Notes for Sixteenth Lecture
Lecture 17, April 9, Richard Gott

You are responsible for material in the book (Time Travel in
Einstein's Universe), even if they are not covered in lecture.  

The lecture covered the following points:

We saw last time that in order to satisfy the postulates of Special
Relativity, time ticks more slowly for clocks moving relative to me.
The ratio of the rate at which clocks tick is given by the Lorentz
Factor, sqrt(1 - v^2/c^2).   Similarly, lengths are contracted by
the same amount.  If an astronaut travels for 500 years at 0.99995 the speed of
light, she travels 500 light years (actually, *slightly* less).  At
that speed, the Lorentz factor is 1/100.  So from her perspective,
only 5 years go by.  She agrees she's traveling at close to the speed
of light, so she sees the distance to be *contracted* to only 5 light
years.  

Notice that this length contraction only happens only *along* the
direction of motion, not perpendicular to the direction of motion. 

You can think of time as the fourth dimension. 
 
Space-Time diagrams and world lines.  There are four dimensions:
length, breadth, thickness and duration.  If you take a movie and cut
it up in frames, and pile them up one on top of another, we have a
space-time diagram, graphing the space dimensions vs. the time
dimension.  Each object in that movie traces out a world line, moving
forward through time.  

To specify a point on the Earth's surface, for example, you need two
coordinates - longitude and latitude. In 4 dimensional space you need
4 to specify an event - this lecture for example takes place in
McDonnell A02 (3 space coordinates), on Thursday April 10th 2008, at
3pm (time coordinate).

Looking at various world lines illustrates how events can appear to
occur simultaneously or not depending on the observer.  

Light travels at the speed of light, and therefore moves at 45 degrees
on a space-time diagram (if we measure time in "length" units by
multiplying by c. Equivalently, if we measure distances in feet, and
time in nanoseconds (ns), the speed of light is one foot per
nanosecond).  All observers can agree on spacetime; when we talk about
events at a given moment in time, we cut slices through spacetime, but
observers at different speeds will slice at different angles.  Thus
observers at different speeds will *disagree* with one another whether
two events are simultaneous or not.  
 
We drew space time diagrams of:
1) The earth moving around the sun - the sun's world line is a straight 
vertical rod, while the earth's worldline is a helix spiraling around
it.

2) The light pulses in a rocket experiment: an astronaut sitting in
the middle of a spacecraft sends 2 light pulses is opposite directions
toward the walls of a 30 feet rocket ship. He is moving at a constant
velocity v=0.8c, and so is his rocket (he is stationary wrt his rocket
at its center). The astronaut observes the 2 pulses reach the opposite
walls at the same time and then come back to the center together. We
see the astronaut moving; his worldline is tilted relative to ours.
We see first one pulse of light hit the back of his rocket (after 5
ns), and 40 ns later the second pulse hits the front. To us the light
pulses hitting the walls are not simultaneous events.  We also
disagree on the time it takes for the pulses - the astronaut thinks it
took 30 ns total for the pulses to hit the walls (15 ns) and come back
(another 15 ns). We think the first pulse hit the back of the rocket
after 5ns and came back to the center after another 45ns (for 50 ns
total). We think the second pulse hit the front of the rocket after 45
ns and the center after another 5 ns (again a total of 50 ns). So we
say 50ns pulse travel time, he says 30ns!  We also disagree on the
length of the rocket - we (stationary) think it is compressed to 18
feet (length contraction), the astronaut thinks it's 30 feet long.

Of course, from the point of view of the astronaut, *we* are moving
relative to him, so he will think that our lengths are contracted. 

  Both us and the astronaut agree that the two light beams come back
  to the astronaut at the same time.  

Although we and the astronaut disagree on the measurements of time,
delta t, and on the measurement of distance, (delta x, delta y, delta
z), we do agree on a combination of them; the space-time separation
of:
  (delta x)^2 + (delta y)^2 + (delta z)^2 - (c delta t)^2

(Think of this as exactly analogous to the Pythagorean Theorem).  

The space-time separation between two objects connected by a
light-beam is always zero.  

  That minus sign (which is needed to make the speed of light look the
  same for all observers) is the origin of the entire distinction
  between time and space in our world.  Space and time are not
  entirely interchangeable! 
 
Also note that space travel at close to the speed of light allows you to 
"visit the future"; time will progress less fast for you than for your
friends back home.   

The twin paradox.
  If one twin, an astronaut, travels to alpha-Centauri, turns around,
and comes back to Earth.  and the other stays home, the traveling twin
will have aged less than the one who stayed at home.  But from the
point of view of the traveling twin, the Earth is the one moving at
high speed.  So what's the point?  The one in the space craft has to
accelerate strongly at the beginning, middle (to turn around) and end
of her journey, while the twin at Earth did not accelerate.  That is,
the one in the spacecraft is *not* in uniform motion the whole time.
We can use space-time diagrams to see what's going on.  The two twins
are not identical; one (the one on the Earth) stays at uniform motion,
while the one in the space craft changed her velocity.

The greatest human time traveler in history was a Russian cosmonaut who
spent more than two years on the Mir space station (travelling at
typically 17,000 miles/hour); he is 1/48 of a second younger than we
would otherwise be. 

Atomic clocks on airplanes have seen this time dilation directly.

Another way to time-travel into the future:  Surround yourself with a
*very* massive shell of matter; the gravitational effect on photons coming
out from you, it turns out, causes your clock to tick slower.
Notes for Eighteenth Lecture