BLACK HOLE EVAPORATION IN A NONCOMMUTATIVE CHARGED VAIDYA MODEL
M. Sharif* W. Javed,
Department of Mathematics, University of the Punjab 54590, Lahore, Pakistan
Received October 25, 2011
We study the black hole evaporation and Hawking radiation for a noncommutative charged Vaidya black hole. For this purpose, we determine a spherically symmetric charged Vaidya model and then formulate a noncommutative Reissner-Nordstrom-like solution of this model, which leads to an exact (t - /¡-dependent metric. The behavior of the temporal component of this metric and the corresponding Hawking temperature are investigated. The results are shown in the form of graphs. Further, we examine the tunneling process of charged massive particles through the quantum horizon. We find that the tunneling amplitude is modified due to noncommuta-tivity. Also, it turns out that the black hole evaporates completely in the limits of large time and horizon radius. The effect of charge is to reduce the temperature from a maximum value to zero. We note that the final stage of black hole evaporation is a naked singularity.
1. INTRODUCTION
The classic concept of a smooth spacetinie manifold breaks down at short distances. Noncommuta-tive geometry offers an impressive framework to investigate the short-distance spacetime dynamics. In this framework, a universal minimal length scale sfo exists (equivalent to the Planck length). In general relativity, the effects of the nonconmmtativity can be taken into account by keeping the standard form of the Einstein tensor and using the altered form of the energy momentum tensor in the field equations. This involves a distribution of point-like structures in favor of smeared objects1). Noncommutative black holes (BH) require an appropriate framework in which the nonconmmtativity corresponds to the general relativity-
Black hole evaporation leads to comprehensive and straightforward predictions for the distribution of emitted particles. However, its final phase is unsatisfactory and cannot be resolved due to the semiclassical representation of the Hawking process. Black hole evapo-
E-mail: msharif.matli'fflpu.edu.pk 11 An object constructed by means of a generalized function p(t,r) is smeared in space and is known as a smeared object. These objects are nonlocal. Smearing cannot change the physical nature of the object but the spatial structure of the object is changed, being smeared in a certain region determined by \/a.
ration can be explored in curved spacetime by quantum field theory but the BH itself is described by a classical background geometry. On the other hand, the final stage of BH decay requires quantum gravity corrections while the semiclassical model is incapable to discuss evaporation. Noncommutative quantum field theory (based 011 the coordinate coherent states) treats the short-distance behavior of point-like structures, where IIlclSS and charg 6 Э.Г6 distributed over a region of size ^/a.
Hawking fl] suggested that the radiation spectrum of an evaporating BH is just like a purely thermal black-body spectrum, i.e., the BH can radiate thermally. Consequently, a misconception [2] was developed with respect to the information loss from a BH, leading to the nonunitary of the quantum evolution2К Accordingly, when a BH evaporates completely, all the information related to matter that has fallen inside the BH is lost. Gibbons and Hawking [3] proposed a formulation to visualize radiation as tunneling of charged particles. In this formulation, radiation corresponds to electron positron pair creation in a constant electric field, with the energy of a particle changing sign as it
2 ' Nonunitary quantum evolution is one of the interpretations of the information paradox to modify quantum mechanics. In a unitary evolution, the entropy is constant with the usual .S-mat-rix, whereas it is not constant in a nonunitary quantum evolution.
crosses the horizon. The total energy of a pair created just inside or outside the horizon is zero when one member of the pair tunnels to the opposite side. Parikh and Wilczek [4] derived Hawking radiation as a tunneling through the quantum horizon on the basis of null geodesies. In this framework, the BH radiation spectrum corrected due to back-reaction effects is obtained. This tunneling process shows that the extended radiation spectrum is not exactly thermal yielding a unitary quantum evolution.
There are two different semiclassical tunneling methods to calculate the tunneling amplitude, which lead to the Hawking temperature. The first method, called the null geodesic method, gives the same temperature as the Hawking temperature. The second method, called the canonically invariant tunneling, leads to a canonically invariant tunneling amplitude and hence the corresponding temperature which is higher than the Hawking temperature by a factor of 2 [5]. It was argued in [6] that a particular coordinate transformation resolves this problem in the semiclassical picture.
Black hole evaporation spectra in the Einstein di-laton Gauss Bonnet four dimensional string gravity model was discussed in [7] using the radial null geodesic method. It was shown that BHs should not disappear and become relics at the end of the evaporation process. The authors of [7] numerically investigated the possibility of experimental detection of such remnant BHs and discussed the IIlclSS loss rate in analytic form. These primordial BH relics could form a part of the nonbaryonic dark matter in our universe.
