When I first encountered the theory
of Special Relativity in a layman’s science book, I was certain it could be
proved wrong. Take the statement
“Moving clocks run slow”. How could two
observers moving past each other both believe that the other’s clock is running
slow? Surely there must be a way for
one observer to jump over to the other observer, compare clocks and verify
which, if any, was *really* running slow.
But when I learned enough to put numbers to the problem, I found out
differently. Following is an example
with numbers.

Lets say that you and I are
snoozing in a spaceship, rockets shut off, way beyond the influence of any gravitating
body. Suddenly the collision avoidance
alarm rings. We rouse ourselves just in
time to see a Chinese spaceship go whizzing by. Our computer dutifully records the time of the event and also
that the Chinese ship is traveling at .8c (that’s 80% the speed of light with
respect to us). Of course the crew in
the Chinese ship is recording that it is *we* who nearly collided with *them*,
callously speeding by at .8c. I tell
you that I *know* their clocks are running only 60% as fast as ours. I also tell you that they *know* our
clocks are running only 60% as fast as theirs.
After a year of hearing me say that, you’ve had enough. No way both crews could be right and you are
going to prove somebody wrong. So you
put one of our clocks in your pocket, get in the shuttlecraft and take off
after the Chinese spaceship. I just
keep making calculations.

Lets see. Since a year has passed, the Chinese ship is now .8 light years
away from us and still receding. To
catch up, you will have to do better than .8c, and for quite a while. Lets say you get the shuttle up to .9c (with
respect to me). You will have to stay
at .9c for 8 years to catch them. At
.9c the rate of the clock in your pocket has slowed to 43.59% of the clocks
still with me. When you finally catch
up with the Chinese spaceship, I calculate the time expired on your clock will
read *1 year with me + 8 years at 43.59%, that’s 1 + 3.487 or 4.487 year*s
*total* since the spaceships passed each other. On the other hand I calculate the clocks on the Chinese spaceship
will show *9 years at 60% or about 5.400 years* *total* since the
passing. Assuming SR theory to be
correct, when you catch them, your clock will read 4.487 years elapsed and
their clock will read 5.400 years elapsed.
Now I just make calculations.
You are in a position to compare clocks and prove the calculations right
or wrong. You don’t need to make any
adjustments that would be required of someone looking through a telescope. You can set aside phrases like “It only
appears that way”. You will be holding
yours and the Chinese clocks in your tight little hands. You don’t have to worry that the clocks
might work differently. Your clock
reads “made in China” too. Lets say you
compare clocks and they indicate what I calculated. What have you proved?
Nothing yet. Not until the
Chinese crew shows you their SR calculations.
In order for Special Relativity to be relative, they have to predict the
same readings. Will they?

Let’s take the point of view of the
crew on the Chinese spaceship. The only
number we have in common is that we both hold that the other ship went by at
.8c. You tell them you started in
pursuit 1 year after the passing. They
conclude you started when their clocks read 1.667 years because your clock was
running only 60% as fast as theirs. By
their reckoning, you were 1.333 light years away when you started in pursuit (x
= vt; .8c times 1.667). And they
observed your pursuit speed at -.357c (Relativistic Velocity Addition of the
American spaceship, .8c, plus the shuttlecraft, -.9c). Again, by their reckoning, it took you 3.734
years to cover the 1.333 light years separation (t = x/v). So, on their clocks, *1.667 years waiting
+ 3.734 years pursuing gives 5.401 years* expired off their clocks when you
meet. That’s the same prediction I made for their clocks give or take.

And what are they going to
calculate for the time on your clock?
During the pursuit you were traveling at -.357c and had to cover 1.333
light years by their reckoning. At .357c,
your clock rate was 93.4% of theirs so they calculate your clock ran off 3.487
years while in pursuit (t’ = (x/v) * 93.4%).
In total, *1 year waiting + 3.488years pursuing gives 4.488 years*
expired off your clock. Same prediction
I made, give or take.

To summarize, both spaceship crews assumed the other ship’s clocks were running slow. Yet both crews predicted the same results when a face-to-face clock comparison was finally made.

t = 1 year

v = .8c

x = v * t = .8c light years

_{}

The other guy’s clock rate = t’/t = .6/1 = .600

t = 8 years

v = .9c

x = v * t = 7.2c light years

_{}

Shuttlecraft clock rate = t’/t = 3.487/8 = .4359

v = .8c

u = -.9c

_{}

time shuttlecraft began journey with respect to American spaceship

t’ = 1 year ago

time shuttlecraft began journey with respect to Chinese spaceship

t = t’/Clock Rate = 1/.6 = 1.667 years ago

x = v * t = .8 (1.667) = 1.333c light years

t = x/v = 1.333c/.357c = 3.734 years

_{}

Clock rate = t’/t = 3.488/3.736 = .934