Time dilation why or how, Special Relativity causes

[ghwellsjr]

It proves my point if that was my point (which it was).

Proper Time is frame invariant. Coordinate Time is frame variant. Time Dilation is the ratio of Coordinate Time to Proper Time and that makes it purely a coordinate effect. Change the coordinates, you change the Time Dilation. I know that you know all this. I just don't agree that your scheme for establishing the events according to one frame using Time Dilation (which makes it frame variant) justifies the conclusion that Time Dilation is not "merely a coordinate effect".

I have made some spacetime diagrams depicting the two frames you described plus one more where the Time Dilation is the same for both worldlines.

Here is your original scenario:

Alice's worldline is the thick blue line and Bob's is red with dots marking off 1-year increments of Proper Time for both of them. Since Bob is moving at 0.866c, the Time Dilation factor for him is 2 while it is 1 for Alice since she is at rest.

Transforming to Bob's rest frame we have your second frame:

Now the Time Dilation factors have interchanged between them since their speeds have interchanged.

Finally, a new frame in which both are traveling at 0.578c and both have a Time Dilation factor of 1.23:

 

[Sugdub]

Wowww! It's clear I'll have difficulties in answering so many strong criticisms at a time. Thanks to all for your inputs anyway. I'll do my best to address the most pressing ones.

"Time Dilation" has a well-defined meaning in SR and I'm troubled that you think it is subject to debate. I also haven't seen you articulate a coherent definition. ... Could you please give what you consider to be the best definition of "Time Dilation" and then we'll see if we agree.
... And as a side note, I'd like you to expand on your statement "the length contraction formula (dealing with the space separation between events)". I have no idea what this means or why you are including it in a discussion of Time Dilation.

Let's consider first the “inertial” clock. It travels from location A to location B at constant speed. The rest frame of this clock is inertial. In this specific frame, A and B are represented by the same space position. Both collocation events of the twins only differ through their time coordinate. The relativistic space-time interval between both events has only a time component, the value of which is equal to the value displayed by the clock when it reaches location B (the elapsed time as measured by the inertial clock).
If projected onto any other inertial frame, this space-time interval, the magnitude of which is invariant, has both space and time components. The collocation events A and B are no longer represented at the same position and the time interval between both collocation events has changed value: it is no longer equal to the value displayed by the inertial clock when it reaches location B (obviously this display is not affected by the decision to represent the motion of the inertial clock in another IRF). The time dilation formula computes the duration calculated by SR for the journey of the inertial clock for any possible IRF, and this value only matches the elapsed time measured by the inertial clock when the selected IRF coincides with the rest frame of this clock.
The above contains my definition of the time dilation formula insofar it transforms the time component of the space-time interval - when expressed in the rest frame of the inertial clock - into the time component of the same space-time interval - when projected onto another inertial frame. It is clear that the length contraction formula plays a symmetrical role in respect to the space component of the same space-time interval, since it computes, in the selected IRF, the (space) distance between the collocation events. Hopefully this is convincing enough for securing the fact that the elapsed time calculated by SR is an IRF-dependent quantity, a coordinate-like quantity, and that changing IRF has no impact on the elapsed time measured and displayed by the clock itself. Time dilation (and the same can be said of length contraction) relates to a change of representation for the unique space-time interval separating two physical events, it has no bearing to any physical effect.

Not really, taking the space-time interval approach, the particular coordinate(s) used to represent the clocks are immaterial. There isn't any such requirement in SR, though it is probably convenient to use a single coordinate system for both.
The magnitude of the predicted time gap is well-approximated by the angle between worldlines on the space-time diagram, in the same sense that the magnitude of the "distance gap" when you add two sides of a triangle and compare it to the hypotenuse depends on the included angle - the angle between the two sides.
The angle between the wolrdlies on a space-time diagram is another way of describing the velocity between the physical objects represented by the worldlines.
Thinking that focuses on the acceleration tends to cause confusion - the actual formula for time dilation and the time gap uses relative velocities, you won't find the acceleration in the calculation at all. From the standpoint of learning about SR as it is taught in textbooks and understood by professionals, the long philosophical arguments that time dilation and/or the time gap "should" depend on acceleration are just a distraction, when one looks at the actual formula written in the textbook, one does not see the acceleration referred to anywhere.

Since some of the quantities we are dealing with are IRF-dependent, it would not make sense to compare them unless they are calculated in the same IRF. Let's come back to the inertial rest frame attached to the inertial clock. We are now going to represent the motion of the “non-inertial” clock in this specific frame. Initially the position of the non-inertial clock coincides with position A (first collocation event). Either its initial speed is zero, in which case only an acceleration applied to the clock will get it moving away from the inertial (rest) clock; or it is different from zero in which case it moves away from position A. In both cases, only an acceleration applied to the moving clock will revert its motion so that it comes back to A for the second collocation event. It is therefore equivalent to state that the second clock is “non-inertial”, that it is “subject to an acceleration for part or all of its journey”, or to state that it “moves away and then comes back to the same position”. Since it leaves and then comes back to the same position, the velocity of the second clock varies alongside its closed loop journey ( = the clock is “non-inertial” or “accelerated”). The occurrence of an acceleration cannot be waived: should you eliminate the acceleration (= the relative speed between clocks is invariant), either the second clock stays at rest collocated with the first one, or it will never come back.
It is effective that the magnitude or intensity of the accelerations involved do not appear explicitly in the equations. I've never claimed it should. But you may also remember that for most presentations of the twins experiment, the “moving clock” goes away at speed v and instantaneously reverts and comes back at speed v. So the knowledge of the acceleration pattern (which is infinite in this case) is not necessary to establish the mean value of the module of the relative speed (v in this case) ((by the way, the qualification "mean" is missing in your input, the angle between worldlines relates to the mean velocity, and this is why, in less peculiar presentations, the acceleration pattern needs to be considered)) , this mean value being the parameter involved in the equations computing the outcome gap. The mean value of the module of the relative speed (which is determined by combining the acceleration pattern and the initial speed) qualifies how much the motion of the second clock differs from the inertial rest state of the first clock. This mean relative speed will in turn determine the gap between both clocks at their second collocation event.

