Exploring Time Dilation: Questions and Insights from a Hobby Mathematician

[bookofproofs]

Hi, I'm a (hobby) mathematician and only an amateur physicist, so maybe below there are only trivia questions. Thank you in advance for conversation and clarification.

All mathematical proofs of "time dilation" I have looked at so far were based on the Pythagorean theorem. In all such proofs, there is a step of "dividing" both sides of the equation by the term (1-v^2/c^2), leading to the Lorentz factor 1/√(1-v^2/c^2), which is the time dilation factor. My questions are:

1) Why can we "without loss of generality" assume that the observed light signal is sent perpendicularly to the direction of movement so that we can rely on the "right angle" and use the Pythagorean theorem in the calculation? My point is that we could also assume a more general case (which I wonder why it should not be possible), in which the light signal would be sent in an arbitrary angle to the direction of movemen. In this case, the proof would require the general "law of cosines" rather than the Pythagorean theorem. But then, the Lorentz factor and time dilation formula would look different, leading to different time dilation effects, wouldn't they? (Actually, I was trying to check out but was not able to simplify my general "law of cosine" factor to the known Pythagorean-case Lorentz factor)?

2) Why is it necessary at all to divide by the term (1-v^2/c^2)? I mean, this needs the assertion that this calculation step only holds, if the velocity v does not equal the speed of light c. By how about the case v=c (e.g. if two photons move relatively to each other and experience a time dilation)? Why is it not more convenient to use the more simple form √(1-v^2/c^2) x t1 = t2 (which would allow the case v=c) instead of the classical Lorentz factor version of time dilation t1=t2 x 1/√(1-v^2/c^2), where we must exclude the case (v=c), because otherwise we would divide by zero?

3) It is known from particle accelerators that the energy needed to speed up massive particles (like protons) grows exponentially as their speed v approaches c from below. Why is this argument often used to "show" that "no massive particle can travel faster than light"? I mean, it only shows that it cannot travel at a speed exactly equal to the speed of light (v=c), but it does not (!) show that it cannot travel "faster" than light.

4) Please note that for the case v > c, the Lorentz factor is still well-defined and produces a complex number (square root of a negative real number). Because the time dilation factor would be a complex number in this case, my question is, could we interpret this as time having "two dimensions" (just like a complex number can be interpreted as a two-dimensional real number)? The first dimension of time would be its "real part" and this can be "observed" for all speeds v < c. The second dimension of time would be its "imaginary part", which would be "observed" in inertial frames of reference moving relatively to each other at a constant speed v > c?

<<mentor note: link to homepage removed>>

 

[Orodruin]

First, I have changed the thread level to I. Using thread level A indicates you have an understanding of the subject at the level of a grad student or higher.

1. You can assume any direction you want. Time dilation follows uniquely only from the orthogonal case. This is not a loss of generality, merely a convenient choice of clock mechanism so that you can compute what you set out to. Also note that the typical light clock arguments made in typical intro courses are quite heuristic and not what you would see in more advanced courses.

2. There is no inertial frame that moves at c relative to another. If something moves at c in one inertial frame it does so in all inertial frames.

3. What you are talking about are tachyons. They come with their entirely own set of problems such as sending messages back in time etc.

4. No. Space time is a 4-dimensional Lorentzian manifold.

 

[Mister T]

1) Why can we "without loss of generality" assume that the observed light signal is sent perpendicularly to the direction of movement so that we can rely on the "right angle" and use the Pythagorean theorem in the calculation?

The derivation of time dilation using the light clock demonstrates the effect that we observe with real clocks. There's no loss of generality. It's not possible to make it more general.

My point is that we could also assume a more general case (which I wonder why it should not be possible), in which the light signal would be sent in an arbitrary angle to the direction of movement.

It is possible to do that and it has been done. It doesn't increase the generality of the derivation, but it does increase the generality of the light beam paths considered in the derivation.

In this case, the proof would require the general "law of cosines" rather than the Pythagorean theorem.

It could certainly be done that way, but it's not required. Note that if you construct a light beam clock that sends the light along a path that's diagonal to the vertical, it will always be possible to find a frame of reference in which the path is vertical. Thus when you look at the light clock from yet a third frame of reference it's easy to compare the two with diagonal paths of differing slopes to the one where the path is vertical.

But then, the Lorentz factor and time dilation formula would look different, leading to different time dilation effects, wouldn't they?

No. As long as you make no errors in your derivation you will get the result that matches what's observed in the real world.

