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  Spring 2012 ASTR 2030 Black Holes: Weekly Summaries

Week 1

• Read Th Prolog.
• 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)
• Project 1: Twin paradox.

Week 4

This was mainly a week of summary and review, in preparation for the midterm. See Review for Th Feb 9 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 — 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.
  • Einstein's 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
• “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.
• 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?

Week 6

• 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
  • Jeffrey E. McClintock et al.'s 2011 drawings of 21 of the 23 known X-ray Binaries containing a Black Hole (pdf).
  • 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.
• Kormendy & Gebhardt's 2001 census of Black Holes in Galactic Nuclei (pdf).
  • 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.
• 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.
• Project 2: River Model of Black Holes.

Week 7

• More about the Schwarzschild black hole — read Th Ch 3, and Ch 6 (especially page 244 and following).
• The Schwarzschild geometry
  • 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)
• Schwarzschild wormhole, white hole
  • The mathematically extended Schwarzschild geometry contains a white hole and a non-traversable wormhole connecting to a parallel universe
  • In reality, the collapse to a black hole of a star does not produce a white hole or wormhole or parallel universe
• The Schwarzschild spacetime diagram seems to show that light rays freeze at the horizon, and never pass through it.
• The Finkelstein spacetime diagram shows that a change of time coordinate allows light rays to fall through the horizon.
  • Watch Schwarzschild morph into Finkelstein.
• A Penrose diagram is a kind of spacetime diagram constructed to make clear the causal structure of a spacetime. In a Penrose diagram
  • Light moves upward at 45° from vertical (just as in a special relativistic spacetime diagram);
  • Points at infinity (infinite distance, or infinite past and future) are contained in the diagram.
• Journey into a Reissner-Nordström (charged) black hole
  • The tension (negative pressure) of the electric field causes a gravitational repulsion which slows the faster-than-light inflow of space inside the outer horizon;
  • Where the inflow of space slows back down to the speed of light, there is an inner horizon;
  • In the idealized mathematical (Reissner-Nordström) solution, the inner horizon is the entrance to a one-way traversable wormhole and white hole connecting to another “universe”;
  • At the instant they pass through the inner horizon, an infaller sees the outside Universe concentrated into an infinitely bright, infinitely blueshifted burst of light.

Week 8

• Read Thorne Ch 13 to find out Thorne's thinking about what happens inside black holes. At the time of writing, Thorne did not know about the all-important “mass inflation” instability discovered by Poisson and Israel in 1990.
• Kerr (rotating) black hole
  • The geometry inside a rotating (Kerr) black hole is similar to that of a charged spherical (Reissner-Nordström) black hole;
  • In a rotating black hole, the centrifugal force slows the inflow of space inside the outer horizon;
  • Where the inflow of space slows back down to the speed of light, there is an inner horizon;
  • In the idealized mathematical (Kerr) solution, the inner horizon is the entrance to a one-way traversable wormhole and white hole connecting to another “universe”;
  • At the instant they pass through the inner horizon, an infaller sees the outside Universe concentrated into an infinitely bright, infinitely blueshifted burst of light.
• The idealized rotating (Kerr) geometry is more complicated than the charged spherical (Reissner-Nordström) geometry in that:
  • The singularity is a ring, not a point;
  • Around the ring singularity is a tunnel containing Closed Timelike Curves: if you go retrograde sufficiently fast around the tunnel, you can go into your own past, and violate causality;
  • The other side of the disk bordered by the singularity is a new piece of spacetime, the antiverse.
• In reality, the concentration of energy at the inner horizon produces an exponentially growing “mass inflation” instability
  • The inflationary instability acts like a gravity-powered particle accelerator, accelerating “ingoing” (positive energy) and “outgoing” (negative energy) particles through each other to energies reaching and exceeding the concentration of energy at the Big Bang;
  • In a realistic black hole, the inflationary instability destroys the inner horizon, and precipitates collapse.
• We did Project 3, critiquing an excerpt from “Under the Night,” the opening episode of the Andromeda sci-fi series.