Hmm. I don't think that there is any kind of consensus about that. Here's an analogy: You have two travelers (on Earth, forget about relativity). One travel goes along a road that goes straight from point A to point B. Another traveler starts off on the same road, and takes an exit, which leads him to another (longer) road that also eventually reaches point B. Obviously, turning the steering wheel caused the second traveler to take an alternative route, but it would be weird to say that turning the steering wheel CAUSED the second route to be longer.
Saying that acceleration caused the age difference is a very weird way of looking at it, in my opinion. Acceleration caused one of the travelers to take a different spacetime path, but acceleration didn't cause that spacetime path to be longer.

I'm not too sure what you mean by "longer": spacewise or timewise? You seem to agree on the first step: the acceleration causes a change of the space-time path. But what if any non-inertial space-time path is shorter (timewise) than the inertial path having the same end-events? Take the formula giving the time gap: the gap is equal to zero iff v equals zero, which indicates that the mean value of the relative speed is zero and therefore that there has been no acceleration: both clocks have remained collocated.

 

[stevendaryl]

I'm not too sure what you mean by "longer": spacewise or timewise?

For the Euclidean analogy, the curved path is longer (in spatial distance). For the Minkowsky case, the "curved" path is shorter (in proper time).

Let me try another analogy: Suppose you have a rubber tube of length 10 inches. You stuff it into a box that is only 5 inches long. Then you can prove that it's necessary to bend the tube to get it to fit into the box. But it would be weird to say that bending the tube is what made it 10 inches long. Saying that accelerating a clock makes its path shorter (in proper time) seems exactly analogous to saying that bending a tube makes it longer.(in spatial distance).

What bending the tube does is to allow a 10 inch tube to fit into a 5 inch box.

 

[PeterDonis]

Time Dilation is the ratio of Coordinate Time to Proper Time and that makes it purely a coordinate effect.

While I agree with this usage, because it's clear and unambiguous, unfortunately this usage is not consistently adopted, either here on PF or in the literature. Many sources, including textbooks and papers, attribute invariant effects like the differential aging of Alice and Bob to "time dilation", as though time dilation were the cause of the differential aging. But a purely coordinate effect can't be the cause of anything; a causal agent has to be something invariant. So whatever these sources are calling "time dilation", it can't just be the coordinate effect you are describing.

Please note that this is an issue of language and pedagogy, not physics. We all agree on the physics; we all agree on the proper time elapsed along Alice's and Bob's worldlines. We all agree on what Alice and Bob actually see (with their eyes or telescopes)--they see the relativistic Doppler shifted signals from each other, according to the standard formula. The only point of discussion is over what English words to use to describe what's going on.

When people are having a disagreement over words, one useful tactic is to taboo the problem terms--force everyone in the discussion to rephrase what they're saying without using the problem terms at all. In this discussion, the problem term is "time dilation"; so it seems to me that it might be a good idea for everybody to restate what they've been saying without using the term "time dilation" at all. Perhaps that might help to focus the discussion.

 

[PeterDonis]

what if any non-inertial space-time path is shorter (timewise) than the inertial path having the same end-events?

In flat spacetime, all of them; the inertial path between two given events (that are timelike separated) is always the longest (in terms of proper time).

In curved spacetime, this is not necessarily true; there can be non-inertial paths between two given events that are longer than an inertial path. However, if that is true, there will always be some other inertial path between those same two events that is longer than the non-inertial ones. (This is because, in curved spacetime, there can be multiple inertial paths between the same two events.)

 

[PeterDonis]

Saying that accelerating a clock makes its path shorter (in proper time) seems exactly analogous to saying that bending a tube makes it longer.(in spatial distance).

I agree that neither of these statements makes sense. at least not on the interpretation you are giving them. But you seem to be implying that "time dilation" is what makes the accelerated clock's path shorter. If I taboo the term "time dilation", per my previous post, the only restatement I can come up with that makes the statement true is that the geometry of spacetime is what makes the accelerated clock's path shorter. So it seems to me that you are implicitly using the term "time dilation" to mean "the geometry of spacetime", whereas ghwellsjr is using it to mean "the ratio of proper time to coordinate time". So you are using the same term to mean different things, which of course is going to impede communication.

 

[stevendaryl]

I agree that neither of these statements makes sense. But you seem to be implying that "time dilation" is what makes the accelerated clock's path shorter. If I taboo the term "time dilation", per my previous post, the only restatement I can come up with that makes the statement true is that the geometry of spacetime is what makes the accelerated clock's path shorter. So it seems to me that you are implicitly using the term "time dilation" to mean "the geometry of spacetime", whereas ghwellsjr is using it to mean "the ratio of proper time to coordinate time". So you are using the same term to mean different things, which of course is going to impede communication.

I'm not exactly sure about the distinction that you're talking about.

To me, time dilation is the fact that

[itex]\delta \tau = \sqrt{1-\frac{v^2}{c^2}} \delta t[/itex]

where [itex]\tau[/itex] is the time showing on a clock, [itex]v[/itex] is the speed of that clock as measured in an inertial, Cartesian coordinate system, and [itex]t[/itex] is the time, as measured in that same coordinate system.

