Time dilation why or how, Special Relativity causes

[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 \Leftrightarrow distance along the x axis
2. Lattitude \Leftrightarrow distance along the t axis
3. Odometer reading \Leftrightarrow clock reading
4. "A ended up at the same longitude as B, but at different lattitudes" \Leftrightarrow "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" \Leftrightarrow "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 e_1 and ends at an event e_2. B starts at the same event, e_1, but ends at a different event e_3.

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

e_3 and e_2 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 e_1 and ends at an event e_2. B starts at the same event, e_1, but ends at a different event e_3.

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

e_3 and e_2 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 x=0 is connected to the point x=L. So space is a circle, rather than a straight line. In this universe, a twin who stays put at x=0 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.

 

[stevendaryl]

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.

You're mixing up different things in your list of phrases. "Proper time" and "proper length" are definitely SR concepts (and they extend to GR). They are not "remnants of a Newtonian ontology" at all.

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”.

I don't think what you're saying is true, at all. Clocks certainly do measure proper time, and proper time is certainly a fundamental concept of SR (and also GR).

 

[PeterDonis]

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.

No, this is not correct. Proper time is a geometric quantity: it is the length along a given timelike curve between two events. It depends on the curve (as well as the chosen events), but it doesn't depend at all on the "representation" we choose for the curve (I'm not sure exactly what you mean by "representation", but I think you mean something like coordinates or parameterization, and proper time doesn't depend on those).

Moreover, its definition refers to a specific frame, the rest frame of the clock

No, it doesn't. You are using the wrong definition; see above for the correct, geometric one, which makes no mention of frames at all.

I think there is no alternative but accepting that a clock measures amounts of "space-time".

If by "amounts of spacetime" you simply mean "the geometric length along a particular timelike curve", then that is what a clock measures. See above.

 

[ghwellsjr]

...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...

Thank you for using the correct term Proper Time. You now need to respond to my questions in post #57 regarding Proper Length and how the length contraction formula is related "the space component of the same space-time interval".

 

[Dale]

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.

I appreciate this response. It is actually relevant to the use of the word "metaphysical". Thank you for that.

Two general thoughts on this:

First, I don't think that an ontology can contradict a formalism even in principle. You always have the flexibility, in any formalism, to label x as "real" and y as "not-real". So the primary question of ontology is formalism-neutral. All that you can talk about is what a given formalism predicts for the outcome of a given experiment. The formalism cannot tell you which parts are "real". Furthermore, you can change formalisms quite easily, and I suspect that very few people believe that in changing a formalism you have changed reality. For example, you can use the Newtonian formalism or the Lagrangian formalism or the Hamiltonian formalism to work the same problem in classical mechanics.

Second, I agree with you completely that many of the wordings are shamelessly taken from Newtonian physics. However, as I said above, there is no ontology defined by the Newtonian formalism either. You are still free to classify things as "real" or "not real" even in Newtonian physics. Furthermore, it is well-known that the same word can have different meanings in different contexts. If you are going to classify a defined term in a theory then you have to use that theory's definition of the term, not some other theory. This does indeed make it more difficult for students to learn.

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.

This is simply incorrect. Proper time is not "an amount of a coordinate quantity". While it is true that you can always build a coordinate system around a given clock's proper time, that does not make the proper time "representation-dependent".

Your same objection, were it correct, would also apply to the Newtonian concept of the length of a ruler which you correctly described above as being independent. You can also build a Newtonian coordinate system around a ruler, but that does not make the concept of the ruler's length a coordinate-dependent quantity.

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.

The solution is simply for you to understand the actual definition as understood by the physics community.

I think there is no alternative but accepting that a clock measures amounts of "space-time".

Physicists clearly already accept that, and that is precisely what they mean when they use the word "proper time".

 

[Sugdub]

Thank you for using the correct term Proper Time. You now need to respond to my questions in post #57 regarding Proper Length and how the length contraction formula is related "the space component of the same space-time interval".

You are perfectly right. Time dilation and length contraction relate to different, exclusive classes of space-time intervals. Thanks for this.

 

[Sugdub]

... Physicists clearly already accept that, and that is precisely what they mean when they use the word "proper time".

Hmmm... Let me express some doubts. Considering the responses proposed in #71, 72, 73 and 74, it appears there is no consensus about the actual meaning of the wording “proper time”. Does it refer ...
1- to an amount of “space-time” (here I mean the compound quantity S which is mathematically described by all SR lectures using a 4-coordinates vector in a manifold, one of these coordinates being “time” and the three other being “space”)?
2- to an amount of “time” (e.g. the “time” component of a space-time interval)?
3- to an amount of “space” (e.g. the “length” of a curve)?
4- ?
Looking at #71, I interpret this as option 2.
In #72, although the spelling looks close to option 3, I think the proposed definition can be better understood alongside option 1 subject to replacing “length” (which is too much “space”-related) with “measure”. Hence the question I raised in #53 about the nature of the “ageing” quantity, but no clear answer so far.
According to #73, it would seem that option 2 is accepted.
Finally the last statement in #74 seems to agree with option 1.

