Entropy spectrum and area spectrum of Kerr black hole in gravity's rianbow

Abstract

Black hole spectroscopy is an important content in the quantum properties of black holes. In this paper, we use the adiabatic invariants of black holes to investigate the entropy spectrum and area spectrum of the Kerr black hole in gravity's rainbow. Firstly, by considering the particles passing through the event horizon, the adiabatic invariance action for the modified Kerr black hole is calculated. Here, the Euclidean coordinate and the period of the Euclidean time of a loop about the event horizon are used. Combined the obtained adiabatic invariants with the Bohr-Sommerfen quantization condition, the equally spaced entropy spectra that are the same as the original Beckenstein spectra are derived. The entropy spectrum of the gravity's rainbow is independent of the test particle energy. Next, using the first law of the black hole thermodynamics and the black hole entropy spectrum, the area spectrum of the modified Kerr black hole is studied. Due to the quantum gravity effect of the gravity's rainbow, the obtained area spectrum is different from the original Beckenstein spectrum. The present area spectrum is non-equidistant and dependent on the horizon area of the black hole. With the decrease of black hole area, the area space gradually turns smaller. When the black hole reaches the minimum area on a Planck scale, the area quantum is zero. Thus the black hole area no longer decrease and a remnant of the black hole radiation appears. In the case of a large black hole, the correction of the area spectrum to the equally spaced spectra can be ignored, and the area spectrum of the Kerr black hole in gravity's rainbow can return to the original Beckenstein spectrum. It is also shown that like the entropy spectrum, the area spectrum of the gravity's rainbow does not depend on the energy of the test particles either. In addition, the entropy of the modified Kerr black hole in gravity's rainbow is discussed by using the first law of the black hole thermodynamics. The black hole entropy with quantum correction items as the area reciprocal to the Beckenstein-Hawking entropy is derived and the relation between the quantum correction items and the area is discussed. In addition, the consistency between the entropy correction and the area correction for the modified black hole is analyzed. The current research supports that in different spacetimes including quantum corrected spacetimes, the black hole entropy spectrum has the universality, but the black hole area spectrum is dependent on the area due to the spacetime quantum properties.