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Classical black-brane and non-commutative geometry
Chaos, Solitons and Fractals 41 (2009) 1518–1519
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Classical black-brane and non-commutative geometry A. Farmany a,b,c,*, S. Abbasi a, A. Naghipour a, A. Nuri b, N. Rahimi b a b c
Department of Chemistry, University of Ilam, Ilam, Iran Young Researchers Club, Azad University of Ilam, Ilam, Iran Michigan Center for Theoretical Physics, Michigan University, United States
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 11 June 2008
Perturbed Numbo-Goto membrane at the equatorial plane of Reissner-Nordstrøm de-Sitter space-time is studied. It is shown that, the space-time geometry of the black-brane is noncommutative. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Non-commutativity in the coordinates of space-time has been the subject of much interesting works over the years. Genuine non-commutative geometry in the string theory and D-branes first pioneered by Seiberg and Witten [1] where the coordinates of D-brane embedding becomes non-commutative. In addition, the non-commutative geometry in the matrix theory described by Connes in the C*-algebra point of view [2], and the non-commutative cantorian geometry is studied carefully by El Naschi [3,4]. Also, the non-commutative quantum mechanics is pointed out in [5,9] In this letter, we consider the non-commutativity of the space-time coordinates of the perturbed Numbo-Goto membrane at the equatorial plane of the Reissner-Nordstrøm de-Sitter space-time [6–8]. 2. Classical membranes A membrane is the time-like hyper-surface (R,cab) embedded in (3 + 1) dimensional space-time (M, glm). Relation between space-time coordinates xl and membrane coordinates na is,
X l ¼ X l ðna Þ;
l ¼ 0; . . . ; 3 a ¼ 0; 1; 2;
ð1Þ
where the induced metric cab on R is,
cab ¼ X la X mb g lm :
ð2Þ
The dynamics of the membrane is described by the Numbo-Goto action as,
S½xl ; xla ¼ r
Z
3
Rd
pffiffiffiffiffiffiffi n c:
ð3Þ l
Using the Gauss-Weingarten equation, Db X la þ Cab X aa X bb ¼ K ab gl we obtain the equation of motion of the membrane as,
K ¼ cab K ab ; l m
ð4Þ
where K ab ¼ X a X b rm gl and rm is the space-time covariant derivative. Note that Eq. (4) cannot be solved analytically but approximately is k = 0.
* Corresponding author. Address: Department of Chemistry, University of Ilam, Ilam, Iran. E-mail address: [email protected] (A. Farmany). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.013
1519
A. Farmany et al. / Chaos, Solitons and Fractals 41 (2009) 1518–1519
The perturbation equation of the membrane is defined by, lm
U þ ð3 R4 Rlm h ÞU ¼ 0;
ð5Þ
where U = nldXl. 2.1. Black-branes and non-commutative geometry The spherically symmetric Reissner-Nordstrøm de-Sitter black-brane is,
cab dna dnb ¼ f dT 2 þ f 1 dR2 þ R2 du2 ;
ð6Þ
e2
where f ¼ 1 2M þ r2 and M, e are the mass and charge, respectively. In the (RNdeS) black-brane framework Eq. (5) becomes, r
U þ
2 e U ¼ 0: r4
ð7Þ
Moreover, & is the d’Alambert on the membrane and is defined by,
1 o pffiffiffiffiffiffiffi ab o ¼ pffiffiffiffiffiffiffi a cc : c on onb
ð8Þ
pffiffiffi Using a suitable separation variable U ¼ ðxðrÞ= rÞ expðiwtÞ expðimuÞ, and Eq. (7) we obtain a Schrödinger equation as, 2
d x þ VðrÞx ¼ w2 x; dr 0 where r0 ¼
R
ð9Þ
1 . e2 12M r þ 2 r
Note that r0 goes from 1 to +1. In this case the potential v(r) approaches zero as r0 goes to ±1. So Eq. (9) can be solved as, 0
0
x ¼ eiwr þ eiwr :
ð10Þ
With boundary condition,
x ¼ fk1 expðiwr 0 Þ for ðr 0 ! þ1Þ and
x ¼ fk2 expðiwr0 Þ for ðr 0 ! 1Þ:
ð11Þ
Let Xl, Xm, be coordinates, with boundary condition (11) Using Eq. (3) we obtain,