Is that a "coordinate effect", or is that a fact about spacetime geometry? On the one hand, it talks about the ratio of proper time to coordinate time, and so seems to be a coordinate effect, but on the other hand, if we rewrite it using [itex]v = \frac{\delta x}{\delta t}[/itex] (just considering 1 spatial dimension, for simplicity), we get

[itex]\delta \tau^2 = \delta t^2 - \frac{\delta x^2}{c^2}[/itex]

which is just an expression of the spacetime metric in inertial cartesian coordinates. So that seems to be about spacetime geometry.

So, I'm not clear about the distinction between spacetime geometry and mere coordinate effects when it comes to time dilation.

 

[PeterDonis]

I'm not clear about the distinction between spacetime geometry and mere coordinate effects when it comes to time dilation.

And to me, that is an indication that "time dilation" is a vague, imprecise term and should be avoided. Notice how the problem disappears when you rephrase things without using that term? You can either focus on ##\delta \tau^2##, which, as you show, is an invariant; or you can focus on coordinate-dependent quantities like ##\delta t##, ##\delta x##, and ##v##. It's perfectly clear which is which, so there's no confusion.

 

[Sugdub]

I agree that neither of these statements makes sense. at least not on the interpretation you are giving them...

Let's come back to facts:

all things equal, the numerical value displayed by the accelerated clock is lower than the numerical value displayed by the inertial clock.

This is experimentally true. How should we understand this?

Often physicists state that “the elapsed time is what gets measured by a clock”. As for any definition, this is a convention which establishes a gateway between our intuitive apprehension of the time flow and some well-defined operational processes. Based on this convention (did I say: “metaphor”?), it appears that the accelerated clock measures less elapsed time than the inertial clock. The experiment cannot sort out whether its path is shorter (time-wise) of if the clock itself runs slower. Anyway it follows that the accelerated twin comes back younger than the static one.

Now try and “read” the same experimental facts using a slightly different convention: “the elapsed time is what gets measured by an inertial clock”. Based on this new convention, it is no longer certain that the accelerated clock measures the elapsed time: one may also envisage that an accelerated clock delivers a biased value for the elapsed time. This could equally be “explained” by a shorter path or a slower clock, no way to distinguish both options experimentally. But then one cannot conclude that the accelerated twin comes back younger: one must envisage that the measurement gets biased.

I have no preference between both options. I only wish to highlight how far statements about time slowing down, clocks being late and people ageing slower are fragile in consideration of the arbitrariness of the conventions (metaphors?) from which they have been derived. This is why I recently called such statements “metaphysical”. May be “metaphoric” is more appropriate, but it does not make a big difference: they are neither true nor false.

 

[PeterDonis]

Often physicists state that “the elapsed time is what gets measured by a clock”. As for any definition, this is a convention

Not in the sense you mean. The actual definition is that the elapsed time measured by a clock between two events--i.e., the directly observed difference in clock readings between those events--is equal to the spacetime path length of the clock's worldline between those events. You can't just arbitrarily change this definition, because without it you have no link between the math of the theory and the experiments that validate it.

In other words, the important thing is not the words "elapsed time" but the specific concept they point at. See below.

The experiment cannot sort out whether its path is shorter (time-wise) of if the clock itself runs slower.

That's because "shorter path length" vs. "clock running slower" are not observables; they're different ways of representing the same observable mathematically. But either way, it's still the same observable, and you have to check it against the theory using the same output from the theory. Jiggering around the internal mathematical tricks you use to get that output from the theory doesn't change that. See below.

Now try and “read” the same experimental facts using a slightly different convention: “the elapsed time is what gets measured by an inertial clock”.

This changes what is meant by the words "elapsed time", but it doesn't change the underlying correspondence between the theory and experiments. All you have done here is make it no longer true that "elapsed time" by your definition is what we compare with experiments. We compare something else with experiments, something called, oh, let's just pick a term at random, "proper time". Your "elapsed time" by this definition is just a different term defined for your convenience, because you have chosen a particular mathematical method for calculating what the theory predicts for the outcomes of experiments. It doesn't change the predictions or how they compare with experiment.

But then one cannot conclude that the accelerated twin comes back younger: one must envisage that the measurement gets biased.

You have this backwards. The traveling twin coming back younger is the direct observable; that doesn't change. What changes is the internal mathematical representation you choose to use to calculate what the theory predicts for that observable, and how that representation might be described in English. If you choose one representation, you might describe it in English as "the traveling twin's path through spacetime is shorter". If you choose another representation, you might describe it in English as "the traveling twin's clock is biased". But these aren't descriptions of different ways the world might be that would account for the same observable; they are descriptions of different mathematical representations you can use to extract the same prediction from the theory.

(When considering these different representations, you also have to consider how well they work for all experimental predictions, not just one. The "spacetime path length" representation works well for all experiments and is the most general one we have; it also has the advantage, as above, of mapping the observable of proper time elapsed on a clock directly to a geometric property of the model, the spacetime path length along a curve. This has yet another advantage, that of generalizing easily to curved spacetime and GR. The "biased clock" representation, by contrast, only works well in flat spacetime, or to an extent in stationary curved spacetimes, and it gets convoluted if here is not an obvious frame, such as that of the stay at home twin, in which to analyze the experiment.)

I only wish to highlight how far statements about time slowing down, clocks being late and people ageing slower are fragile in consideration of the arbitrariness of the conventions (metaphors?) from which they have been derived.

I think you may be mistaken as to what is arbitrary here. Once again, the difference in aging between the twins is the direct observable; it's not arbitrary, it's a fact. To the extent that statements about "time slowing down" or "aging slower" are just referring to that fact, they are not arbitrary. Such statements are only arbitrary to the extent that they refer to internal mathematical representations that could be used in calculating the same prediction, as above.

