Entropy of Black-Branes System and T-Duality

General Relativity and Gravitation (2011) DOI 10.1007/s10714-010-1128-8

RESEARCHARTICLE

Wung-Hong Huang Entropy of black-branes system and T-duality

Received: 23 May 2010 / Accepted: 17 November 2010 c Springer Science+Business Media, LLC 2010

Abstract A general black-branes system under the T-duality transformation could produce a smeared system with different dimensional black branes. We ﬁrst use some simple examples to see that both systems have the same entropy and then present a rigorous method to prove this general property. Using the property we could easily know the entropy of some complex black-brane systems. Keywords Black-branes, T-duality

1 Introduction

T-duality is a symmetry of string theory relating small and large distances. It does not shown in point particle theory, indicating that the extended object of strings will experience the spacetime in dramatically different way from that in the point object of particle. Especially, it relates different string theories to each other, which preceded the second superstring revolution [1]. T-duality could also relate a Dp brane to another Dp±1 brane. The property had been used to find many supergravity backgrounds which are dual to different kind of field theories. For example, Maldacena and Russo [2; 3] had found the supergravity background which duals to the non-commutative gauge theory through T-duality. They also discussed the thermodynamics of near-extremal D3-branes with B fields and found that the entropy and other thermodynamic quantities are the same as those of the corresponding D3-branes without B fields. This means that the thermodynamics of non-commutative gauge theory is the same as that in the commutative gauge theory [2; 3; 4; 5; 6]. Using the T-duality and twist one can construct the supergravity backgrounds which dual to the finite temperature non-commutative dipole field theories [7; 8;

W.-H. Huang Department of Physics National Cheng Kung University Tainan, Taiwan whh- [email protected] 2 W.-H. Huang

9; 10]. The spacetime found for the case of N = 2 theory is [10] dz2 ds2 = f (r)−1/2 −h(r)dt2 + dx2 + dy2 + 10 1 + B2U2 sin2 θ " 1/2 −1 2 2 2 2 2 2 2 2 2 + f (r) h(r) dr + r dθ + cos θdφ + sin θ dχ1 + cos χ1dχ2 (1) 2 2 4 2 2 2 !# 2 2 r B sin θ cos χ1dχ2 + sin χ1dχ3 +sin χ1dχ − , 3 1 + r2B2 sin2 θ 2 2 2 2 r Bsin θ cos χ1dχ2 + sin χ1dχ3 2Φ 1 Bzχ = − e = , i 1 + r2B2 sin2 θ 1 + r2B2 sin2 θ 4 r4 f (r) = + N h(r) = − 0 in which 1 r4 and 1 r4 . Using the resulting metric and dilaton field we can find the entropy S of the solution with a help of symbolic algebra calculation in a computer. The result shows that it is the same as that without dipole field strength B fields. The spacetime for the case of N = 1 and N = 0 have also been found by us in [10] (which is too lengthy to be cited in here). After a computer calculation we also find that the system has a same entropy as that without dipole field strength. It is known that a black-branes system under the T-duality transformation will become another smeared system with different dimensional black branes. Above property leads us to suspect that both systems have a same value of entropy in the general case and we will prove this property in this short paper. Historically, Suzuki [11] had shown the property in the system of the non- extremal intersecting D-branes. In this paper we present a more straightforward method to prove this property in a general black-brane system which has non-zero NS-NS B field.

2 Entropy of black-brane systems and T-duality

To begin with, we quote the formula of T-duality [12; 13; 14; 15; 16]. After the T-duality on z coordinate the metric and dilaton field become: −2φ˜ −2φ g˜zz =1/gzz, g˜µz =Bµz/gzz, g˜µν =gµν −(gµzgνz−BµzBνz)/gzz e =gzz e . (2) Step 1 : First, let us consider the simplest case of black D3 brane which has the metric 2 −1/2 2 2 2 2 ds10 = f (r) −h(r)dt + dx + dy + dz 1/2 −1 2 2 2 + f (r) h(r) dr + r dΩ5 , (3) 1 It has zero value of dilaton field. As the entropy density is s = 4 A we can calculate the horizon area A from above metric and find the entropy density of black D3 brane

