Today's lecture covered the following points:

A derivation of E=mc^2. 

Consider an object at rest, of mass m, which emits two photons in
opposite directions.  They have equal and opposite momenta, of
magnitude h nu/c, where nu (the greek letter that looks a little like
the letter 'v') is the frequency of the photon, and h is Planck's
constant.  They each have energy h nu.

  Now look at the same situation in a reference frame where the
original object appears to be moving at some slow speed v.  The two
photons are now each Doppler-shifted, so that their frequencies are
now nu (1 + v/c) and nu (1 - v/c), respectively.  (This is just the
simple Doppler shift that we saw a few weeks ago; in class, Prof. Gott
actually derived these formulae using space-time diagrams).

  So the net momentum of the two is no longer zero, but the difference
of these two expressions (times h/c), namely 2 h nu v/c^2.  2 h nu is
the energy of these photons (remember, there is two of them), so we
can write this as E v/c^2.

  Momentum is conserved, so that extra momentum has to come from our
object which gave rise to the photons in the first place.  All we did
was switch reference frames from our original situation, so its speed
v doesn't change.  Its momentum was mv, and if v doesn't change, only
the mass m could change.

  That is, delta m v = E v/c^2, or E = delta m c^2.  Voila!  The
energy of the photons came from a change in mass of the object from
which they were generated, and we've derived Einstein's most famous
equation.

  Note: you are *not* responsible for the details of this derivation.
It uses some notions from physics which we've covered lightly, if at
all.  In particular, this is the first time you've seen that photons
can carry momentum.

  Remember, c is a huge number, so a little amount of mass gives you
  an *enormous* amount of energy out.  We saw this already, for
  example, when we talked about the energy released when 0.7% of the
  mass of protons is converted to energy when it is fused to make
  Helium nuclei in the cores of stars.  

  That's it for special relativity.  We're going to go on to discuss
  gravity.  We saw Newton's viewpoint: the force pulling an apple from
  a tree is the same gravity that's keeping the moon in its orbit.
  Indeed, from Kepler's Third Law, Newton determined that the force of
  gravity between two objects of distance d was proportional to
  1/d^2.  Newton did this work while still a student at Cambridge.
  The lecture went on to describe some of Newton's history at
  Cambridge, his rivalry with Huygens and interactions with Halley,
  his statue with the quote "Qui genus humanum ingenio superavit" (the
  superior genius of humankind), and so on.  

  Now we go on to general relativity, which is Einstein's theory of
  gravity.  Einstein once said that he had only three good ideas in
  his life: they are:
    -Special relativity
    -The relationship between the energy and frequency of a photon, E = h nu
    -The Principle of Equivalence. 

 Let's now explain the latter: 

  In Newtonian physics, all objects fall at the same rate in the Earth's
  gravitational field, because their inertial mass (that which goes into F=ma)
  and gravitational mass (that which goes into F = GMm/r^2) are the same.
  Einstein thought there was something deep here.

  Consider dropping two balls in outer space inside an accelerating
spacecraft.  The balls won't move (no force on them), but the floor
will come up to meet them.  Einstein's principle of equivalence ("the
happiest idea of his life"): there is *no* way to tell the difference
between the accelerating spacecraft and a gravitational field
(assuming you can't look out the window).  Not only do these two
situations *look* the same, they *are* the same, says Einstein.

  Think also of an elevator on Earth, in free fall (you cut the cord).
Everything is falling together; you are weightless (until you hit the
ground).  This is how they filmed the movie Apollo 13: put everyone
into an airplane, and cut the engines, letting it fall to Earth (don't
worry; they turned the engines on before hitting the ground).

  This turns out to work only if you can allow both space and time to be
curved.  (In the case of a uniform gravitational field, it turns out
you don't need that curvature, but as soon as the gravitational field
points in different directions in different points in space (like
around the Earth), curvature is needed).  Consider a ball tossed up;
it makes a parabolic trajectory.  In space it goes, say, a few feet,
while in time, it travels a few seconds; when converting to distance
units (a few light seconds), the curvature in space *and* time is
quite small.  

  An analogy is the curved surface of the Earth.  Keep yourself on the
surface, and you see that the relationship between radii and
circumferences of circles is not like on a flat piece of paper.  A
straight line (read "geodesic", or "great circle") on a globe is
actually curved if we look at it from afar.  In particular, if we look
at a flat map, like in an Atlas, a great circle appears curved.  The
surface of the Earth does not satisfy Euclidean geometry; the shortest
line between two points is not a straight line in the usual sense of
the term.  (Note that if we step back, and look at the Earth as a
sphere in three dimensions, then it does obey Euclidean geometry.  The
deviation from Euclidean geometry occurs when we restrict ourself to
the two-dimensional surface of the Earth.)

  Einstein: what we call the gravitational force is simply objects moving in
straight lines (a geodesic) in a curved space-time. 

  So the helical worldline of the Earth in orbit around the Sun is a
geodesic; it is as straight as you can be in the curved spacetime
around the Sun.  It doesn't *look* very straight at all.  But remember
that in one year, it moves 1 light-year in the time direction, but
only in a circle of radius 8 light minutes in the space directions.
So it is a very long and stretched out helix; it indeed is pretty
close to a straight line when plotted in space-time.

  So Einstein concludes that gravity causes light to move in curved paths.

  Einstein had the idea for all this in 1907; it took him 8 years to
work out all the rather nasty math.  He was looking for a theory that
agreed with Newton in the limit of relatively small masses and low
speeds. Poor Einstein had to teach himself about a tremendously
complicated mathematical structure called the "Reimann Curvature
Tensor", which has no less than 20 independent components.

  In 1915, he came up with his equations, which look like this:
R_{mu nu} - 1/2 g_{mu nu} R = 8 pi T_{mu nu}
(No, you are *not* responsible for this equation!), and made two
calculations: the orbit of Mercury should not travel in a perfect
ellipse, but precess by 43 arcsec per century (in beautiful agreement
with observations), and light from a star close to the Sun, observed
during an eclipse, would appear to be displaced by 1.74 arcsec.  A
pair of expeditions carried out in 1919 confirmed this latter
prediction, convincing the world that Einstein's theory was right.
This was the beginning of his time of fame. 

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