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ASTR 3740 Relativity & Cosmology Spring 2000. Problem Set 4.
Due Wed 8 Mar
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1. Trajectories of particles in the Schwarzschild geometry
In this problem you will find it helpful to visit John Walker's
Orbits in Strongly Curved Spacetime.
Another possible resource is
Peter Musgrave's
Black Holes with Java.
The most fun part of the sites are the Java applets,
so you will probably want to seek out a Javaenabled machine,
although you can also use John Walker's site without Java.
Try the
Physics 2000 help site
for advice on how to enable Java on a PC or Mac.
In what follows, the time t, radial coordinate r,
polar angle q, and azimuthal angle f
are the usual Schwarzschild coordinates in the Schwarzschild metric
(with c = 1 as usual)
d s^{2} = 
æ ç
è

1  
r_{s}
r

ö ÷
ø

d t^{2}  
d r^{2}

 r^{2} ( d q^{2} + sin^{2} q d f^{2} ) 
 (1.1) 
with r_{s} the Schwarzschild radius
Without loss of generality,
the trajectory of a particle falling freely in the Schwarzschild geometry
may be taken to lie in the equatorial plane,
q = p/2.
For a particle of finite (nonzero) mass,
the trajectory satisfies the equations
æ ç
è

d r
d s

ö ^{2} ÷ ø

+ V_{eff}^{2} = E^{2} 
 
where s is the proper time of the particle,
and E and L are constants,
the particle's energy and angular momentum per unit mass.
The quantity V_{eff} is the effective potential given by
V_{eff}^{2} = 
æ ç
è

1  
r_{s}
r

ö ÷
ø

æ ç
è

1 + 
L^{2}
r^{2}

ö ÷
ø

. 
 (1.4) 
(a) Check
Are John Walker's equations the same as the ones given above
(aside from possible differences in notation)?
(b) Velocity at infinity
Argue from equations (1.3) that
relative to the rest frame of the Schwarzschild geometry,
the radial velocity v_{r} and transverse velocity v_{^}
of the particle at extremely large distances from the Schwarzschild geometry,
r ® ¥,
are related to E and L by
v_{r}^{2} º 
æ ç
è

d r
d t

ö ^{2} ÷ ø

= 1  
1
E^{2}

 
L^{2}
E^{2} r^{2}

 (1.5) 
v_{^} º 
r d f
d t

= 
L
E r

 (1.6) 
(note that L can be extremely large at large r,
so L / r is not necessarily zero in the limit
r ® ¥).
Hence show that
the velocity
v_{¥} º (v_{r}^{2} + v_{^}^{2})^{1/2}
of the particle as
r ® ¥
is related to its energy E by
E = 
1
(1  v_{¥}^{2})^{1/2}

. 
 (1.7) 
What does it mean if E < 1?
(c) Extrema of the effective potential
Find the radii at which the effective potential V_{eff}
is a maximum or a minimum,
i.e. d(V_{eff}^{2}) / dr = 0,
as a function of angular momentum L.
You should find that extrema exist only if the absolute value
L
of the angular momentum exceeds a certain critical value
L_{c}.
What is that critical value?
(d) Sketch
Sketch what the effective potential looks like for values of L
(i) less than, (ii) equal to,
(iii) greater than the critical value L_{c}.
Make sure to label the axes clearly and correctly.
Describe physically, in words, what the possible orbital trajectories
are for the various cases.
(e) Circular orbits
Circular orbits, satisfying dr / ds = 0,
occur where the effective potential
is a minimum (stable orbit) or a maximum (unstable orbit).
Show (from your equation for the extrema of the effective potential) that the
angular momentum L of a particle in circular orbit at radius
r satisfies
L = 
r
æ ç
è

2 r
r_{s}

 3 
ö ^{1/2} ÷ ø



 (1.8) 
and hence show also that the energy E in this circular orbit is
E = 
2^{1/2} (r  r_{s})
[ r ( 2 r  3 r_{s} ) ]^{1/2}

. 
 (1.9) 
(f) Orbital period
Show that the orbital period t,
as measured by an observer at rest at infinity,
of a particle in circular orbit at radius r is given by
Kepler's 3rd law
(yes, it's true even in the fully general relativistic case!)
G M t^{2}
(2p)^{2}

= r^{3} . 
 (1.10) 
[Hint:
The time measured by an observer at rest at infinity is just
the Schwarzschild time t.
Argue that the azimuthal angle f evolves according to
d f
d t

= 
L ( r  r_{s} )
E r^{3}

. 
 (1.11) 
The period t is the time taken for f
to change by 2p.]
(g) Infall time
Calculate the proper time s for a particle with
L = 0 and E = 1 to fall from a finite radius r
to the singularity at zero radius.
What is the physical significance of the choice L = 0
and E = 1?
[Hint:
Write down the equation for dr / ds
for L = 0 and E = 1,
and then solve it.]
(h) Infall time  numbers
Use your answer to part (g) to show that the proper time
to fall from the Schwarzschild radius
r = r_{s} to the singularity
(for L = 0 and E = 1) is, in units including c,
Evaluate your answer, in seconds,
for the case of a black hole of mass
2.4×10^{6} M_{sun},
such as may be in the center of our Galaxy, the Milky Way.
[Constants:
c = 299,792,458 m s^{1};
G = 6.67259×10^{11} m^{3} kg^{1} s^{2};
1 M_{sun} = 1.99×10^{30} kg.]
2. Photons in the Schwarzschild geometry
The orbit equations (1.3) would appear to break down
for photons, which have zero mass,
hence infinite energy per unit mass E
(cf. equation [1.7] for v_{¥} = 1)
and infinite angular momentum per unit mass L.
Another way of looking at this is that photons
follow null geodesics, ds = 0,
so that s, which does not change, is not a very useful time coordinate
for expressing the equations of motion of photons.
The difficulty is cured by introducing
an ``affine parameter'' l = Es,
which functions as a good scalar coordinate along null geodesics.
In terms of the affine parameter l,
the equations of motion (1.3) for freely falling massless particles,
such as photons, become
æ ç
è

d r
d l

ö ^{2} ÷ ø

+ V_{eff}^{2} = 1 
 
where
L º L / E
is the photon's angular momentum per unit energy,
and V_{eff} º V_{eff} / E is the effective potential given by
V_{eff}^{2} = 
æ ç
è

1  
r_{s}
r

ö ÷ ø

L^{2}
r^{2}

. 
 (2.2) 
(a) Circular orbits
Circular orbits,
occur where the effective potential
V_{eff}
(or equivalently its square)
is a minimum (stable orbit) or a maximum (unstable orbit).
At what radius can photons orbit in circles?
Is the orbit stable or unstable?
(b) Shine a light out of a black hole
A person who has fallen inside the horizon of a black hole
(but is not dead yet)
shines a light beam radially outward
(so the photons have zero angular momentum,
L = 0).
What happens to the outward going photons?
Do they approach the horizon and freeze there?
Do they fall to the singularity?
[Hint:
I want you to explain what happens both physically and mathematically.
It is important to worry about signs.
From equations (2.1),
derive an equation for dr / dt for radially moving photons.
The result should be the same as what is written on page 2.30 of the notes,
and if you like you can quote the solution of the integral from there.
Now interpret the equations physically.]
File translated from T_{E}X by T_{T}H, version 2.01.