S. Hamid Mehdipour- Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole

8/3/2019 S. Hamid Mehdipour- Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole

http://slidepdf.com/reader/full/s-hamid-mehdipour-parikhwilczek-tunneling-as-massive-particles-from-noncommutative 1/6

Commun. Theor. Phys. (Beijing, China) 52 (2009) pp. 865–870c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 5, November 15, 2009

Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild

Black Hole

S. Hamid Mehdipour∗

Islamic Azad University, Lahijan Branch, P.O. Box 1616, Lahijan, IRAN

(Received November 28, 2008)Abstract In this paper, we apply the tunneling of massive particle through the quantum horizon of a Schwarzschild

black hole in noncommutative spacetime. The tunneling effects lead to modified Hawking radiation due to inclusion

of back-reaction effects. Our calculations show also that noncommutativity effects cause the further modifications to

the thermodynamical relations in black hole. We calculate the emission rate of the massive particles’ tunneling from a

Schwarzschild black hole which is modified on account of noncommutativity influences. The issues of information loss

and possible correlations between emitted particles are discussed. Unfortunately even by considering noncommutativity

view point, there is no correlation between different modes of evaporation at least at late-time. Nevertheless, as a result

of spacetime noncommutativity, information may be conserved by a stable black hole remnant.

PACS numbers: 04.70.-s, 04.70.Dy, 11.10.NxKey words: quantum tunneling, hawking radiation, noncommutative spacetime, black hole entropy, informa-

tion loss paradox

1 IntroductionOver three decades ago, Stephan Hawking[1] found

that the radiation spectrum is almost like that of a blackbody, and can be described by a characteristic Hawkingtemperature with a purely thermal spectrum given byT H = c3κ/2πkBG, utilizing the procedure of quantumfield theory in curved spacetime (κ is the surface grav-ity that demonstrates the strength of the gravitationalfield near the black hole surface) and this yields to non-unitarity of quantum theory where maps a pure state to amixed state. It has been proposed that Hawking radiationcan be extracted from the null geodesic method suggestedby Parikh and Wilczek.[2] In their method, they take theback reaction effects into consideration and present a lead-ing correction to the probability of massless particles tun-neling across the horizon. The tunneling process clari-fies that the extended radiation spectrum is not preciselythermal which leads to unitarity. Even though the formof correction is not sufficient by itself to recover informa-tion because of its failure to have correlations betweenthe tunneling probability of different modes in the blackhole radiation spectrum. Recently, Nicolini, Smailagic,and Spallucci (NSS)[3] derived that black hole in a newmodel of noncommutativity does not allow to decay lowerthan a minimal mass M 0, i.e. black hole remnant. If we

really believe the idea of stable black hole remnants due tothe fact that there are some exact continuous global sym-metries in nature,[4] and also do not find any correlationsbetween the tunneling rates of different modes in the blackhole radiation spectrum, then these leave only one possi-bility: the information stays inside the black hole and canbe retained by a stable Planck-sized remnant. Although,this issue is then accepted if information conservation isreally conserved in our universe. Thus we proceed ourwork with hope that this model of noncommutativity can

provide a way to explain how the black hole decays, par-ticularly in its final stages.

The NSS model of noncommutativity of coordinatesthat is carried on by the Gaussian distribution of coherentstates, is also consistent with Lorentz invariance, Unitar-ity, and UV-finiteness of quantum field theory.[5−7] More-over, noncommutativity of spacetime is an innate prop-erty of the manifold by itself even in absence of gravityand some kind of divergences which emerge in generalrelativity and black hole physics, can also be deleted byit. Then with hope to cure the divergences of evapora-tion process of black hole physics we apply both back-reaction and noncommutativity effects to proceed the ra-diative process. The plan of this paper is the following.In Sec. 2, we perform a brief discussion about the exis-tence of black hole remnant within the noncommutativecoordinate fluctuations at short distances (noncommuta-tive inspired Schwarzschild solutions). In Sec. 3, a detailedcalculation of quantum tunneling near the smeared quan-tum horizon by considering a new model of noncommu-tativity is provided. The tunneling probability at whichmassive particles tunnel across the event horizon is calcu-lated and its applicability for the Schwarzschild black holeis discussed. In Sec. 4, we find the analytic form of themodified (noncommutative) entropy and test our results

approximately. In Sec. 5, the problem of lost informationand possible correlations between emitted particles are in-vestigated. And finally the paper is ended with summary(Sec. 6).