2) Why is it necessary at all to divide by the term (1-v^2/c^2)? I mean, this needs the assertion that this calculation step only holds, if the velocity v does not equal the speed of light c. By how about the case v=c

That's not a case. The speed $v$ is the relative speed of two reference frames. There is never a possibility that it will equal $c$.

3) It is known from particle accelerators that the energy needed to speed up massive particles (like protons) grows exponentially as their speed v approaches c from below. Why is this argument often used to "show" that "no massive particle can travel faster than light"?

Because it confirms a conclusion that was already arrived at. The second postulate tells us that the speed of light is the same in all reference frames. From that you can immediately conclude that it's not possible for a reference frame to move at the speed of light. There is a reference frame in which that proton is at rest, thus it's not possible for the proton to move at the speed of light.

I mean, it only shows that it cannot travel at a speed exactly equal to the speed of light (v=c), but it does not (!) show that it cannot travel "faster" than light.

That argument doesn't, but there are others that do.

4) Please note that for the case v > c, the Lorentz factor is still well-defined and produces a complex number (square root of a negative real number).

But the factor was derived using reference frames moving at speeds less than $c$. Thus it doesn't apply in the case you mention.

 

[bookofproofs]

You can assume any direction you want. Time dilation follows uniquely only from the orthogonal case

My question was "why you can assume any direction you want". Take the relativistic length contraction as an example. It does not happen in "any direction you want" but only in the direction of movement. So I just do not get it, why the direction of movement is important for length contraction (I do not doubt it!), but for some miraculous reason it should not be important for time dilation (which I doubt!).

The derivation of time dilation using the light clock demonstrates the effect that we observe with real clocks. There's no loss of generality. It's not possible to make it more general.

I know about the experiments with cesium clocks moved in airplanes around the world or about the measurements of time decay of cosmic muons "confirming" the simplified Pythagorean time dilation formula. All these experiments involve complex systems comprised of myriads of particles moving at different directions relatively to the observer. Imagine a "thought experiment" with an inertial frame moved at a constant speed along a straight line with thousands of light clocks in it sending they light rays in all possible diagonal directions and an observer from outside the frame measuring the "average" time dilations. Why do we exclude the possibility that the real life experiments only confirm the "average" time dilation of all these clocks? If you know about experiments, in which laser beams have been directed a) perpendicularly to the direction of movement and b) diagonally to the direction of movement and c) always confirmed the simple "perpendicular special case" time dilation formula, I would appreciate it if you could post me a link or a reference to such a publication.

Note that if you construct a light beam clock that sends the light along a path that's diagonal to the vertical, it will always be possible to find a frame of reference in which the path is vertical. Thus when you look at the light clock from yet a third frame of reference it's easy to compare the two with diagonal paths of differing slopes to the one where the path is vertical.

Maybe I do not get the same picture in my mind like you do. If I understand you correctly, then you are saying that you can rotate the observer to another reference frame, from which she will see a vertical light ray. But note that if the path is diagonal to the vertical, then it is also diagonal to the direction of movement of the light clock's reference frame. Of course it is easy to choose a third frame, from which the path will become vertical, but then the "observed" direction of movement and the "observed" vertical path will still form no right angle, but an acute or obtuse one. In an extreme case, the light ray of the light clock would be parallely (at 0 degree angle) to the direction of movement, in which case the length contraction effect becomes apparent.

The second postulate tells us that the speed of light is the same in all reference frames. From that you can immediately conclude that it's not possible for a reference frame to move at the speed of light. There is a reference frame in which that proton is at rest, thus it's not possible for the proton to move at the speed of light.

Is there an inertial frame in which a photon (instead of a proton) is at rest? If not (according the second postulate) then why is it possible to put a photon detector at some space "coordinates", at which the photon reaches the detector? Maybe I'm wrong, but I interpret the second postulate like this: there is exactly one inertial frame, in which a photon is at rest - its the photon itself. However, relatively to any other inertial frame, it moves at the constant speed c. Such an interpretation of the second postulate makes sense for me.

There is no inertial frame that moves at c relative to another.

How about my interpretation of the second postulate just like I explained it above, in which each photon has its own inertial frame in which it rests, but it moves at c relative to any other photon?

No. Space time is a 4-dimensional Lorentzian manifold.

Yes, this is the standard space time model. Unfortunately, this model does not explain what happens inside black holes, dark mater, dark energy or other phenomena like Einstein's "spooky action at a distance" (quantum entanglement). I'm not saying that time "having two dimensions" would explain all these open questions. I'm just saying that we should dare to ask ourselves some other "weird" questions, if "normal" questions lead to unsatisfactory answers.

