Physics Procedia 22 (2011) 537 – 542

1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Garry Lee.doi:10.1016/j.phpro.2011.11.083

Available online at www.sciencedirect.com

Available online at www.sciencedirect.com

PhysicsProcedia

Physics Procedia 00 (2011) 000–000

www.elsevier.com/locate/procedia

2011 International Conference on Physics Science and Technology (ICPST 2011)

The Radius of the Schwarzschild Black Hole in Noncommutative Space with GUP

Fatemeh Ahmadia,a*, Fatemeh Valia aDepartment of Physics, Shahid Rajaee Teacher Training University, Lavizan, Tehran 16788, Iran

Abstract

Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so called Generalized Uncertainty Principle (GUP) which predict the existence of a minimum observable length or a maximum observable momentum. On the other hand noncommutativity also corresponds to a minimum measurable length. So we expect that there is a relation between GUP and noncommutativite space in the range of the Planck scale. Here, we are going to derive a relation between the noncommutative coordinate and a commutative coordinate space with GUP. Then we use this new relation to compute the event horizon of a quantum Schwarzschild black hole in the Plank scale. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [ICPST 2011] Keywords: Noncommutativity; Generalized Uncertainty Principle; Schwarzschild Black Hole.

1. Introduction

Heisenberg obtained the Uncertainty Principle on very general grounds, using only the quantization of the electromagnetic radiation field. He did not consider gravitational effects in his uncertainty relation, as is usually assumed to be negligible. However, at increasingly large energies the gravitational interaction is more and more important. Various approaches to Quantum Gravity (such as String Theory, Doubly Special Relativity theories, as well as Black Hole Physics) suggest that near the Planck scale, the Heisenberg Uncertainty Principle should be modified. The modified Uncertainty Principle is called Nomenclature

* Fatemeh Ahmadi. Tel.: +0982122970005; fax: +0982122970005. E-mail address: fahmadi@srttu.edu

© 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Garry Lee.

538 Fatemeh Ahmadi and Fatemeh Vali / Physics Procedia 22 (2011) 537 – 5422 F. Ahmadi, F. Vali / Physics Procedia 00 (2011) 000–000

GUP Generalized Uncertainty Principle

NC Noncommutative

GeV Giga electron Volt

Lpl Planck length

Mpl Planck mass

Generalized Uncertainty Principle (GUP). These corrections are generically quite small to be measurable. However they could signal a new intermediate length scale between the electroweak and the Planck scale [1].

On the other hand, there are arguments in String Theory suggesting that spacetime must change its nature at short distances of the order of the Planck length [2]. Since localization in sharp regions might induce the formation of closed horizons, spacetime must become noncommutative to prevent the instability of the geometric background [3]. Noncommutativity restricts measurement of space which can be interpreted as the emergence of a natural effective ultraviolet cut off, a minimal observable length, which regulating not only gravity but any quantum field theory [4]. A Noncommutative (NC) spacetime which may model a noncommutative space can be described by the Moyal algebra [5]. Noncommutativity eliminates point-like structures in favor of smeared objects in flat spacetime. Based on this idea, several attempts have been performed to find modification of Bekenstein-Hawking formalism of black hole thermodynamics within NC geometry [6].

In brief word, quantum gravity has an uncertainty principle which prevents one from measuring positions to better accuracies than the Planck length: the momentum and energy required to make such a measurement will itself modify the geometry at these scales (DeWitt, 1962) [7]. So we expect that in the fundamental space, Plank scale, both momentum and space change. Thus, in this article we tend to use the GUP in the Moyal algebra to derive a new relation between the commutative and NC coordinates. Finally, we use the new relation to compute the event horizon of a quantum Schwarzschild black hole in probably the fundamental structure. Here we use the natural system, where 1==== BkcG .

Fatemeh Ahmadi and Fatemeh Vali / Physics Procedia 22 (2011) 537 – 542 539 F. Ahmadi, F. Vali / Physics Procedia 00 (2011) 000–000 3

2. The Noncommutative Space

The NC spaces are characterized by the fact that, their coordinate operators satisfy [xi,xj]=iθij, (1)

where θij is a real antisymmetric matrix of dimension (length)2. If this matrix is constant, then the above commutators generate a Heisenberg algebra [8].

There is an isomorphism between the commutative and the NC spaces. By using the Weyl map, which

is ,e)p(f

~pd

)())x(f(W)x(f

jj xip

/ ∫π

== 3232

1 (2)

where )p(f~ is the usual Fourier transform of f(x)[9]

,e)x(xfd

)()p(f

~ jj xip

/ ∫π

= −3232

1 (3)

the relation between the NC and commutative coordinates can be obtained .pxx j

ijii θ+=21 (4)

where

,]p,p[ ji 0= .i]p,x[ ijji δ= (5)

On the other hand, the commutation relations in the commutative space and in quantum scales much bigger than the Plank scale, are

,]x,x[ ji 0= ,]p,p[ ji 0= .i]p,x[ ijji δ= (6)

On the other hand, various theories of Quantum Gravity suggest the following commutation relation between position coordinates and momenta [10]

)],ppp()

ppp

p([i]p,x[ jiijji

ijijji 322 +δα++δα−δ= (7)

where 0=]x,x[ ji , 0=]p,p[ ji (via the Jacobi identity), ∑= =3

12

i ii ppp , plM/0α=α , GeV/lcM plpl

192 10≈= Planck energy, and mlpl3510−≈ Planck length. It is assumed that the

dimensionless parameter 0α is of the order of unity, in which case the α dependent terms are important only when energies (momenta) are comparable to the Planck energy (momentum), and lengths are comparable to the Planck length. The following definitions,

