Exercise Solutions
794
CHAPTER 16: General Covariance
16.1 From Eqs. (16.25) and (16.26) with dummy indices suitably relabeled we have
A0
0
;
A0 D A;
@x @x
@2 x
C A 0
@x 0 @x 0
@x @x 0
!
0
@2 x @x 0
@x @x @x
C0
@x 0 @x 0 @x
@x @x 0 @x
D A;
@x @x @x 0
Exercise Solutions
914
CHAPTER 30: Newtonian Gravity and Cosmology
30.1 The redshift z is related to the scale factor a(t ) by z = (a0 a)/a, where a0 a(t0 )
denotes the scale factor at the present time. Expanding the scale factor to rst order in time
arou
Exercise Solutions
911
CHAPTER 29: The Hubble Expansion
29.1 Assume a single luminosity bin i, with the number of galaxies in the bin given by
ni n(Li ). Out to a distance r, the volume of space in 4 steradians is
V (4 sr) = 4
so the volume in 1 sr is
V (
898
Exercise Solutions
CHAPTER 19: Neutron Stars and Pulsars
19.1 The Schwarzschild radius is rs = 2M , where M is the mass. Assuming gravity to
pack neutrons down to a hardcore radius r0 0.5 1013 cm, the radius of the neutron
star will be R r0 A1/3 , whe
Exercise Solutions
963
CHAPTER 17: The General Theory of Relativity
17.1 Take the line element in the form
ds2 = e dt 2 + e dr2 + r2 (d 2 + sin2 d 2 )
where = (r, t ) and = (r, t ). The corresponding metric is
g = diag (e , e , r2 , r2 sin2 )
g = g1 = dia
Exercise Solutions
801
CHAPTER 18: Black Holes
18.1 It is convenient to introduce an exponential parameterization B.r/ e .r/ and
A.r/ e .r/, so that the metric is
1
1
22
2
g D diag e ; e ; r ; r sin
g D diag
e ;e ; 2;
:
r r 2 sin2
The unknown functions
Exercise Solutions
940
CHAPTER 31: Friedmann Cosmologies
31.1 (a) The Friedmann equations for P = 0 and k = 0 are
3a2 = 8 G a2
+ 3
a
a
= 0.
Rewrite the rst equation as
8 G
adt
3
and the second is clearly satised by the condition (t )a(t )3 = 0 , where 0
Exercise Solutions
792
CHAPTER 15: The Principle of Equivalence
15.3 The particle created at z2 has mass m D h =c 2 , where h is Plancks constant and
is the frequency of the photon. Upon dropping to z1 in the gravitational eld, the energy
is mc 2 C mg.z2
Astro 616
Fall, 2013
Dr. Guidry
Final Test Solutions
1. Equation (30.43) follows directly from inserting Eqs. (30.41)(30.42) into Eq.
(30.40). From z a0 a1 1, and (30.43) for a,
2
z = 1 + H0 (t t0 ) 1 H0 q0 (t t0 )2
2
1
1.
Then Eq. (30.44) follows from a
Astro 616
Fall, 2013
Dr. Guidry
Final Test
Do all problems. Points for each problem given in parentheses. You may not get
direct help from others, or consult solution sheets from my classes in earlier years.
1. Verify that Eqs. (30.43), (30.44), and (30.4
Astro 616
Fall, 2013
Dr. Guidry
Midterm Test Solutions
r
1. The coordinate systems are related by d t = dT and d r = dr v dT , which when
2 = d t 2 + d r 2 expressed in the freely falling coordi
substituted into the metric ds
nates gives
ds2 = 1
2M
dT 2
Astro 616
Fall, 2013
Dr. Guidry
Midterm Test
Do all problems. Points for each problem given in parentheses. You may not get
direct help from others, or consult solution sheets from my classes in earlier years.
1. Construct the Schwarzschild line element u
Exercise Solutions
922
CHAPTER 32: The Big Bang
32.1 From Table 32.1, at a temperature of 109 K there are 141 neutrons and 859 protons
for every 1000 nucleons in the Universe (corresponding to a neutron/proton ratio of 141/859
= 0.164). How many 4 He nucl