2. A (Very) Short Course in Relativity In 1905 Einstein published the Special Theory of Relativity (along with photoelectric effect proving light was a photon; and Brownian motion proving atoms exist!) An improvement on Newton’s laws of motion when things move close to c Key postulates: 1) speed of light is constant in a vacuum and the same in all directions; and nothing can go faster than light 2) equations of physics should be the same in all inertial frames (those moving relatively with constant velocities)--the Principle of Relativity TOGETHER THESE LEAD TO IMPORTANT RESULTS: 3) There is no absolute frame of reference -- no preferred observer AND 4) Space and time can’t be considered independently: we have SPACE-TIME: different observers, different values

3. Proof of Constancy of c Michelson & Morley (1887) used an interferometer to see how much faster light was moving with and against the earth’s motion Answer: NO DIFFERENCE!

4. Adding Velocities Relativistically

5. Lorentz Contraction and Time Dilation A moving object appears shorter A moving clock appears to tick slower Lorentz factor, 

6. Special Relativity Works! E=mc 2 : tested in nuclear fission and fusion Lifetimes of cosmic ray muons: they decay in 2.0 microseconds at rest, but travel big distances, implying longer lives (like 44  s) in our frame if they move at 0.999c. Effective mass increases from rest mass as v  c: m eff =  m So it’s harder to accelerate a particle that is moving faster (a = F/m eff ), explaining why so much energy is needed in cyclotrons and other “atom smashers”.

7. GENERAL RELATIVITY In 1916 Einstein published the final form of the General Theory of Relativity Equivalence between gravity and acceleration: you are weightless in a plummeting elevator Improves on Newtonian gravity and motion laws when masses are big

8. Space-Time Warped Near Masses In GR, matter warps space-time, so that the straightest and shortest path (geodesic) looks like a curve to us. Mass tells space how to curve. Space tells matter how to move. Analogy: weight on a tight rubber sheet depresses it, so a ball is deflected

9. General Relativity Works Too! GR predicts that light will appear to bend as it follows a curved path near a mass Measure small displacement of stellar positions near Sun during a solar eclipse (done in 1919): 1.75” at limb Made Einstein world famous since it agreed very nicely!

10. Other Tests of GR Mercury’s perihelion was found to advance some 574”/century but planetary perturbations explained only 531”/cent GR perfectly explained the excess 43”/century Later tests: radar ranging to planets; Global Positioning Satellite (GPS) system; dragging of inertial frames by rotating earth (Gravity Probe B)

11. Gravity Waves: a GR Prediction Gravity radiates energy away as waves, causing orbits to shrink: perfect fit to binary pulsar orbit decay (Noble Prize to Hulse and Taylor in 1993) Detectors (LIGO now; LISA in space planned) may “see” : NS-NS mergers, NS-BH collisions, Supernova explosions; providing a new “window on the universe” (not photons or neutrinos or cosmic rays)

12. BLACK HOLES A part of space-time divorced from the rest of the universe. Not even light can escape if emitted too close to a black hole (BH); inside event horizon or Schwarzschild radius.

13. General Relativity and BHs A BH is a singularity: finite amount of mass at a point, so Density there is (nominally) INFINITE The BH is surrounded by an event horizon or infinite redshift surface or Schwarzschild radius So a BH with Earth’s mass has R S = 1 cm! 100,000,000 M sun BH has R s = 300,000,000 km or 3x10 8 km = 10 -5 parsec = 1000 light-seconds

14. Too much mass in too little volume! Warping of space-time can be so severe that the region effectively pinches off Space-time curvature becomes extremely strong in the vicinity of a BH’s event horizon

15. If the Sun shrank into a black hole, its gravity would be different only near the event horizon Black holes don’t suck!

16. Light waves take extra time to climb out of a deep hole in spacetime leading to a gravitational redshift

17. Redshifted Emission Photons lose energy as they climb out of the gravitational pit established by a BH. We observe longer (redder) wavelengths (lower frequencies) compared to those emitted. Time freezes for a distant observer watching something fall past event horizon

18. Black Hole Applets Escape Velocity and Radius Schwarzschild Radii and Mass Time Near BH Spacetime Orbits

19. Black Holes have no Hair! A BH is characterized by only: 1.Mass 2.Electric charge (astrophysically unimportant) 3.Angular momentum (spin)  ergosphere

20. Rotating Black Holes A rotating (Kerr) BH will have a SMALLER EVENT HORIZON than the same mass non-rotating (Schwarzschild) BH. BUT, outside the Event Horizon there will be an ellipsoidal STATIONARY LIMIT: inside of it, everything MUST rotate w/ BH; outside the Stationary Limit, a powerful enough rocket could stand still. The region between the Event Horizon and the Stationary Limit is called the ERGOSPHERE: (it is sort of donut shaped) In principle (and maybe in practice too!)

21. More About Kerr BH’s In principle (and maybe in practice too!) the ROTATIONAL ENERGY of a BH can be EXTRACTED by PARTICLES or MAGNETIC FIELDS that penetrate the ERGOSPHERE (Penrose effect). A way to make a great garbage disposALL plus power plant! If the SPIN of a BH is too large it could become a NAKED SINGULARITY, with no EVENT HORIZON; but the COSMIC CENSORSHIP HYPOTHESIS argues this never happens and BH's stay clothed with horizons.

22. Tidal Stretching & Hawking Radiation Large gravity differences (tides): “toothpaste tube effect” Quantum gravity effect: Hawking temperature T=h/16  2 kGM=6  10 - 8 K(M  /M) Hawking power: L  R 2 T 4  M 2 /M 4  1/M 2 Incredibly small if BH mass > 10 17 g (rules out stars/galaxies)

23. It’s Hard to Find Black Holes They don’t emit (significant) radiation Light bending means they don’t even show up as dark spots:  Unless distance is close to R S, gravity is close to that of a regular star of the same mass

24. Origin of Black Holes Collapse of very massive stars (>30 M  ) can lead to BHs of ~3-25 M  (neutron stars must have masses below about 2 M  ). A NS could accrete more gas from a binary companion, kicking it over the upper mass limit Collapse of densest regions of forming galaxies, either directly or through merger of stars in dense clusters can yield BHs with M > 1000 M . Quantum fluctuations in the early universe could give primordial BHs of a wide range of masses.