PHY-105
Assignment #6 and 7
Due before 11:59pm Wednesday, 27 April 2011
Problem 1 [4 pts]
As discussed in class, the Schwarzschild radius defines the Event Horizon of an object if all the mass
of the object were located at a single point (a singularity) in space. It is the radius inside which
nothing can be transmitted to the universe outside. Calculate the Schwarzschild radius for (a) a
black hole of mass 40 times that of our Sun, (b) the Sun, (c) the Earth, and, (d) a proton.
Problem 2 [7 pts]
Consider a white dwarf of mass 2 solar masses, and radius R = 3 × 108 m. Using the Schwarzschild
metric calculate at the surface of the white dwarf: (a) The gravitational refshift, (b) the time
dilation, and, (c) the radius of curvature of the path of an initially horizontally travelling photon.
(This last part is a bit tricky, but it will show you how much a photon’s path deviates from a straight
line (which has infinite radius of curvature)).
Problem 3 [7 pts]
As we have seen, the space-time around a spherically symmetric object of mass M is given by the
Schwarzschild metric. We should be able to relate the results predicted by the Schwarzschild metric
to Newtonian gravity. For example, consider a satellite of mass m cicularly orbiting a planet of mass
M at a distanceqr from the planet’s center. We know from Newtonian gravity that the speed of this
satellite is v = GM/r. Using the Schwarzschild metric and the fact that the path (geodesic) of an
object in a curved space-time is that for which ∆s is an extremum (that is d(∆s) = 0, show that
General Relativity gives the same result (albeit in a much more difficult way for this simple case!).
Problem 4 [7 pts]
The so-called “null geodesic” is the path taken by a photon in a curved space-time, and is defined by
ds = 0. Consider a black hole with a Schwarzschild radius rs . Consider a photon leaving us (in an
assumed flat space-time) moving towards the black hole. Use the Schwarzschild metric to show that
the time it takes the photon to reach rs is infinite. That is, the photon will never reach the event
horizon! This is the same fate that we’ll observe for any object, and is why we can think of a black
hole as a “star frozen in time” – it will take an infinite amount of time (as observed by us) for the
star to collapse to the event horizon.
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Problem 5 [10 pts]
(a) Show that the proper distance from the event horizon to a radial coordinate r is given by:
1 + (1 − rsch /r)1/2
rsch 1/2 rsch
) +
ln
∆L = r(1 −
r
2
1 − (1 − rsch /r)1/2
!
This shows that one cannot interpret the coordinate r as a distance in a curved space-time (think
about this). As a hint to solving this, use the Schwarzschild metric, assuming that the end points of
the proper distance are measured at the same time.
(b) Show that as r → ∞, ∆L → r. That is, far from the Black Hole, the radial coordinate r can be
treated as a distance.
Problem 6 [6 pts]
Text problem 5.2 (for properties of the Sun see for example: http://en.wikipedia.org/wiki/Sun)
Problem 7 [6 pts]
Text problem 5.4
Problem 8 [3 pts]
How do the values obtained in problems 6 and 7 compare to those obtained using a White Dwarf as
the starting point for a Neutron Star (see class notes)? Why wouldn’t they necessarily agree ?
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Problem 9 [25 pts]
In class we obtained the following differential equation that we need to solve to get the scale factor
as a function of time, R(t):

1 dR

R dt
!2

8
− πGρ R2 = −kc2
3
where R and ρ are both functions of t.
(a) Show that we can rewrite this in terms of R and t only thus:
dR
dt
!2
1
8
− πGρ0 = −kc2
3
R
where ρ0 is a constant, the average density of the present universe.
(b) From this equation, verify the following solutions for R(t) for a flat, closed, and open universe:
2/3 3
R(t) =
2
2/3
(for k = 0)
1 Ω0
(1 − cos x)
2 Ω0 − 1
where, t =
Ω0
1
(x − sin x)
2H0 (Ω0 − 1)3/2
1 Ω0
(cosh x − 1)
2 1 − Ω0
where, t =
Ω0
1
(sinh x − x)
2H0 (1 − Ω0 )3/2
R(t) =
R(t) =
t
tH
(for k > 0)
(for k < 0)
For the flat universe case explicitly solve the differential equation for R above, but for the closed and
open universe cases it is sufficient to simply substitute these solutions into the differential equation
to show they do in fact solve it.
(c) Sketch the form of these solutions on a plot of R(t) versus t.
(d) Show that from these solutions we can obtain the time(age) of the universe as functions of the
redshift z thus (this is a bit messy, sorry!):
t(z)
2
1
=
tH
3 (1 + z)3/2

(for k = 0)
Ω0 z − Ω0 + 2
Ω0
t(z)
cos−1
−
=
3/2
tH
2(Ω0 − 1)
Ω0 z + Ω0

t(z)
Ω0 z − Ω0 + 2
Ω0
− cosh−1
+
=
3/2
tH
2(1 − Ω0 )
Ω0 z + Ω0
q
2 (Ω0 − 1)(Ω0 z + 1)
q
Ω0 (1 + z)


2 (1 − Ω0 )(Ω0 z + 1)
Ω0 (1 + z)
(for k > 0)


(for k < 0)
where Ω0 is the density parameter of the present universe, as defined in class.
(e) For density parameters of Ω0 = 1, Ω0 = 2, and Ω0 = 0.5, calculate the age of the universe for
each as a ratio of tH (tH = 13.6 billion years for h = 0.72).
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Problem 10 [10 pts]
(a) For the closed universe scenario derive an expression for the maximum “radius” (Rmax ) of the
universe.
(b) If the density of the universe was twice the critical density, ρc , by what factor would the universe
expand beyond its present radius, and what would be the time between the “Big Bang” and the “Big
Crunch” ?
Problem 11 [5 pts]
Assuming that our knowledge of particle physics is uncertain at energies greater than about 103 GeV ,
find the earliest time t at which we can have confidence in the physics that existed in the early
universe. How much smaller was the universe at that time than it is now ?
Problem 12 [10 pts]
Given that Helium nuclei were being formed when the temperature of the Universe was about 109 K,
estimate the time(t), scale factor (R(t)), and redshift (z), of the Universe when this was occuring.
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