Problems
305
using (10.149). This is a spectacular result. We have recovered the BekensteinHawking formula (10.149) for the entropy of the extreme black hole (10.145)
from counting the microstates that make up this black hole. We assumed that
V 1/4 « R in order to simplify the counting of states but this is not essential. Since
the number of states (10.152) does not change when the radii are continuously
varied, we know that the entropy is given by (10.153) for all values of R. Also,
the states were counted in the limit gsql « 1 and gsqS « 1 where we have
D-branes in flat space, whereas the black-hole solution (10.145) is valid only
when gsql > I and gsqs > I so that higher-order curvature corrections are
negligible. However, because these are extreme solutions, they are protected by
their supersymmetry and we may assume that the calculated degeneracy will not
undergo renormalization by quantum effects.
This five-dimensional RN black hole utilizes three non-zero U (I) charges-the two R-R charges ql and qS, and the momentum n/R in the internal xldirection-and this is the minimum number needed to get a finite area with a
regular horizon. In four dimensions, a minimum of four non-zero charges is
needed. The result can be generalized to near-extreme black holes. in which case
the entropy becomes a function of the mass of the black hole as well as its four
charges. We shall not pursue this further. The interested reader is referred to one
of the excellent reviews [23-25] in the literature.
10.10 Problems
1. Show that a particle on a radial timelike geodesic r = R(t) in a Schwarzchild
spacetime falls from rest at r = R(O) > 2M to R = 0 in proper time
-3/2
(
)
'l' = re M 1 - ./1 - 2M R(O)
.
2. Verify that the vector I'" normal to the event horizon of a Schwarzchild black
hole has components (10.11) in Eddington-Finkelstein coordinates.
3. Verify that the Kerr-Newman metric reduces to the Reissner-NordstrOm
metric when the angular momentum J = O.
4. Verify Smarr's formula and (10.44) for the Kerr-Newman metric where the
surface gravity is K = (r + - r _) /2(rl + a 2 ) and the co-rotating electrostatic
potential is H = (;f+!2)'
5. The surface gravity K can be calculated directly using the formula
K 2 - - -~ I X"';v X I
",;v r=rH
where X is a timelike Killing vector normal to the horizon (and normalized so
that X 2 = I at spacelike infinity) and the serni-colon indicates the covariant
derivative. Using the metric (lO.7) and the Killing vector X'" = 8~, verify
that X is a unit timelike Killing vector and that the previous formula is
satisfied by K = 1/4M, as required.
305
using (10.149). This is a spectacular result. We have recovered the BekensteinHawking formula (10.149) for the entropy of the extreme black hole (10.145)
from counting the microstates that make up this black hole. We assumed that
V 1/4 « R in order to simplify the counting of states but this is not essential. Since
the number of states (10.152) does not change when the radii are continuously
varied, we know that the entropy is given by (10.153) for all values of R. Also,
the states were counted in the limit gsql « 1 and gsqS « 1 where we have
D-branes in flat space, whereas the black-hole solution (10.145) is valid only
when gsql > I and gsqs > I so that higher-order curvature corrections are
negligible. However, because these are extreme solutions, they are protected by
their supersymmetry and we may assume that the calculated degeneracy will not
undergo renormalization by quantum effects.
This five-dimensional RN black hole utilizes three non-zero U (I) charges-the two R-R charges ql and qS, and the momentum n/R in the internal xldirection-and this is the minimum number needed to get a finite area with a
regular horizon. In four dimensions, a minimum of four non-zero charges is
needed. The result can be generalized to near-extreme black holes. in which case
the entropy becomes a function of the mass of the black hole as well as its four
charges. We shall not pursue this further. The interested reader is referred to one
of the excellent reviews [23-25] in the literature.
10.10 Problems
1. Show that a particle on a radial timelike geodesic r = R(t) in a Schwarzchild
spacetime falls from rest at r = R(O) > 2M to R = 0 in proper time
-3/2
(
)
'l' = re M 1 - ./1 - 2M R(O)
.
2. Verify that the vector I'" normal to the event horizon of a Schwarzchild black
hole has components (10.11) in Eddington-Finkelstein coordinates.
3. Verify that the Kerr-Newman metric reduces to the Reissner-NordstrOm
metric when the angular momentum J = O.
4. Verify Smarr's formula and (10.44) for the Kerr-Newman metric where the
surface gravity is K = (r + - r _) /2(rl + a 2 ) and the co-rotating electrostatic
potential is
5. The surface gravity K can be calculated directly using the formula
K 2 - - -~ I X"';v X I
",;v r=rH
where X is a timelike Killing vector normal to the horizon (and normalized so
that X 2 = I at spacelike infinity) and the serni-colon indicates the covariant
derivative. Using the metric (lO.7) and the Killing vector X'" = 8~, verify
that X is a unit timelike Killing vector and that the previous formula is
satisfied by K = 1/4M, as required.