General Relativity

Problems

practice

1. How much older is your head than your feet?
2. Write something.
3. Write something.
4. Finding the volume of a hypersphere should be something like finding the surface area of an ordinary sphere. Derive the equations for…
   1. the surface area of an ordinary sphere of radius R and
   2. the volume of a hypersphere of radius R
   Say you looked out into space and saw a galaxy 4.5 billion light years away that turned out to be the Milky Way as it was 4.5 billion years in the past. (This is about the time that the Earth was forming.) Given this hypothetical, hyperspherical universe, determine…
   3. its radius of curvature (in light years) and
   4. its volume (in cubic light years)

numerical

1. Gravitational waves from some orbiting astronomical objects (like the Earth orbiting the Sun) have frequencies that are best stated with nanohertz as the unit (1 nHz = 10⁻⁹ Hz or one billionth of a hertz).
   1. What is the frequency of the gravitational waves produced when the Earth orbits the Sun in nanohertz?
   2. What is the period of a nanohertz gravitational wave in years?
   Explain the reasoning behind your answers to these two questions. "I googled it" is not a form of reasoning.
2. Gravitational waves are disturbances in the gravitational field that propagate at the speed of light and are formed whenever objects with mass accelerate.
   1. A black hole is the remnant of a very massive star that has collapsed to the point where nothing, not even light, can escape it. The first direct detection of gravitational waves was the merger of two black holes in a distant galaxy by the Laser Interferometer Gravitational-Wave Observatory in 2015. LIGO's two detectors simultaneously observed an 8 cycle, 0.2 s gravitational chirp that swept upward in frequency and amplitude from 35 to 250 Hz as the black holes spiraled into one another — something like a giant toilet flushing. Given this information, determine…
      1. the shortest wavelength detected in the chirp
      2. the longest wavelength detected in the chirp
      3. the average frequency of the chirp
      4. Why isn't the answer to the previous question equal to the average of the initial and final frequencies of the chirp?
   2. A pulsar is a highly magnetized, burned out star (a bit more massive than the Sun) that has collapsed down to the size of a city, is made up almost entirely of neutrons, and is spinning as fast as a kitchen blender. The North American Nanohertz Observatory of Gravitational Waves (NANOGrav) is a consortium of scientists whose goal is to detect gravitational waves with frequencies of about 10 nanohertz using precisely timed millisecond pulsars as clocks. Given this information, determine…
      1. the frequency of a 1.5578 ms pulsar
      2. the period of a 10 nHz gravitational wave in…
         1. seconds
         2. years
      3. What kind of astronomical phenomena might emit ~10 nHz gravitational waves?