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Strings at Strong Coupling


g is a dimensionless coupling constant where $\alpha ' = l_s ^2 $(length2). This series is valid for smalll g and breaks down when g is large, eg. black holes or low energy case. Before considers strings at strong coupling let look at a simple black hole, Schwarzschild black hole.

Schwarzschild black hole

\begin{displaymath}ds^2 = -\left( 1-\frac{r_0}{r}\right) dt^2 + \frac{dr^2}{1-\f...
...}} + r^2 \left(
d\theta ^2 + \sin ^2{\theta} \phi ^2 \right)
\end{displaymath} (1)

The horizon is $r_0 = 2GM_{\rm {b.h.}}$, which's obtained from -gtt=1-r0 /r by letting r $\rightarrow$$\infty$ and expanding in term of 1/r. The first order is 2V(r), where V(r) = GM/r. The black hole radiates giving a Hawking temperature, ie.

\begin{displaymath}T_H = \frac{\hbar}{4\pi}h'(r_0 ) = \frac{1}{4\pi r_0},
\end{displaymath} (2)

where h(r) = 1- r0 /r and set $\hbar$=1. The entropy is

\begin{displaymath}S = \frac{A_H}{4G},\ A_H = \rm {area\ of\ horizon}.
\end{displaymath} (3)

From the second law of thermodymaics
dU = TdS (4)
$\displaystyle T_H dS_{\rm {b.h.}}$ = $\displaystyle \frac{1}{4\pi r_H }d( \frac{4\pi r_H ^2}{4G} )$  
  = $\displaystyle d(\frac{r_H}{2G}) = d M_{\rm {b.h.}}$ (5)

which is agree with the second law. From classical thermodymaics a sphere will radiate with energy density, u
u $\textstyle \varpropto$ T4  
U = $\displaystyle uV \varpropto T^4 R^3 .$  

and from schwarzschild black hole
r0 $\textstyle \varpropto$ $\displaystyle M_{\rm {b.h.}} = U \varpropto T^4 R^3 \ \rm {but}$  
T $\textstyle \varpropto$ $\displaystyle \frac{1}{\sqrt{r_0}}$  
S $\textstyle \sim$ $\displaystyle \frac{U}{T} \sim T^3 V = r_0 ^{-3/2}r_0 ^3 \sim r_0 ^{3/2} \sim M_{\rm {b.h.}}^{3/2}
<< M_{\rm {b.h.}}^2$ (6)

which is disagree with the second law. Next try to reconcile by considering strings. From Hamiltonian eigenvalue,
H $\textstyle \sim$ $\displaystyle \alpha ' p^2 + N$  
M2 $\textstyle \sim$ $\displaystyle \frac{N}{\alpha '} = \frac{N}{l_s ^2 },\ \alpha '\ =\ l_s ^2 .$  

At high excited strings, N >> 1,

#1#2#3#4#5 @font#1#2pt #3#4#5
...}{\updefault}The strings looks like ramdom walks with N steps}}}}}

Strings live in D dimensions, therefore there are D possible directions for strings to move to or $D\times D\times \cdots $, for n steps, Dn possiblities, ie.

S $\textstyle \sim$ $\displaystyle \ln{D^n} = n\ln{D} \sim n$  
$\displaystyle \rm {total\ mass}\ \ M$ = $\displaystyle nm_0 \sim \frac{n}{l_s} \sim \frac{S}{l_s},\ \rm {then}$  
S $\textstyle \sim$ Mls (7)

but black holes have a strong coulping g or strong gravitational interaction. Therefore let take the strong coupling g into consideration. From classical physics, we know that G is a constant and has unit time2 and from strings G$\sim$ $g^2\alpha '$$\sim$ g2 ls 2(in everyday life experience, curvature <1/ls making Einstien's equation still valid). With G$\sim$ g2 ls 2 constant, there are two possiblities.

$\ \ \ \ \ $ (i) If g small, then string perturbation thoery is still valid.

$\ \ \ \ \ $ (ii) If g large(ls small) or M$\sim$ $\frac{\sqrt{N}}{l_s}$ increase, then ls small enough ls$\sim$r0 and $S_{\rm {bh.}}$$\sim$ $\frac{r_0 ^2}{G}$$\sim$ $\frac{1}{g^2}$. Then S $\rightarrow$ $S_{\rm {b.h.}}$ and

$\displaystyle S_{\rm {b.h.}}^{1/2}MG^{1/2}$ $\textstyle \sim$ $\displaystyle S_{\rm {b.h.}} \ \ \rm {or}$  
$\displaystyle S_{\rm {b.h.}}$ $\textstyle \sim$ M2. (8)

