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Teaching Black Holes

This teaching guide explores the concept of black holes, focusing on the horizon and the curvature of spacetime. It includes PDF notes, spacetime diagrams, and explanations using the Schwarzschild metric and curved spacetimes.

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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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