Week 9

• Read Thorne Ch 8 for the history of the discovery of x-ray binaries, and Ch 9 for the history of the discovery of supermassive black holes.
• If you could see in x-rays:
  • The brightest objects in the x-ray sky are x-ray binaries, concentrated in the plane of the Milky Way, especially towards the center of the Milky Way. The brightest of x-ray binaries is Sco X-1, which is a neutron star of 1.4 solar masses in orbit around a normal (main sequence) star of 0.4 solar masses. There are a few bright supernova remnants, notably Cas A, also in the plane of the Milky Way.
  • Away from the plane of the Milky Way, the brightest x-ray objects are mostly Active Galactic Nuclei at the centers of galaxies such as M87, thought to be powered by a supermassive black hole.
• Active Galactic Nuclei — a generic term for a variety of phenomena seen at the centers of galaxies
  • Bright.
  • Variable — on timescales of days to years.
  • Non-stellar spectrum — radio, x-rays, gamma-rays; optical emission lines.
  • Jets, often relativistic — superluminal motion, one-sided.
  • Some AGN, known as blazars (after the prototype, BL Lac), have featureless spectra variable down to hour timescales — interpreted as we are looking down the maw of a relativistic jet which is pointed directly at us, and therefore brightened, blueshifted, and speeded up.
  • Maser emission from M106 reveals a thin, warped accretion disk around a 3.9 × 10⁷ solar mass black hole.
• Quasars are the most extreme (luminous) form of Active Galactic Nuclei
  • 3C273, the nearest bright quasar, the first quasar discovered, by Maarten Schmidt in 1963.
  • Name quasar shortened from quasi-stellar-object — an unresolved, star-like source with a non-stellar spectrum and a large redshift.
  • Large redshift, from the expansion of the Universe, indicates that quasars are at cosmological distances.
  • Quasars were most abundant at a redshift of 2, when the Universe was younger.
  • Intensely luminous, as much as 1000 times the brightness of their parent galaxy.
  • Typically variable, on timescales of days; this indicates that they are less than lightdays in size.
• Gravity is the perpetual motion machine that drives the Universe.
  • When you remove energy from a gravitating system, it contracts, and gets hotter (particles move faster).
  • Examples of gravity power in the Universe:
    • A protostar contracts, heats up to the point where it can fuse H (hydrogen).
    • When a main sequence star exhausts H at its center, the He (helium) core contracts and heats up, enabling H to burn in a shell around the core, and causing the star to bloat into a luminous red giant.
    • When the Fe (iron) core of an evolved massive star collapses, the gravitational energy released powers a supernove.
    • The gas in an accretion disk gets faster and hotter as it spirals inward on to a neutron star or black hole. Near the central compact object, the disk reaches relativistic temperatures, and emits x-rays.
    • Twin jets, often relativistic, emerge from the vortical opening along the spin axis of the accretion disk.
    • The ultimate state of gravitational contraction is a black hole.
• Thursday was a review session. See Review for Tue Mar 20 midterm for questions to study.

Week 10

• There was a midterm on Tue Mar 20.
• On Th we watched a NOVA PBS documentary “Monster of the Milky Way”

Week 11

Spring break. Love it.

Week 12

This week was about how to script a movie. We watched excerpts from two movies, “Contact”, and “Walt Disney's The Black Hole”.

• 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 13

• Read Thorne Ch 4 in preparation for Project 4: The Mystery of White Dwarfs.

What happens to cold matter when you compress it very prodigiously?

• The second half of Thorne Ch 5 discusses the behavior of cold matter at very high density.
• 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.

Week 14

Supernovae

• Supernova. A star that for a while becomes comparable in brightness to its parent galaxy.

