Black hole spectroscopy via adiabatic invariant in a quantum corrected spacetime

Using the modified Kunstatter method, which employs as proper frequency the imaginary part instead of the real part of the quasinormal modes, the entropy spectrum and area spectrum of the modified Schwarzschild black holes in gravity’s rainbow are investigated. In the current study, two cases of modified dispersion relations concerning energy dependent and energy independent speed of light are considered. The entropy spectra with equal spacing are derived in these two cases. Furthermore, the obtained entropy spectra are independent of the energy of a test particle and are the same as the one of the usual Schwarzschild black hole. Also, the same area spectrum formulas are obtained in these different dispersion relations. However, due to the quantum effect of spacetime, the obtained area spectra are not equally spaced and are different from the one of the usual Schwarzschild black hole. Besides, in these two cases, the same black hole entropy formulas with logarithmic correction to the standard Bekenstein–Hawking area formula are obtained by the adiabatic invariant. The form of area spacing formulas and entropy formulas are independent of the particle’s energy, but the area spacing and entropy can have energy dependence through the area.