Transcript Black Holes: The half hour tour

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?
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
A picture is worth
(over!!) 1000 words… Spacetime diagrams!
Spacetime Diagrams
A better scale
Particles and information travel inside the “light cone.”
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)
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!!!!
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.31:919-944,1999
e-Print Archive: gr-qc/9806123 .
Flat Spacetime
 Down
Center
Particles and information travel inside the “light cone.”
Up 
The same flat plane
from another perspective
• Particles and
information
must stay on
the surface…..
and within
light cone.
Close-up of simple star: (r,t)-plane
large r
r=0
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
large r
r=0
Light ray has to follow spacetime, takes a little
longer to get out.
Up, Down, and Time
for a black hole
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.
Light trapped! (Horizon)
Star
Light escapes!
(No 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 hold
up 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/~marolf
2. Gen.Rel.Grav.31:919-944,1999
e-Print Archive: gr-qc/9806123