Download Cosmology and Relativity - Problem Set 4 for Assignment | ASTR 3740 and more Assignments Astronomy in PDF only on Docsity!
ASTR 3740 Relativity & Cosmology Spring 2004. Problem Set 4. Due Wed 3 Mar
Warning: this problem set is quite lengthy, so please do not wait until the last day to start it.
- 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.to/gravitation/orbits/.
Another possible resource is Peter Musgrave’s “Black Holes with Java” at
http://www.astro.queensu.ca/∼musgrave/cforce/blackhole.html.
The most fun part of the sites are the Java applets, so you will probably want to seek out a Java-enabled machine, although you can also use John Walker’s site without Java. Try
http://www.Colorado.EDU/physics/2000/java help.html
for advice on how to enable Java on a PC or Mac. In what follows, the time t, radial coordinate r, polar angle θ, and azimuthal angle φ are the usual Schwarzschild coordinates in the Schwarzschild metric (with c = 1 as usual)
ds^2 =
( 1 − rs r
) dt^2 − ( dr^2 1 − rs r
) (^) − r^2
( dθ^2 + sin^2 θ dφ^2
) (1.1)
with rs the Schwarzschild radius rs = 2GM. (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, θ = π/2. For a particle of finite (nonzero) mass, the trajectory satisfies the equations ( 1 −
rs r
) (^) dt ds
= E
r^2 dφ ds
= L (1.3)
( dr ds
) 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 Veff is the effective potential given by
V (^) eff^2 =
( 1 − rs r
) ( 1 +
L^2
r^2
)
. (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 vr ≡ dr/dt and the transverse velocity v⊥ ≡ rdφ/dt (the ≡ sign means “is defined to be equal to”) of the particle at extremely large distances from the Schwarzschild geometry, r → ∞, are related to E and L by
v r^2 = 1 −
E^2
L^2
E^2 r^2
v⊥ =
L
Er
(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∞ ≡ (v^2 r + v^2 ⊥)^1 /^2 of the particle as r → ∞ is related to its energy E by
E =
(1 − v^2 ∞)^1 /^2
What does it mean if E < 1?
(c) Extrema of the effective potential
Find the radii at which the effective potential Veff 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 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 and correctly. Describe physically, in words, what the possible orbital trajectories are for the various cases. [Hint: You will need to experiment with different choices of axes to make the graph look good. I found it clearer not to start the effective potential at zero. For cases (i) and (iii), values near the critical value Lc showed the distinction most clearly.]
(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
- 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” λ = Es, which functions as a good scalar coordinate along null geodesics. In terms of the affine parameter λ, the equations of motion (1.3) for freely falling massless particles, such as photons, become ( 1 − rs r
) (^) dt dλ
r^2 dφ dλ
= L (2.1)
( dr dλ
) 2
where L = L/E is the photon’s angular momentum per unit energy, and Veff = Veff /E is the effective potential given by
Veff^2 =
( 1 − rs r
) L 2
r^2
(a) Circular orbits
Circular orbits, occur where the effective potential Veff (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?