xl ¼
Z
X l ðr; sÞ dr;
xm ¼
Z
X m ðr; sÞ dr:
ð12Þ
So we obtain,
½xl ; xm ¼ 2pihlm ;
ð13Þ
where h is a matrix. Eq. (13) shows the non-commutativity in the coordinates of the (RNdeS) black-brane space-time. 3. Conclusion To conclude, we have derived the perturbation equation to the curved black-brane coordinates. According to Eq. (13) we found that in the perturbed curved black-brane the space-time geometry is non-commutative. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Seiberg N, Witten E. String theory and noncommutative geometry. JHEP 1999;09:032. Connes A. Non-commutative geometry. New York: Academic Press; 1994. El Naschi MS. On the uncertainty of Cantorian geometry and the two slit experiment. Chaos, Solitons & Fractals 1998;9(3):517–29. El Naschi MS. Penrose Universe and Contorian spacetime as a model for noncommutative quantum geometry. Chaos, Solitons & Fractals 1998;9:931–3. Chaichian M, Sheikh-Jabbari MM, Tureanu A. Hydrogen atom spectrum and Lamb shift in noncommutative QED. Phys Rev Lett 2001;86:2716. Higaki S, Ida D, Ishibashi A. Instability of a membrane intersecting a black hole. Phys Rev D 2001;63:025002, and references therein. Garriga J, Vilenkin A. Quantum fluctuations on domain walls, strings, and vacuum bubbles. Phys Rev D 1991;44:1007. Gregory R, Padilla A. Braneworld instantons. Class Quant Grav 2002;19:279. Farmany A. Spinning particles and non-commutative geometry. Phys Scripta 2005;72:353.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Classical black-brane and non-commutative geometry A. Farmany a,b,c,*, S. Abbasi a, A. Naghipour a, A. Nuri b, N. Rahimi b a b c
Department of Chemistry, University of Ilam, Ilam, Iran Young Researchers Club, Azad University of Ilam, Ilam, Iran Michigan Center for Theoretical Physics, Michigan University, United States
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 11 June 2008
Perturbed Numbo-Goto membrane at the equatorial plane of Reissner-Nordstrøm de-Sitter space-time is studied. It is shown that, the space-time geometry of the black-brane is noncommutative. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Non-commutativity in the coordinates of space-time has been the subject of much interesting works over the years. Genuine non-commutative geometry in the string theory and D-branes first pioneered by Seiberg and Witten [1] where the coordinates of D-brane embedding becomes non-commutative. In addition, the non-commutative geometry in the matrix theory described by Connes in the C*-algebra point of view [2], and the non-commutative cantorian geometry is studied carefully by El Naschi [3,4]. Also, the non-commutative quantum mechanics is pointed out in [5,9] In this letter, we consider the non-commutativity of the space-time coordinates of the perturbed Numbo-Goto membrane at the equatorial plane of the Reissner-Nordstrøm de-Sitter space-time [6–8]. 2. Classical membranes A membrane is the time-like hyper-surface (R,cab) embedded in (3 + 1) dimensional space-time (M, glm). Relation between space-time coordinates xl and membrane coordinates na is,
X l ¼ X l ðna Þ;
l ¼ 0; . . . ; 3 a ¼ 0; 1; 2;
ð1Þ
where the induced metric cab on R is,
cab ¼ X la X mb g lm :
ð2Þ
The dynamics of the membrane is described by the Numbo-Goto action as,
S½xl ; xla ¼ r
Z
3
Rd
pffiffiffiffiffiffiffi n c:
ð3Þ l
Using the Gauss-Weingarten equation, Db X la þ Cab X aa X bb ¼ K ab gl we obtain the equation of motion of the membrane as,
K ¼ cab K ab ; l m
ð4Þ
where K ab ¼ X a X b rm gl and rm is the space-time covariant derivative. Note that Eq. (4) cannot be solved analytically but approximately is k = 0.
