horizon that represent the energy and angular momentum coming into and going out of the K(A)dS black hole; the fluxes can be obtained from the solution of the scalar field at the outer horizon considering separation and normalization

Summary

Solution to scalar field equation

We investigate the infinitesimal changes in the K(A)dS black hole when the external energy and the angular momentum are transferred from the scattering of a complex scalar field. As shown below, the effects from the mass and non-minimal coupling terms are removed from the solution of the scalar field at the outer horizon. The equations of motion are similar to those of the massive scalar field with minimal coupling in the K(A)dS black hole. When energy and angular momentum fluxes are obtained at the limit of the outer horizon, the radial solution plays an important role; otherwise, the contribution of the θ-directional solution in eq (3.5) will be reduced to unity in the fluxes by the normalization condition: Θ2(θ)dΩ = 1. In the non-rotating case, a = 0, the θ-directional equation becomes that of the Schwarzschild-dS black hole; λ = l(l + 1) of the angular momentum number [56].

Thermodynamics under scattering of scalar field

Superradiance

Summary