Contradictions?

[Zeno]

Here's a spaceship with Sally in the corner....

         /\       |  ------------------->  V
       /    \     |
     /        \   |
   /            \ |
/                Sally                     John

Sally is in the lower right corner of the spaceship with 2 flashlights. One flashlight is aimed straight up at a mirror which reflects the light back down towards Sally. This is a light clock with the light bouncing straight up and down. She has another flashlight aimed up and to the left towards a mirror. So this is another light clock but at an angle. The ship is moving to the right with velocity V according to John who is outside the ship. This ship's velocity is such that from John's point of view the light from the angled flashlight isn't going at an angle but is going straight up and down because the rightward velocity of the ship exactly matches the leftward movement of the light.

Upward flashlight: Sally sees the light up and down. John sees it at an angle.
Angled flashlight: Sally sees the light at an angle. John sees the light up and down.

The 2 situations are reversed.

If T and t represent time for the 2 different reference frames then using the pythagorean theorem we can derive 2 contradictory time dilation formulas:

delta T = delta t/((1 - (v/c)^2)^.5)
delta t = delta T/((1 - (v/c)^2)^.5)

 

[Janus58]

Here's a spaceship with Sally in the corner....

         /\       |  ------------------->  V
       /    \     |
     /        \   |
   /            \ |
/                Sally                     John

Sally is in the lower right corner of the spaceship with 2 flashlights. One flashlight is aimed straight up at a mirror which reflects the light back down towards Sally. This is a light clock with the light bouncing straight up and down. She has another flashlight aimed up and to the left towards a mirror. So this is another light clock but at an angle. The ship is moving to the right with velocity V according to John who is outside the ship. This ship's velocity is such that from John's point of view the light from the angled flashlight isn't going at an angle but is going straight up and down because the rightward velocity of the ship exactly matches the leftward movement of the light.

Upward flashlight: Sally sees the light up and down. John sees it at an angle.
Angled flashlight: Sally sees the light at an angle. John sees the light up and down.

The 2 situations are reversed.

If T and t represent time for the 2 different reference frames then using the pythagorean theorem we can derive 2 contradictory time dilation formulas:

delta T = delta t/((1 - (v/c)^2)^.5)
delta t = delta T/((1 - (v/c)^2)^.5)

I'm not sure why you are going to all this trouble about light being aimed at a angle. All you have to do is give both Sally and John their own light clock. Each will measure the light from their own clock as going straight up and down and the light from the other light clock as traveling at an angle. They also will both conclude that the other light clock is ticking slower than their own. This is not a contradiction but just a consequence of Relativity. Time dilation is reciprocal.
But time dilation is just one part of Relativity, for a complete picture of what is happening with Relativity, you also have to account for Length contraction and the Relativity of simultaneity.
So for example, the fact that Sally and John disagree as to who's clock is ticking slower is never an issue as long as they keep a constant velocity with respect to each other. As under this scenario they could only meet up once. If they are to meet up more than once, one, the other, or both has to change their velocity, and how this velocity change takes place will determine which of the two's (if either) has accumulated less time between their two meetings.

 

[Zeno]

Sally's angled flashlight is at such an angle that the light follows the same angled path right to left as the path of the light from the upward flashlight as seen by John as Sally moves left to right and Sally's rightward movement exactly matches the leftward movement of the light.
Therefore:
Sally's view of the up and down movement of the light from her upward flashlight exactly matches John's view of the up and down movement from Sally's angled flashlight.
Sally's view of the movement of the angled light from her angled flashlight exactly matches John's view of the movement of the light from Sally's upward flashlight as Sally moves left to right.

You don't see a problem here?

 

[Halc]

Sally's angled flashlight is at such an angle that the light follows the same angled path right to left as the path of the light from the upward flashlight as seen by John as Sally moves left to right and Sally's rightward movement exactly matches the leftward movement of the light.
Therefore:
Sally's view of the up and down movement of the light from her upward flashlight exactly matches John's view of the up and down movement from Sally's angled flashlight.
Sally's view of the movement of the angled light from her angled flashlight exactly matches John's view of the movement of the light from Sally's upward flashlight as Sally moves left to right.

Correct, which is why Janus said that you can avoid the complication by just letting John hold the other flashlight. The light takes the same path at the same speed regardless of who shines the light.

You don't see a problem here?

Not at all. Again, as was said above, this is just a consequence of relativity.

 

[Zeno]

I will try to illustrate my point with a specific example. Sally is moving to the right at .6c. The height of her spaceship is .8 light-seconds. If Sally has a light clock with the light bouncing straight up and down the light will make a 3-4-5 right triangle from the viewpoint of John. If the change in time for Sally is delta T_o and the change in time for John is delta T then the following equation can be derived:

delta T = delta T_o/((1-.6^2)^.5)

Now Sally has a light clock but this time she is holding a flashlight at an angle of 53.13 degrees above the horizontal and pointed to the left. Now the leftward movement of the light exactly matches the rightward movement of the spaceship from John's viewpoint. Now the light is bouncing straight up and down from the viewpoint of John and the light is making a 3-4-5 right triangle from viewpoint of Sally. If the change in time for Sally is delta T_o and the change in time for John is delta T then the following equation can be derived:

delta T_o= delta T/((1-.6^2)^.5)

The 2 equations are in direct contradiction to each other.