* Corresponding author. Address: Department of Chemistry, University of Ilam, Ilam, Iran. E-mail address: [email protected] (A. Farmany). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.013
1519
A. Farmany et al. / Chaos, Solitons and Fractals 41 (2009) 1518–1519
The perturbation equation of the membrane is defined by, lm
U þ ð3 R4 Rlm h ÞU ¼ 0;
ð5Þ
where U = nldXl. 2.1. Black-branes and non-commutative geometry The spherically symmetric Reissner-Nordstrøm de-Sitter black-brane is,
cab dna dnb ¼ f dT 2 þ f 1 dR2 þ R2 du2 ;
ð6Þ
e2
where f ¼ 1 2M þ r2 and M, e are the mass and charge, respectively. In the (RNdeS) black-brane framework Eq. (5) becomes, r
U þ
2 e U ¼ 0: r4
ð7Þ
Moreover, & is the d’Alambert on the membrane and is defined by,
1 o pffiffiffiffiffiffiffi ab o ¼ pffiffiffiffiffiffiffi a cc : c on onb
ð8Þ
pffiffiffi Using a suitable separation variable U ¼ ðxðrÞ= rÞ expðiwtÞ expðimuÞ, and Eq. (7) we obtain a Schrödinger equation as, 2
d x þ VðrÞx ¼ w2 x; dr 0 where r0 ¼
R
ð9Þ
1 . e2 12M r þ 2 r
Note that r0 goes from 1 to +1. In this case the potential v(r) approaches zero as r0 goes to ±1. So Eq. (9) can be solved as, 0
0
x ¼ eiwr þ eiwr :
ð10Þ
With boundary condition,
x ¼ fk1 expðiwr 0 Þ for ðr 0 ! þ1Þ and
x ¼ fk2 expðiwr0 Þ for ðr 0 ! 1Þ:
ð11Þ
Let Xl, Xm, be coordinates, with boundary condition (11) Using Eq. (3) we obtain,
xl ¼
Z
X l ðr; sÞ dr;
xm ¼
Z
X m ðr; sÞ dr:
ð12Þ
So we obtain,
½xl ; xm ¼ 2pihlm ;
ð13Þ
where h is a matrix. Eq. (13) shows the non-commutativity in the coordinates of the (RNdeS) black-brane space-time. 3. Conclusion To conclude, we have derived the perturbation equation to the curved black-brane coordinates. According to Eq. (13) we found that in the perturbed curved black-brane the space-time geometry is non-commutative. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Seiberg N, Witten E. String theory and noncommutative geometry. JHEP 1999;09:032. Connes A. Non-commutative geometry. New York: Academic Press; 1994. El Naschi MS. On the uncertainty of Cantorian geometry and the two slit experiment. Chaos, Solitons & Fractals 1998;9(3):517–29. El Naschi MS. Penrose Universe and Contorian spacetime as a model for noncommutative quantum geometry. Chaos, Solitons & Fractals 1998;9:931–3. Chaichian M, Sheikh-Jabbari MM, Tureanu A. Hydrogen atom spectrum and Lamb shift in noncommutative QED. Phys Rev Lett 2001;86:2716. Higaki S, Ida D, Ishibashi A. Instability of a membrane intersecting a black hole. Phys Rev D 2001;63:025002, and references therein. Garriga J, Vilenkin A. Quantum fluctuations on domain walls, strings, and vacuum bubbles. Phys Rev D 1991;44:1007. Gregory R, Padilla A. Braneworld instantons. Class Quant Grav 2002;19:279. Farmany A. Spinning particles and non-commutative geometry. Phys Scripta 2005;72:353.