• The first half of Thorne Ch 5 discusses supernovae.
• History.
  • 1054. Chinese astronomer Yang Wei-te reported a "guest star" (supernova) in constellation Taurus, which was recorded in the Annals of the Sung Dynasty.
  • 1931. Fritz Zwicky points out that some novae (new stars) are exceptionally bright. He calls them "supernovae". He begins a solo campaign to discover supernovae.
  • 1932. James Chadwick discovers the neutron.
  • 1933. Baade & Zwicky make their famous prediction that supernovae produce neutrons stars.
  • 1940s. Baade and others measure the expansion rate of Messier 1, the Crab nebula, and show that it must have originated in an explosion around 1054. They conclude that the Crab nebula is the remnant of the Supernova of 1054.
  • 1967. Jocelyn Bell discovers a pulsar, with a precise period of 1.337301 seconds, then three more. The first few pulsars are given the name "LGM" (Little Green Men). Observations that pulsars pulse over a wide range of radio frequencies ruled out the possibility of the source being intelligent life.
  • 1968. American astronomers discover a 0.033 second pulsar in the Crab nebula. The period is too short and too regular to be anything other than a rotating neutron star.
  • Today. More than 1300 pulsars are now known in our Galaxy, with periods ranging from 0.00156 to several seconds.

• A pulsar is a rotating, magnetized neutron star.
  • Pulsar animation by Paul Bourke, Swinburne Centre for Astrophysics and Supercomputing.
  • Listen to the sounds of pulsars.

• There are two kinds of supernova:

  Type:	Core collapse supernova	Thermonuclear supernova
  Spectra:	Show H lines	Show no H lines
  Where:	In star-forming regions, spiral arms	Anywhere in a galaxy
  ⇒	Young, massive star (> 8 solar masses)	Old white dwarf

• Core collapse supernova
  • Most stars spend most of their lives fusing Hydrogen (H) to Helium (He) at their centers. These are main sequence stars. The Sun is a main sequence star.
  • When stars exhaust H at their center, the He core contracts, H burns in a shell around the core, the envelope expands, and the star becomes a Red Giant. In the Red Giant phase, the star develops a strong wind.
  • For stars less massive than about 8 suns, the wind eventually drives off the entire envelope. For a while, the star is seen as a Planetary Nebula (nothing to do with planets), containing a small hot star inside a glowing gaseous nebula. The central star cools to become a White Dwarf.
  • For stars more massive than about 8 suns, the He core burns to heavier elements. The core develops a complicated multi-layered structure. At the center is iron (Fe), which contains no more nuclear fuel — Fe is the most tightly bound of all nuclei.
    

    Credit: Mark Whittle

  • The inert Fe core is held up by electron degeneracy pressure. When its mass has increased to 1.4 solar masses, the Chandrasekhar limit, the core collapses to a neutron star. Electrons and protons are pressed together into neutrons, releasing an explosion of neutrinos:
    

    e + p → n + ν

    The collapsing envelope of the star bounces off the neutron star core, producing a Supernova.
  • A core-collapse supernovae is powered by the gravitational energy released by the collapse of the core.
  • Computer simulation by Adam Burrows.
  • For stars more massive than about 25 or 30 suns, material falling back on to the newly formed neutron star takes the neutron star over its 3 solar mass limit, causing it to collapse to a black hole.
• Thermonuclear (Type Ia) supernova
  • The light from a thermonuclear supernova is produced by the radioactive decay of ⁵⁶Ni through ⁵⁶Co to ⁵⁶Fe.
    
  • A carbon-oxygen white dwarf accretes from a companion star.
  • When its mass has increased to 1.4 solar masses, the Chandrasekhar limit, the white dwarf starts to collapse.
  • Carbon ignites under pyconuclear (cold, dense) conditions close to the center.
  • Because the pressure is electron degeneracy pressure, the increase in temperature does not initially change the pressure. The temperature skyrockets, causing the nuclear burning to be explosive.
  • The nuclear burning front is extremely sensitive to the precise conditions, and develops in a complicated way: computer simulation of a thermonuclear supernova explosion.
  • The white dwarf incinerates and explodes, leaving nothing behind.

