Physics 616

  • Prof. Andrew W. Steiner
    (or Andrew or "Dr. Steiner")
  • Office hour: 103 South College, Thursday 11am
  • Email:
  • Homework: Electronically as .pdf
  • You may work with each other on the homework, but you must write the solution in your own words
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Why is there a gap between the disk and the neutron star?

Methods in GR

Isometries and Killing Vectors

Dark Stars in Newtonian Gravity

Orbits in GR

Orbits in GR II

Orbits in Schwarzchild Spacetime

  • Scattering orbits are not parabolas as in Newtonian gravity
  • "Plunging orbits" do not appear in Newtonian gravity
Guidry ch. 18

Circular Orbits

Innermost stable circular orbits

Stable Circular Orbits

  • For a particle in a stable circular orbit, solve $$ r = \frac{\tilde{L}}{2 M} \left[ 1 + \left( 1 - \frac{12 M^2}{\tilde{L}^2} \right)^{1/2}\right] $$ for $\tilde{L}^2$, giving $$ \tilde{L}^2 = \frac{M r}{1-3 M/r} $$ and then using $\tilde{E}^2=\tilde{V}^2$ we know the energy $$ \tilde{E}^2 = \left( 1-\frac{2 M}{r}\right)^2 \left( 1-\frac{3 M}{r}\right)^{-1} $$
  • Compute angular velocity by first computing $$ \frac{d \phi}{d \tau} = \frac{\tilde{L}}{r^2} \quad \mathrm{and} \quad \frac{dt}{d \tau} = \frac{\tilde{E}}{1-2 M/r} $$ and then taking the ratio $$ \frac{dt}{d \phi} = \left(\frac{r^3}{M}\right)^{1/2} $$
  • (Same as Newtonian expression)

Approximate Newtonian Orbits

Perhelion Advance

Perhelion Advance II

Mercury and Earth

Clemence (1947)

Tests of GR

Kramer et al. (2006)
  • Larger periastron precession in double NS systems (16 degrees per year)
  • This system, PSR J0737-3039A/B
  • Double pulsar systems allow observation of GR corrections
  • Shapiro delay: radio pulses delayed by curved space-time
  • Tests of GR have always passed
  • See also Ch. 19 in Guidry's book

Group Work

Gravitational Deflection of Light

Gravitational Deflection of Light II