*g*** is a dimensionless coupling constant where **
**(length**^{2}**).
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.
**

**
**

(1) |

(2) |

(3) |

dU |
= | TdS |
(4) |

= | |||

= | (5) |

u |
T^{4} |
||

U |
= |

r_{0} |
|||

T |
|||

S |
(6) |

H |
|||

M^{2} |

**
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**

**
**

**Strings live in D dimensions, therefore there are D possible directions for strings to move to or
**
**, for n steps, ***D*^{n}** possiblities, ie.
**

S |
|||

= | |||

S |
Ml_{s} |
(7) |

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

** (ii) If g large(***l*_{s}** small) or ***M*
** increase, then ***l*_{s}**
small enough ***l*_{s}*r*_{0}** and **
**.
Then ***S*
** and
**

M^{2}. |
(8) |

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

ds^{2} |
= | (9) | |

f(r) |
= | (10) | |

h(r) |
= | ||

2GM |
= | (11) | |

T_{H} |
= | (12) | |

A_{H} |
= | (13) | |

Q |
= | (14) | |

n |
= | (15) |

p |
= | ||

Q |
= |

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

ds^{2} |
= | (16) | |

h(r) |
= | ||

= | f_{1} (r)f_{2} (r)f_{3} (r)f_{4} (r) |
(17) | |

Q_{i} |
= | (18) | |

= | (19) | ||

S |
= | (20) | |

T_{H} |
= | (21) | |

dU |
= | (22) |

**If consider in an extremmal case, with fixed ***Q*_{i}**, ***r*_{0}
0**, then
**
** and
**

= | |||

S |
= | ||

= | (23) | ||

M |
(24) | ||

T_{H} |
= | (25) |

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

**
**

**
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**

*R*_{i}**, ***i*** = 1,...,6, compactified dimension index. D6-brane is on **
(*x*^{1} ,..,*x*^{6} )** with ***Q*_{1}**,
D2-brane is on **
(*x*^{1} ,*x*^{2} )** with ***Q*_{2}**, and D5-brane is on **
(*x*^{2} ,..,*x*^{6} )** with ***Q*_{3}**, where ***p*_{i}** = ***n*/*R*_{i}**.
And count the states on the branes. The entropy and **
** will be
**

S |
= | (26) | |

= | |||

= | |||

= | |||

= | |||

G |
= | (27) |

**This is agree with the result from GR.
**