 

[maline]

I agree with OP- something is missing. Relativity means it is possible to describe all phenomena in terms of anyone IRF, without reference to other frames. In the twin paradox scenario, seen from the inertial frame, the accelerated clock comes back showing a lower number. This is equal to the sum of the proper times over the various inertial parts of its trip, but this statement involves reference to those frames. If we keep our perspective firmly in the original IRF, what we have is a mechanical object that did something different than it would have done had it stayed put. Explaining this from this perspective requires getting into the clock's innards and describing how the increased momenta of its parts and the slowing of information transfer in the direction of motion physically does make the clock run slower. Very complicated and not at all useful, but true nonetheless.

 

[PeterDonis]

This is equal to the sum of the proper times over the various inertial parts of its trip, but this statement involves reference to those frames.

No, it doesn't. You can calculate these proper times in the stay at home twin's frame; there's no need to adopt any other frames.

If we keep our perspective firmly in the original IRF, what we have is a mechanical object that did something different than it would have done had it stayed put. Explaining this from this perspective requires getting into the clock's innards and describing how the increased momenta of its parts and the slowing of information transfer in the direction of motion physically does make the clock run slower.

Sure, this can be done. For a light clock it is straightforward.

However, by itself it doesn't solve the twin paradox, because the traveling twin can perform the same analysis while he is outbound, using his outbound IRF, or inbound, using his inbound IRF. This analysis will tell him that the stay at home twin's clock is running slower than his, because of exactly the same mechanical effects. So you have to add something else to the analysis (or use a completely different analysis, such as the one based on observed Doppler shifts) to explain why the traveling twin's clock is the one that reads less elapsed time when the two twins meet up again.

 

[maline]

as long as you stick to one IRF, you're fine. if you take the "outbound" frame, then the traveler had to move very fast to catch up with the receding earth, giving him the stronger time dilation. It's easy to check the math on that.

 

[PeterDonis]

as long as you stick to one IRF, you're fine.

Meaning, you can pick whichever IRF you like, as long as you use the same one to analyze the entire experiment? Yes, this is true, you can construct the "moving clocks run slower because their motion changes their mechanical behavior" explanation in any IRF, as long as you stick to the same one. And of course you can use any IRF you like to calculate the correct prediction for the twin paradox scenario. Nobody is disputing that.

However, your claim amounts to saying that something "real", the mechanical behavior of a clock, depends on which IRF you choose. This is really a matter of terminology, not physics, but saying that the moving clock "physically runs slower" is going to cause more problems than it solves, at least if the history of threads on this topic here on PF is any indication.

 

[Dale]

I only wish to highlight how far statements about time slowing down, clocks being late and people ageing slower are fragile in consideration of the arbitrariness of the conventions (metaphors?) from which they have been derived. This is why I recently called such statements “metaphysical”.

What is your aversion to calling them what they are: "coordinate dependent" or "frame variant"?

You seem oddly insistent on using an incorrect term when a correct one is available. Particularly given that all of your justifications focus on coordinate dependence and none on metaphysics. Why then go out of your way to discard the standard term you have justified and use a term that you have not justified?

 

[maline]

I'm pretty sure the mechanical behavior perspective is what OP wanted to know about. It is a physical phenomenon in some IRF's, and the only problems it causes are people getting angry at things that sound unfamiliar. I think OP already got scared off the thread by all the "it's that way because you can prove it from the postulates" garbage.
BTW, I'm not saying this is the best answer to what causes time dilation. Looking at things in terms of proper time is simpler and also philosophically more satisfying. But I do think you need to see how things work in one IRF in order to get a good sense of what combining them in Minkowski geometry means, and also just for the theory to be consistent. I think the consistence requirement is what was bothering OP.

 

[PeterDonis]

It is a physical phenomenon in some IRF's

This doesn't make sense, at least not with the usual usage of the term "physical phenomenon", which implies something invariant, i.e., not dependent on which IRF you choose.

I do think you need to see how things work in one IRF in order to get a good sense of what combining them in Minkowski geometry means

This is really a matter of pedagogy; since you appear to agree that the proper time viewpoint, which basically means the spacetime geometry viewpoint, is a better one, the question is what is the best way to get there. IMO it's not fruitful to look at Minkowski geometry as "combining" all the different possible IRFs. Would you think it appropriate to describe the geometry of the Earth's surface as "combining" all the different possible coordinate charts (Mercator, stereographic, latitude/longitude, etc.) that you could use on it?

and also just for the theory to be consistent

Here I disagree. The theory of SR can be formulated in terms of Minkowsi geometry without using the concept of IRFs at all. The only reason we use IRFs is to try to satisfy our pre-relativistic intuitions that things should "work" a certain way. The OP's question that started this thread is an example: his pre-relativistic intuitions make him believe that there must be some mechanical process that "makes" the moving clock slow down. The idea that it's all just geometry--that if the clock is moving relative to him, he is seeing it "at an angle" in spacetime, and therefore its space and time look different, just as the shape of a coin seen at an angle in ordinary space looks different than if it is seen face on--simply doesn't occur to him, because he's so used to thinking of space and time as separate and only space as having geometry, not spacetime.

Once you understand the geometric viewpoint, it becomes clear that there is no mechanical process that slows down the moving clock; and this actually makes sense in terms of other intuitions we have. For example, if there is indeed some mechanical process affecting the moving clock, we would expect to be able to measure it somehow--put a strain gauge on the clock to measure the stresses caused by the mechanical process that is slowing it down and contracting its length. But there is no such measurement that you can make: the moving clock feels no internal stresses, and there is no indication from anything within the clock itself that it is moving at all. (Of course this is just the principle of relativity, but it's important to remember that that principle predates SR: it was first explicitly stated, AFAIK, by Galileo.)