2 Noncommutative Schwarzschild Black Hole

There exist many formulations of noncommuta-tive field theory based on the Weyl–Wigner–Moyal ∗-product[8] that lead to failure in resolving of some impor-tant problems, such as Lorentz invariance breaking, loss

∗E-mail: [email protected]



866 S. Hamid Mehdipour Vol. 52

of unitarity, and UV divergences of quantum field the-ory. But recently, Smailagic and Spallucci[5−7] explaineda fascinating model of noncommutativity, the coordinatecoherent states approach, that can be free from the prob-lems mentioned above. In this approach, the particle massM , instead of being quite localized at a point, is describedby a smeared structure throughout a region of linear size√

θ. In other words, we shall smear the point mass Dirac-

delta function utilizing a Gaussian function of finite width.So the mass density distributions of a static, sphericallysymmetric, particle-like gravitational source will be givenby a Gaussian distribution of minimal width

√θ, instead

of a delta function distribution, with the following equa-tion

ρθ(r) =M

(4πθ)3/2e−r

2/4θ . (1)

The line element which solves Einstein’s equations in thepresence of smeared mass sources can be obtained as

ds2 = −

1− 2M θr

dt2 +

1− 2M θ

r

−1

dr2 + r2dΩ22 , (2)

where the smeared mass distribution can implicity begiven in terms of the lower incomplete Gamma function,

M θ =

r0

ρθ(r)4πr2dr =2M √

πγ 3

2,

r2

4θ

≡ 2M √π

r2/4θ0

t1/2e−tdt . (3)

(Throughout the rest of this work natural units are usedwith the following definitions; = c = kB = 1). In thelimit θ → 0, one recovers the complete Gamma functionΓ(3/2), and M θ → M as expected. Then noncommuta-tive (modified) Schwarzschild solution reduced to the com-mutative (ordinary) one. The line element (2) character-

izes the geometry of a noncommutative black hole. Theradiating behavior of such modified Schwarzschild blackhole can now be investigated and can easily be shown byplotting g00, as a function of r, for different values of M (hereafter, for plotting the figure we set the value of thenoncommutativity parameter equal to unity, θ = 1). Fig-ure 1 shows that coordinate noncommutativity yields tothe existence of a minimal non-zero mass which black hole(due to Hawking radiation and evaporation) can shrink toit.

The event horizon of this line element can be foundwhere g00(rH) = 0, that is implicity written in terms of the upper incomplete Gamma function as

rH = 2M θ(rH) = 2M

1− 2√π

Γ3

2,

r2H

4θ

. (4)

The noncommutative Schwarzschild radius versus themass can approximately be calculated by setting rH = 2M into the upper incomplete Gamma function as

rH = 2M

Erf M √

θ

− 2M √

πθexp

−M 2

θ

. (5)

For very large masses, the Erf function tends to one andsecond term will exponentially be reduced to zero andone retrieves the classical Schwarzschild radius, rH ≈ 2M .

Now we want to calculate the Hawking temperature dueto investigation of radiating treatment of such noncom-mutative black hole,

T H =1

4π

dg00dr

r=rH

= M Erf(rH/2

√θ)

2πr2H

− exp(−r2H

/4θ)

4(πθ)3/2

rH +

2θ

rH

. (6)

For the commutative case, M/√θ →∞, one recovers theclassical Hawking temperature, T H = 1/8πM . The nu-merical calculation of the modified Hawking temperatureas a function of the mass is presented in Fig. 2. In thismodified (noncommutative) version, not only T H does notdiverge at all but also it reaches a maximum value beforedropping to absolute zero at a minimal non-zero mass,M = M 0 ≈ 1.9, that black hole shrink to it.

