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