Various noncommutative models in terms of coordinate coherent states that satisfy the Lorentz invariance, unitarity, and UV finiteness of quantum field theory-were found in [8]. A generalized noncommutative metric that does not allow a BH to decay below a minimal nonzero mass M0, i.e., the BH remnant mass, was derived in [9]. The effects of noncommutative BHs have been studied [10, 11] and consistent results were found. The evaporation process stops when a BH approaches a Planck-size remnant with zero temperature. Also, it does not diverge but rather reaches a maximum value before shrinking to the absolute zero temperature, which is an intrinsic property of the manifold. Some other authors [12] also explored information loss problem during BH evaporation.
Quantum corrections to the thcrmodynamical quantities for a Bardeen charged regular BH were investigated in [13] using the quantum tunneling approach over semiclassical approximations. In a recent work [14], the effects of noncommutativity on
the thermodynamics of this BH were discussed. The tunneling of massive particles through the quantum horizon of the noncommutative Schwarzschild BH was analyzed in [15] and the modified Hawking radiation, thcrmodynamical quantities, and emission rate was derived. Stable BH remnants and the information loss issues were also discussed there. The effects of smeared mass were studied in [17] with the conclusion that information might be saved by a stable BH remnant during the evaporation process. In [17], this work was extended to a noncommutative Reissner Nordstrom (RN) BH and the emission rate consistent with a unitary theory was determined. The same author [18] also formulated a noncommutative Schwarzschild-like metric for a Vaidya solution and analyzed three possible causal structures of the BH initial and remnant mass. He also studied the tunneling of charged particles through the quantum horizon of the Schwarzschild-like Vaidya BH and evaluated the corresponding entropy.
The purpose of this paper is two-fold. First, we formulate a noncommutative RN-like solution of the spherically symmetric charged Vaidya model. Second, we investigate some of its features. In particular, we explore the BH evaporation and Parikh Wilczek tunneling process. The paper is organized as follows. In Sec. 2, we solve the coupled field equations for the spherically symmetric charged Vaidya model. The effect of the noncommutative form of this model is investigated in the framework of coordinate coherent states in Sec. 3. Here, an exact (f — r)-dopondont RN-like BH solution is obtained. In Sec. 4, we find the behavior of the temporal component of this solution and also discuss the BH evaporation in the limits of large time and charge. In Sec. 5, we study the Parikh Wilczek tunneling for such a Vaidya solution and the Hawking temperature in the presence of a charge. The tunneling amplitude at which massless particles tunnel through the event horizon is computed. Finally, the conclusions are given in the last section. Throughout the paper, we set h = c=G= 1.
2. CHARGED VAIDYA MODEL
This section is devoted to the formulation of a spherically symmetric charged Vaidya model in the RN-like form using the procedure given in [19]. We skip the details of the procedure because they are already available and use only the required results. The spherically symmetric Vaidya-form metric is given by Eq. (2.34) in [19]:
ds2 = -evll'r)dt-2 + e'lil'r)dr2 + rW, (1)
where
pi'iM-) =
M
-ti(t,r) 2M
<m2 = <ie2 + sm20<Z02,
M(t,r) is a slowly varying mass function, and \(M) depends on the details of the radiation. The corresponding field equations are [19]
//.' = 8 nrT,,.
1 - e'1
r
1 - e'1
v' = 8Trre'1-l'Tu 11 = SnrTtr,
1 - e-'1 + ^re-i'iii1 - v') - ir2i?<°) =
(2)
(3)
(4)
where
= -SwTaa = —_(l _ +p x
= 8nToo = ^^. (5) sin" 0
-(/-! + !')/2
(ßeUl-
(yl(M>->DI2)'] . (6)
The dot and prime respectively denote derivatives with respect to time and r. We note that Eqs. (2) and (4) represent the respective Hamiltonian and momentum constraints [20]. Equations (2) and (3) lead to
1 1 - e'1
while Eqs. (5) and (C) yield
T-\TV — G ' J- ££ •
(7)
(8)
For the spherically symmetric Vaidya metric of form (1), we define by adding charge Q(t,r)
as follows:
_/j(/;r) = 1 _ 2.1/(7. r) + Q2(t,r)
0)
/ /~
Using the procedure in [19], we can deduce from the field equations that
T'rJ1'-'1^2+T', = 0. (10)
Also, using Eqs. (2), (4), (8) and
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