Fig. 1 g00 versus the radius r for different values of massM . The figure shows the possibility of having extremal

configuration with one degenerate event horizon whenM = M 0 ≈ 1.9 (i.e., the existence of a minimal non-zero mass), and no event horizon when the mass of theblack hole is smaller than M 0. Also as figure shows, thedistance between the horizons will increase by increasingthe black hole mass (two event horizons).

Fig. 2 Black hole temperature, T H, as a function of M .The existence of a minimal non-zero mass and disappear-ance of divergence are clear.



No. 5 Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole 867

3 Parikh–Wilczek Tunneling as MassiveParticles

We are now ready to discuss the quantum tunnelingprocess in the noncommutative framework. To describenoncommutative quantum tunneling process, where a par-ticle moves in dynamical geometry and pass through thehorizon without singularity on the path we should usea coordinates system that, unlike Schwarzschild coordi-

nates, are not singular at the horizon. A especially con-venient choice is Painleve coordinate,[9] which is obtainedby definition of a new noncommutative time coordinate,

dt = dts +

√2M θr

r − 2M θdr = dts + dtsyn , (7)

where ts is the Schwarzschild time coordinate, and

dtsyn = −g01g00

dr . (8)

Note that only the Schwarzschild time coordinate is trans-formed. Both the radial coordinate and angular coordi-nates remain the same. The noncommutative Painlevemetric now immediately reads

ds2

= g00dt2

+ 2g01dtdr + g11dr2

+ g22dϑ2

+ g33dϕ2

= −

1− 2M θr

dt2 + 2

2M θ

rdtdr + dr2

+ r2(dϑ2 + sin2 ϑdϕ2) . (9)

It should be stressed here that the relation (8), in accordwith Landau’s theory of the synchronization of clocks,[10]

allows us to synchronize clocks in any infinitesimal radialpositions of space (dϑ = dϕ = 0). Since the tunnelingphenomena through the quantum horizon i.e. the barrier

is an instantaneous procedure it is important to considerLandau’s theory of the coordinate clock synchronizationin the tunneling process. The mechanism for tunneling

through the quantum horizon is that particle anti-particlepair is created at the event horizon. So, we have two eventsthat occur simultaneously; one event is anti-particle andtunnels into the barrier but the other particle tunnels outthe barrier. In fact, the relation (8) mentions the differ-ence of coordinate times for these two simultaneous eventsoccurring at infinitely adjacent radial positions. Further-more, the noncommutative Painleve–Schwarzschild metricexhibits the stationary, non-static, and neither coordinatesingularity nor intrinsic singularity. Since we shall ob-tain the radial geodesic of the massive particle which isdifferent with massless one, we follow the noncommuta-tive method with massless particle’s tunneling across thepotential barrier in one of our new papers[11] but now ex-tended to the case of massive particle.

According to the non-relativistic quantum theory, deBroglie’s hypothesis and the WKB approximation, it canbe easily proved that the treatment of the massive parti-cle’s tunneling as a massive shell is approximately derivedby the phase velocity v p of the de Broglie s-wave whoserelationship between phase velocity v p and group velocityvg is given by[12−14]

v p = r =1

2vg . (10)

In the case of dϑ = dϕ = 0, according to the relation (8),the group velocity is

vg = −g00g01

. (11)

Thus, the outgoing motion of the massive particles takesthe form

r = − g002g01

=r − 2M θ

2√

2rM θ. (12)

