A (Very) Short Course in Relativity

1. 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

2. 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!

3. Adding Velocities Relativistically

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

5. Special Relativity Works! • E=mc2 : 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:meff = m • So it’s harder to accelerate a particle that is moving faster (a = F/meff), explaining why so much energy is needed in cyclotrons and other “atom smashers”.

6. 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

7. 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

8. 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!

9. 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)

10. 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)

11. 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.

12. 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 RS= 1 cm! 100,000,000 Msun BH has Rs = 300,000,000 km or 3x108km = 10-5parsec = 1000 light-seconds

13. 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

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

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

16. 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

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

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

19. 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!)

20. 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.

21. Tidal Stretching & Hawking Radiation • Large gravity differences (tides): “toothpaste tube effect” • Quantum gravity effect: Hawking temperature T=h/162kGM=610-8K(M/M) • Hawking power: LR2T4M2/M41/M2 • Incredibly small if BH mass > 1017g (rules out stars/galaxies)

22. 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 RS, gravity is close to that of a regular star of the same mass

23. 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.