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 ,
and azimuthal angle
are the usual Schwarzschild coordinates in the Schwarzschild metric
(with *c* = 1 as usual)

(1) |

(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, .
For a particle of finite (nonzero) mass,
the trajectory satisfies the equations

(3) | ||

(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 *v*_{r} and transverse velocity
of the particle at extremely large distances from the Schwarzschild geometry,
,
are related to *E* and *L* by

(5) |

(6) |

(7) |

**(c) Extrema of the effective potential**

Find the radii at which the effective potential
is a maximum or a minimum, i.e.
,
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.
Describe physically, in words, what the possible orbital trajectories
are for the various cases.

**(e) Circular orbits**

Circular orbits, satisfying
,
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

(8) |

(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!)

(10) |

(11) |

**(g) Infall time**

Calculate the proper time 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 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*,

(12) |

**(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.