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  Spring 2005 ASTR 2030 Black Holes: Weekly Summaries

Week 1

• What is a black hole?
• Event horizon - Schwarzschild radius
• Singularity
• 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"
• Relativistic effects of moving through a scene at near the speed of light:
  • Aberration: ahead appears fisheyed, behind zoomed
  • Color: blueshift (higher energy photons) ahead, redshift (lower energy photons) behind
  • Brightness: brighter ahead, dimmer behind
  • Time: faster ahead, slower behind
• 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?
• 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.
• Lorentz transformations between space and time.

Week 3

• Simultaneity
• Time dilation; spacetime diagram illustrating time dilation
• Spacetime diagram illustrating Lorentz contraction
• Light travel time
• How a scene actually looks seen at close to the speed of light
• Celestial sphere with observer at center ® celestial ellipsoid with observer at focus
• Twin paradox

Week 4

This was mainly a week of summary and review, in preparation for the midterm.

• 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.
• 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.
• Falling into a Black Hole: approach; orbit; singularity
  • What happens when you watch someone else fall into a black hole?
  • What happens when you fall into a black hole?
• The River Model of Black Holes.

Week 6

• 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
• Collapse to a black hole
• Schwarzschild wormhole, white hole
• The Postulates of General Relativity:
  1. Spacetime forms a 4-dimensional continuum (postulate carried over from Special Relativity).
  2. 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).
  3. 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.
  4. Einsteins equations (1915):
     

     Gmn  =  8p 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.

Week 7

This was mainly a week of summary and review, in preparation for the midterm.

• We watched "Under the Night", the first episode of the "Andromeda" series, and did a group project on it.

Week 8

This week was about how to script a movie. We watched excerpts from two movies, "Contact", and "Walt Disney's The Black Hole". At the end of the week we did an in class group project to critique Chapter 4 of Thorne as if it were a story.

Week 9

• Why did Eddington find white dwarfs paradoxical?
• Why did Eddington resist the idea that white dwarfs could not be more massive than 1.4 suns?
• What happens to cold matter when you compress it very prodigiously?
• A solid or liquid is almost incompressible because the atoms are tightly packed together, in a regular lattice in a solid, irregularly in a liquid.
• Atoms resist compression because electrons satisfy an exclusion principle: you can't pack electrons together more closely than a wavelength.
• Heisenberg's uncertainty principle:

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

  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.

• Dick McCray's hypertext includes chapters on Stellar evolution, Novae and supernovae, and Neutron stars and black holes.
• Evolution of stars
  • 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.
  • 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 + n

    The collapsing envelope of the star bounces off the neutron star core, producing a Supernova.
• Core collapse supernovae
  • A core-collapse supernovae is powered by the gravitational energy released by the collapse of the core.
  • computer simulation by Adam Burrows
• Pulsars - rotating magnetized neutron stars
  • The discovery of neutron stars and pulsars. More than 1300 pulsars are now known in our Galaxy, with periods ranging from 0.00156 to several seconds.
  • Baade & Zwicky's (1934) famous prediction of neutron stars as the remnants of supernovae
  • Jocelyn Bell (1967) discovered pulsars
  • Discovery (1968) of a 0.033 second pulsar in Messier 1, the Crab Nebula, the remnant of the Supernova of 1054 AD. The pulsar period was too fast and too regular to be anything but a rotating neutron star.

Week 10

• 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 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.
• 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).
• Gravity power:
  • A protostar contracts, heats up to the point where it can fuse H.
  • When a main sequence star exhausts H at its center, the He 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 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.
• Eddington's paradox: If a star, when it runs out of fuel, heats up, how can it ever cool down?

• 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.
• 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.
• 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.
  • 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.
• 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.

Week 11

Spring Break. Yikes we needed that.