So whatever it is that causes the traveling twin's clock to show less elapsed time than the stay-at-home twin's clock, it can't be "time dilation due to relative motion" viewed as a mechanical process in any IRF, because, as above, there is no such thing; the belief that there "must" be is just our pre-relativistic intuitions leading us astray. It has to be something that equally affects anything following the same path through spacetime as the clock follows; i.e., spacetime geometry.

 

[Sugdub]

Thanks for your input. I see a lot of interesting comments there. I noted in particular your definition of the “elapsed time”, according to which a clock measures a (composite) space-time quantity. As a minimum, its naming convention is misleading and hides a lot of non-intuitive consequences. I can't remember having seen this definition before. May be I just overlooked it … but I'm certainly not the only one. Still the question arises as to whether this definition should also apply to non-inertial clocks and why. It would be interesting to continue digging into this in a dedicated thread. However we'll go into circles unless we address your comments in the reverse order.

... I think you may be mistaken as to what is arbitrary here. Once again, the difference in aging between the twins is the direct observable; it's not arbitrary, it's a fact. ...

This I disagree with. What is directly observable is twofold: the display of the static clock on the one hand, and the display of the accelerated clock on the other hand. Whereas both clocks have been synchronised at their first collocation event, they display different values at their subsequent collocation event.

You did not challenge my statement whereby “all things equal, the numerical value displayed by the accelerated clock is lower than the numerical value displayed by the inertial clock.” is “experimentally true”, however your input contradicts mine insofar:

• you are silent about the conceptual difference between the values displayed by the clocks and the “ageing of the twins” which you did not define; obviously some form of translation code is required which is not given a priori; is the “ageing” also a compound of space and time?
• you seem to admit as uncontroversial (together with everybody else) that it makes sense to subtract (in the mathematical sense) the value displayed by the accelerated clock from the value displayed by the static clock. Are these quantities of a same nature? Which kind of clock could have measured this “time gap” which relates to a single event (values displayed by two different clocks at the second collocation event)? Is this really a “time” gap?
  

 

[PeterDonis]

I noted in particular your definition of the “elapsed time”, according to which a clock measures a (composite) space-time quantity. As a minimum, its naming convention is misleading and hides a lot of non-intuitive consequences.

I only used the term "elapsed time" because you did; the standard term (which I also used) is "proper time". From now on, to avoid confusion, I'll use "proper time" and drop the term "elapsed time" altogether, since it appears to cause problems for you.

What is directly observable is twofold: the display of the static clock on the one hand, and the display of the accelerated clock on the other hand. Whereas both clocks have been synchronised at their first collocation event, they display different values at their subsequent collocation event.

This is what I meant by "difference in aging", so I don't think we disagree about the observable, at least not this part of it (see below). Strictly speaking, in accordance with what I said above about terminology, it should be called "difference in proper time".

you are silent about the conceptual difference between the values displayed by the clocks and the “ageing of the twins” which you did not define; obviously some form of translation code is required which is not given a priori

Unless you have some evidence that biological aging, for example, proceeds at a different rate than clocks tick (and nobody has ever found any such evidence), then the difference in clock readings will be the same as the difference in aging (hair turning gray, etc.) between the twins. At any rate, that is the assumption on which SR is based: that the rate of all processes is given by the proper time along their worldlines. "Clocks" are merely convenient devices for measuring this proper time; they do not define what it is.

you seem to admit as uncontroversial (together with everybody else) that it makes sense to subtract (in the mathematical sense) the value displayed by the accelerated clock from the value displayed by the static clock. Are these quantities of a same nature?

Of course. They're both clock readings. Why would one be "of a different nature" than the other? Assume that both clocks are of identical physical construction, and that both include identical diagnostics to monitor their function, and that both of their diagnostics read "everything normal" throughout their trips. Exactly what do you think would be "of a different nature" about their respective readings at the end of the trips?

At any rate, in SR, proper times are all "of the same nature", regardless of which worldline they are measured along. Experimentally, this assumption seems to work extremely well, so I'm not sure why you would want to drop it.

Which kind of clock could have measured this “time gap” which relates to a single event (values displayed by two different clocks at the second collocation event)? Is this really a “time” gap?

"Time gap" is your own personal term, as far as I can tell, not a standard term in SR. I'm not sure what you mean by it, but if it means anything other than just the observed difference in readings, I don't see how it's relevant.

 

[stevendaryl]

Thanks for your input. I see a lot of interesting comments there. I noted in particular your definition of the “elapsed time”, according to which a clock measures a (composite) space-time quantity. As a minimum, its naming convention is misleading and hides a lot of non-intuitive consequences.

You think that "elapsed time" is misleading? How is it misleading? What non-intuitive consequences are you thinking of?

The point about the phrase "elapsed time" is that it is the time on the clock since some specific event. So it's the difference of two clock values. The so-called "clock hypothesis" of SR claims that elapsed time is the same as proper time.

The geometric view of proper time makes the relationship between coordinates, acceleration, etc., pretty clear. The one thing that is not explained by it is the indefinite metric of spacetime. I don't know if there is a good explanation for why that should be the case, except that spacetime without an indefinite metric wouldn't have any "time" in it.

 

[PeterDonis]

The one thing that is not explained by it is the indefinite metric of spacetime.

As you note, this is required in order to have a metric with "time" in it. More precisely, you need an indefinite metric in order to model the physical fact that timelike intervals and spacelike intervals are fundamentally different kinds of things (this is manifested by the fact that you measure one with a clock and the other with a ruler). A positive definite metric can only model one kind of interval.

 

[ghwellsjr]

Wowww! It's clear I'll have difficulties in answering so many strong criticisms at a time. Thanks to all for your inputs anyway. I'll do my best to address the most pressing ones.