If we suppose that t increases towards the future thenthe above equations will be modified by the particle’sself-gravitation effect. Kraus and Wilczek[15] studiedthe motion of particles in the s-wave as spherical mass-less shells in dynamical geometry and developed self-gravitating shells in Hamiltonian gravity. Further elab-orations was performed by Parikh and Wilczek.[2] In thispaper, we want to develop their work to noncommuta-tive coordinate coherent states and massive particles. Wekeep the total ADM mass (M ) of the spacetime fixed, andpermit the hole mass to fluctuate, because we take into ac-count the response of the background geometry to an emit-

ted quantum of energy E which moves in the geodesics of a spacetime with M replaced by M −E . Thus we shouldreplace M by M −E both in Eqs. (9) and (12).

Since the characteristic wavelength of the radiation isalways haphazardly small near the horizon due to the in-finite blue-shift there, so that the wave-number reachesinfinity and the WKB approximation is reliable close tothe horizon. In the WKB approximation, the probabilityof tunneling or emission rate for the classically forbiddenregion as a function of the imaginary part of the particle’saction at stationary phase would take the form

Γ ∼ exp(−2 Im I ) . (13)

To calculate the imaginary part of the action we consider aspherical shell to consist of components massive particleseach of which travels on a radial timelike geodesic, so thatwe will use these radial timelike geodesics like an s-waveoutgoing positive energy particle which pass through thehorizon outwards from rin to rout to compute the Im I , asfollows

Im I = Im

routrin

prdr = Im

routrin

pr0

dp′rdr , (14)

we now alter the integral variable from momentum in favorof energy by using Hamilton’s equation r = (dH/dpr)|r,where the Hamiltonian is H = M −E ′, hence the r integral

can be done first by deforming the contour,

Im I = Im

M −E

M

routrin

dr

rdH

= Im

E0

routrin

2

2r M θ(M − E ′)

r − 2 M θ(M −E ′)dr(−dE ′) . (15)

The r integral has a pole at the horizon where lies alongthe line of integration and this yields to (−πi) times theresidue.

Im I = Im

E0

4πi M θ(M − E ′) dE ′ , (16)





No. 5 Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole 869

and loop quantum gravity, the entropy of black hole hasbeen achieved by direct microstate counting as follows (inunits of the Planck scale),

S QG = 4πM 2 + α ln(16πM 2) + O 1

M 2

. (24)

It was recently suggested by the authors of Ref. [19] thatthe Planck scale corrections to the black hole radiationspectrum via tunneling can be written as

Γ ∼ exp(∆S QG) = exp

S QG(M −E ) − S QG(M )

=

1− E

M

2αexp

−8πM E

1− E

2M

. (25)

Since, loop quantum gravity anticipates a negative valuefor α (see e.g. Ref. [20]) which yields to diverge the emis-sion rate when E → M that leads to no suppressing theblack hole emission (Although the suppression can onlyoccur when α > 0, which is not recommended at least byloop quantum gravity). But our outcome is actually sen-sible, comparing the noncommutative result for the emis-sion rate, Eq. (23), with the quantum gravity result of

Ref. [19], Eq. (25), shows that the noncommutative resultis reasonably successful in ceasing the black hole emissionwhen (M − E ) → M 0. In fact, the cases (M − E ) < M 0are the noncommutativity-forbidden regions that the tun-neling particle can not be traversed through it. Therefore,the limit (M −E ) → 0 can not be applied by our processbecause of existence of non-vanishing mass at final phaseof black hole evaporation.

Behavior of the entropy S NC, Eq. (22), as a function of the mass is depicted in Fig. 3. As Fig. 3 shows, at the finalstage of the black hole evaporation, the black hole ceasesto radiate and its entropy reaches zero and the existence

of a minimal non-zero mass with approximation is againclear.In the large mass regime, i.e. M/

√θ ≫ 1 and E ≪ M

Eq. (23) leads to

∆S NC = −8πM E + 4πE 2 − 12√

πθE exp−M 2

θ

, (26)

which one recovers the standard Bekenstein-Hawking En-

tropy plus θ-corrections.

