302
Black holes in string theory
that is needed to complete the resemblance to (lO.lOI) is a further warping
approximating H- I in the xO-direction. In any case, in its present form,
the metric does not have a finite horizon 'area', because gll vanishes at the
horizon: it requires a further warping approximating H. In fact, the missing
ingredient is supplied by adding momentum in the x I-direction. Since the ex treme
solution is boost invariant, we cannot add momentum simply by boosting it.
The generalization arises [21] because the solutions have a null-bypersurface
orthogonal isometry which permits the replacement
(dx~2 _ (dx l )2 -+ (dx 0 )2 _ (dx l )2 + Hp(r)(dx 0 _ dx l )2
(l0.144)
for an arbitrary harmonic function Hp(r). This is exactly what happens if we
do a Kaluza-Klein reduction along the x I-direction but with a non-zero offdiagonal component in the metric gOl/gll = Hp(r)-I (see exercise 7). Indeed,
the resulting metric can be obtained from a non-extreme metric by this means:
the dilaton and gauge potential form fields are unaffected. The final form of the
metric is
ds 2 = HS-I/4ii~3/4[(dxO)2 _ (dx l )2 + Hp(dx 0 - dx l )2]
_ HS-I/4iill/4[(dx 2 )2 + ... + (dx S )2] _ H:'4iill/4[dr 2 + r2dn~].
(10.145)
We may write the hannonic functions in the form
2
_ r.
L2
r2
H. = 1 +'2
H5 = I +-f Hp=..J:..
(10.146)
r
r
r2
where, using (10.106), (10.131) and (10.140),
2
2 I (27r #)4 cl
rf = q. g,a' (27r #)4 L~ = qsg,a' rp = ng,a V. "2
V4
4
R
(10.147)
with R the radius of the circle upon which the x l-coordinate is compactified, and
n an integer specifying the (right-moving) momentum P = n/ R on the circle.
Adding this momentum breaks a further half of the supersymmetries leaving a
total of four preserved supersymmetry charges. This corresponds to.N = I
supersymmetry in five dimensions. The 'area' A H of the event horizon is defined
as the (eight-dimensional) volume of the time slice at the horizon. Taking the
limit r -+ 0 from above, this gives a product of factors, one from each of the
three disjoint pieces of the metric:
AH = [(r~3/4L;I/4rp )27r R][(r:/ 4 L;1/4)4V4)[(r:/ 4 L~/4)327r2]
= 47r 3 RV4r.Lsrp.
(10.148)
The surface gravity and, hence, the black-hole temperature is zero, as might be
expected from an extreme black hole (see exercise 7). Using the (generalized)
Black holes in string theory
that is needed to complete the resemblance to (lO.lOI) is a further warping
approximating H- I in the xO-direction. In any case, in its present form,
the metric does not have a finite horizon 'area', because gll vanishes at the
horizon: it requires a further warping approximating H. In fact, the missing
ingredient is supplied by adding momentum in the x I-direction. Since the ex treme
solution is boost invariant, we cannot add momentum simply by boosting it.
The generalization arises [21] because the solutions have a null-bypersurface
orthogonal isometry which permits the replacement
(dx~2 _ (dx l )2 -+ (dx 0 )2 _ (dx l )2 + Hp(r)(dx 0 _ dx l )2
(l0.144)
for an arbitrary harmonic function Hp(r). This is exactly what happens if we
do a Kaluza-Klein reduction along the x I-direction but with a non-zero offdiagonal component in the metric gOl/gll = Hp(r)-I (see exercise 7). Indeed,
the resulting metric can be obtained from a non-extreme metric by this means:
the dilaton and gauge potential form fields are unaffected. The final form of the
metric is
ds 2 = HS-I/4ii~3/4[(dxO)2 _ (dx l )2 + Hp(dx 0 - dx l )2]
_ HS-I/4iill/4[(dx 2 )2 + ... + (dx S )2] _ H:'4iill/4[dr 2 + r2dn~].
(10.145)
We may write the hannonic functions in the form
2
_ r.
L2
r2
H. = 1 +'2
H5 = I +-f Hp=..J:..
(10.146)
r
r
r2
where, using (10.106), (10.131) and (10.140),
2
2 I (27r #)4 cl
rf = q. g,a' (27r #)4 L~ = qsg,a' rp = ng,a V. "2
V4
4
R
(10.147)
with R the radius of the circle upon which the x l-coordinate is compactified, and
n an integer specifying the (right-moving) momentum P = n/ R on the circle.
Adding this momentum breaks a further half of the supersymmetries leaving a
total of four preserved supersymmetry charges. This corresponds to.N = I
supersymmetry in five dimensions. The 'area' A H of the event horizon is defined
as the (eight-dimensional) volume of the time slice at the horizon. Taking the
limit r -+ 0 from above, this gives a product of factors, one from each of the
three disjoint pieces of the metric:
AH = [(r~3/4L;I/4rp )27r R][(r:/ 4 L;1/4)4V4)[(r:/ 4 L~/4)327r2]
= 47r 3 RV4r.Lsrp.
(10.148)
The surface gravity and, hence, the black-hole temperature is zero, as might be
expected from an extreme black hole (see exercise 7). Using the (generalized)