Gravity and Strings

584 String black holes in four and five dimensions
The mass formula should be compared with that of the extreme BH Eq. (20.27): branes
and antibranes contribute equally to it, and there seems to be no interaction/binding energy
between them.
The entropy takes the form
√ND5 S = 2π − √ND1 ¯ND5 − √NW ¯ND1 −
¯NW . (20.35)
d = 4 Black holes from intersecting branes. In [549] a stringy model based on a system of
intersecting D6, D2, S5A, and W in the configuration
0 1 2 3 4 5 6 7 8 9
D6 + + +++++−−−
S5 + + ++++∼−−−
D2 + + ∼∼∼−+−−−
W + →∼∼∼∼∼−−−
(20.36)
was proposed in order to describe d = 4 BHs. In the extreme limit (which had been constructed
before in [606, 680]) it has a regular horizon and moduli that are regular there (if
the D2 were placed in directions 012, then the moduli would be singular at the horizon).
The construction of the d = 10 solution is similar to that of the solution associated with the
d = 5BHwhich we have discussed in detail and we will therefore omit unnecessary details
here.
The black intersecting solution is
d ˆs 2 s = H − 1 2
D6 H − 1
2
D2 H −1
W Wdt2
1 − HW dy + αW(H −1
W − 1)dt2
− H − 1 2
D6 H 1 2
D2 [(dy2 ) 2 + (dy 3 ) 2 + (dy 4 ) 2 + (dy 5 ) 2 ]
− H − 1 2
D6 H − 1 2
D2 HS5(dy 6 ) 2 − H 1 2
D6 H 1 2
D2 HS5[W −1 dr 2 + r 2 d 2 (2) ],
e−2 ˆφ −2 ˆφ0 = e H 3 2
D6H − 1 2
D2
Ĉ (3) ty 1 y 6 = αD2e − ˆφ0
Hi = 1 + ri
r , α2 i
H −1
S5 ,
ˆB (6) ty 1 ···y 5 = αS5e −2 ˆφ0
H −1
D2 − 1 , Ĉ (7) ty 1 ···y 6 = αD6e − ˆφ0
−1
HS5 − 1 ,
−1
HD6 − 1 ,
=+1, i = D6, D2, S5, W,
W = 1 + ω
r , ω= ri(1 − α2 i ), i = D6, D2, S5, W,
(20.37)
We dimensionally reduce this solution in three steps: first on S 1 (y 1 ), then on T 4