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O Great One
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- TL;DR Summary
- The famous derivation of the time dilation equation can be derived in 2 contradictory ways.
It is possible to derive 2 contradictory time dilation equations. The first paragraph below describes the situation with Sally aiming a flashlight straight up and down so that Sally sees the light moving straight up and down and John is outside the spaceship and sees the light forming a triangle with the floor of the spaceship. The second paragraph describes Sally aiming a flashlight towards the left while the spaceship moves to the right. Now the situation is exactly reversed. Sally sees the light forming a triangle with the floor and John sees the light bouncing straight up and down.
Sally is in a moving spaceship. John is outside the spaceship.
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.
Sally is in a moving spaceship. John is outside the spaceship.
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.