Strings at Strong Coupling

g is a dimensionless coupling constant where (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

 (1)

The horizon is , which's obtained from -gtt=1-r0 /r by letting r 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.

 (2)

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

 (3)

From the second law of thermodymaics
 dU = TdS (4) = = (5)

which is agree with the second law. From classical thermodymaics a sphere will radiate with energy density, u
 u T4 U =

and from schwarzschild black hole
 r0 T S (6)

which is disagree with the second law. Next try to reconcile by considering strings. From Hamiltonian eigenvalue,
 H 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 , for n steps, Dn possiblities, ie.

 S = S 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 g2 ls 2(in everyday life experience, curvature <1/ls making Einstien's equation still valid). With G 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 increase, then ls small enough lsr0 and . Then S and

 M2. (8)

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

 ds2 = (9) f(r) = (10) h(r) = 2GM = (11) TH = (12) AH = (13) Q = (14) 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
 p = Q =

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

 ds2 = (16) h(r) = = f1 (r)f2 (r)f3 (r)f4 (r) (17) Qi = (18) = (19) S = (20) TH = (21) dU = (22)

If consider in an extremmal case, with fixed Qi, r0 0, then and

 = S = = (23) M (24) TH = (25)

As r0 0, TH 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 will be

 S = (26) = = = = G = (27)

This is agree with the result from GR.