String/Brane charge & the non-integral dimension

Transcription

1 Jian-Xin Lu (With Wei and Xu) The Interdisciplinary Center for Theoretical Study (ICTS) University of Science & Technology of China September 28, 2012

2 Introduction

3 Introduction Found four laws of black hole mechanics resembling the four laws of thermodynamics for stationary black hole: Four laws of black hole Four laws of thermodynamics 0-th: constant κ at horizon, 1-th: dm = κ 8π da + ΩdJ + ΦdQ, 2-nd: da dt 0, 3-rd: no black hole for κ = 0, uniform T dm = T ds + ΩdJ + ΦdQ ds dt 0 T = 0 cannot be reached In the above, A = area of horizon and κ = the surface gravity of horizon. We also take k = = c = G = 1.

4 Introduction Bekenstein (72) proposed: S BH A, T BH κ (1.1) while Hawking (74) went a step further to show a black hole radiates and has a temperature T BH = κ ( = cκ ). (1.2) 2π 2πk This gives a precise identification: S = A ( ) = kc3 A 4 4G (1.3) Quantum Thermodynamics! Part of Quantum Gravity!

5 Introduction While with the above a black hole appears as a well-defined thermodynamical system, there exists a serious issue for asymptotically flat black hole with such an interpretation. For example, a Schwarzschild black, S BH = 4πM 2, T BH = 1 8πM (1.4) with M the ADM energy carried by the black hole. This system is actually thermodynamically unstable (the specific heat C < 0)!!!

6 Introduction T(r) = T BH (1 r h /r) 1/2 r Black Hole Horizon r h T(r ) = T BH = 1 8πM

7 Introduction So in order to give a proper consideration of asymptotically flat black hole thermodynamics, we need first to suitably stabilize the black hole thermally. In other words, we need to consider ensembles that include not only the black hole under consideration but also its environment. Further, as self-gravitating systems are spatially inhomogeneous, any specification of such ensembles requires not just thermodynamic quantities of interest but also the place at which they take specific values.

8 Introduction r B Black Hole Event Horizon r h Cavity Black hole (r h ) placed in a cavity (r B ) with fixed T and V.

9 Introduction define: Local stability condition: x = r h r B < 1, q = Q r B < x, β B = 1 T B, β(x) = 1 T (x), b = β B 4πr B, b q (x) = β(x) 4πr B. (1.5) df dx ( b b q (x)) = 0 b = b q ( x) (EOS) (1.6) d 2 F dx 2 db q( x) > 0 (1.7) x= x d x local minimal free energy the negative slope of b( x)

10 Chargeless case (Schwarzschild black hole)

11 Charged case (Reissner-Nordström) There exists a critical charge q c = 5 2( x c = 5 2 5, b c = 0.429) and we actually have three cases to consider: q < q c, there exists a unique temperature T t (q) for each given q at which there exists a first order phase transition between a small and large black holes. We have a line of this first-order phase transition, depending on q < q c and ending at a second-order phase transition point at q = q c ; q = q c, this is a second-order critical point at which there exists no distinction between small and large black holes. The critical exponent can be read from c v (T T c ) 2/3 as 2/3; q > q c for each given temperature T there exists a unique global stable black hole with size r + = r B x.

12 Charged case(reissner-nordström) b q (x) q < q c q = q c q > q c bt bc b x 1 x 2 q x c 1 x The typical behaviors of b(x) vs x for q < q c, q = q c, q > q c. x

13 Van der Waals isotherm

14 Charged p-brane The spatial dimensions transverse to the p-brane is D d = d + 2 and note 1 d 7. (D - d)-dimensions p-brane p-dimensions

15 Phase structure and transition

16 Phase structure Summary The d > 2 branes have the similar phase structure as the charged black hole in canonical ensemble, for example, both have van der Waals-Maxwell like phase structure when q < q c. Note the reduced critical quantities are completely determined by d. While the d = 1 brane phase structure resembles that of chargeless black hole instead. The d = 2 case serves as a boardline between d > 2 and d = 1 in that its phase structure resembles that of d = 1 case with no first order phase transition line ending on a critical point while it has also critical quantities (q c, b c ).

17 Critical quantities The relevant quantities at the critical point can be calculated explicitly for each allowed value of d ( d = D d 2, d = p + 1) as: d q c b c p (D = 10) p = p = p = p = p = p = 0

18 Phase structure and transition One natural question is: what means can be used to modify the underlying phase structure of a given type of branes? The simplest for this goal is to consider D5 (or NS5)-branes in D = 10 since they are on the borderline of different types of phase structure as leaned from the above.

