[Dark Tofu]

Ok, so I understand SR pretty well and all, but I have a problem with time dilation. There is something that I don't understand.

The formula for time dilation is:

t ' = t / ( sqrt( 1 - ( v^2 / c^2) ) )

Where:
t ' is the time of the relatively stationary observer
t is the time of the relatively moving observer
v is the velocity of the relatively moving observer
c is the speed of light
and sqrt is the square root function (didn't know how to do it on computer...)

So, assume that I am not accelerating in any direction (deep space where gravity has essentially no effect). From my frame of reference I seem to be at rest (assume I'm in a spaceboat!). Assume also that someone else is flying past me at a constant (close to the speed of light) velocity (he's in a spacey ship!). Now suppose that both I and the other person have clocks with us so that we can measure how much time passes. (Yes, the classic, easy problem.)

Let's assume that the speed of the spaceship is 2/3 c and he is measuring out 100 seconds. So to solve the equation, it would be:
t ' = (100) / ( sqrt( 1 - ( (2/3c)^2 / c^2 ) )
If you do the math, you will get somewhere around 130 seconds as the time measured by us (in our spaceboat). Now, I would like to look at the problem from the other side. Let us get into the spaceship and let him get in the boat, only we'll perform the experiment in reverse. So we start going, but suddenly we realize, from our ship, he is the one moving, not us. We are relatively stationary(spaceship) and he is relatively moving (spaceboat). We quickly run and perform calculations really quickly. We know he will measure 130 seconds on his spaceboat, so we use the equation to solve for our time:
t ' = (130) / ( sqrt( 1 - ( (2/3c)^2 / c^2 ) )
The answer comes out to be around 170 seconds. We (spaceship) are measuring 170 seconds when the spaceboat is measuring 130 seconds. The spaceboat is measuring 130 seconds when the spaceship is measuring 100 seconds. The spaceship is measuring 170 and 100 seconds at the same time! This is a logical contradiction! 100 =/= 170! Please explain this to me.

[Fred_space]

Yea i came across this in an exam last year, and spent ages trying to work out what was going on, you've forgotten that in the your original question that the moving spaceship is measuring out the 100seconds, so really your equation for the second part should read:

130 = t / ( sqrt( 1 - ( (2/3c)^2 / c^2 ) )

As you were applying the time measure by the observer to the moving ship.