
Review Article


Gup and Spectrum of Quantum Black Holes


A. Farmany,
H. Noorizadeh
and
S.S. Mortazavi



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 BekensteinMukhanov 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.





Received: October 06, 2010;
Accepted: November 18, 2010;
Published: February 25, 2011


INTRODUCTION
In canonical quantum gravity, the character of Hawking radiation is modified
when quantum gravity effects are properly taking into account, even for nonrotating,
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 AdlerChenSantiago 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 energytime uncertainty. Then,
we calculate spectral lines and linewidth of quantum black hole.
A fundamental frequency from energy spacing between consecutive levels: In canonical quantum gravity the area of a nonrotating neutral black hole is quantized as (with G = c = 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) = e^{α(n1)/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):
A quantum black hole can decays during interval of observer time Δl by
a sequence of integers {n_{1}, n_{2}, ..., n_{j}} of
length j. During Δl, the black hole first jumped down to n_{1}
elementary levels in one ago, then n_{2} level, etc. In this process,
black hole emits a quantum of some species of energy ,
then a quantum of energy , etc. Each one of j quanta carries the energy .
In average, during Δl, the mass of black hole decreases (Bekenstein
and Mukhanov, 1995) by:
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:
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 energytime 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 (AmelinoCamelia 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):
where,
is the Planck length. Using relation (5) it is easy to obtain a similar relation
between timeenergy. Dividing both side of relation (5) by c (speed of light)
reads (Farmany et al., 2007):
Relation (6) reads:
where,
is the Planck time. We use the natural units ,
however, we restore occasionally t', in important formulae (7) for the sake
of clarity. Using this approximation, the uncertainty in timeenergy reads (Farmany
et al., 2007):
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: This leads to an uncertainty in the energy as follow: The eigenfunctions and eigenvalues 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 eigenstate to another. Here we calculate the modified energy levels by solution of the timedependent Schrödinger equation. The timedependent Schrödinger equation is: In a special case when H doesn’t depend explicitly on time, general solution of Eq. 11 is: where, u_{n} is the eigenfunction of H with energy E_{n},
Note that a_{n} doesn’t depend on time. The equation of motion
for the Schrodingerwave function reads:
or,
In harmonic oscillator, the potential energy is ,
we can write:
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):
Combining Eq. 16 and 17 we obtain the
modified energy levels of
system based on the generalized uncertainty principle. To obtain the eigenvalue
and eigenfunctions of the energy we solve the modified Schrödinger equation:
The solution of Eq. 18 is obtained in (Hossenfelder
et al., 2003; Kempf et al., 1995;
Dadic et al., 2003) in:
Equation 19 shows the energy levels of quantum black hole.
Using w_{n} = w (E_{n}) and comparing Eq. 2
with 19 we can write:
Equation 20 is the frequency of the spectral lines for the
nth level energy of the quantum black hole. Let we calculate the linewith
of the quantum black hole. The total uncertainty in the frequency (halfintensity
line width or just halfwidth) is due to lifetime effects.
is the sum of upper and lower state of energy:
Note that we used the natural unite ,
so h = 2π. To calculate the linewidth of the quantum black hole spectral
line, one can take into account ,
so, .
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.

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