A question about relativity theory?

[John Richard]

I have a question about the time dilation predictions of relativity theory that is keeping me up at night. It goes as follows:

The attached diagram helps!

We imagine 3 independent frames of reference and label them X, Y, Z.
X represents the point of origin, classically considered to be planet Earth. Y and Z are two projectiles, classically considered to be space ships. Y and Z are dispatched simultaneously from the point of origin X. They both accelerate, Y up to 0.45 times light speed and Z up to 0.9 times light speed.

From the perspective of frame X, the time dilation between itself and Y will be slightly less than 12%, and between itself and Z, the dilation will be slightly less than 230%.
From the perspective of Y, the time dilation between itself and Z will be slightly less than 12%.

So, between X and Y the dilation is 12%. Between Y and Z, the dilation is also 12% because the relative differentials between X and Y, and, between Y and Z are identical. And yet, according to relativity theory, the time dilation between X and Z will be 230%. The imbalance in the dilation differentials creates the contradiction.

Classically stated, we have; Y being 12% younger than X, and Z being 12% younger than Y. But, Z would also be 230% younger than X, and would have to be two different ages simultaneously.

The mathematical situation created by the theory, effectively overlays exponential proportionality onto linear proportionality. Naturally this creates an imbalance in the weight of the values when analysed as above. Furthermore, since the theory states that the effects are dependant upon the frame of reference of the observer, then the zero point of the exponential of dilation, must always be set at the observer’s frame of reference with respect to those further to the right, and to some calculated median along the exponential of dilation, with respect to frames laying to its left.

This creates conceptual difficulties arising from the implication, that as an observer moves to the left, or to the right, along the velocity line, (that's the blue line in the attached diagram). They are somehow able to alter the velocity dependant physical effects of time dilation, with respect to any other, stable frames of reference, anywhere along the line.
This raises some extremely bizarre questions; how does the physical universe know where the frame of reference is along the line? How does it alter the effects of time dilation to suit? And how do we cope with the further implication that pure awareness is able to alter the physical effects of time dilation?

I am more than ready to be shown the light (pun intended) but it has me perplexed to say the least!

I thank in advance, anyone willing to put me straight.

 

[Doc Al]

We imagine 3 independent frames of reference and label them X, Y, Z.
X represents the point of origin, classically considered to be planet Earth. Y and Z are two projectiles, classically considered to be space ships. Y and Z are dispatched simultaneously from the point of origin X. They both accelerate, Y up to 0.45 times light speed and Z up to 0.9 times light speed.

OK. I assume those speeds are with respect to frame X. That means: The speed of frame Y is 0.45c with respect to frame X. And: The speed of frame Z is 0.9c with respect to frame X. Realize that this implies that the speed of frame Z with respect to frame Y is not 0.45c, but closer to 0.76c. (Velocities must be added using the relativistic addition of velocity formula.)

From the perspective of frame X, the time dilation between itself and Y will be slightly less than 12%, and between itself and Z, the dilation will be slightly less than 230%.

I think you mean that the time dilation factor between frame X and frame Y is 1.12. (Each will measure the other's clock to be running slower by that factor.) Between X and Z, the factor will be about 2.3.

From the perspective of Y, the time dilation between itself and Z will be slightly less than 12%.

Between frames Y and Z the factor will be about 1.53.

So, between X and Y the dilation is 12%. Between Y and Z, the dilation is also 12% because the relative differentials between X and Y, and, between Y and Z are identical. And yet, according to relativity theory, the time dilation between X and Z will be 230%. The imbalance in the dilation differentials creates the contradiction.

Classically stated, we have; Y being 12% younger than X, and Z being 12% younger than Y. But, Z would also be 230% younger than X, and would have to be two different ages simultaneously.

So far, we're just talking about the rate at which moving clocks are measured to run slow. To talk about the "age" of something, we'd have to also worry about other relativistic effects, such as the relativity of simultaneity and length contraction.

If you want to be puzzled, there's no need for such an elaboration construction. Just consider frames X and Y. Frame X observers say that Y's clocks are slow by a factor of 1.12--but frame Y observers say the same thing about frame X clocks! The effect is completely symmetric.

The only way to understand what's going on--and to realize that there's no contradiction--is to carefully analyze just how one frame would measure the rate of a moving clock. Generally, more than one clock is required and that involves notions of simultaneity and distance--which are also frame dependent.

The mathematical situation created by the theory, effectively overlays exponential proportionality onto linear proportionality. Naturally this creates an imbalance in the weight of the values when analysed as above. Furthermore, since the theory states that the effects are dependant upon the frame of reference of the observer, then the zero point of the exponential of dilation, must always be set at the observer’s frame of reference with respect to those further to the right, and to some calculated median along the exponential of dilation, with respect to frames laying to its left.

Sorry, but I can't parse that paragraph.

This creates conceptual difficulties arising from the implication, that as an observer moves to the left, or to the right, along the velocity line, (that's the blue line in the attached diagram). They are somehow able to alter the velocity dependant physical effects of time dilation, with respect to any other, stable frames of reference, anywhere along the line.
This raises some extremely bizarre questions; how does the physical universe know where the frame of reference is along the line? How does it alter the effects of time dilation to suit? And how do we cope with the further implication that pure awareness is able to alter the physical effects of time dilation?

What affects the measured rates of clocks is relative motion. But the relative motion of frame Y does not "physically" effect the operation of the clocks in frame X. That would be crazy indeed, since there could be an infinite number of frames moving with respect to X. As far as frame X is concerned, nothing that frame Y (or Z or any other frame) does affects the operation of its clocks. The effect is a purely "kinematic" one, due to the structure of space and time itself.

 

[John Richard]

Thank you very much for your reply, I appreciate it greatly.

I am obviously not sufficiently aware of the principals of relativistic addition of velocities!

If we agree that Z is doing 0.95c and Y is doing 0.45c, how does relativity affect the differential?

The paragraph you cannot parse will only make sense if the velocity differentials are linear.

Hope your willing to continue helping

Thanks

 

[JesseM]

Thank you very much for your reply, I appreciate it greatly.

I am obviously not sufficiently aware of the principals of relativistic addition of velocities!

If we agree that Z is doing 0.95c and Y is doing 0.45c, how does relativity affect the differential?

The paragraph you cannot parse will only make sense if the velocity differentials are linear.

Hope your willing to continue helping

Thanks

See this page on relativistic velocity addition. If X sees Z moving at 0.95c, and Y sees X moving at -0.45c (since X sees Y moving at 0.45c in the positive direction), then according to the formula given on that page (with Z substituted for A, X substituted for B, and Y substituted for C), Y sees Z moving at (-0.45c + 0.95c)/(1 - 0.45*0.95) = 0.5c/0.5725 = about 0.87c. This is a little different than what Doc Al said, but he was responding to a quote where you had Z moving at 0.9c rather than 0.95c relative to X, in which case the formula gives (-0.45c + 0.9c)/(1 - 0.45*0.9) = 0.45c/0.595 = about 0.76c.

 

[John Richard]

Sorry, I made a mistake in my reply, I did mean 0.9c.

Thanks for the link, I think I know now where I was going wrong, I was stuck with the assumption that it was all about the speed of transmission of information and had not taken proper account of the kinematics. thank you both.

 

[Dale]

Classically stated, we have; Y being 12% younger than X, and Z being 12% younger than Y. But, Z would also be 230% younger than X, and would have to be two different ages simultaneously. (emphasis added)

The key point and one of the most difficult concepts to master in relativity is the relativity of simultaneity. Z will not be two different ages simultaneously, because simultaneity can only refer to a single reference frame.