"Time Dilation" has a well-defined meaning in SR and I'm troubled that you think it is subject to debate. I also haven't seen you articulate a coherent definition. ... Could you please give what you consider to be the best definition of "Time Dilation" and then we'll see if we agree.
... And as a side note, I'd like you to expand on your statement "the length contraction formula (dealing with the space separation between events)". I have no idea what this means or why you are including it in a discussion of Time Dilation.

Let's consider first the “inertial” clock. It travels from location A to location B at constant speed. The rest frame of this clock is inertial. In this specific frame, A and B are represented by the same space position. Both collocation events of the twins only differ through their time coordinate. The relativistic space-time interval between both events has only a time component, the value of which is equal to the value displayed by the clock when it reaches location B (the elapsed time as measured by the inertial clock).
If projected onto any other inertial frame, this space-time interval, the magnitude of which is invariant, has both space and time components. The collocation events A and B are no longer represented at the same position and the time interval between both collocation events has changed value: it is no longer equal to the value displayed by the inertial clock when it reaches location B (obviously this display is not affected by the decision to represent the motion of the inertial clock in another IRF). The time dilation formula computes the duration calculated by SR for the journey of the inertial clock for any possible IRF, and this value only matches the elapsed time measured by the inertial clock when the selected IRF coincides with the rest frame of this clock.
The above contains my definition of the time dilation formula insofar it transforms the time component of the space-time interval - when expressed in the rest frame of the inertial clock - into the time component of the same space-time interval - when projected onto another inertial frame.

While the above contains all true statements, I think you are getting yourself into trouble by focusing on the space-time interval as evidenced by what you say next:

It is clear that the length contraction formula plays a symmetrical role in respect to the space component of the same space-time interval, since it computes, in the selected IRF, the (space) distance between the collocation events.

That is not clear to me. Let me repeat the two diagrams from post #3. The first one shows a clock at rest in an IRF:

The second diagram shows the clock traveling at 60%c to the left, with gamma equal to 1.25:

Could you please show the length contraction formula and how it relates to the space-time interval and where it is evident in the diagram(s).

Hopefully this is convincing enough for securing the fact that the elapsed time calculated by SR is an IRF-dependent quantity, a coordinate-like quantity, and that changing IRF has no impact on the elapsed time measured and displayed by the clock itself.

It would be helpful if you would use two different standard terms instead of "elapsed time" for both. Then your sentence would read:

"Hopefully this is convincing enough for securing the fact that the elapsed Coordinate Time calculated by SR is an IRF-dependent quantity, a coordinate-like quantity, and that changing IRF has no impact on the elapsed Proper Time measured and displayed by the clock itself."

Time dilation (and the same can be said of length contraction) relates to a change of representation for the unique space-time interval separating two physical events, it has no bearing to any physical effect.

While I agree with your conclusions (that Time Dilation and Length Contraction have no bearing to any physical effect), I still don't understand the rest of your sentence regarding the space-time interval. Please explain in detail. Don't assume that anything is obvious.

 

[nitsuj]

you are silent about the conceptual difference between the values displayed by the clocks and the “ageing of the twins” which you did not define; obviously some form of translation code is required which is not given a priori; is the “ageing” also a compound of space and time?

Unless you have some evidence that biological aging, for example, proceeds at a different rate than clocks tick (and nobody has ever found any such evidence), then the difference in clock readings will be the same as the difference in aging (hair turning gray, etc.) between the twins. At any rate, that is the assumption on which SR is based: that the rate of all processes is given by the proper time along their worldlines. "Clocks" are merely convenient devices for measuring this proper time; they do not define what it is.

Apparently there is a twin paradox scenario where the "at home" twin was cryogenically frozen to near absolute zero!

Remarkable yes, anyways that at home twin was thawed out upon the traveling twins return home. Of course the "at home" twin aged less than the traveling twin, dispite being older; in turn the previously synchronized clock on the outside of his cryogenic chamber had accumulated more ticks than the traveling twin.

 

[PeterDonis]

that at home twin was thawed out upon the traveling twins return home. Of course the "at home" twin aged less than the traveling twin, dispite being older

This one is easy to respond to: just cryogenically freeze the traveling twin at the start as well, and unfreeze him at the end. Then he will have aged even less than the at home twin (and of course the relative readings on the clocks outside their respective cryogenic chambers will be related as in the usual twin paradox).

 

[nitsuj]

This one is easy to respond to: just cryogenically freeze the traveling twin at the start as well, and unfreeze him at the end. Then he will have aged even less than the at home twin (and of course the relative readings on the clocks outside their respective cryogenic chambers will be related as in the usual twin paradox).

ahaha true true, apperently it is not possible to "freeze out" thermodynamics completely (to your point about an age difference existing even if "frozen") otherwise it could idealized so there is no difference in biologicaly age. Just though it a neat twist to help differentiate differential aging from differences in measured proper time.

 

[phyti]

For the Euclidean analogy, the curved path is longer (in spatial distance). For the Minkowsky case, the "curved" path is shorter (in proper time).

Let me try another analogy: Suppose you have a rubber tube of length 10 inches. You stuff it into a box that is only 5 inches long. Then you can prove that it's necessary to bend the tube to get it to fit into the box. But it would be weird to say that bending the tube is what made it 10 inches long. Saying that accelerating a clock makes its path shorter (in proper time) seems exactly analogous to saying that bending a tube makes it longer.(in spatial distance).

What bending the tube does is to allow a 10 inch tube to fit into a 5 inch box.

Assume a clock runs at the same rate independently of its speed. Clock A moving at a constant speed of .6c and clock B moving at a constant speed of .8c, travel a straight line distance of 12 light sec. Starting at t=0, at the finish line, A reads 20 sec, B reads 15 sec. Since t=x/v, an inverse relation of t to v, the faster clock reads less time than the slower clock. Applying SR corrections, the difference is even greater. The confusion of longer line, shorter time, results from interpreting the Minkowski graphic as a 2D geometric road map, which it is not.

