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Teaching Black Holes
Donald Marolf, UCSB
July 20, 2006
• SR, GR, & Cosmo
• One semester, 20-30 students
• Only calculus as a pre-requisite
GR can be taught at many levels….
My context:
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
What is a black hole?
What is a horizon?
Physics First! (Hartle, Taylor, Schutz…)
1. With the Schwarzschild metric
2. Without!
With Special Relativity: accelerated frames!
(e.g., Taylor & Wheeler…..)#2 also of some use in public lectures
Spacetime Diagrams
A picture is worth (over!!) 1000 words… Spacetime diagrams!
A better scaleParticles and information travel inside the “light cone.”
Flat spacetime: FBF sBsF
s+L - s = sL s/c2
Equivalence Principle: s
(d/ds)ln(s)
Some quantitative info
I. With the Schwarzschild metric:
ds2 = -(1-Rs/r) dt2 + (1-Rs/r)-1 dr2 + r2 d2
(r) = infinty (1-Rs/r)1/2
~ c2/s + small corrections…
Just like flat spacetime!!!!
Near Horizon:
• 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.31:919-944,1999 e-Print Archive: gr-qc/9806123 .
II. Without the Schwarzschild metric (as an equation)
Flat SpacetimeParticles and information travel inside the “light cone.”
Up Down
Cen
ter
The same flat plane from another perspective
• Particles and information must stay on the surface….. and within light cone.
Down
Up
Close-up of simple star: (r,t)-plane
r = 0
large r
Free fallers fall toward r=0.Effect is stronger near source.
Star not itself freely falling --- some force holds it up!
Star emits a ray of light
r = 0
large r
Light ray has to follow spacetime, takes a little longer to get out.
Up, Down, and Time for a black hole
DownUp
Up
Up
A light ray (45o):
Directed “Up”-wards,
but never gets far away…The horizon!!!
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!!!!
Black Hole vs. Star
Light escapes!(No Horizon)Light trapped! (Horizon)
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.
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.
To see,boost with surface!
• Follow gray dot through time.
• Stay in rest frame of dot.
• Curvature increases and quickly becomes large!
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.31:919-944,1999 e-Print Archive: gr-qc/9806123