19 D1/D5 (F/NS5) The D1/D5 (F/NS5) system: d = 2 delocalized D1 brane 5 brane

20 D1/D5 (F/NS5) The EOS b = bq1,q5 ( x), (2.1) 1/2 1 x ( ) q 2 1 q ( /x ) 2 1/2 b q1,q5 (x) = 1 1 x 1 1 x 1 q 2 x1/2 5 2/x 1 q2 5 x 2. Note (2.2) q 5 < x < 1. (2.3)

21 D1/D5 (F/NS5) Consider q 1 = 0, ( ) 1/2 b 0,q5 (x) = b q5 (x) = 1 1 x 2 x1/2. (2.4) 1 q2 5 x Note q 5 < x < 1. (2.5)

22 D1/D5 (F/NS5) Now consider instead q 5 = 0, [ b q1,0(x) = 1 2 (1 x + ] 1/2 x 2 + 4q 2 x)1/2 1 (1 x). (2.6) 2 Note 0 < x < 1. (2.7)

23 D1/D5 (F/NS5) Choose the proper variable y as y = x + x 2 + 4q1 2 (1 x) < 1, (2.8) 2 and x = 0 gives y = q 1, x = 1 gives y = 1, so now q 1 < y < 1. b q1,0(y) = 1 2 y1/2 1 y 1 q2 1 y 1/2. (2.9)

24 D1/D5 (F/NS5) So the delocalized D-strings (F-strings) have the same phase structure as the D5-branes (NS5-branes)!

25 D1/D5 (F/NS5) This symmetry in phase must imply the existence of a variable f with which b q1,q 5 (f) = b q5,q 1 (f), (2.10) i.e., symmetric with respect to q 1 and q 5. This can be achieved via x = 1 f + (1 f) 2 + 4q5 2f, (2.11) 2

26 D1/D5 (F/NS5) b q1,q 5 (f) = 1 ( ) [ f 1/2 1 f + ] 1/2 (1 f) 2 + 4q1 2f 2 1 f 2 [ 1 f + ] 1/2 (1 f) 2 + 4q5 2f, (2.12) 2 where 0 < f < 1. (2.13)

27 D1/D5 (F/NS5) If either q 1 or q 5 is zero, the corresponding system behaves like the case of simple black 5-brane. b q1,q 5 (f 0) 0, b q1,q 5 (f 1) q i, (i = 1 or 5) (2.14) In other words, the delocalized charged 1-branes along 4-spatial directions behave like a charged 5-brane in phase structure. When both q 1 and q 5 are non-zero, b q1,q 5 (f 0) 0, b q1,q 5 (f 1), (2.15) therefore this system appears to mimics a system of simple black p-branes with d > 2.

28 D1/D5 (F/NS5) So the presence of q 1 effectively increases the transverse dimensions to the D5 (NS5) branes! To be precise, the increase of transverse dimensions is directly related to q 1c, instead of q 1 itself.

29 Critical quantities The relevant quantities at each critical point defined by 2/3 f c 1 can be calculated explicitly as: f c q 1c q 5c b c

30 D1/D5 (F/NS5) Now there exists a critical line of second-order transition determined by q 2 1c = (1 + f c) 2 [(2 f c ) 2 ± (2 f c )(3f c 2)] 2 4f c (10 5f c + f 2 c ) 2 (1 f c) 2 4f c, q 2 5c = (1 + f c) 2 [(2 f c ) 2 (2 f c )(3f c 2)] 2 4f c (10 5f c + f 2 c ) 2 (1 f c) 2 where 4f c (2.16) 2/3 < f c < 1. (2.17) For comparing with 5-branes, we choose the plus sign in q 5c and the minus sign in q 1c abive. Then 0 < q 1c < 6/16, 6/16 < q5c < 1/3. (2.18)

31 Phase structure and transition

32 Discussions The charge q 1 (q 1c ) of delocalized D1 (F) effectively increases the transverse dimensions of D5 (NS5) from the d = 2 to a d only slightly greater than 2 ( d > 2) in the thermodynamical sense and so we find a way of generating spatial dimension using charge which appears to be new. The generated dimension can be continuous or discrete, depending on that the charge is continuous or quantized. So we find the familiar phenomena as in usual critical phenomenon where the spatial dimension can be continuous.

33 THANK YOU!