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• Problem Set
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Week 8 - Black Holes

Problem Set

Problem 1 - Photon Orbit

1. Start with the Schwarschild metric. We want a circular orbit so we will set dr=0, d (theta)=0 and theta=π/2. What is ds² for a photon (a photon travels along a null geodesic)? Solve for (dφ/dt)².
2. dφ/dt is simply Ω for the photon orbit. Kepler's third law works in the Schwarschild spacetime for circular orbits. Solve for M.

Answer for Problem 1

1. ds=0 for a photon, so we get

   / | \

   dφ d t

   \2 | /

   =

   1 - 2M/r r2

2. Ω² = M/r³. This gives M = r - 2 M or r=3 M

Problem 2 - Kepler's Law

1. Let's suppose that the particle at one moment is just going around the center of the black hole so the velocities in the r and theta directions vanish and we'll take theta=π/2 (the equatorial plane).

   In this situtation u^t and u^φ are the only components of the four velocity that don't vanish and Γ^r_tt and Γ^r_φφ are the only Christoffel symbols that don't vanish. Write out the geodesic equations.

2. We would like for the velocity to be constant around the circular orbit so we would like the first term in the geodesic equation to vanish. Solve for Ω = u^φ/u^t in terms of the non-vanishing Christoffel symbols.
3. The two non-vanishing Christoffel symbols are
   

   &Gammartt =

   (r - 2 M) M r3

   and &Gamma^r_φφ = (2 M - r) sin² theta. What is Ω in terms of M and r?
4. Substitute your value of Ω into the Schwarzschild metric and calculate ds² along the circular orbit. Over what range of radii can a material object (a toaster, UBC undergrad etc.) travel in a circular orbit around a Schwarzschild black hole.

Answer for Problem 2

1. The geodesic equation gives

   d ut d τ

   =

   d utheta d τ

   =

   d uφ d τ

   = 0

   and

   d ur d τ

   +

   &Gammartt ut ut+ &Gammarφφ uφ uφ= 0

2. Because the first term vanishes, the second term must vanish as well this gives

   Ω2 =	&Gammartt &Gammarφφ

3. Ω² = M/r³. Kepler would be pleased.
4. ds² = ( 1 - 3 M/r ) dt²
   If r>3M then ds² is positive which is what you need for a toaster to orbit the black hole.