The lecture begun with the introduction of 
the Map of the Universe (in pdf)  constructed by Prof. Gott and
Mario Juric. A version that can be viewed in the browser is found  here
 (click on the image if your browser shrinks the picture). 
Note that this is a logarithmic map -- it goes in factors of 10
in distance starting from the Earth. Make sure to study it!
A version of this map can also be seen on the carpet on the floor of
 Peyton Hall. 

In special relativity, the interval between two events in spacetime
is:
  ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

The minus sign means that two events connected by a lightbeam has an
interval of zero.  

Consider a 3-dimensional space-time: two coordinates of space, and one
of time.  "Flatland".  One can work out the equations of general
relativity in this geometry.  It turns out the solutions for a point
mass in flatland are essentially equivalent to those for cosmic
strings in our 4-dimensional world.   Interestingly, two point masses
don't attract one another in flatland. 

  How about more dimensions?  In 1926, Kaluza and Klein said, suppose
there was an extra dimension of space.  I.e., 3 normal space
dimensions, one time dimension, and one more space dimension that's
curled up like a soda straw.  This extra dimension could be *very*
small, indeed truly microscopic.  Indeed, Kaluza-Klein suggested that
this last dimension have a radius of 10^-31 cm.  When one applies
General Relativity to this problem, lo and behold, they found 4-D
general relativity, plus the equations of electromagnetism (i.e.,
Maxwell's equations).  This seems a unification of general relativity
and electromagnetism; cool!  Unfortunately, there is no testable
difference between this and the standard picture, so there is no way
to test whether the idea was right.  That is, it is indistinguishable
from our standard picture.

  There are other forces: weak forces (responsible for some kinds of
radioactivity) and strong forces (the latter holds nuclei together, as
we've seen), in addition to gravity and electromagnetism.

  In the 1970's, electromagnetism and the weak force were unified;
i.e., a theory was found which explains the two in a single framework,
which had predictions which were later experimentally verified.  But
getting all four together in one unified theory (a holy grail of
modern physics) remains beyond our reach.  The current best hope is
something called string theory, which really is an extension of Kaluza
and Klein's ideas.  In modern string theory, they want 10(!)
dimensions of space (and one of time).  All but 3 of the space
dimensions are "compactified", as a 7-dimensional very small doughnut.
So an electron is not a point, but it is a loop of string (closely
analogous, but *much* smaller, to the cosmic strings discussed last
time).  People are working on this; it is not yet clear whether this
will work.  No testable predictions have come out of all this work
yet, so we don't know yet if it is right.

The structure of Einstein's Equations:
  They relate the curvature of spacetime (through something called the
  Reimann Curvature Tensor) to the mass, pressure, and energy at every
  point in space (the "stress-energy tensor").  

  So much for the very small.  Now let's talk about the very large.
So can we apply the ideas of general relativity to the universe as a
whole?  Newton thought about this problem.  He imagined an infinite
universe of stars (he didn't know about galaxies yet), and saw that at
each point in space, there were equal numbers of stars in each
direction; the gravitational forces cancelled, and so there was no net
force: such a universe was static, and had existed forever.  In
general relativity, things are more complicated: the mass of galaxies
causes space to curve, and so things can't be static.  Einstein
recognized this, and imagining that the universe "must be static",
decided that his equations were wrong.  He added another term to his
equations of general relativity, to balance the apparent tendency of
the universe to contract under gravity; this term was called 'the
cosmological constant'.  We will see in the next lecture that this
term is important, but not for the reasons Einstein envisioned
them. Our modern way to look at it is to put this term on the
right-hand side of the equation, as part of the stress-energy tensor:
it is an energy density, plus negative pressure associated with empty
space.  (Huh?  Negative pressure?  What's empty space doing with a
pressure?  We'll discuss these questions next lecture; in the
meantime, remember that pressure causes a force only when there are
*differences* in pressure from one region to another.  There are 15
pounds per square inch of atmospheric pressure on our bodies, but we
don't feel it, because there is an equal and opposite pressure inside
our bodies to counteract it.  This negative pressure "stuff" is the
same everywhere, so it doesn't push things around.  However, pressure
*can* cause gravitational effects, indeed, because it is negative, it
makes gravity *repulsive*; again, we'll see what this does in the next
lecture).   We give this extra negative pressure stuff the name "dark energy".  

