ASTR 3740 Relativity & Cosmology Spring 1998. Problem Set 2.
Due Tue 10 Mar

Warning: this problem set may take you several hours to complete, so please do not wait until the last day to start it.

I encourage you to take some time, while you are on the web, exploring about black holes and relativity. A good place to start is the relativity links attached to the ASTR 3740 homepage. I would love to know about any good links you find which are not already linked there.

Trajectories of particles in the Schwarzschild geometry

In this problem you will find it helpful to visit John Walker's web site at http://www.fourmilab.ch/gravitation/orbits/. The most fun part of the site is the Java applet, so you will probably want to seek out a Java-enabled machine, although you can also use the site without Java. Try http://www.colorado.edu/physics/2000 for advice on how to enable Java on a PC or Mac.

In what follows, the time t, radial coordinate r, polar angle theta, and azimuthal angle phi are the usual Schwarzschild coordinates in the Schwarzschild metric (with c = 1 as usual)
d s^2 = [1 - (r_s/r)] d t^2 - d r^2 / [1 - (r_s/r)]
 - r^2 ( d theta^2 + sin^2 theta d phi^2 ) (1)
with rs the Schwarzschild radius
r_s = 2 G M (2)

Without loss of generality, the trajectory of a particle falling freely in the Schwarzschild geometry may be taken to lie in the equatorial plane, theta = pi/2. For a particle of finite (nonzero) mass, the trajectory satisfies the equations
 
d t / d tau = E /[1 - (r_s/r)]
 r^2 d phi / d tau = L (3)
(d r / d tau)^2 + V_eff^2 = E^2
where tau 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 = [1 - (r_s/r)] [1 + (L^2/r^2)] (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 (3) that relative to the rest frame of the Schwarzschild geometry, the radial velocity vr and transverse velocity v_\perp of the particle at extremely large distances from the Schwarzschild geometry, r -> infinity, are related to E and L by
v_r^2 = (d r / d t)^2 = 1 - 1/E^2 - L^2/(E^2 r^2) (5)
v_\perp = r d phi / d t = L/(E r) (6)
(note that L can be extremely large at large r, so L/r is not necessarily zero in the limit r -> infinity). Hence show that the velocity v_\infty = (v_r^2 + v_\perp^2)^{1/2} of the particle as r -> infinity is related to its energy E by
E = 1 / (1-v_\infty^2)^{1/2} (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) / d r = 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 Lc. 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 Lc. Make sure to label the axes clearly. Describe physically, in words, what the possible orbital trajectories are for the various cases.

(e) Circular orbits

Circular orbits, satisfying d r / d tau = 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 / [(2 r/r_s) - 3]^{1/2} (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} (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 / (2 pi)^2 = r^3 (10)
[Hint: The time measured by an observer at rest at infinity is just the Schwarzschild time t. Argue that the azimuthal angle phi evolves according to
d phi / d t = L ( r - r_s ) / (E r^3) (11)
The period t is the time taken for phi to change by 2 pi.]

(g) Infall time

Calculate the proper time tau 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 d r / d tau 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 = rs to the singularity (for L = 0 and E = 1) is, in units including c,
tau = 4 G M / (3 c^3) (12)
Evaluate your answer, in seconds, for the case of a black hole of mass 106 solar mass, such as may be in the center of our Galaxy, the Milky Way. [Constants: c = 299,792,458ms-1; G = 6.67259נ10-11m3kg-1s-2; 1 solar mass = 1.99נ1030kg.]

(i) Suggestions

Briefly, do you have any suggestions for how John Walker might improve this web page (this one in particular, not the 100s of others he has)? Please be polite and helpful, and offer reasoned arguments rather than opinions.



Andrew Hamilton
2/26/1998