S NC = 4πM 2 + 12√

πθM exp−M 2

θ

. (27)

Fig. 3 Black hole entropy, S NC, as a function of M . Notethat the figure is plotted approximately by Eq. (22).

5 Possible Correlations Between EmittedParticles

In this time we want to demonstrate that there areno correlations between emitted particles even with theinclusion of the noncommutativity corrections at least atlate-times. (However, there might be short-time correla-tions between the quanta emitted early and the quantaemitted later on that decay to zero after the black hole isequilibrated at late-times.) This means it can be exhib-ited that the probability of tunneling of two particles of energy E 1 and E 2 is precisely similar to the probability

of tunneling of one particle with their compound energies,E = E 1 + E 2, i.e.

∆S E1+∆S E2

= ∆S (E1+E2) ⇒ χ(E 1+E 2; E 1, E 2) = 0 , (28)

where ∆S E1≡ ∆S (M, E = E 1), ∆S E2

≡ ∆S (M =M − E 1, E = E 2), ∆S (E1+E2) ≡ ∆S (M, E = E 1 + E 2).Thereby, the emission rate for a first quanta emitted, E 1,yields

∆S E1= 4π

(M − E 1)2 − 3

2θ

Erf M − E 1√

θ

+ 12

√πθ(M −E 1)exp

− (M − E 1)2

θ

− 4π

M 2 − 3

2θ

Erf M √

θ

− 12

√πθM exp

−M 2

θ

, (29)

and so the emission rate for a second quanta emitted, E 2, is written by

∆S E2= 4π

(M − E 1)− E 2

2− 3

2θ

Erf (M − E 1)−E 2√

θ

+ 12

√πθ((M −E 1)−E 2)exp

− ((M − E 1)−E 2)2

θ

− 4π

(M −E 1)2 − 3

2θ

Erf (M −E 1)√

θ

− 12

√πθ(M − E 1)exp

− (M −E 1)2

θ

, (30)

finally, the emission rate for a single quanta emitted with the same total energy, E , is given by

∆S (E1+E2) = 4π

M − (E 1 + E 2)2 − 3

2θ

Erf M − (E 1 + E 2)√

θ

+ 12

√πθ(M − (E 1 + E 2))

× exp−

M − (E 1 + E 2)2

θ

− 4π

M 2 − 3

2θ

Erf M √

θ

− 12

√πθM exp

−M 2

θ

. (31)



870 S. Hamid Mehdipour Vol. 52

It can be easily confirmed that these probabilities of emis-sion are uncorrelated. On the other hand, the statisti-cal correlation function, χ(E ; E 1, E 2) is zero which leadsto the independence between different modes of radiationduring the evaporation. Hence, in this method the form of the corrections as back-reaction effects even with consid-ering the noncommutativity corrections are not adequateby themselves to retrieve information because there are no

correlations between different modes at least at late-timesand information does not come out with the Hawking ra-diation (for reviews on resolving the so-called information

loss paradox , see Refs. [21–23]). But the noncommutativ-ity effect is adequate by itself to preserve information dueto the fact that in the noncommutative framework blackhole does not evaporate completely and this leads to theexistence of a minimal non-zero mass (e.g., a Planck-sized

remnant containing the information) which black hole canreduce to it. So information might be preserved in rem-nant. Indeed, it is not conceivable to date to give a clearanswer to the question of the black hole information para-dox and this is reasonable because there is no complete

self-consistent quantum theory of evaporating black holes.

6 Summary

We summarize this paper with some significant re-marks. In this paper, generalization of the standardHawking radiation via tunneling through the event hori-zon based on the solution of Eq. (13) within the contextof coordinate coherent state noncommutativity has beenstudied and then the new corrections of the emission ratebased on noncommutative framework has been achieved.In this study, we see that there are not any correlationsbetween the tunneling rates of different modes in the blackhole radiation spectrum at least at late-times. In our opin-ion, if we truly have faith in the idea of stable black holeremnants due to the fact that there are some exact con-tinuous global symmetries in nature.[4] Then we shouldaccept that the information stays inside the black holeand can be retained by a stable Planck-sized remnant.

