Abstract
We studied the thermodynamics and spectroscopy of a 4-dimensional, z=0 Lifshitz black hole (Z0LBH). Using Wald’s entropy formula and the Hawking temperature, we derived the quasi-local mass of the Z0LBH. Based on the exact solution to the near-horizon Schrödinger-like equation (SLE) of the massive scalar waves, we computed the quasi-normal modes of the Z0LBH via employing the adiabatic invariant quantity for the Z0LBH. This study shows that the entropy and area spectra of the Z0LBH are equally spaced.
Highlights
Ever since the publication of the seminal papers of Bekenstein and Hawking [1,2,3], it has been known that black hole (BH) entropy (SBH) should be quantized in discrete levels as discussed in detail by Bekenstein [4,5,6,7]
After getting NH form of the Schrodinger-like equation (SLE), we show that the radial equation is reduced to a confluent hypergeometric (CH) differential equation [36]
The quantum spectra of the Z0LBH were studied using the Maggiore’s method (MM), which is based on the adiabatic invariant quantity (3)
Summary
Ever since the publication of the seminal papers of Bekenstein and Hawking [1,2,3], it has been known that black hole (BH) entropy (SBH) should be quantized in discrete levels as discussed in detail by Bekenstein [4,5,6,7]. Following the seminal works of Bekenstein, new methods have been developed to derive the entropy/area spectra of the numerous BHs (see [12] and references therein). Bohr-Sommerfeld quantization rule [15] states that Iadb acts as a quantized quantity (Iadb ≃ nħ) when the highly excited modes (n → ∞) are considered In such a case, the imaginary part of the frequency dominates the real part of the frequency (ωI ≫ ωR), implying that Δω ≃ ΔωI. To analyze the physical features of the z = 0 Lifshitz BH (Z0LBH) geometry, we first calculate its quasi-local mass MQL [27] and temperature via Wald’s entropy [28] and statistical Hawking temperature formula. QNM calculations of the Z0LBH must be performed in order to implement the MM successfully. The imaginary part of the QNMs is used in (3), and the quantum spectra of entropy and area of the Z0LBH are obtained.
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