Teaching Black Holes Donald Marolf, UCSB July 20, 2006

SR, GR, & Cosmo One semester, 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

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

Flat spacetime:  F    B  F  s B  s F  s+L -  s =  s L  s /c 2 Equivalence Principle:  s  (d/ds)  ln  (s) Some quantitative info

I. With the Schwarzschild metric: ds 2 = -(1-R s /r) dt 2 + (1-R s /r) -1 dr 2 + r 2 d  2  (r) =  infinty (1-R s /r) 1/2  ~ c 2 /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: ,1999 e-Print Archive: gr-qc/ II. Without the Schwarzschild metric (as an equation)

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

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  Down Up  A light ray (45 o ): 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/c 2. Crease becomes sharper. At r=2MG/c 2, 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: Gen.Rel.Grav.31: ,1999 e-Print Archive: gr-qc/