),pp(pp ii

20

200 21 α+α−= ,xx ii 0= (8)

satisfy the equation (7), with ix0 , ip0 satisfying the canonical commutation relations (6). ip0 is the canonical momentum( ii x/ip 00 ∂∂−= ). Here ip0 is the momentum at low energies and ip (the modified

540 Fatemeh Ahmadi and Fatemeh Vali / Physics Procedia 22 (2011) 537 – 5424 F. Ahmadi, F. Vali / Physics Procedia 00 (2011) 000–000

momentum) is the momentum at higher energies [10]. By substituting the canonical momentum with the modified one in the Weyl map, and using

,

)x(g)xx())x(g(

x∑

′−δ

=δ0 0

0 (9)

(where 00 ≠′ )x(g and 0x is the solution of g(x)=0 ), we can obtain the following relation ,p

ppx

x jij

ii θ−

α+α−=

21

21 20

20

(10)

this NC coordinate satisfies the relation (1), to )(O 3α and )(O θα3 . The commutation relation between ixand jp (where jj pp = ) is

],p,x[

pp]p,p

ppx

[]p,x[ jijj

iji

ji 20

20

20

20 21

121

21 α+α−=θ−

α+α−= (11)

.)ppp()

ppp

p(pp

i]p,x[ jiijji

ijijji⎭⎬⎫

⎩⎨⎧

+δα++δα−δα+α−

= 321

2220

20

(12) Here we obtain a relation which relates the coordinate in commutative space near the Plank scale to

one in NC space at plank scale. In the next section we use (10) to calculate the event horizon of the Schwarzschild black hole in the NC space with GUP.

3. The Event Horizon of the Schwarzschild Black Hole in the Framework

Here, we want to obtain the Schwarzschild radius of the black hole by using (10). The Schwarzschild metric in the NC space can be written as follows [11,12]

),dsind(rrrdrd

)r(fdt)r(fsdsd 2222 1

ϕθ+θ−−= (13)

where )r(f is

rrM)r(f 21−= (14)

and, r satisfies (10). The horizon of the NC Schwarzschild black hole satisfies the condition 000 =g , so

,

rrM 021 =− (15)

by using (10) we have

(16)

for simplicity we define 20

20 21 pp α+α−=β . By expanding the denominator of the fraction, we have

2M1- =0,ix 1 x 1j iki( - θ p )( - θ p )ij k2 2 2 22 21-αp +2α p 1-αp +2α p0 0 0 0

Fatemeh Ahmadi and Fatemeh Vali / Physics Procedia 22 (2011) 537 – 542 541 F. Ahmadi, F. Vali / Physics Procedia 00 (2011) 000–000 5

,),(O)

r

ppx

r

px(

rM k

jiikijij

ji

082

121 33222 =θαα+β

θθ−β

θ+β− (17)

where k

ijkij θε=θ21 . Using the identity s

jkr

sr

kj

iksijk δδ−δδ=εε [12], we can rewrite (17) as follows

,)).p(p(M.LMrMr 0

16422 3222223 ≅βθ−θ+

θβ−β− (18)

where ji

ijkk pxL ε= , jj ppp =2 and j

jθθ=θ2 . Supposing that noncommutativity, just exists in the x-y plane, (all components of θ except 3θ is zero) then (18) leads to

,)pp(ML

MrMr 0164

22 323

22

3223 ≅β−θ

+θ

β−β− (19)

since 23

22

21

2 pppp ++= , we can write 22

21

23

2 pppp +=− . Since the angular momentum of the Schwarzschild black hole is zero, the third term vanishes. Thus up to first order of noncommutativity parameter, Schwarzschild geometry doesn't change. By defining a and b

,r)pp(Ma sβ−=α+α−−= 0

20

20 212 (20)

,)pp(r)pp(Mb s

322

21

2

032

221

2

3216β

+θ=β+

θ= (21)

where sr0 is the Schwarzschild radius in the ordinary space, the equation (19) can be rewritten as follows

.barr 023 ≅+−+ (22)

This is a cubic equation which has 2 imaginary roots and a real root [13]. The real root is the event horizon of the Schwarzschild black hole in commutative space with GUP

,)a()ba(ba)a()ba(baar

//

s

31

6233

31

6233

34

272

21

22734

272

21

2273

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+++−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−+++−−≅ (23)

to have a real solution it’s essential that the square root to be positive or zero. Therefore, we can obtain a lower limit for the noncommutativity parameter

.

)pp(r s

22

21

0

32

38

+≥θ (24)

Now, since ba >> the above equation can be approximated as ,

ab

abars 5

3

2 23

−−−≅ (25)

by substituting (20) and (21) in (24), the Schwarzschild radius can be obtained, .

r)pp(rr

sss β

+θ−β≅

0

22

21

2

0 32 (26)

It can be verified that when θ (commutative space with GUP), (25) reduces to β≅ ss rr 0 and as

0→α ( 1→β ), the classical Schwarzschild radius would be regained.

4. Conclusions

542 Fatemeh Ahmadi and Fatemeh Vali / Physics Procedia 22 (2011) 537 – 5426 F. Ahmadi, F. Vali / Physics Procedia 00 (2011) 000–000

In this paper we derive a relation between the NC coordinate space and the commutative one at Planck scale by substituting GUP instead of the Heisenberg Uncertainty Principle. Then, by considering a Schwarzschild black hole at this space, we obtain its event horizon. We find a nonzero lower limit for NC parameter, which depends on the classical Schwarzschild radius and momentum of the black hole. It shows that a black hole cause a NC geometry around itself.

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