The Dirac Equation in the Kerr-de Sitter Metric

Abstract

We consider a Fermion in the presence of a rotating black hole immersed in a universe with positive cosmological constant. After deriving new formulae for the event, Cauchy and cosmological horizons we adopt the Carter tetrad to separate the aforementioned equation into a radial and angular equation. We show how the Chandrasekhar ansatz leads to the construction of a symmetry operator that can be interpreted as the square root of the squared total angular momentum operator. Furthermore, we prove that the the spectrum of the angular operator is discrete and consists of simple eigenvalues and by means of the functional Bethe ansatz method we also derive a set of necessary and sufficient conditions for the angular operator to have polynomial solutions. Finally, we show that there exist no bound states for the Dirac equation in the non-extreme case.

Fate of strong cosmic censorship conjecture in presence of higher spacetime dimensions

“…Moreover, the presence of timelike and angular Killing vectors imply that the Dirac field can be separated as follows : Ψ(t, r, θ, φ) = e −iωt e imφ R(r) ⊗ S(θ), where S(θ) = (S + (θ) S − (θ)) T and R(r) = (R + (r) R − (r)) T , where suffix '+' and '−' corresponds to s = 1/2 and s = −1/2 respectively. This decomposition results in a set of coupled equations in both angular and radial part which can be written as follows [30] ∆…”

Dirac fermion, cosmological event horizons, and quantum entanglement

“…(45) can be studied using the usual Newman-Penrose basis, e.g. [49] and references therein. However, in order to define the aforementioned rigidly rotating states near the horizon, it is a bit convenient to go to a diagonal basis as follows.…”

“…The angular functions S ± (λ, θ) are spin-1/2 weighted spheroidal harmonics with eigenvalues λ, e.g. [49] and references therein. They are normalised as,…”