Spring 2025 ASTR 3740 Homepage

Spring 2025 ASTR 3740 General Relativity Midterm Review

This midterm review sheet is posted at http://jila.colorado.edu/~ajsh/courses/astr3740_25/review_mid2.html

The midterm will test the material covered during weeks 5 to 10 which is to say, general relativity and black holes. Obviously, there is a non-trivial overlap with special relativity, so you should not forget what you learned during weeks 1 to 4. Please write your own answers, not something copied from elsewhere. If you do copy something, please quote it, and give the reference. Please write clear, grammatically correct sentences.

You agree to respect the CU Honor Code. If you use AI or other outside resources, you must credit the source. If you use language from AI, you must justify that use, using logic or other more credible resources.

You can download the test from the link in canvas. You must upload your answers back to canvas no later than two hours after you downloaded the test. Canvas will record when you download the test, and when you upload your answer. The midterm is due in canvas no later than class Wed Apr 2.

You get −1 point for each time you misspell Schwarzschild.

Review Questions

1. Observational evidence for Black Holes.
   • Formation in core-collapse supernovae
   • X-ray binaries
   • Active Galactic Nuclei (e.g. quasars)
   • Accretion disks
   • Jets
   • Lots of mass in a small space
   • Sgr A*
   • Masses of BHs from motions of stars and gas around supermassive black holes
   • Gravitational waves from merging compact objects
   • Event Horizon Telescope observations of M87* and SgrA* black holes

2. Postulates.
   • Existence of local inertial (free-fall) frames
   • (Weak/Strong) Principle of Equivalence
   • Einstein's equations:
     

     (Compressive, non-tidal, part of) Curvature = Energy-momentum density

3. Consequences of the Equivalence Principle.
   • Gravitational redshift
   • Gravitational bending of light
   • Gravity ≡ curvature of spacetime

4. Metric and curvature.
   • Metric
   • Proper time, distance
   • Geodesics
   • Relation between curvature and tidal forces
   • A gravitational wave is a wave of curvature; what does that mean?

5. Schwarzschild Geometry.
   • What does it describe?
   • What do the various parts of the metric mean?
   • Embedding diagram
   • Spacetime diagram
   • Finkelstein coordinates
   • Kruskal coordinates
   • Penrose diagram
   • River model
   • Orbits
   • What do things falling in look like from outside?
   • What do things look like if you fall in?
   • Collapse to a black hole
   • Wormhole, white hole, parallel universe, and problems therewith

6. No Hair Theorem
   • What does it state?
   • When does it apply? When does it not apply?

7. Reissner-Nordström Geometry.
   • What does it describe?
   • Similarities, differences vs. Schwarzschild
   • Outer, inner horizons
   • Penrose diagram
   • River model
   • Why are the vacuum solutions inconsistent?
   • What does an infaller experience at the inner horizon?
   • Mass inflation

8. Kerr-Newman Geometry.
   • What does it describe?
   • Similarities, differences vs. Reissner-Nordström
   • Oblate spheroidal geometry
   • Ergosphere
   • Ring singularity
   • Antiverse (r < 0)
   • Sisytube
   • What does an infaller experience at the inner horizon?

9. Gravitational Waves.
   • Hulse-Taylor binary pulsar
   • What is LIGO (Laser Interferometer Gravitational-Wave Observatory)? How does it work?
   • GW150914: the first detection of gravitational waves from the merger of two black holes
   • What do waveforms of merging black holes look like? Why?
   • GW170817: fhe first detection of gravitational waves from the merger of two neutron stars
   • 90 gravitational wave merger events observed to date (2015-2020)

10. Hawking Radiation.
    • Quantum mechanical origin
    • Black body spectrum
    • Wavelength ≈ horizon size
    • Evaporation
    • Irrelevant to astronomical black holes

11. Event Horizon Telescope.
    • What is the EHT (Event Horizon Telescope)? How does it work?
    • Compare the EHT image of Sgr A* to that of M87*. How and why do they differ?
    • How are the masses of the Sgr A* and M87* black holes measured from the image?
    • In what way(s) do the images confirm (or not) general relativity?

Updated 2025 Mar 19