Week 12

• You watched the "Black Holes and Relativity" Planetarium show at Fiske, in two episodes. Much of what you saw recapitulated things you have read in the book or heard in the lectures. However, Fiske showed you some things hard to see other than in a planetarium.
• The x-ray sky is very different from the visible sky. X-rays are high energy photons, so when you see an x-ray source in the sky, it inevitably represents some energetic and violent process going on.
• X-ray sources on the sky are distributed in two ways:
  • X-ray sources in the plane of the Milky Way, especially towards the center of the Milky Way. Most of these are x-ray binary systems, though a few are supernova remnants.
  • X-ray sources out of the plane of the Milky Way. These are almost all Active Galactic Nuclei.

Week 13

• Webster Cash lectured on MAXIM (Micro-Arcsecond X-ray Imaging Mission), and on his latest concept, the New Worlds Imager.
• Cash's theme was: We will see solar systems (with telescopes) long before we visit them (with spacecraft).
• Air absorbs x-rays, so x-ray telescopes must observe from space.
• Air blurs visible light (to about an arcsecond), so to see at high angular resolution an optical telescope must also observe from space.

• Recall the no-hair theorem, that an isolated BH is characterized by just 3 things, its mass, charge, and spin.
• Charged black holes
  • The Reissner-Nordström (RN) geometry describes the geometry of empty space around a spherical, electrically charged, black hole. The RN geometry assumes that the electric charge of the black hole is concentrated at a point at the central singularity.
  • Real black holes probably have very little electric charge. But the geometry of a charged black hole is qualitatively similar to that of a rotating black hole, so models of black holes often use charge as a surrogate for spin (as in the BHFS).
  • The RN geometry is, surprisingly, gravitationally repulsive in its core.
  • The gravitational repulsion causes a suite of remarkable phenomena: the inside of the RN black hole has a wormhole which connects to a white hole which connects to a new universe. The new universe connects via another blackhole-wormhole-whitehole to yet another universe, and so on.
  • The RN geometry is however inconsistent: it assumes that the BH is empty, yet the repulsive core will prevent mass or charge from falling on to the central singularity, so in reality the BH could not be empty.
  • The river model of the RN geometry describes how space falls into the RN black hole, turns around, and flies back out through a white hole into a new universe. Horizons, outer and inner, occur wherever the velocity of space hits the speed of light.
  • The RN geometry proves to be violently unstable at its inner horizon. The RN geometry predicts that light that falls into the black hole will pile up at the inner horizon, producing an infinitely energetic shell, contradicting the hypothesis that the RN geometry is empty. An infaller passing through the inner horizon sees this light as an infinitely bright flash of light. The concentration of light and matter at the inner horizon destabilizes the inner horizon, changing the geometry drastically.
  • In a real black hole, the instability at the inner horizon has the consequence that the region from the inner horizon inward would contain a dense, relativistic plasma, which would fry an infaller.
• Rotating black holes
  • The Kerr-Newman (KN) geometry describes the geometry of empty space around a spinning, possibly charged, black hole.
  • Real black holes probably spin.
  • Centrifugal force causes the KN geometry to be gravitationally repulsive in its core.
  • The phenomenology of the KN geometry is quite similar to that of the RN geometry: the gravitational repulsion causes the KN geometry to have both inner and outer horizons, and blackhole-wormhole-whitehole connections to new universes.
  • The KN geometry is, again, inconsistent, because the black hole cannot be empty in its core if the core is repulsive.
  • The singularity of the KN geometry forms a ring, kept open by the centrifugal force.
  • The outer horizon and inner horizon are confocal ellipsoids, with the ring singularity at the focus.
  • The other side of the disk bounded by the ring singularity is a new region, the "antiverse", at negative radius. In the antiverse, the black hole appears to have negative mass.
  • In the antiverse, circles around the axis near the disk are timelike. These circles form "closed timelike loops" (CTLs), where time keeps repeating itself.
  • The inner horizon is, as in the charged black hole, violently unstable.
  • As in a charged black hole, the instability causes the region inside the inner horizon to contain a dense, relativistic plasma, which would fry an infaller.

