Hi Qihan and Wufan,
Special relativity predicts that clocks which are traveling
quickly will run slower than those which are stationary. This is a
difficult prediction to reconcile with our intuition, because one
observer will think a second person's clocks are slow if the second
person is moving, but according to the second person, the first person
is doing the moving, and hence *his* clock should run slow. Puzzling
over who's right in this case leads to the twin paradox.
The special theory of relativity starts with the hypotheses that
1) Space is the same everywhere and in all directions
2) The laws of physics are the same everywhere and to all
observers, even if they are moving at constant velocities relative to
each other.
(If the observers are accelerating, then the rules are broken, and that's what solves the twin paradox).
One set of laws of physics are the laws of electricity and
magnetism. Out of what we know from electricity and magnetism, we can
predict the speed of light. If the speed of light is something
resulting from the laws of physics in one frame, it must be the same in
all frames. This is very hard to match with intuition, because we are
used to the velocities of objects in one frame add to the relative
velocity of the frames to get the velocities of the same objects as
seen by another observer with another point of view. But the physics
has to be the same regardless of the point of view of the observer.
Imagine if you will a contraption consisting of two mirrors facing
each other. They're perfectly shiny, so a pulse of light will bounce as
many times as you like from one mirror to the other and back again. The
time it takes for the pulse of light to go from one mirror to the other
is fixed by the distance between them and the speed of light, and so
this contraption functions as a clock.
Let's view this clock from the point of view of an observer
traveling perpendicular to the path of the light pulse (if the clock is
bouncing light vertically, say the observer is traveling horizontally
past it). According to this observer, as the clock flies past, the
light pulse bounces off of the bottom mirror at an angle so that it
gets to the top mirror just when the top mirror makes it to the same
location. The light pulse makes the same angle with the top mirror and
heads for a spot where the bottom mirror hasn't gotten to yet, but will
get to by the time the light pulse gets there.
According to this observer, the length of the path the light pulse
takes is longer than the distance between the mirrors because the light
beam travels diagonally instead of straight up and down. But the light
pulse has the same speed in both frames, and so it must take longer in
this observer's frame than it does to an observer resting on the clock.
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One of Einstein's great advances here was that this isn't just a
property of this kind of clock, it is a general property of the time
coordinate in moving frames.
"To study the properties of time, one must study the properties of clocks."
Of course the twin paradox is resolved because one twin
accelerates when he turns around to go home. During this turnaround
process, his stay-at-home twin's clock changes rapidly in the moving
twin's accelerating frame, and then runs slow again on the return trip,
but still ends up with much more time on it than the traveling twin's
clock.
Tom
(published on 10/22/2007)
