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  Fall 2009 ASTR 2030 Black Holes: Weekly Summaries

Week 1

• What is a black hole?
  • An object whose gravity is so strong that not even light can escape;
  • A region of spacetime where space is falling faster than light.
• Event horizon — Schwarzschild radius.
  • The place (surface) from within which light cannot escape.
• Singularity
  • The center of the black hole, where the spacetime curvature becomes infinite;
  • A place where space and time as we know them come to an end.
• Astronomical evidence exists for two kinds of black holes:
  • Stellar-sized black holes, observed in X-ray binary systems;
  • Supermassive black holes, observed at the centers of galaxies.
• Black hole bends light around it.
• Tidal forces tear infaller apart:
  • well outside the horizon of a stellar-sized black hole;
  • well inside the horizon of a supermassive black hole.
• Time dilation allows you to travel “faster than light”.
• Stable and unstable orbits around a black hole.
• Object falling into a black hole appears to an outside observer to redshift and freeze at the horizon.
• Nothing special appears happens as you yourself fall through the horizon.

Week 2

• History of the development of Special Relativity — read Th Ch 1.
• Look at Special Relativity.
• What is an inertial frame?
  • A frame with respect to which unaccelerated objects move in a straight line at constant velocity
• 4 postulates of Special Relativity
  • Spacetime is 4D (space 3D + time 1D);
  • Globally inertial frames exist;
  • The speed of light c is constant in any inertial frame;
  • The Principle of Relativity: absolute spacetime does not exist.
• c = constant is the nub of SR.
• Spacetime diagrams.
  • fill in, label, and describe in your own words this spacetime diagram illustrating simultaneity and time dilation (pdf)
  • fill in, label, and describe in your own words this spacetime diagram illustrating Lorentz contraction (pdf)
• Lorentz transformations between space and time.
• Simultaneity.
• Time dilation.

Week 3

• How a scene actually looks seen at close to the speed of light.
• Light travel time.
  • When you move through a scene at near the speed of light, what you actually see is modified by the light travel time. You don't directly see the Lorentz contraction and time dilation apparent on spacetime diagrams, but rather Lorentz contraction and time dilation modulated by light travel time delays.
• Relativistic beaming:
  • Aberration: the scene is concentrated ahead, expanded behind;
  • Color shifts: the scene is blueshifted ahead, redshifted behind;
  • Brightness changes: the scene is brightened ahead, dimmed behind;
  • Time changes: the scene is speeded up ahead, slowed down behind.
• The rules of 4-dimensional perspective (pdf)
• Twin paradox.

Week 4

This was mainly a week of summary and review, in preparation for the midterm. See Review for Th Sep 17 midterm for questions to study.

• Quasars are unresolved (“quasi-stellar”), extremely bright (as much as 1000 times as luminous as a whole galaxy) objects seen at the centers of galaxies. Typically they have large redshifts, indicating that they are moving rapidly away from us, which in turn indicates that they are far away. They are thought to be powered by supermassive black holes at the centers of galaxies.
• The quasar 3C273 and its jet
• Two indications that the jet is moving at almost the speed of light:
  • The jet is superluminal &mdash blobs in the jet are observed to move several times faster than the speed of light.
  • We see only one jet, the jet coming towards us, which is relativistically brightened; there is undoubtedly a second jet going in the opposite direction, which we do not see because it is relativistically dimmed.

Week 5

• History of the development of General Relativity, and Einstein's rejection of Black Holes — read Th Ch 2, 3
• The Postulates of General Relativity:
  • Spacetime forms a 4-dimensional continuum (postulate carried over from Special Relativity).
  • Existence of locally inertial frames:
    In a small neighborhood of any point of spacetime, there exist locally inertial, or free-fall, frames with respect to which the rules of Special Relativity apply (unaccelerated objects move in straight lines at constant velocity; the speed of light is constant).
  • Strong Principle of Equivalence of gravity and acceleration (1907):
    The rules of physics in a gravitating frame are the same as those in an accelerating frame.
  • Einsteins equations (1915):
    

