Abstract: The evaporation of quantum black holes would leave very distinctive imprints on the detectors and spectrum of such black holes could be obtained. To study the quantum gravity effects on the black hole spectrum, one can take into account the generalized uncertainty principle. In this paper, employing the Bekenstein-Mukhanov approach, the spectrum of a quantum black hole is obtained. It is shown that the energy spacing between consecutive levels for MJ>>h is corresponding to a fundamental frequency.
INTRODUCTION
In canonical quantum gravity, the character of Hawking radiation is modified when quantum gravity effects are properly taking into account, even for non-rotating, neutral, and very massive black hole with respect to the Planck scale. To study the quantum gravity effects on a quantum black hole, Adler et al. (2001) used the generalized uncertainty principle. They showed that in the presence of generalized uncertainty principle, the radiation temperature of black hole is modified. Other approaches of Adler-Chen-Santiago proposal are obtained in Nouicer (2007), Myung et al. (2007), Farmany et al. (2008), Dehghani and Farmany (2009) and Farmany and Dehghani (2010). In this letter, we concentrate on the quantum gravity effects of a quantum black hole. First, we begin with a fundamental frequency from energy spacing between consecutive levels. Then we consider the relation between generalized uncertainty principle and energy-time uncertainty. Then, we calculate spectral lines and line-width of quantum black hole.
A fundamental frequency from energy spacing between consecutive levels: In canonical quantum gravity the area of a non-rotating neutral black hole is quantized as (with G = c = 1):
| (1) |
where, n is the energy level. Thermal character of black hole radiation is entirely due to degeneracy of levels and same degeneracy becomes manifest as black hole entropy (Bekenstein, 2002; Jiang et al., 2010; Majhi, 2010; Banerjee et al., 2010; Jadhav and Burko, 2009; Drasco, 2009; Van Den Broeck and Sengupta, 2007; Dappiaggi and Raschi, 2006; Dreyer et al., 2004; Setare, 2004a, b; Bekenstein and Mukhanov, 1995) Setting (n) as the multiplicity of degeneracy, Bekenstein and Mukhanov (1995) found that in the level n = 1, g (1) = 1, in this level (n = 1) the black hole entropy is zero. Here a general form of multiplicity degenerate (energy level) is g (n) = eα(n-1)/4 where, αInk and k = 2, 3, 4,... the energy spacing between consecutive levels for M>>h corresponds to a fundamental frequency (Bekenstein and Mukhanov, 1995):
| (2) |
A quantum black hole can decays during interval of observer time Δl by
a sequence of integers {n1, n2, ..., nj} of
length j. During Δl, the black hole first jumped down to n1
elementary levels in one ago, then n2 level, etc. In this process,
black hole emits a quantum of some species of energy
| (3) |
Since the main value of j is Δt/lτ where, τ is a survival timescale, both could be determined. This decreasing of black hole mass is radically different from one obtained in the standard discussion of hawking radiance. Bekenstein and Mukhanov (1995) obtained the mean time γ between quantum leaps as:
| (4) |
Bekenstein and Mukhanov argues that “this (spectroscopy of quantum black hole) to be possible to test quantum gravity with black hole well above Planck scale”.
Relationship between generalized uncertainty principle and energy-time uncertainty: Generalized uncertainty principle has been the subject of interesting works over the years. In these works, modification of usual uncertainty relation at microphysics is obtained (Amelino-Camelia et al., 2006; Adler and Santiago, 1999; Adler et al., 2001; Farmany and Dehghani, 2010; Farmany, 2010; Farmany et al., 2008; Farmany et al., 2007; Hossenfelder et al., 2003):
| (5) |
where,
| (6) |
Relation (6) reads:
| (7) |
where,
| (8) |
Frequency of spectra: Observation of spectrum of any quantum black hole would immediately make quantum gravity effects well above the Planck scale. Relation (8) is quadratic in ΔE:
| (9) |
This leads to an uncertainty in the energy as follow:
| (10) |
The eigen-functions and eigen-values of energy operator play an important role in our calculations. The physical measurements often involve determination of energy (or radiation frequency) emitted or absorbed by system that makes a transition from one energy eigen-state to another. Here we calculate the modified energy levels by solution of the time-dependent Schrödinger equation. The time-dependent Schrödinger equation is:
| (11) |
In a special case when H doesn’t depend explicitly on time, general solution of Eq. 11 is:
| (12) |
where, un is the eigen-function of H with energy En,
| (13) |
Note that an doesn’t depend on time. The equation of motion for the Schrodinger-wave function reads:
| (14) |
or,
| (15) |
In harmonic oscillator, the potential energy is
| (16) |
From Eq. 5 we can obtain a relation between momentum and coordinate as (Hossenfelder et al., 2003; Kempf et al., 1995; Dadic et al., 2003):
| (17) |
Combining Eq. 16 and 17 we obtain the
modified energy levels
| (18) |
The solution of Eq. 18 is obtained in (Hossenfelder et al., 2003; Kempf et al., 1995; Dadic et al., 2003) in:
| (19) |
Equation 19 shows the energy levels of quantum black hole. Using wn = w (En) and comparing Eq. 2 with 19 we can write:
| (20) |
Equation 20 is the frequency of the spectral lines for the
n-th level energy of the quantum black hole. Let we calculate the line-with
of the quantum black hole. The total uncertainty in the frequency (half-intensity
line width or just half-width) is due to lifetime effects.
| (21) |
Note that we used the natural unite
CONCLUSION
The character of Hawking radiation is modified when quantum gravity effects are properly taking into account, even for very massive black hole. In this viewpoint, decreasing of black hole mass is radically different from one entertained in the standard discussion. Thermal character of radiation is entirely due to degeneracy of levels. Same degeneracy becomes manifest as black hole entropy. Bekenstein and Mukhanov calculated that, in the level n = 1, the black hole entropy is zero. The energy spacing between consecutive levels fo MJ>>h is corresponding to a fundamental frequency.