 

[Orodruin]

My question was "why you can assume any direction you want".

Because it is a clock, it must behave in a certain way. Consider the case when you have two clocks following each other - one which is orthogonal and one which is not. They tick at the same rate in the rest frame and therefore must do so in all frames. However, if you do not like clocks to derive time dilation, do it from the Lorentz transformations and definition of simultaneity. I never talk about light clocks in my master level SR course. Light clocks are more appropriate for beginner's courses.

How about my interpretation of the second postulate just like I explained it above, in which each photon has its own inertial frame in which it rests, but it moves at c relative to any other photon?

You cannot do this. By definition light signals travel at c - this is what the second postulate says, there is no room for interpretation. There is no inertial frame where a light signal is at rest (please keep photons out of this - this is not quantum physics).

I'm just saying that we should dare to ask ourselves some other "weird" questions, if "normal" questions lead to unsatisfactory answers.

There are good ways and bad ways of doing this.

Unfortunately, this model does not explain what happens inside black holes, dark mater, dark energy

It does have a prediction for what happens inside black holes away from the singularity. That it is unobservable is another matter. GR says nothing about what dark matter consists of, but it also says nothing about what normal matter consists of. For this you need quantum field theory. GR does allow a description of both dark matter and dark energy in terms of some of their properties as continuous distributions, namely their equations of state relating their pressure to their energy densities.

like Einstein's "spooky action at a distance" (quantum entanglement)

Why do you think this is something that needs explaining? It does not allow you to send information or in any other way violate relativity.

 

[Mister T]

Maybe I do not get the same picture in my mind like you do. If I understand you correctly, then you are saying that you can rotate the observer to another reference frame, from which she will see a vertical light ray.

No. What I'm saying is that you can construct a light clock by mounting a laser pointer on the floor and aiming it diagonally upward. To some observer moving in a horizontal direction, in the same vertical plane as the beam's path, the beam's path will be vertical.

For example, call yourself Observer A and in a frame of reference in which you are at rest construct a light clock so that the tilt of the beam is 45°. Observer B can move at a speed of $0.5 c$ relative to you and observe, in his rest frame, that the beam's path is vertical. Next imagine Observer C moving at a speed of $0.8 c$ relative to B. He will observe the beam's path to tilted at an angle of about 51°.

Now that you have three different views of your light clock. By comparing A's and C's drawings of what they see, you can carry out the analysis that you asked about in your original post. Neither A's nor C's beams follow vertical paths, yet you can easily show, by comparing to B's, that you get the same formula for time dilation.

 

[Mister T]

I'm not saying that time "having two dimensions" would explain all these open questions. I'm just saying that we should dare to ask ourselves some other "weird" questions, if "normal" questions lead to unsatisfactory answers.

You can imagine a group of theoretical physicists sitting around a white board and writing on it what you call weird questions. Then going down the list and answering each one. The answers that lead to things that don't match what's observed get crossed off the list. Those that remain get pursued. The equivalent of this is done, and has been done throughout the history of physics.

 

[Janus]

Hi, I'm a (hobby) mathematician and only an amateur physicist, so maybe below there are only trivia questions. Thank you in advance for conversation and clarification.

All mathematical proofs of "time dilation" I have looked at so far were based on the Pythagorean theorem. In all such proofs, there is a step of "dividing" both sides of the equation by the term (1-v^2/c^2), leading to the Lorentz factor 1/√(1-v^2/c^2), which is the time dilation factor. My questions are:

1) Why can we "without loss of generality" assume that the observed light signal is sent perpendicularly to the direction of movement so that we can rely on the "right angle" and use the Pythagorean theorem in the calculation? My point is that we could also assume a more general case (which I wonder why it should not be possible), in which the light signal would be sent in an arbitrary angle to the direction of movemen. In this case, the proof would require the general "law of cosines" rather than the Pythagorean theorem. But then, the Lorentz factor and time dilation formula would look different, leading to different time dilation effects, wouldn't they? (Actually, I was trying to check out but was not able to simplify my general "law of cosine" factor to the known Pythagorean-case Lorentz factor)?