    Gmn  =  8pG Tmn Geometry (curvature) of spacetime  =  energy-momentum content of spacetime

• Consequences of the Principle of Equivalence:
  • Gravitational redshift;
  • Gravitational bending of light;
  • Spacetime is curved.
• River Model of BHs

Week 6

• Rosalba Perna lectured on Gamma Ray Bursts (pdf).
• We did Project 2 on the River Model of BHs

Week 7

• Astronomical black holes — read Th Ch 8, 9
  • The observational evidence for stellar-mass black holes:
    • The compact object in an x-ray binary system can be no larger in size than a white dwarf;
    • If the mass of the compact object is measured from the orbital dynamics to be more than 3 solar masses (larger than the maximum possible mass of a white dwarf or neutron star), then astronomers conclude that it must be a black hole.
  • The observational evidence that the centers of galaxies contain supermassive black holes:
    The most direct evidence is:
    • A lot of mass (inferred from velocities of stars or gas)
    • in a small space (inferred from imaging).
    Supporting evidence is:
    • Active Galactic Nuclei.
• X-ray binary systems
  • J. Orosz's (2002) Inventory of Black Hole Binaries.
  • X-ray binaries consist of a low or high mass main sequence losing gas on to an accretion disk around a neutron star or black hole.
  • Simulation of a binary star produced from Rob Hynes' binsim sofware.
  • If the mass of the neutron star or black hole is more than 3 solar masses (the maximum theoretical mass of a neutron star), then it is inferred to be a black hole. Cygnus X-1, a 5.6 day x-ray binary, is the most famous example.
  • In some cases the compact object is an x-ray pulsar. Hercules X-1 is the most famous example.
  • In some case the x-ray binary emits a jet; such x-ray binaries are called microquasars.
  • John Hawley's magnetohydrodynamic computer simulation of a jet from an accretion disk.
• The supermassive black hole at the center of the Milky Way
  • Observations of the motions of stars, both angular motions on the sky, and radial motions from spectroscopy (redshifts and blueshifts), indicate the presence of a 4 × 10⁶ solar mass black hole at the center of our own Galaxy, the Milky Way.
• John Kormendy's 2001 census of Supermassive Black Holes in Galactic Nuclei
  • Observations of the centers of nearby galaxies indicate the presence of a large unseen gravitational mass in a small region — it must be a black hole.
  • Every galaxy large enough to have a bulge appears to have a central black hole.
  • Black Hole masses range from millions (10⁶) to billions (10⁹) of solar masses.
  • M87, the huge galaxy at the center of the Virgo cluster, at the center of the Local Supercluster of galaxies, contains the most massive black hole known in the local Universe, 3 × 10⁹ solar masses.
  • Maser emission from M106 reveals a thin, warped accretion disk around a 3.9 × 10⁷ solar mass black hole.

• Schwarzschild black hole — read Th Ch 3, and Ch 6 page 244 and following.
• “No-hair” theorem: An isolated black hole rapidly (on few light crossing times) evolves to a state characterized by just 3 properties:
  • Mass
  • Electric charge
  • Angular momentum (spin)
  It loses its "hair" by gravitational radiation.
• A Schwarzschild black hole is the simplest kind of black hole, one with mass, but no electric charge, and no spin.
• Falling into a Black Hole: approach; orbit; singularity; or alternatively, Journey into a Schwarzschild black hole
  • What happens when you watch someone else fall into a black hole?
  • What happens when you fall into a black hole?
• The Schwarzschild geometry for a non-rotating, uncharged black hole
  • Embedding diagram of the Schwarzschild geometry
• Intervals of spacetime can be either timelike, or lightlike, or spacelike:
  • Timelike = possible worldline of a particle (traveling normally, at less than the speed of light)
  • Lightlike = possible worldline of light
  • Spacelike = a possible “now-line”: a line between two events that are simultaneous with respect to some observer (a particle would have to travel faster than light to follow such a line)

Week 8

• Schwarzschild wormhole, white hole
• Collapse to a black hole
• Penrose diagrams
• Journey into a Reissner-Nordström (charged) black hole
• Kerr (rotating) black hole

Week 9

• We did Project 3, critiquing an excerpt from “Under the Night”, the opening episode of the Andromeda sci-fi series.
• Review for Th Oct 22 midterm.