Gravitational waves

• Read Thorne Ch 10 for the history of gravitational waves, which are still yet to be discovered.
• Gravitational waves
  • General relativity predicts that masses that accelerate non-uniformly will produced gravitational waves, which propagate at the speed of light.
  • The gradual decrease in the orbital period of the Hulse-Taylor binary pulsar is consistent with the loss of energy by gravitational waves predicted by general relativity.
  • Werner Benger's video of two black holes merging, producing gravitational waves.
  • The Caltech-Cornell group's visualizations of black holes merging in the first ever detection of gravitational waves by LIGO.
  • A passing gravitational wave produces an oscillating tidal force, which could be detected by an oscillation in the distance between freely-falling masses.
  • To date no gravitational waves have been detected directly.
  • LIGO (Laser Interferometer Gravitational-wave Observatory) has two detectors, one in Washington State, the other in Louisiana. Ligo is as yet too insensitive to detect gravitational waves, but a 2010-2013 upgrade, Advanced LIGO, may permit the first detection of gravitational waves in a few years.
  • NASA's ambitious possible future mission LISA (Laser Interferometer Space Antenna), consisting of three spacecraft in a triangle a million kilometers on a side, should easily dectect gravitational waves from merging black holes and other objects.

Week 15

• Project 5: Compact Binary.
• Gravitational slingshot.

Hawking radiation

• See Hawking radiation. Thorne Ch 12 discusses Hawking radiation.
• The 2nd law of thermodynamics asserts that “entropy always increases” in time. Entropy is a quantitative measure of disorder. The 2nd law is thought to be one of the most fundamental laws of physics.
• As of 1970, it appeared that black holes violated the 2nd law, because stuff that fell through the horizon, carrying entropy with it, simply disappeared.
• Hawking (1970) showed that when two black holes merge, the area of the horizon must increase: the merged black hole's horizon area must exceed the sum of the two original black holes' horizon areas.
• Jacob Bekenstein (1972) noticed a formal analogy between the equations of the thermodynamics and those of black holes, with entropy being proportional to the area of the black hole's horizon.
• Stephen Hawking (1974) made his famous breakthrough, the theoretical discovery of the Hawking effect. Using sophisticated ideas of quantum field theory in curved spacetime, he showed that black holes have a finite temperature, and they radiate. The analogy with thermodynamics was real, not a coincidence. The entropy of a black hole equals 1/4 of its horizon area in Planck units (c = G = ħ = 1).
• How is Hawking radiation produced? Quantum mechanically, virtual pairs of particles can pop out of the vacuum. The black hole swallows one of the particles (which effectively carrys negative energy), leaving the other particle (carrying positive energy) to go off to the outside world.
• Hawking radiation has a black body spectrum.
• The characteristic wavelength of Hawking radiation is roughly equal to the diameter of the black hole. Thus if you could see a black hole by its Hawking radiation, it would look fuzzy, a quantum mechanical object.
• The Hawking temperature and luminosity of astronomical black holes is tiny. The radiation observed from near astronomical black holes is emitted by hot gas from an accretion disk, not Hawking radiation.
• The Hawking temperature and luminosity of so-called mini-black holes is much larger. Stephen Hawking hypothesized that mini-black holes might be created in the Big Bang, but there is no observational evidence for their existence.
• A mini-black hole with mass less than the mass of a mountain can evaporate in the age of the Universe. A mini-black hole of 1000 tons will evaporate in 1 second, in a burst of gamma rays and high energy particles. From a human point of view this is a large explosion, but astronomically it is a tiny explosion, far less energetic than a nova, supernova, or gamma-ray burst.

Week 16

• Review for Sat May 5 Final.

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