1 5 5 s = f (r ) 4 r Ω , (4) 4 h h 5 Entropy of black-branes system and T-duality 3

in which rh is the horizon radius. To proceed, we may consider all the (D-1) space dimensions to be compact (with periodic boundary condition–box normalization). Of these some may be large corresponding to eventual non-compact directions, while the others may be small and correspond to small compact dimensions, but the conclusions are inde- pendent of these details. A p-brane in this space has (D-1-p) transverse directions. So it’s entropy density per unit p-volume is given by the ‘area’ of a (D-2-p)- dimensional surface. The total entropy, therefore, is this density integrated over the p-volume of the brane. This is given by the square-root of the determinant of the (D-2)-metric with time and radial coordinates removed. The key computation of the paper is that of the (D-2)-dimensional area and show that it is invariant under T-duality.

Thus, the total entropy SD3 is Z √ 1 1 5 S = s g g g dxdydz = f (r ) 2 r Ω L L L D3 xx yy zz 4 h h 5 x y z 1 Z p = |gD−2| ∏dxi, (5) 4 i=1 in which Lx is the length of the compacted coordinate x, for example. gD−2 repre- sents the determinant of metric without t and r coordinates. After the T-duality on z coordinate the metric becomes

2 −1/2 2 2 2 ds10 = f (r) −h(r)dt + dx + dy 1/2 2 −1 2 2 2 −2φ −1/2 + f (r) dz + h(r) dr + r dΩ5 ,e = f (r) , (6) which describes black D2-brane smeared along z-direction in string frame, in which the dilaton ﬁeld is not zero. However, the area is to be computed in the Einstein metric and this has a factor of the dilaton compared to the string metric. More precisely, the line elements in string frame and Einstein frame has relation 1 2 2φ 4 2 dsS = e dsE . It is crucial to take into account this dilaton-ﬁeld factor to prove the invariant property of entropy in this paper. The corresponding metric in the Einstein frame is therefore

2 −1/8h −1/2 2 2 2 ds10 = f (r) f (r) −h(r)dt + dx + dy

1/2 2 −1 2 2 2i + f (r) dz + h(r) dr + r dΩ5 . (7)

It is interesting to see that the metric of a black D0-brane which is uniformly smeared along x, y, z is described by [17; 18]

2 −1/2 2 ds10 = f (r) −h(r)dt 1/2 2 2 2 −1 2 2 2 + f (r) dx + dy + dz + h(r) dr + r dΩ5 , (8) e−2φ = f (r)−3/2.

With a T-duality transformation on the directions x, y we deduce the solution for black D2-brane smeared along the z-direction, which is also described by (6). 4 W.-H. Huang

Now, from (7) we can ﬁnd the entropy density of black smeared D2 brane

1 9 5 s = f (r ) 8 r Ω L , (9) 4 h h 5 z

Thus, the total entropy SD2 is Z √ 1 1 5 S = s g g dxdy = f (r ) 2 r Ω L L L D2 xx yy 4 h h 5 x y z 1 Z p = |gD−2| ∏dxi, (10) 4 i=1 which is equal to (5) and both system have the same value of |gD−2|. Therefore, in the following we will investigate the invariant property of |gD−2| itself. How the intrinsic cancelation mechanism, which leads to the invariant property, occurs in the mathematic calculation could be seen in the next example. Step 2 : Next, we consider the more general system with the D dimensional background described by the following metric and dilaton ﬁeld

D 2 2 2 −2φ 2 −2φ D−2 2 dsS = ∑ Aidxi , e , ⇒ dsE = e dsS, (11) i=1

2 2 in which dsS is the line element in the string frame metric and dsE is that in the Einstein frame. It follows that ! ! 2 D−2 D−2 D−2 −2φ D−2 −4φ |gD−2| = e ∏ Ai = e ∏ Ai . (12) i=1 i=1