Week 10

This week was about how to script a movie. We watched excerpts from two movies, “Contact”, and “Walt Disney's The Black Hole”. Read Thorne Ch 4 in preparation for Project 4 next week.

• A well-structured movie has 3 parts, a beginning, a middle, and an end:
  1. The first part of the movie should set up the "World", or genre, of the story. The story could be a cartoon, a cowboy film, an action movie, or whatever, but once the world has been set up, the movie should remain consistent to that world. While setting up the "World", the movie should introduce the hero, the villain, and any other key characters. The movie is often more interesting if the hero has flaws and the villain has qualities.
  2. The main part of the movie should be characterized by Conflict. The hero should encounter problems of one kind or another. A novel can develop characters by getting inside their heads and telling you what the characters are thinking, but that is not possible in a movie. Movies develop characters by facing them with problems, and seeing how the characters respond to those problems. The hero should overcome the problems encountered. Often overcoming the problems generates new problems, so that there is a cycle between encountering problems and overcoming them.
  3. In its final part, the movie should build up to a Climax. The climax should be the most exciting part of the movie. The movie should end with closure, a resolution of some kind. If you have a budget, spend it here. Sometimes the movie ends with the climax; other times there is an epilogue after the climax. Sometimes the ending is perfect like a fairy tale, but it can be more interesting if the ending has a twist or is not perfect.

Week 11

• What happens to cold matter when you compress it very prodigiously? Read Thorne Ch 5.
  • Mass-radius relation for cold matter (pdf).
• A solid or liquid is almost incompressible because the atoms are tightly packed together. In a solid, the atoms form a regular lattice, while in a liquid the atoms are irregularly stacked.
• Atoms resist compression because electrons satisfy an exclusion principle: you can't pack electrons together more closely than a wavelength.
• Like any particle, an electron is both wave and particle:
  

  Particle property   Wave property E = h   f energy =  Planck's constant  × frequency p = h / l momentum =  Planck's constant  / wavelength

  For a non-relativistic electron, momentum p equals mass times velocity, p = m v.
• The wavelength of electrons in atoms is set by their orbital velocities. The characteristic atomic velocity of electrons is 1/137 of the speed of light:
  

  vatom » (1/137) c

  The (1/137) factor is the “fine-structure constant”, which describes the characteristic strength of electromagnetism.
• To compress atoms requires compacting electrons to the point where their velocities exceed the atomic velocity c/137. This requires a pressure exceeding the pressure at the center of Jupiter.
• Thus planets of Jupiter's mass or less are made of incompressible liquid or solid; this is why the moons and planets in the solar system all have about the same density, and more massive planets have larger radii.
• At pressures higher than the pressure at the center of Jupiter, electrons are squeezed out of atoms, and form a compressible electron degenerate gas.
• Thus cold objects more massive than Jupiter are held up by electron degeneracy pressure. They are effectively cold white dwarfs.
• An electron degenerate gas is so compressible that larger mass white dwarfs have smaller radii.
• The white dwarf sequence terminates when the electrons have been compressed to the point that their velocities are relativistic (almost the speed of light). This happens at the Chandrasekhar limit of 1.4 solar masses.
• If mass is added to a white dwarf to bring it above 1.4 solar masses, then the white dwarf will collapse.
• A white dwarf that exceeds the Chandrasekhar limit will collapse to a neutron star.
• A neutron star is held up in part by the degeneracy pressure of neutrons, and in part by the strong (nuclear) force.
• The neutron star sequence terminates at an upper mass limit of about 3 solar masses.
• A neutron star whose mass exceeds about 3 solar masses will collapse to a black hole.

• At the end of the week we did Project 4, to critique Chapter 4 of Thorne as if it were a story.

Week 12

Supernovae