 

[stevendaryl]

Assume a clock runs at the same rate independently of its speed.

Let me stop you right there. What does that mean? The only way that we can measure the "rate" of a clock is by comparing it with other clocks that are at the same location. It's very much like two cars that take different paths to go between city A and city B. If the two cars have different odometer readings, you can't say that one car's odometer is running fast or running slow, unless you have an independent way of knowing the distances the two cars traveled.

You can only say that a car's odometer is running fast or slow compared with an idealized, perfect odometer that took the same route.

Clock A moving at a constant speed of .6c and clock B moving at a constant speed of .8c, travel a straight line distance of 12 light sec. Starting at t=0, at the finish line, A reads 20 sec, B reads 15 sec.

If A and B don't end up at the same points in spacetime, then I don't see what kind of comparison is being made here. A and B travel for different amounts of time, in different directions in spacetime, and they end up at different locations in spacetime. There is nothing mysterious about that. It's as if A and B both started in Chicago. One traveled for 330 miles, and winded up in Des Moines, Iowa. The other traveled for 400 miles and winded up in Minneapolis. There is nothing to explain if they end up in different locations after having traveled for different distances.

I know, you say that the difference is that in the case you're talking about, A and B winded up in the same spatial location. In the case I'm talking about, they end up at the same longitude (approximately).

The analogies are

1. Longitude [itex]\Leftrightarrow[/itex] distance along the x axis
2. Lattitude [itex]\Leftrightarrow[/itex] distance along the t axis
3. Odometer reading [itex]\Leftrightarrow[/itex] clock reading
4. "A ended up at the same longitude as B, but at different lattitudes" [itex]\Leftrightarrow[/itex] "A ended up at the same x-location as B, but at different t-locations"
5. "A's odometer reading is different than B's" [itex]\Leftrightarrow[/itex] "A's clock reading is different than B's"

Since t=x/v, an inverse relation of t to v, the faster clock reads less time than the slower clock. Applying SR corrections, the difference is even greater. The confusion of longer line, shorter time, results from interpreting the Minkowski graphic as a 2D geometric road map, which it is not.

What you're calling "confusion" is the geometric way of looking at SR, which has proved enormously successful. You're confused about SR, but there is no confusion about the geometric description. SR uses a manifold with a metric, just like road maps. It's a different type of metric, it's an indefinite metric instead of a Euclidean metric, but mathematically, they are very similar. All the paradoxes of SR completely vanish in the geometric view.

Now, what the geometric view doesn't explain is WHY spacetime has an indefinite metric. But neither does regular space have an explanation for why it should have a Euclidean metric.

I would say that the geometric view completely does away with the confusion.. It doesn't do away with the mystery of SR, but there's no escaping that.

 

[phyti]

stevendaryl

You are reading, but not comprehending.

A and B travel the same course, just as all racers run from start line to finish line. The winner is the one who does it in the least time. A master clock at the finish line is used

to synchronize and and compare. If x is constant in t=x/v, t is inversely proportional to v. The math explains why the longer line in the (x, ct) graphic represents less time. There is no need of an odometer analogy.

The geometric (x, ct) graphic is confusing to the unfamiliar, who attempt to interpret it using the x=vt mode. Obviously the confusion has not been eliminated, since the same questions are still being asked.

The op never got an answer ln terms he could understand, and the thread became a debate about semantics.

 

[Dale]

A and B travel the same course

That is a frame variant statement, if you boost the scenario then A and B no longer travel the same path. Not that there is anything wrong with that, but I am not sure if you realize that. Stevendaryl is trying to explain the invariant geometry.

 

[stevendaryl]

stevendaryl

You are reading, but not comprehending.

A and B travel the same course, just as all racers run from start line to finish line..

No, they don't. Not from the point of view of 4-D spaceTIME. A starts at an event [itex]e_1[/itex] and ends at an event [itex]e_2[/itex]. B starts at the same event, [itex]e_1[/itex], but ends at a different event [itex]e_3[/itex].

In your example,
[itex]e_1[/itex] has coordinates [itex]x=0, t=0[/itex]
[itex]e_2[/itex] has coordinates [itex]x=12, t=20[/itex]
[itex]e_3[/itex] has coordinates [itex]x=12, t=15[/itex]

[itex]e_3[/itex] and [itex]e_2[/itex] are NOT the same endpoint, in spacetime. They have the same x-coordinate, in the same way that Minneapolis and Des Moines have the same longitude. But they have different t-coordinates.

Having the same x-coordinate is not physically significant. Different frames have different conventions for when two events have the same spatial location.

 

[phyti]

No, they don't. Not from the point of view of 4-D spaceTIME. A starts at an event [itex]e_1[/itex] and ends at an event [itex]e_2[/itex]. B starts at the same event, [itex]e_1[/itex], but ends at a different event [itex]e_3[/itex].

In your example,
[itex]e_1[/itex] has coordinates [itex]x=0, t=0[/itex]
[itex]e_2[/itex] has coordinates [itex]x=12, t=20[/itex]
[itex]e_3[/itex] has coordinates [itex]x=12, t=15[/itex]

[itex]e_3[/itex] and [itex]e_2[/itex] are NOT the same endpoint, in spacetime. They have the same x-coordinate, in the same way that Minneapolis and Des Moines have the same longitude. But they have different t-coordinates.

Having the same x-coordinate is not physically significant. Different frames have different conventions for when two events have the same spatial location.

The x distance is significant if you are comparing times at the same spatial location. There wasn't any attempt to make the finish for both, the same event, which would be impossible, given the setup.
It doesn't matter anyway.