After T-duality on the x1 coordinate the metric and dilaton ﬁeld become

D 2 2 2 2 dx1 −2φ˜ −2φ 2 −2φ D−2 2 ds˜S = ∑ Aidxi + , e = A1 e , ⇒ ds˜E = A1 e ds˜S. i6=1 A1 (13)

It follows that ! ! 2 D−2 D−2 D−2 −2φ D−2 2 −4φ 1 |g˜D−2| = A1e ∏ Ai = A1e 2 ∏ Ai . (14) i6=1 A1 i=1

2 Above calculations tell us that the extra factor in the dilaton term, i.e. A1, just be 1 canceled by the extra factor in the metric term, i.e. 2 . The intrinsic cancelation A1 mechanism therefore leads to the invariant property of |gD−2| once |gD−2| contains the coordinate which T-duality performs. Notice that |gD−2| is the corresponding determinant of the metric without the coordinates x2 and x3 which can be the arbitrary coordinates except x1. Step 3 : Finally, consider the most general black-branes system which is described by the metric gµν and a dilaton ﬁeld φ. After the T-dual transformation Entropy of black-branes system and T-duality 5

on z coordinate we have the new metricg ˜µν and a dilaton ﬁeld φ˜. With the help of transformation relation (2) we now have to investigate the following determinant

g˜i j g˜iz gi j − (gizgiz − BizBiz)/gzz Biz/gzz ≡ (15) g˜ g˜ B /g 1/g iz zz iz zz zz Using a simple property of matrix determinant relation (15) becomes

g˜i j g˜iz gi j − gizgiz/gzz Biz/gzz gi j − gizgiz/gzz 0 = = g˜ g˜ 0 1/g 0 1/g iz zz zz zz

gi j − gizgiz/gzz giz/gzz gi j giz/gzz 1 gi j giz = = = . 0 1/g g /g 1/g 2 g g zz iz zz zz gzz iz zz (16)

Notice that the above determinant transformation is performed in the string 1 frame. Now, the extra factor 2 in above result will be canceled by that from the gzz dilation ﬁeld if we consider the determinant |gD−2| in Einstein frame, as could be easily seen from (13). Thus we have shown that the entropy of the T-duality transformed black-branes system, which being a smeared lower dimensional black- branes system, is the same as the original system.

3 Discussions

Let us make following comments to conclude this paper. 1. As the entropy in statistics is related to the number of the microstate it is rea- sonable to belive that the number of the microstate does not be changed once the space was performed T-duality transformation. Thus the entropy of the T-duality transformed black-branes system is the same as the original system. However, this argument could only be directly applied to the p-branes system such as the Strominger-Vafa model [19], in which the entropy could be calculated directly by counting the degeneracy of BPS soliton bound states in the p-branes system. 2. The steps to construct the Maldacena and Russo [2; 3] model are performing a T-dual, then a rotation and finally take a T-dual. As the rotation does not modify the volume the final value of |gD−2| is therefore invariant. Thus the entropy is invariant. When this procedure could be used to found the corresponding supergravity background for arbitrary gauge theory (that in [2; 3] is the N = 4 theory) then we may conclude that the arbitrary gauge theory has a same entropy as its corresponding noncommutatitive part of gauge theory. 3. In constructing the dual gravity of noncommutatitive dipole theory [7; 8; 9; 10] we perform a T-dual, then a twist and finally take a T-dual. As the twist does not modify the volume the entropy is also invariant. Using the property the entropy associated with (1) could be known without complex computer calculations. 4. Finally, the property derived in this paper could be the corresponding determinant of the metric without the two arbitrary coordinates, which in evaluating the black brane entropy are the time t and radius r coordinates. Thus, besides the 6 W.-H. Huang entropy there are many other invariant quantities under the T-duality transformation. The physical meaning of theses quantities are unknown as yet.

Acknowledgements The author thanks Kuo-Wei Huang for his encouragement and interesting discussion. We are supported in part by the Taiwan National Science Council.

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