  Anyway, back to Einstein.  He imagined a uniform, static universe.
The mass in the universe caused space to be curved like a 3-sphere
(the three-dimensional surface of a four-dimensional sphere).  This is
the 3-D analog of the 2-d surface of an ordinary sphere.  That is, the
2-d surface of an ordinary sphere has the equation x^2 + y^2 + z^2 =
r^2; the 3-D surface of a 3-sphere, similarly, has the equation 
x^2 + y^2 + z^2 + w^2 = r^2.  

A 3-sphere has 3 dimensions; go a long distance in any one direction,
and come back to where you started (again, think of the analogy of the
surface of an ordinary sphere).  Such a model is called 'closed'; it
is finite in volume, but has no edge (there is no outside).

When we talk about a three-sphere as the surface of a four-dimensional
sphere, or draw the expanding universe as a football (see below), we
find ourselves representing curvature by "embedding" the universe in
larger-dimensional representations that obey our familiar Euclidean
geometry.  The four-dimensional space doesn't exist in any sense.  The
interior of the football doesn't exist; this is just a mathematical
device that helps us visualize curvature.

  Friedmann went back to Einstein's original equations, (forgetting
  the cosmological constant), he found solutions in which the universe
  starts in a point (a Big Bang), it then expands, reaches a maximum,
  and then starts collapsing again to a "Big Crunch".  The mutual
  attraction between galaxies pull them back together.   So the
  prediction is that the universe should be either expanding or
  contracting.  The spacetime diagram of this universe looks like a
  football.  

  In 1929, Hubble discovers the expansion of the universe, as we
  discussed in an earlier lecture.  So Friedmann was right! Einstein
  realized that the cosmological constant was a terrible blunder, and
  dropped the idea like a hot potato. 

  What happens to a light wave in an expanding universe?  The waves
  expand with the universe itself.  That is, the wavelengths get
  longer with time.  This is the redshift that we discussed earlier.  

  There was a Big Bang, a time when the galaxies were all on top of
  each other.  There was no center in space in this universe; every
  point looks like the center, even at the Big Bang itself. 

  The early universe was hot.  Expanding things cool, and the universe
  is therefore cooler than it used to be.  The heat of the universe
  will give rise to black-body radiation, which we might actually be
  able to observe. 

  Gamow thought about this in the 1940's.  He realized that the early
universe was *very* hot, so hot that even atoms, even atomic nuclei,
couldn't exist.  But as the Universe cools, various nuclear reactions
would take place to create atomic nuclei.  Gamow and his students
Hermann and Alpher attempted to use this to explain the full range of
atomic elements observed in the present-day universe.  At the same
time, Fred Hoyle (who was very skeptical about the Big Bang idea) said
that all the elements are instead created in stars.  It turns out both
were right, and both were wrong.  In the early universe, one naturally
gets out the helium nuclei (remember, it is 24% of the atoms in the
universe), and deuterium too ("heavy hydrogen", a proton and a neutron
in the nucleus), a bit of Lithium (#3 on the periodic table), and not
much more.  Elements on the rest of the periodic table get created in
the interior of stars, as we saw earlier.  

  Gamow and his collaborators, in trying to explain the amount of
helium we saw in the present-day universe, realized that it would
depend on the balance between the expansion rate and the temperature
of the early universe.  They therefore calculated the temperature.
Again, an expanding gas cools, so they could extrapolate forward to
ask, what is the temperature of the universe today?  Their answer was
about 5 degrees Kelvin.  That is, they predicted that we should see
blackbody radiation of this temperature coming to us in all
directions. 

  In the 1960's, Bob Dicke, Jim Peebles, and David Wilkinson of
Princeton went through a similar calculation, and came to the same
conclusion (they were unaware of Gamow's earlier work).  Realizing
that a 5 degree blackbody will emit most strongly at microwave
wavelengths, they started building a microwave antenna (on the rooftop
of Guyot Hall!) to detect this.  However, Penzias and Wilson of Bell
Labs (30 miles from here!) serendipitously discovered the black-body
radiation predicted by the hot Big Bang idea. This is one of our
strongest confirmation that the Big Bang idea is correct.  

 
Notes for Twenty-second Lecture 

© Copyright 2008-9 J. Richard Gott III and Michael A. Strauss