Years ago, I ran across someone who tried to argue the same thing, that using a different angle would result in a different value for time dilation. What he failed to take into account was length contraction. Imagine your initial light source is inside a spherical mirror. The light goes out, hits the inside of the Spherical surface and returns to the center. The thing to remember is that according to a frame in which this arrangement is moving with respect to, the Mirror will not be a sphere, instead, it will be an oblate spheroid with its short dimension laying along the line of motion. It you now calculated how long it takes for the light emitted from the center to return to the center as measured in this frame, you will get the same answer no matter what direction you fired the light. (I actually proved this mathematically for the above individual, but will not repeat it here as it takes a long series of steps). You get the same Lorentz factor, no matter what. The only real reason that we use the perpendicular direction for the light in the common light clock example is so we do not have to account for the length contraction effect.

3) It is known from particle accelerators that the energy needed to speed up massive particles (like protons) grows exponentially as their speed v approaches c from below. Why is this argument often used to "show" that "no massive particle can travel faster than light"? I mean, it only shows that it cannot travel at a speed exactly equal to the speed of light (v=c), but it does not (!) show that it cannot travel "faster" than light.

As noted earlier >c particles are hypothetically possible, and are known as tachyons. Particles that always travel at <c speeds are called tardyons. The trick is that c provides a barrier between the two. Tardyons can never be accelerated to become tachyons, and tachyons can never be slowed down to become tardyons. One the peculiar properties of Tachyon is that the slower they move, the more energy they have. You have to pump energy into a tachyon to slow it down towards the speed of light. Thus if a tachyon gives up energy in some manner, it speeds up. If it has a charge, this causes it to emit EMR, which robs it of more energy, which speeds it up, which causes it to emit EMR... You end up with a run-away process with the tachyon accelerating to higher and higher speeds. There is also the causality problems already mentioned. The tachyon might turn out to one of those things that the math allows for, but reality doesn't.

 

[PeroK]

@bookofproofs there is a simple thought experiment to show that there can be no length contraction in a direction perpendicular to the relative motion. It's far from a miracle!

You might like to think about that and work it out for yourself. Then you don't have to take anybody's word for it.

 

[bookofproofs]

The thing to remember is that according to a frame in which this arrangement is moving with respect to, the Mirror will not be a sphere, instead, it will be an oblate spheroid with its short dimension laying along the line of motion. It you now calculated how long it takes for the light emitted from the center to return to the center as measured in this frame, you will get the same answer no matter what direction you fired the light. (I actually proved this mathematically for the above individual, but will not repeat it here as it takes a long series of steps).

Janus, this is the kind of argument I have been looking for my first question. Could you please provide me with your mathematical proof? Not because I'm still not convinced, but because I would like to use your proof as a "lemma" to complete the mathematical proof of time dilation on my homepage. If you do not like the idea to repeat all steps in this thread, please send it to me in a bilateral conversation. Note: Understanding length contraction in the direction of motion is no problem for me. I'm only having problems with understanding time dilation in "any direction" of movement in relation to the observer.

As noted earlier >c particles are hypothetically possible.

Can you recommend me some good book, in which the tachyon hypothesis is "mathematically" derived?

The answers that lead to things that don't match what's observed get crossed off the list. Those that remain get pursued.

Nothing could be more true than that. Thank you all for your answers.

There is no inertial frame where a light signal is at rest (please keep photons out of this - this is not quantum physics).

It (fourdimensional spacetime model) does have a prediction for what happens inside black holes away from the singularity. That it is unobservable is another matter.

I think, observability is key in both, quantum physics (think of Schrödinger's cat) and special /general relativity (think of observing simultaneity of clocks). I suppose the real cause of why you are asking me to "keep photons out of this" is that inertial reference frames are only defined for special / general relativity, but not for the micro cosmos, in which quantum effects take place? But I do not like to start another thread about this :-)

 

[Orodruin]

I suppose the real cause of why you are asking me to "keep photons out of this" is that inertial reference frames are only defined for special / general relativity, but not for the micro cosmos, in which quantum effects take place?

No. I am asking you to keep photons out of this because photons are not the small balls of light many people make them out to be. They are the quanta of the quantised electromagnetic field in QED and are among the most complicated things you can encounter in quantised Abelian gauge theory. Relativity in itself is a classical theory.

That being said, there is absolutely no problem in defining inertial frames microscopically. In fact, most QFT computations are done assuming a flat Minkowski space background.

 

[Janus]

Janus, this is the kind of argument I have been looking for my first question. Could you please provide me with your mathematical proof? Not because I'm still not convinced, but because I would like to use your proof as a "lemma" to complete the mathematical proof of time dilation on my homepage. If you do not like the idea to repeat all steps in this thread, please send it to me in a bilateral conversation. Note: Understanding length contraction in the direction of motion is no problem for me. I'm only having problems with understanding time dilation in "any direction" of movement in relation to the observer.

