"What goes up must come down": this is not always true!  Throw
something up fast enough, and it can escape Earth's gravity.  The
escape speed that a rocket must attain to escape the gravity of an
object of mass M and radius R is:

V_escape = (2 G M/R)^{1/2}

  For the Earth, V_escape is 25,000 miles/hour. 

  So for a given mass, the smaller R is, the bigger V_escape is.  What
if you compress something to the point that V_escape = the speed of
light?  At this point, nothing can escape (because nothing can go
faster than the speed of light), and the object simply collapses to a
point (nothing can hold it up against gravity).  A black hole.  

  The Schwarzschild radius; think of this as the effective radius of a black
hole; the point of no return :

  R_Sch = 2 G M/c^2

  The material of the black hole is an infinitely small point in the
center, a so-called singularity. (Well, quantum effects may mean that
it has a microscopic, but finite size).

  For 1 Earth mass, this is a bit under one centimeter.  For a 10
solar mass star, this works out to 30 km.  The biggest black holes we
know are about 3 billion solar masses in the centers of massive
galaxies, with a Schwarzschild radius of 9 billion kilometers, or
about 60 AU.  Such black holes power quasars; as gas falls into a
black hole, it tends to swirl around, and glow brightly before it
falls beyond the event horizon.

  Consider two observers (professor and grad student): one safely
outside the black hole, and one falling in.  The one falling in sends
out messages.  Any message sent after the person passes the event
horizon (i.e., the point at 1 Schwarzschild radius) will never make it
vout.  Those that are sent just before this point of no return take a
*long* time to make it out.  The strong gravitational force delays
signals falling in, so that a far-away observer will never see
something fall all the way in.  In particular, a message or signal
sent out *after* you cross the event horizon will *never* make it out!
This is why a black hole is black; no photons ever come out from it!
The event horizon is the horizon beyond which you can't see anything.  

  Even worse, as you fall in, the gravitational force on your feet is
*much* greater than that on your head; you get torn to shreds by the
differential tidal force.  "Spaghettification".  This process is
very quick (0.09 sec); it would be a painless death. 

In 1916, Karl Schwarzschild (who was the father of Martin
Schwarzschild, an eminent astronomer who was a member of our
department until his death a decade ago) worked out an exact solution
for the curvature of space-time around a point mass in General
Relativity.  This laid the basis for our modern understanding of black
holes.  

  Dr. Gott illustrated the propagation of light signals and people
falling into black holes with a series of space-time diagrams (not
reproduced here).  One trick that physicists play is to do various
transformations of these space-time diagrams to try to better
understand what's going on.  We play a similar game when we make a map
of the Earth on a flat piece of paper; we've distorted the
instrinsically curved surface of the Earth, but for a Mercator
projection, at least, directions (East, West, North, South), are
preserved in a simple way.

  For understanding the geometry of space-time in the interior of
black holes, it's useful to transform a space-time diagram to a
coordinate system in which infalling observers appear to be in a
constant position, moving forward in time (this was worked out by
Martin Kruskal of Princeton in 1965; he passed away last year).  In
contrast, the professor staying at constant radius has a 
world line that is hyperbolic (she needs to fire her rockets, and thus
accelerate, to stay at a constant radius).  When one does the Kruskal
coordinate transformation, the singularity at the center of a black
hole looms inevitably in your future, once you pass the event horizon.

  To visualize things further, take a slice of a space-time diagram at
a constant time, to see what the geometry of space looks like.  A
black hole looks flat (Euclidean) far from the black hole, but funnels
down to a long cylinder at the black hole itself.

If a black hole is rotating, then things get yet more interesting (as
first worked out by Kerr in the early 1960's).  The singularity is no
longer a point in this case, but a ring.  If you fall into such a
black hole, falling into the singularity is not inevitable.  Indeed,
it may well be that if you could step *through* that ring, you could
come out into another universe.  Indeed, you can get a whole series of
universes.  To visualize all this, we do another mapping of the
space-time diagram, called a Penrose diagram. The special property of
this representation of space-time is that it brings a point at
infinity to a finite point on the diagram.

However, the infinity of time outside a black hole gets compressed to
a finite time inside a black hole, which means that you will receive
an infinite amount of radiation, infinitely blue shifted, from the
outside in a finite time; you'll get fried!  (On the other hand, you
can watch the entire future of the universe inside the black hole).
Indeed, you can get a singularity from all this radiation, making it
difficult to get through the ring.  If you can avoid this little
problem (perhaps quantum mechanical effects will save us, but to
figure this out, we'd need a theory of quantum gravity, which we don't
yet have), it may well be possible (in principle) that you can step
through the ring singularity to another universe.

John Wheeler of Princeton was the fellow who coined the term 'Black
Hole'.  Sadly, he passed away just last year, at the age of 96.  Among
his famous students was Richard Feynman.  One of the things they
postulated is that a positron (a positively charged version of an
electron) is an electron going backwards in time.  They thought they
could use this to state that *all* electrons in the universe are the
same electron!  It doesn't work, as it would require there to be
roughly the same number of positrons and electrons in the universe,
which turns out not to be the case: electrons are ubiquitous, and
positrons are quite rare.

  Another student of Wheeler was Jacob Bekenstein.  Entropy is a
measure of the disorder of an object; a basic law of thermodynamics is
that the entropy of the universe is always increasing (think of mixing
cold milk into hot coffee; separating the two after the fact would
require an input of a lot of energy).  Does this law of increasing
entropy hold true if I threw my coffee-milk mixture into a black hole?
Bekenstein thought about this, and realized that black holes
themselves have entropy; this entropy is related to the surface area
of its event horizon.

Imagine lowering a mass down into a black hole at the end of a long
cable.  You can tap the energy of the mass falling down the
gravitational field (essentially all of the mc^2 of the mass).  When
the cable finally snaps, the black hole increases slightly in energy
and entropy.  Indeed, one can relate the entropy to the information
enclosed in the black hole (here, information is a formal concept
related to how many bits, in the computer science sense, it would take
to describe the whole physical situation).  

 But entropy is always associated with a finite temperature, so it
would give off black-body radiation.  So do black holes give off black
body radiation?  Steven Hawking thought about this.  Consider the
space right at the edge of the event horizon.  A pair of particles is
formed out of the vacuum; an electron and a positron. Normally, the
two would immediately annihilate, but perhaps one of them will fall
into the black hole, and the other would escape.  The net effect is
for the black hole to give off black-body radiation (albeit very
slowly), and thus slowly lose energy and thus mass.  So black holes
can evaporate on *very* long timescales.  Thus for the supermassive
black hole at the center of a big quasar, 3 billion solar masses, it
will evaporate on a timescale of 10^{94} years.  No wonder no-one has
ever actually seen Hawking radiation!

© Copyright 2009 J. Richard Gott and Michael A. Strauss