Next let consider Kaluza-Klein black hole with the metric and following properties

ds2 = $\displaystyle -\frac{1}{\sqrt{f}}(1-\frac{r_0}{r})dt^2 + \sqrt{f}\frac{dr^2}{1-\frac{r_0}{r}} +
\sqrt{f}r^2(d\theta ^2 + sin^2 \theta d\phi ^2 )$ (9)
f(r) = $\displaystyle 1+\frac{r_0 sinh^2 \gamma}{r}$ (10)
h(r) = $\displaystyle \frac{1}{\sqrt{f}}(1-\frac{r_0}{r})$  
  $\textstyle \approx$ $\displaystyle \left( 1- \frac{r_0 sinh^2 \gamma }{2r}\right)(1-\frac{r_0}{r})$  
  $\textstyle \approx$ $\displaystyle 1 - \frac{r_0}{r}\left( 1+\frac{sinh^2 \gamma}{2}\right)$  
2GM = $\displaystyle r_0 \left( 1+\frac{sinh^2 \gamma}{2}\right)$ (11)
TH = $\displaystyle \frac{h'(r_0)}{4\pi} = \frac{1}{4\pi r_0 \sqrt{f}} = \frac{1}{4\pi r_0 cosh\gamma}$ (12)
AH = $\displaystyle 4\pi\sqrt{f}r_0 ^2 = 4\pi r_0 ^2 cosh\gamma$ (13)
Q = $\displaystyle \frac{r_0}{2G}sinh(2\gamma )$ (14)
n = $\displaystyle \frac{r_0 R}{2G}sinh(2\gamma )$ (15)

where Q, a charge, comes from momentun in the extra dimension and n, a number of state, relates to momentum and the extra dimension radius, R, by
p = $\displaystyle \frac{n}{R}$  
Q = $\displaystyle \frac{p}{2} = \frac{n}{2R}.$  

The 10 dimensional Kaluza-Klein metric, reduced to 4-D, and its properties are

ds2 = $\displaystyle -h(r)dt^2 + \frac{dr^2}{h(r)} + \sqrt{\Delta}r^2 (d\theta ^2 + sin^2 \theta d\phi ^2 )$ (16)
h(r) = $\displaystyle \frac{1}{\sqrt{\Delta}}(1-\frac{r_0}{r})$  
$\displaystyle \Delta$ = f1 (r)f2 (r)f3 (r)f4 (r) (17)
Qi = $\displaystyle \frac{\lambda _i r_0}{G}sinh(2\gamma _i )$ (18)
$\displaystyle M_{\rm {b.h.}}$ = $\displaystyle \frac{r_0}{8G}\prod _{i=1}^4 cosh(2\gamma _i )$ (19)
S = $\displaystyle \frac{A_H}{4G} = \frac{\pi r_0 ^2}{G} \prod _{i=1}^4 cosh\gamma$ (20)
TH = $\displaystyle \frac{1}{4\pi r_0} \prod _{i=1}^4 \frac{1}{cosh \gamma _i}$ (21)
dU = $\displaystyle TdS + \sum_i \Phi _i dQ_i .$ (22)

If consider in an extremmal case, with fixed Qi, r0 $\rightarrow$0, then $\gamma _i$ $\rightarrow$$\infty$ and

$\displaystyle \sinh(2\gamma _i )$ $\textstyle \sim$ $\displaystyle \frac{e^{2\gamma _i}}{2}$  
$\displaystyle \frac{GQ_i}{\lambda _i}$ $\textstyle \sim$ $\displaystyle \frac{r_0}{2} e^{2\gamma _i}$  
$\displaystyle \sqrt{r_0}e^{\gamma _i}$ = $\displaystyle \sqrt{\frac{2GQ_i}{\lambda _i}}$  
S = $\displaystyle \frac{\pi r_0 ^2}{G} \prod _i \frac{e^{\gamma _i}}{2}$  
  = $\displaystyle \frac{\pi G}{4} \prod _i \sqrt{ \frac{Q_i}{\lambda _i}}$ (23)
M $\textstyle \sim$ $\displaystyle \frac{1}{8} \sum _{i=1} ^4 \frac{Q_i}{\lambda _i}$ (24)
TH = $\displaystyle \frac{r_0 ^2}{4\pi r_0}\prod _{i=1}^4 \frac{2}{e^{\gamma _i}\sqrt{r_0}}.$ (25)

As r0 $\rightarrow$0, TH $\rightarrow$0 or no radiation but S still constant, ie. the black hole is still stable(no quatum corrections needed) or no g expansion or this is non-perturbative result.

From string theory, six dimensions are compactified. Let try to do different combinations of D-brane.

#1#2#3#4#5 @font#1#2pt #3#4#5

Ri, i = 1,...,6, compactified dimension index. D6-brane is on (x1 ,..,x6 ) with Q1, D2-brane is on (x1 ,x2 ) with Q2, and D5-brane is on (x2 ,..,x6 ) with Q3, where pi = n/Ri. And count the states on the branes. The entropy and $\lambda _i$ will be

S = $\displaystyle 2\prod _{i=1}^4 \sqrt{Q_i}$ (26)
$\displaystyle \lambda _1$ = $\displaystyle \frac{gl_s ^7}{8R_1 ...R_6}$  
$\displaystyle \lambda _2$ = $\displaystyle \frac{gl_s ^3}{8R_1 R_2}$  
$\displaystyle \lambda _3$ = $\displaystyle \frac{g^2 l_s ^6}{8R_2 R_3 R_4 R_5 R_6}$  
$\displaystyle \lambda _4$ = $\displaystyle \frac{R_2}{8}$  
G = $\displaystyle \frac{g^2 l_s ^8}{8R_1 ...R_6}.$ (27)

This is agree with the result from GR.

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