Here's the basic proof. assume the initial light emitted is emitted at an angle alpha with respect to the line of relative motion, The top image below shows the light traveling out to a radius of R and returning, for anyone at east with respect to the circle, the time for the round trip is 2R/c. The red and green line show the horizontal and vertical components of the light's path.
Now consider the outbound path of the light as seen by someone for which the circle is moving from left to right a velocity v(second image). The light is emitted but it has to chase after the circle. The circle is also length contracted to an oval by a factor of 1/gamma. The dotted line oval represents the location of the oval at emission, and the solid line one its position upon reaching the edge. If t₁ is the time it takes for the light to reach the edge of the circle, then vt₁ is the distance traveled by the oval and vt₁+ R cos(alpha)/gamma is the horizontal component of the light's path. The vertical component is the same as was measured in the rest frame of the circle. The distance ct₁ is the distance the light travels. From the above, we can solve for t₁

Lastly we consider the return or reflected path(third image). Now the center of the oval (the light source) is rushing towards the reflected light. Now the horizontal component is the difference between Rcos(alpha) and vt₂. Again, with this information you can find t₂.
t₁+t₂ is the total time for the round trip.

If you do the math above, you will find that 2Rc will differ from t₁+t₂ by the Lorentz factor. The angle alpha makes no difference.

 

[bookofproofs]

This is sublime, thank you.

 

[pixel]

...Consider the case when you have two clocks following each other - one which is orthogonal and one which is not. They tick at the same rate in the rest frame and therefore must do so in all frames...

Why?

 

[Ibix]

Why?

Set the two clocks to tick at the same frequency and phase in their rest frame and with one end co-located. Rig a bomb to blow if the light pulses in the two clocks ever return at different events. In the rest frame it is obvious that there is no boom today, nor boom tomorrow. There is never boom tomorrow.

A change of reference frame cannot change this in an internally consistent theory.

 

[pixel]

Ibix: OP questioned the generality of the usual time dilation derivation, where the light clock operates perpendicular to the direction of relative motion, i.e., would the observer moving relative to the rest frame of the clock measure a different time dilation if the clock operated at a different angle? I'm not sure how your example removes this possibility. Whatever the moving observer measured, he would know how to transform those measurements to the rest frame of the clocks and agree that there should be no boom. I'm not questioning the result as the more general derivation of time dilation by Lorentz transformation is independent of any clock mechanism assumptions, just trying to understand your no doubt correct example.

 

[Ibix]

If you obtain a different time dilation factor using the diagonal clock then, viewed from a moving frame, the diagonal clock would tick at a different rate from the vertical clock. That's the implication of different rates from differently oriented clocks.

If you co-locate one end of both clocks to remove any ambiguity about simultaneity then it is immediately obvious that, if the above were true, the pulses would return simultaneously in one frame but not in others. This can have physical consequences (e.g. the bomb) just from a frame change. That doesn't make sense.

 

[pixel]

Okay, so we use the fact that if two events are simultaneous in one frame and at the same location in that frame, then they must be simultaneous in any other moving frame. Thanks.

 

[Orodruin]

if two events are simultaneous in one frame and at the same location in that frame,

... then they are not different events, but the same event ...

 

[bookofproofs]

Okay, so we use the fact that if two events are simultaneous in one frame and at the same location in that frame, then they must be simultaneous in any other moving frame. Thanks.

Indeed, this is the probably most simple argument, why the perpendicular case is not a loss of generality. Consider a light clock which is sending simultaneously two different light beams to its spherical mirror surface: one which is perpendicular to the direction of movement and one at an arbitrary angle relatively to this direction. For an observer in the frame, the light of both beams is reflected and returns simultaneously to the center, just like it is the case for any observer outside the frame. Therefore, the observed time it takes for both beams to reach the spherical mirror (or to return to the center) is the same for any observer outside the frame. Thus, the perpendicular beam can be used without loss of generality to calculate the time dilation effect. Thank you guys for this insight.

 

[robphy]

@bookofproofs , you can see this in action by clicking on my avatar, which is actually an animated gif of a circular light clock (rather than a spherical one). It uses essentially the same construction given by @Janus , but done on a spacetime diagram (rather than a spatial diagram).

 

[bookofproofs]

cool