Week 14

This week was on Hawking radiation.

• Hawking radiation is quantum mechanical radiation from black holes. 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 15

This week was on cosmology. Cosmology is one of the major applications of general relativity, the other major application being black holes. A third major application, gravitational waves, will become important in the next two decades with the advent of gravitational wave astronomy.

To allow you time to prepare thorougly for the final, none of the material presented in the last two weeks (15 and 16) of the semester will be tested on the final. In particular, Cosmology will not be tested on the final.

• The Standard Model of Cosmology.
• Hubble's law v = H₀ d, recession velocity = Hubble's constant H₀ times distance.
  • Implies that the Universe is expanding, and that there was a Big Bang.
  • H₀ determines the age of the Universe, t » 1/H₀ (the equality is not exact because of the deceleration or acceleration of the Universe).
• W = (actual density of the Universe)/(critical density of the Universe).
  • Through Einstein's equations of general relativity, W determines the curvature, and also the fate, of the Universe.
  • Open, flat, and closed geometries of the Universe.
• The Cosmic Microwave Background (CMB) is the radiation remnant of the primeval hot Big Bang fireball.
• Observationally, the Cosmic Microwave Background:
  • Has an exquisite black body spectrum, with a temperature of 2.726 Kelvin.
  • Is almost uniform on the sky.
  • Shows a "dipole" distortion from the motion of the Sun through the CMB, at 365 km/s.
  • After subtraction of the dipole distortion, the residual temperature fluctuations are tiny, a few parts in 10⁵.
  • The power spectrum of fluctuations shows a pattern of acoustic peaks in remarkable agreement with the predictions of the theory of Inflation.
• Theoretically, the Cosmic Microwave Background:
  • Supports the idea that there was a hot Big Bang.
  • The CMB cools as the Universe expands, the wavelengths of CMB photons being stretched (redshifted) by the expansion of the Universe.
  • Its uniformity and black body spectrum tell us that the Universe used to be much simpler when it was young.
  • Comes to us from the Epoch of Recombination, when the Universe was 370,000 years old, and the temperature was 3000 Kelvin.
  • At Recombination, hydrogen and other elements combined from an ionized plasma of nuclei and free electrons to a neutral gas of atoms. As a result, the Universe changed from being opaque to transparent, allowing the CMB to propagate freely to us from that time.

• The theory of Inflation postulates that the during its earliest moments (the first 10^-33 seconds or so), the mass-energy of the Universe was dominated by vacuum energy. Vacuum energy is gravitationally repulsive, and would cause the Universe to expand exponentially (inflate), doubling in radius every 10^-35 seconds.
• Inflation solves all of the following problems:
  • The Horizon Problem. How come regions of the CMB more than 2° apart have the same temperature, even though they were causally disconnected at the time of Recombination?
  • If gravity is always attractive, then why is the Universe expanding?
  • How come the Universe is so flat?
  • How come the Universe is so large?
  • Where did the mass-energy content of the present day Universe come from?
  • What produced the small fluctuations that grew by gravity into galaxies and stars today? Answer: quantum fluctuations of the vacuum.
• Inflation predicts several key observed features of fluctuations in the temperature of the CMB:
  • A scale-invariant spectrum of fluctuations at large scales (much larger than the horizon size at Recombination).
  • A regular sequence of acoustic peaks.
  • Flat curvature.
  • A random (Gaussian) noise pattern.
• Inflation ends when the vacuum energy decays into other forms of energy, namely matter and radiation. This is where matter and radiation in today's Universe came from.
• The vacuum energy that powers inflation is thought to be the energy of the unification of forces. Specifically, the energy is thought to be associated with the hypothesized unification of the electroweak and color (or strong) forces into a GUT (Grand Unified Theory) Force. The energy scale is not accessible to present day particle accelerators, so the nature of the unification is not well understood.

Week 16

• Review week.

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