 

[stevendaryl]

The x distance is significant if you are comparing times at the same spatial location

But "the same spatial location" is not physically meaningful. Different coordinate systems give different answers to the question "Do these two events happen at the same spatial location?"

 

[Sugdub]

Hmm. I don't think that there is any kind of consensus about that... Saying that acceleration caused the age difference is a very weird way of looking at it, in my opinion. Acceleration caused one of the travelers to take a different spacetime path, but acceleration didn't cause that spacetime path to be longer.

I'm afraid the “cause and effect” relationship is not properly spelled out: there are two facets of the same relationship. On the ontological side, the difference between the values displayed by both clocks is due to the constraint which imposes different paths through space-time. On the mathematical formalism side, this constraint gets represented by an acceleration applied to one of the clocks, and of course it triggers the difference in measurement outcomes. The acceleration and the change of space-time path are two equivalent descriptions of the cause.

 

[Sugdub]

What is your aversion to calling them what they are: "coordinate dependent" or "frame variant"?
You seem oddly insistent on using an incorrect term when a correct one is available. Particularly given that all of your justifications focus on coordinate dependence and none on metaphysics. Why then go out of your way to discard the standard term you have justified and use a term that you have not justified?

It took me some time until I could formulate an answer to this. Indeed I have a strong “aversion” to wordings such as “proper time, proper length, clock slowing down” or “running slow”, being “late”... and several more. On second thoughts, I think these wordings are remnants of an ontology - the former Newtonian ontology - which contradicts the SR formalism.
We would normally expect that a measurement process reveals the value of a quantity attached to “something” that has an objective existence independent from our investigation process and from our formal representation schemes. According to the pre-relativistic ontology, a clock has a “period” which is an objective attribute to its physical description, it measures amounts of “time”; a ruler has a “length” and measures amounts of “space”. Although the ruler can be mathematically described in a 3-dimensional coordinate system, the measurement process does not deal with its “coordinates” - which are frame variant quantities - it deals with the “length”. In this ontology, the “length” has a logical precedence over the space coordinates. Not only its value is not frame-variant, but the concept itself is independent from the representation scheme: there is no privileged frame for the 3-dimensions representation of space related quantities.
According to the SR ontology, “space-time” takes over the logical precedence over “space” and “time”. The SR formalism makes clear that “space” and “time” become coordinate-like quantities invoked by the mathematical representation of space-time in a 4-dimensions manifold. Both loose their former status as ontological concepts, they become part of the representation scheme. It can't be true any more that clocks and rulers respectively measure amounts of “time” and amounts of “space”. The ontology must change. So what do they measure?
According to (peculiar) presentations of the SR theory often displayed in PF, a clock is now assigned a “proper period” and it measures amounts of “proper time”. The value of the “proper time” equals the extremal value taken by the “time” component of the “space-time” interval between two events in the clock's worldline, i. e. when this interval gets projected onto the 4-dimensions rest frame of the clock. Whereas this effectively makes the “proper time” invariant in value, it remains in essence a representation-dependent concept, an amount of “time”, an amount of a coordinate quantity. Moreover, its definition refers to a specific frame, the rest frame of the clock, and this contradicts the fundamental principle of SR insofar there is a privileged frame.
For me the “proper time” and “proper period” do not belong to the ontological domain: actually these wordings point to amounts of a coordinate quantity. It can't be true that clocks measure amounts of “time” and neither of “proper time”.
It appears that the revolution initiated by Einstein (in the sense given by T Kuhn who described the structure of scientific revolutions), further completed by Minkovski in the mathematical representation domain, failed short of properly addressing the compulsory revolution of the corresponding ontological domain. It is clear to me that in the SR context a clock does not measure amounts of “time” any more. The “proper time” obviously constitutes an attempt to recover an acceptable ontological view, however it fails for the reason I explained above. Whereas it remains the duty of physicists to propose a way forward, I think there is no alternative but accepting that a clock measures amounts of "space-time".

 

[stevendaryl]

I'm afraid the “cause and effect” relationship is not properly spelled out: there are two facets of the same relationship. On the ontological side, the difference between the values displayed by both clocks is due to the constraint which imposes different paths through space-time. On the mathematical formalism side, this constraint gets represented by an acceleration applied to one of the clocks, and of course it triggers the difference in measurement outcomes. The acceleration and the change of space-time path are two equivalent descriptions of the cause.

In flat spacetime, it happens to be true that

• there is only one inertial spacetime path connecting two events, and
• that path has the greatest proper time.

Those two facts do suggest that acceleration somehow causes clocks to run slower. However, those two facts are only true in flat spacetime. In curved spacetime, you can certainly have multiple inertial paths connecting the same two events, and they will not have the same proper time. You can also have an inertial path that has a shorter proper time than an accelerated path. So in curved spacetime, the explanation that "acceleration causes clocks to run slower" is definitely not available. Since SR is a limiting case of GR, it really doesn't make sense to attribute the difference in proper time to acceleration in that case, either. In my opinion.

For the simplest example of a spacetime with multiple inertial paths connecting the same two events, you can consider a "cylindrical" universe in which the point [itex]x=0[/itex] is connected to the point [itex]x=L[/itex]. So space is a circle, rather than a straight line. In this universe, a twin who stays put at [itex]x=0[/itex] will have a longer proper time than a twin who travels inertially all the way "around" the universe back to the start. Neither experiences acceleration, yet their clocks don't agree. This universe is almost SR, in that for any experiment taking place within a region that doesn't go all the way "around" the universe, it's indistinguishable from SR.

Thinking of acceleration as the cause of the difference sends you down a dead-end path. Perhaps it works for SR, but it has to be completely tossed out when you go on to study GR.