When I first encountered the theory of Special Relativity in a laymanís science book, I was certain it could be proved wrong.† Take the statement ďMoving clocks run slowĒ.† How could two observers moving past each other both believe that the otherís 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 (thatís 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, youíve 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%, thatís 1 + 3.487 or 4.487 years 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 donít need to make any adjustments that would be required of someone looking through a telescope.† You can set aside phrases like ďIt only appears that wayĒ.† You will be holding yours and the Chinese clocks in your tight little hands.† You donít have to worry that the clocks might work differently.† Your clock reads ďmade in ChinaĒ 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?
Letís 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. Thatís 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 (tí = (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 shipís 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 guyís clock rate = tí/t = .6/1 = .600
t = 8 years
v = .9c
x = v * t = 7.2c light years
Shuttlecraft clock rate = tí/t = 3.487/8 = .4359
v = .8c
u = -.9c
time shuttlecraft began journey with respect to American spaceship
tí = 1 year ago
time shuttlecraft began journey with respect to Chinese spaceship
††††††††††††††††††††††† t = tí/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 = tí/t = 3.488/3.736 = .934