Contradictory time dilation equations

[O Great One]

TL;DR

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.

 

[Dale]

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

Your geometry doesn’t work out here. Write down the path of the light, you will see that it does not and cannot go straight up and down for John. It can go straight up (assuming your angle is correct), but not straight down.

 

[O Great One]

The situation is exactly reversed. If Sally has mirrors in the top and bottom of her ship the rightward movement of her ship is the same speed as the leftward movement of the light. The means that John sees the light bouncing straight up and down. The rightward movement of the ship is .6c the leftward movement of the light is .6c.

 

[Dale]

The situation is exactly reversed.

No, it is not. You should actually do the math.

There is no angle at which Sally can tilt a light clock so that the light path goes up and down for John

 

[O Great One]

It's a 3-4-5 triangle. The leftward speed of the light is (3/5)c. The rightward movement of Sally is (3/5)c. Why don't you do the math to show that it's wrong?

There is no angle at which Sally can tilt a light clock so that the light path goes up and down for John

Really? Are you sure?

 

[Dale]

Why don't you do the math to show that it's wrong?

That isn’t the way these things work. Extraordinary claims require extraordinary evidence. You are making the extraordinary claim that the time dilation formula is not self consistent. So it is up to you to deliver the extraordinary evidence.

Really? Are you sure?

100%

If I have time tomorrow I may prove it, but that is simply that I enjoy the math and not me accepting the burden of proof which remains on you.

 

[O Great One]

If Sally wasn't moving to the right and she aimed her flashlight upward and to the left at an angle of 53.13 degrees with the horizontal John would see the light moving up the hypotenuse of the 3-4-5 triangle at a speed of c and the direct leftward movement at a speed of .6c. Since Sally's speed to the right is .6c, John just sees the light bouncing straight up and down.

 

[PeterDonis]

@O Great One, you might want to read up on the relativistic aberration of light:

https://en.wikipedia.org/wiki/Relativistic_aberration

 

[Dale]

Since Sally's speed to the right is .6c John just sees the light bouncing straight up and down.

Up, yes, but not down. Remember, the light in Sally’s light clock must return to her (otherwise it won’t be her clock). It cannot do that by going straight down.

Again, you need to fully do the math. Your evidence is not extraordinary like your claim. If you had actually done the math then you would have seen that the light didn’t return to Sally.

 

[O Great One]

I don't know what you mean by "it must return to her". The light is moving away from her and bouncing between the ceiling and floor at an angle. This has nothing to do with the aberration of light.

 

[Dale]

I don't know what you mean by "it must return to her". The light is moving away from her and bouncing between the ceiling and floor at an angle.

A light clock works by emitting light, letting it travel to a mirror, and then having it return to the emitter. That is one tick of the clock. If the light does not return then it is not a light clock.

That is the standard meaning of the term. You can tilt it at any angle, but the light must return to the emitter for the clock to tick.

 

[O Great One]

A clock works by cyclical events. There is no need for the light to return to the emitter. Sally could still use the hits on the ceiling and floor as ticks.

 

[Dale]

A clock works by cyclical events. There is no need for the light to return to the emitter.

If it doesn’t return then obviously it is not cyclical.

Sally could still use the hits on the ceiling and floor as ticks.

That would not constitute a clock at rest in Sally’s frame, and certainly would not be described as Sally’s light clock. The “ticking” would be going away from her.

At this point we will close this thread. Please read up on light clocks. We are willing to help if you are confused, but in keeping with the forum rules it cannot be based on incorrect assertions and unfounded extraordinary claims of a flaw in relativity. It must be approached as the learning questions of a student.

 

[PeterDonis]

Sally could still use the hits on the ceiling and floor as ticks.

No, she can't, because she can't be at both the ceiling and the floor. She can only be in one spatial location, and can only use hits at that spatial location as ticks of the clock. That's the definition of a light clock.