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

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

The horizon is $r_0 = 2GM_{\rm {b.h.}}$, which's obtained from -g_tt=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.

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

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

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

From the second law of thermodymaics

dU = TdS (4)

$$T_H dS_{\rm {b.h.}} \frac{1}{4\pi r_H }d( \frac{4\pi r_H ^2}{4G} )$$ =

$$d(\frac{r_H}{2G}) = d M_{\rm {b.h.}}$$ = (5)

which is agree with the second law. From classical thermodymaics a sphere will radiate with energy density, u

$$\varpropto$$ u T⁴

$$uV \varpropto T^4 R^3 .$$ U =

and from schwarzschild black hole

$$\varpropto M_{\rm {b.h.}} = U \varpropto T^4 R^3 \ \rm {but}$$ r₀

$$\varpropto \frac{1}{\sqrt{r_0}}$$ T

$$\sim \frac{U}{T} \sim T^3 V = r_0 ^{-3/2}r_0 ^3 \sim r_0 ^{3/2} \sim M_{\rm {b.h.}}^{3/2} << M_{\rm {b.h.}}^2$$ S (6)

which is disagree with the second law. Next try to reconcile by considering strings. From Hamiltonian eigenvalue,

$$\sim \alpha ' p^2 + N$$ H

$$\sim \frac{N}{\alpha '} = \frac{N}{l_s ^2 },\ \alpha '\ =\ l_s ^2 .$$ M²

At high excited strings, N >> 1,

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

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.

$$\sim \ln{D^n} = n\ln{D} \sim n$$ S

$$\rm {total\ mass}\ \ M nm_0 \sim \frac{n}{l_s} \sim \frac{S}{l_s},\ \rm {then}$$ =

$$\sim$$ S Ml_s (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 l_s ²(in everyday life experience, curvature <1/l_s making Einstien's equation still valid). With G$\sim$ g² l_s ² constant, there are two possiblities.

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

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

$$S_{\rm {b.h.}}^{1/2}MG^{1/2} \sim S_{\rm {b.h.}} \ \ \rm {or}$$

$$S_{\rm {b.h.}} \sim$$ M². (8)

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

$$-\frac{1}{\sqrt{f}}(1-\frac{r_0}{r})dt^2 + \sqrt{f}\frac{dr^2}{1-\frac{r_0}{r}} + \sqrt{f}r^2(d\theta ^2 + sin^2 \theta d\phi ^2 )$$ ds² = (9)

$$1+\frac{r_0 sinh^2 \gamma}{r}$$ f(r) = (10)

$$\frac{1}{\sqrt{f}}(1-\frac{r_0}{r})$$ h(r) =

$$\approx \left( 1- \frac{r_0 sinh^2 \gamma }{2r}\right)(1-\frac{r_0}{r})$$

$$\approx 1 - \frac{r_0}{r}\left( 1+\frac{sinh^2 \gamma}{2}\right)$$

$$r_0 \left( 1+\frac{sinh^2 \gamma}{2}\right)$$ 2GM = (11)

$$\frac{h'(r_0)}{4\pi} = \frac{1}{4\pi r_0 \sqrt{f}} = \frac{1}{4\pi r_0 cosh\gamma}$$ T_H = (12)

$$4\pi\sqrt{f}r_0 ^2 = 4\pi r_0 ^2 cosh\gamma$$ A_H = (13)

$$\frac{r_0}{2G}sinh(2\gamma )$$ Q = (14)

$$\frac{r_0 R}{2G}sinh(2\gamma )$$ n = (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

$$\frac{n}{R}$$ p =

$$\frac{p}{2} = \frac{n}{2R}.$$ Q =

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

$$-h(r)dt^2 + \frac{dr^2}{h(r)} + \sqrt{\Delta}r^2 (d\theta ^2 + sin^2 \theta d\phi ^2 )$$ ds² = (16)

$$\frac{1}{\sqrt{\Delta}}(1-\frac{r_0}{r})$$ h(r) =

$$\Delta$$ = f₁ (r)f₂ (r)f₃ (r)f₄ (r) (17)

$$\frac{\lambda _i r_0}{G}sinh(2\gamma _i )$$ Q_i = (18)

$$M_{\rm {b.h.}} \frac{r_0}{8G}\prod _{i=1}^4 cosh(2\gamma _i )$$ = (19)

$$\frac{A_H}{4G} = \frac{\pi r_0 ^2}{G} \prod _{i=1}^4 cosh\gamma$$ S = (20)

$$\frac{1}{4\pi r_0} \prod _{i=1}^4 \frac{1}{cosh \gamma _i}$$ T_H = (21)

$$TdS + \sum_i \Phi _i dQ_i .$$ dU = (22)

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

$$\sinh(2\gamma _i ) \sim \frac{e^{2\gamma _i}}{2}$$

$$\frac{GQ_i}{\lambda _i} \sim \frac{r_0}{2} e^{2\gamma _i}$$

$$\sqrt{r_0}e^{\gamma _i} \sqrt{\frac{2GQ_i}{\lambda _i}}$$ =

$$\frac{\pi r_0 ^2}{G} \prod _i \frac{e^{\gamma _i}}{2}$$ S =

$$\frac{\pi G}{4} \prod _i \sqrt{ \frac{Q_i}{\lambda _i}}$$ = (23)

$$\sim \frac{1}{8} \sum _{i=1} ^4 \frac{Q_i}{\lambda _i}$$ M (24)

$$\frac{r_0 ^2}{4\pi r_0}\prod _{i=1}^4 \frac{2}{e^{\gamma _i}\sqrt{r_0}}.$$ T_H = (25)

As r₀ $\rightarrow$0, T_H $\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

R_i, 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 p_i = n/R_i. And count the states on the branes. The entropy and $\lambda _i$ will be

$$2\prod _{i=1}^4 \sqrt{Q_i}$$ S = (26)

$$\lambda _1 \frac{gl_s ^7}{8R_1 ...R_6}$$ =

$$\lambda _2 \frac{gl_s ^3}{8R_1 R_2}$$ =

$$\lambda _3 \frac{g^2 l_s ^6}{8R_2 R_3 R_4 R_5 R_6}$$ =

$$\lambda _4 \frac{R_2}{8}$$ =

$$\frac{g^2 l_s ^8}{8R_1 ...R_6}.$$ G = (27)

This is agree with the result from GR.