April 15

Chapter 6 covers up to the end of this week, so you should read it!


1. We started off making the point that because everything heavier than
helium formed in stars and not in the early universe, and your body is
made up of H, O, C, N, Ca (and others), every non-H nucleus in your body
used to be in a star!


2. Summary of stellar remnants:

   When a star can't generate any more energy, it eventually becomes:
 	white dwarf	supported by electron degeneracy	max 1.4M_O
	neutron star	supported by neutron degeneracy		max 2.5M_O

   What happens if the mass is higher? Gravity is stronger than the outward
   pressure, so it must keep collapsing!


3. General Relativity

   To discuss black holes, we need to use Einstein's theory of gravity,
   called GENERAL RELATIVITY. Newton's theory of gravity is a good description
   except in very strong gravitational fields... for example, Mercury's orbit. 
   But Mercury is very close to the most massive thing around, and the
   correction is still very small there, so Newton's theory is still valid most
   of the time.

   	Newton						Einstein

     F = G M m / R^2				No force, but space-time
     						is curved by mass

   What do we mean by "curved" space? From the law of inertia, objects
   move in straight lines if you don't apply a force. In curved space,
   "straight" lines aren't!

      Imagine you stand at the equator of the Earth and walk straight along
      it. Even though you walk "straight", because you are on a curved
      surface your path curves around. This is exactly analagous.

   What happens to light? It has no mass, so according to Newton's theory,
   there is no force and no effect. But in GR, it follows "straight" lines
   in curved space, so it's affected by gravity! This is called GRAVITATIONAL
   LENSING, and has been observed (eg. the cluster of galaxies I showed
   you).


4. Escape Velocity

   If I throw a piece of chalk up in a gravitational field, it rises,
   loses energy, slows down, turns around when it runs out of energy,
   and falls back down where I miss it when I try to catch it. The faster
   I throw it up, the higher it gets before turning around. If I threw
   it fast enough, it could completely escape from the Earth's gravity.
   That speed is called the ESCAPE VELOCITY.

   	v_esc = sqrt( 2 G M / R )

   At the surface of the Earth, v_esc = 11 km/s = 40,000 km/hr. If you
   give the chalk less than v_esc, it comes back. If you give it more,
   it escapes.


   	Q. What if v_esc > c (the speed of light)?

	A. Anything that moves slower than c (ie. EVERYTHING) can't escape!


   We call this a BLACK HOLE.


5. Schwarzschild Radius

   We can turn this around and ask how small you would need to squish an
   object of a given mass for it to be a black hole. This is called the
   SCHWARZSCHILD RADIUS:

   	R_S = 2 G M / c^2     and is about 3km per solar mass

   Some sample numbers:

     Object		Mass		R_S

     Jeremy		55kg		10^-26cm (that's 100 billion times
	 					smaller than an atomic nucleus)
     Earth		6x10^24kg	1cm
     Sun		1 M_O		3km
     neutron star	1.5 M_O		4km
     stellar mass BH	10 M_O		29km
     super massive BH	1 million M_O	4 R_O

   Neutron stars are only 12km anyway - only 3 times bigger than they'd need
   to be to turn them into black holes! That's why it's unlikely that there's
   a force able to stop things bigger than a NS from collapsing - it would
   have to kick in very quickly after the NS limit to prevent a BH from
   forming.

   We also call R_S the EVENT HORIZON. It's a "horizon" because you can't
   see past it - no signals or information or matter or light or ANYTHING
   can come to us from inside the event horizon.


6. "Hair"

   Because of the event horizon, there are only 3 BH properties we can find out
   about:

   	Mass	from how strong the gravity is
  	Spin	from the shape of the event horizon (remember: we expect
	    		black holes to be spinning for the same reason that
			neutron stars spin, only faster!)
	Charge	from how charged particles react (BUT: it's hard to imagine
	    		how you would get a BH with any net charge)

   The fact that we can't determine how the mass is distributed within
   the event horizon, what sort of stuff fell into it, or ANYTHING else,
   is called the NO HAIR THEOREM.


We also started discussing what happens when we watch someone fall into
a black hole, but that will fit better in the next set of notes. Stay
tuned...