Acknowledgment

I would like to thank K. Nozari for careful reading of the preliminary version and a long-time collaboration on

this area.

References

[1] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.

[2] M.K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000)5042; [arXiv:hep-th/9907001].

[3] P. Nicolini, A. Smailagic, and E. Spallucci, Phys. Lett. B632 (2006) 547; [arXiv:gr-qc/0510112].

[4] J.D. Bekenstein, Phys. Rev. D 5 (1972) 1239.

[5] A. Smailagic and E. Spallucci, J. Phys. A 36 (2003) L467;

[arXiv:hep-th/0307217].[6] A. Smailagic and E. Spallucci, J. Phys. A 36 (2003) L517;

[arXiv:hep-th/0308193].

[7] A. Smailagic and E. Spallucci, J. Phys. A 37 (2004) 7169;[arXiv:hep-th/0406174].

[8] H. Weyl, Z. Phys. 46 (1927) 1; E. Wigner, Phys. Rev. 40

(1932) 749; J.E. Moyal, Proc. Camb. Phil. Soc. 45 (1949)99.

[9] P. Painleve, Compt. Rend. Acad. Sci. (Paris) 173 (1921)677.

[10] L.D. Landau and E.M. Lifshitz, The Classical Theory of

Field , Pergamon Press, London (1975) p. 254.

[11] K. Nozari and S.H. Mehdipour, Class. Quant. Grav. 25

(2008) 175015; [arXiv:0801.4074].[12] J. Zhang and Z. Zhao, Nucl. Phys. B 725 (2005) 173.

[13] J. Zhang and Z. Zhao, J. High Energy Phys. (JHEP) 0510

(2005) 055.

[14] Q.Q. Jiang, S.Q. Wu, and X. Cai, Phys. Rev. D 73 (2006)064003; [arXiv:hep-th/0512351].

[15] P. Kraus and F. Wilczek, Nucl. Phys. B 433 (1995) 403;[arXiv:gr-qc/9408003]; P. Kraus and F. Wilczek, Mod.Phys. Lett. A 9 (1994) 3713; [arXiv:gr-qc/9406042].

[16] E. Keski-Vakkuri and P. Kraus, Nucl. Phys. B 491 (1997)249; [arXiv:hep-th/9610045].

[17] S. Massar and R. Parentani, Nucl. Phys. B 575 (2000)333; [arXiv:gr-qc/9903027].

[18] M.K. Parikh, Int. J. Mod. Phys. D 13 (2004)2351; [arXiv:hep-th/0405160]; M.K. Parikh, [arXiv:hep-th/0402166].

[19] A.J.M. Medved and E. Vagenas, Mod. Phys. Lett. A20 (2005) 1723; [arXiv:gr-qc/0505015]; M. Arzano, A.Medved, and E. Vagenas, J. High Energy Phys. (JHEP)0509 (2005) 037; [arXiv:hep-th/0505266].

[20] K.A. Meissner, Class. Quant. Grav. 21 (2004) 5245;[arXiv:gr-qc/0407052].

[21] J. Preskill, An international symposium on Black Holes,

Membranes, Wormholes and Superstrings, Houston Ad-

vanced Research Center , 16-18 January 1992, eds. SunnyKalara and D.V. Nanopoulos, Singapore World Scientific,

Singapore (1993) p. 22, [arXiv:hep-th/9209058].[22] D.N. Page, Phys. Rev. Lett. 71 (1993) 3743; [arXiv:hep-

th/9306083].

[23] J.G. Russo, [arXiv:hep-th/0501132].

Documents

S. Hamid Mehdipour- Parikh–Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole