Today's lecture covered the following points:

Albert Einstein is a *really* smart guy, and his work was truly
important.  

Einstein's Theory of
General Relativity was a reworking of Newton's Law of Gravity. A prediction
of the theory is that light passing close to the limb of the Sun is
bent by the Sun's gravity.  This can be, and was, checked by carefully
measuring the position of stars near the Sun during a solar eclipse.
Einstein's prediction was confirmed, and once his theory of gravity
had beaten Newton's, he became a world-famous person.  

  Newton's most famous equations: 
     F = ma (how force relates to acceleration)
     F = -GmM/r^2 (gravitational force between two masses)

  Einstein's most famous equations:
     E = m c^2 (mass and energy are equivalent)
     E = h nu (light comes in packets called photons; this relates
          their energy and frequency)

And Einstein lived in Princeton!  He worked at the Institute for
Advanced Study.  His house, 112 Mercer Street, is a private residence
(go by and take a look at it!).  Frank Wilczek lived there, and he and
the current resident (an economist) both won the Physics Nobel Prize.  Not bad.

Newton invented the milling on the rims of coins.  
Einstein wrote the letter to Roosevelt which led to the development of
the atomic bomb (the Manhattan Project).  

  In the 1870's, Maxwell put together our modern understanding of the
interaction of electricity and magnetism.  The basic input:
   1.  The force between two charges, is, like gravity, inversely proportional
to the distance between them (Coulomb's law).  Opposite charges attract, but,
unlike gravity, like charges repel.
   2.  A moving charge gives rise to a magnetic field. The
ratio of magnetic and electric forces is given roughly by v^2/c^2, where v is
the speed of the charges, and c is the speed of light.
   3.  If an electric field changes with time, it will give rise to a magnetic
field.
   4.  If a magnetic field changes with time, it will give rise to an electric
field.

Putting these into mathematical form (Maxwell's Equations), Maxwell was able to
solve the equations to find a prediction of a travelling wave of electric and
magnetic fields, which moves at 300,000 km/s; the speed of light.  This
traveling wave *is* light.  

  The speed of light was measured by Roemer in 1676, using careful measurements
of the orbits of Jupiter's moons.  Their eclipses behind Jupiter are a sort of
clock; that clock varied by 16 minutes depending on the relative position of
Jupiter and Earth, due to the varying delays in the light signal arriving to
us.

  In 1728, Bradley measured the "aberration of starlight", whereby the position
a star appears in the sky varies by 20 arcsec, due to the fact that we're
travelling around the Sun at 30 km/s.  

  As a teenager, Einstein thought about Maxwell's equations.  Suppose we rode
alongside the light wave (necessarily at the speed of light).  The electric and
magnetic fields are *not* varying, so they can't create each other.  This
sounds like a fundamental inconsistency in the theory.

  In thinking about these various puzzlements, Einstein came up with his two
fundamental postulates of Relativity (where Relativity refers to the fact that
all that matters is *relative* motion):

  -The effects of the laws of physics should look the same to every
observer in uniform motion (i.e., motion at a constant speed in a
constant direction, without turning), no matter what speed you're
moving at.  Thus there is no absolute reference frame which we can
state is at rest, and all that counts is relative motion.

  -The speed of light through empty space should be the same as
  witnessed by every observer in uniform motion.  (Otherwise, you
  could discover that you are moving relative to an absolute reference
  frame, in violation of the first postulate.)

  Newton would have been quite happy with the first postulate.  The
second postulate, which is needed to make Maxwell's interpretation of
the nature of light self-consistent, is much more non-intuitive.  It
says that even if you travel in the same direction as a light beam,
at, say, 2/3 the speed of light, the speed of that light beam will be
the same as measured by you as someone standing still.

  One immediate implication: you cannot go faster than the speed of
light.  The first postulate of special relativity says that if you're
moving at constant velocity, there is no experiment you could do that
would demonstrate your speed (i.e., all speeds are relative).  If you
went faster than the speed of light in a spaceship, a beam of light
would never hit the far wall (the far wall is outrunning the light
beam), which would indicate to you that you're going faster than the
speed of light.  You would be able to tell that you're moving in an
absolute sense, in violation of the first postulate.

  Next implication: clocks tick slower when moving.  

  Consider a light clock: a closed cell with mirrors on both ends; light
bounces back and forth between the two mirrors, clicking at each bounce so as
to give the beat of the clock.  Hold this clock vertically, and start counting
beats.  The light travels at the speed of light.

  Now travel forward at very high speed.  Seen from an observer at rest, the
light beam makes a zig-zag path through space.  It travels at the same speed
(by the second postulate), so the clock is ticking *slower*.
Moreover, by the first postulate, you can't do an experiment to
distinguish the fact that you're moving, which says that all clocks
inside the rocket, including the biological clock of the astronaut,
must agree with one another.  

The time the light takes to make a round trip in the clock of length l when it
is at rest with respect to the observer is t0 = 2*l/c. 

The time it takes the clock to tick when it is travelling at a speed v
perpendicular to its length is 
t1 = 2*d/c = 2*l/(c*sqrt(1-(v/c)^2)) = t0/sqrt(1-(v/c)^2). We see the clock
ticking more slowly.  So the measurement of time is different for us
sitting still, and the observer moving fast.  If the speed is small,
much less than speed of light, then the correction effect is tiny, and
time changes very little relative to those of us at rest.  

We'll see next time that this leads to the most famous equation,
E=mc^2.  

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