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Teaching Black Holes

Teaching Black Holes. Donald Marolf, UCSB July 20, 2006. GR can be taught at many levels…. My context:. SR, GR, & Cosmo One semester, 20-30 students Only calculus as a pre-requisite. Goals:. Excite Students!! Recruit Majors!! What is a horizon? What is an expanding universe?.

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Teaching Black Holes

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  1. Teaching Black Holes Donald Marolf, UCSB July 20, 2006

  2. GR can be taught at many levels…. My context: • SR, GR, & Cosmo • One semester, 20-30 students • Only calculus as a pre-requisite Goals: • Excite Students!! Recruit Majors!! • What is a horizon? • What is an expanding universe? PDF notes (300+ pages) at http://www.physics.ucsb.edu/~marolf

  3. What is a black hole? What is a horizon? Physics First! (Hartle, Taylor, Schutz…) • With the Schwarzschild metric • Without! With Special Relativity: accelerated frames! (e.g., Taylor & Wheeler…..) #2 also of some use in public lectures

  4. A picture is worth (over!!) 1000 words… Spacetime diagrams! Spacetime Diagrams

  5. A better scale Particles and information travel inside the “light cone.”

  6. Some quantitative info Flat spacetime: aF/aB = tB/tF = sB/sF ts+L - ts = tsL as/c2 Equivalence Principle: as = (d/ds)ln t(s)

  7. I. With the Schwarzschild metric: ds2 = -(1-Rs/r) dt2 + (1-Rs/r)-1 dr2 + r2 dW2 t(r) = tinfinty (1-Rs/r)1/2 Near Horizon: a ~ c2/s + small corrections… Just like flat spacetime!!!!

  8. II. Without the Schwarzschild metric (as an equation) • Examine and interpret pictures of curved spacetimes. • Physics first!!! Give them a picture! Embed (r,t) plane in 2+1 Minkowski space • Approach provides some insight with or without explaining how these solutions are generated. • For details, see Gen.Rel.Grav.31:919-944,1999e-Print Archive: gr-qc/9806123 .

  9. Flat Spacetime Particles and information travel inside the “light cone.” Center  Down Up 

  10. The same flat plane from another perspective • Particles and information must stay on the surface….. and within light cone.  Down Up 

  11. Close-up of simple star: (r,t)-plane Star not itself freely falling --- some force holds it up! large r r = 0 Free fallers fall toward r=0.Effect is stronger near source.

  12. Star emits a ray of light large r r = 0 Light ray has to follow spacetime, takes a little longer to get out.

  13. Up, Down, and Time for a black hole Up  • Down Up  A light ray (45o): Directed “Up”-wards, but never gets far away… Up  The horizon!!!

  14. More views of the Horizon: • Yellow rays don’t fly away. Remain `at the same place’ but `directed outward.’ • All information which enters is trapped inside!!!!

  15. Black Hole vs. Star Light escapes!(No Horizon) Light trapped! (Horizon)

  16. Approaching a black hole • Make star smaller but keep total mass fixed. Star approaches Schwarzschild radius r=2MG/c2. • Crease becomes sharper. • At r=2MG/c2, would require infinite force to holdup star. Star collapses uncontrollably.

  17. Where is the singularity? • Singularity inside and in future. • Hard to see ‘cause surface strongly boosted there. • Moves at nearly light speed. Makes surface look flat, but in reality strongly curved! Similar to `headlight effect.’ • Strong boost also brings`far future’ to finite proper time! • Proper time to `top’ is finite along surface.

  18. To see,boost with surface! • Follow gray dot through time. • Stay in rest frame of dot. • Curvature increases and quickly becomes large!

  19. Summary • General Relativity predicts black holes when large masses are compressed to small size. • Spacetime becomes highly curved, and a horizon forms. • A horizon is just a sphere of outward-directed light rays that “don’t make any progress” due to the curvature of spacetime. • Since information cannot flow faster than light, any info that enters must remain inside. • References:1. http://www.physics.ucsb.edu/~marolf2. Gen.Rel.Grav.31:919-944,1999e-Print Archive: gr-qc/9806123

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