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Black holes in string theory
6. Calculate the Schwarzchild radius for a black hole having mass M = M0'
Calculate also its temperature and entropy.
7. By choosing new coordinates p = Jr - r + and 'l' = it. show that near
the horizon the line element for the Reissner-Nordstrom black hole may he
written in the form
_ds 2 ::::: Ap2 d1'2 + 4A -I dp2 + r2 dQ~
(l0.IS4)
where A == (r + - r _) 1 r!. Hence. show that l' is an angular coordinate with
period 411' A-I and. therefore. that the temperature of the extreme black hole
is zero.
8. Evaluate the extrinsic curvature Kab of the Schwarzchild metric on a
spherical shell of radius R and verify equations (I 0.6S) and (10.66).
9. Show that the N = 2 supersymmetry algebra (10.91) can be written in the
form (10.92).
10. Express the function f(w) given in (10.82) in terms of the Dedekind eta
function
00
11(1') == e un / 12 n (I - e2n'im'l').
m:ol
Using the property
11(-1/1') = (-i1')1/2 rJ (1')
show that
( -211')1/2
( 11'2) (411'2)
f(w) = lnw w- I / 24 exp 61nw f Inw .
Hence. show that
f(w)"" A(I - w)-1/2 exp ( 6(llr~ w») for W"" I
and find the power-law correction to the exponential dependence given in
(10.89).
11. For the extreme Reissner-Nordstrom black hole with the metric (10.94).
show that near the horizon the metric approximates Ad S2 x S2.
12. Show that the partition function Z(w) defined in (10. ISO) and (10.1SI) has
the asymptotic behaviour
Z(w) .... exp (qIQS1r2) forw"" I
I-w
and. hence. verify the degeneracy (10.152) of states having momentumnl R.