I have made clear that I consider option 1 is the only one viable: a clock measures the amount of “space-time” “crossed” along a “path” linking a pair of time-like physical events. An inertial clock will deliver the lowest value since it travels along a geodesic curve. A non-inertial clock will deliver an higher value. The value of S can by defined as the integral over the curved path of the dS element which defines the measure of an infinitesimal space-time interval. It is a compound quantity which contributing coordinates are “time” and “space” related, however it is clear that contrary to its coordinates, dS is frame-invariant whilst being path-dependent. Both characteristics are required in order to account for the objective nature of the outcome of a physical measurement process involving a clock, whilst ensuring the varaibility of this outcome in response to any physical constraint applied to the clock which forces it to deviate from a geodesic path. This is why I suggested that a clock measures amounts of “space-time”. In the rest frame of an inertial clock, only the “time” coordinate contributes to the value of dS. However, due to the equivalence of all inertial frames, that does not imply that what gets measured by a clock is an amount of “time”: although the numerical values are equal, the ontological status of the “time” and “space-time” concepts is different.
I'm afraid I can't go further until what is referred to under “proper time” gets clarified.

 

[ghwellsjr]

Thank you for using the correct term Proper Time. You now need to respond to my questions in post #57 regarding Proper Length and how the length contraction formula is related "the space component of the same space-time interval".

You are perfectly right. Time dilation and length contraction relate to different, exclusive classes of space-time intervals. Thanks for this.

I'm perfectly right about what? I said in post #57, "I think you are getting yourself into trouble by focusing on the space-time interval" and yet you continue to do so. It's no wonder you can't figure out what Proper Time is and I haven't seen how you relate Proper Length to the spacetime interval. If you would explain your concept of Proper Length, I think you would see that your notions are misguided.

 

[stevendaryl]

Hmmm... Let me express some doubts. Considering the responses proposed in #71, 72, 73 and 74, it appears there is no consensus about the actual meaning of the wording “proper time”. Does it refer ...
1- to an amount of “space-time” (here I mean the compound quantity S which is mathematically described by all SR lectures using a 4-coordinates vector in a manifold, one of these coordinates being “time” and the three other being “space”)?
2- to an amount of “time” (e.g. the “time” component of a space-time interval)?
3- to an amount of “space” (e.g. the “length” of a curve)?

There is no ambiguity about the definition of "proper time". It's given by: \tau = \int \sqrt{|g_{\mu \nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}|} d\lambda for a parametrized path \mathcal{P}(\lambda) described in coordinates by x^\mu(\lambda). The physical meaning is that if the parametrized path is the path taken by an idealized clock, then \tau is the elapsed time on that clock. There is complete consensus about these two claims.

The integral giving \tau makes use of coordinates, but the value is independent of which coordinates you use. You could state the definition more abstractly by:

\tau = \int \sqrt{g(U,U)} d\lambda

where U is the tangent vector to the path \mathcal{P}(\lambda). But to actually compute that integral, you have to choose a coordinate system.

 

[stevendaryl]

I have made clear that I consider option 1 is the only one viable: a clock measures the amount of “space-time” “crossed” along a “path” linking a pair of time-like physical events. An inertial clock will deliver the lowest value since it travels along a geodesic curve. A non-inertial clock will deliver an higher value.

You've got that exactly backwards. An inertial path has the greatest value for proper time (at least in SR).

 

[Dale]

Let me express some doubts. Considering the responses proposed in #71, 72, 73 and 74, it appears there is no consensus about the actual meaning of the wording “proper time”.

Looking at the responses in 71, 72, 73, and 74 it appears that there is complete consensus about the meaning of "proper time". Your doubts seem to be based on a simple misunderstanding on your part.

That is not particularly surprising, all of us are trying to express the math in English, and many things get garbled in the translation. However, the meaning is clear in the math. Please review post 78. To see the complete link between your post 76 and stevendaryl's post 78 please note that $ds^2=g_{\mu\nu} dx^{\mu} dx^{\nu}$.

There is no ambiguity, proper time is an invariant measure of the spacetime interval along a timelike path. That is what clocks measure.

 

[PeterDonis]

Looking at the responses in 71, 72, 73, and 74 it appears that there is complete consensus about the meaning of "proper time"...proper time is an invariant measure of the spacetime distance along a timelike path. That is what clocks measure.

Since I made one of the posts under discussion, I'll chime in here as well to say I agree with all of this.

 

[HeyBert]

I understand the theory of special relativity and the mathematics which support it. I even understand that the time dilation has been proven. Therefore I am going to ask a question which on first blush may appear that I disagree with it but that is not the case. The question I can not seem to find an answer to anywhere is the why or how time dilation occurs? I am not seeking an example of where and when it occurs but the cause of time slowing as you accelerate the time tracking device.

That question is currently driving me nuts and I was hoping someone might have the answer as to why or how time slows as it measured during acceleration...

Ask why the grass is green in Special Relativity...get a comprehensive explanation of Global Warming and Biological Evolution, but not why the grass is green in Special Relativity.
The answer to your question lies within Einstein's second postulate as it applies to the light clock (http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/lightclock.swf). At rest, the light travels between a fixed path distance between the mirrors causing the light clock to tick. In motion, the light must traverse a greater path distance between the mirrors, due to motion of the clock as a result of simple geometry, thereby taking a greater amount of time to traverse the path between the mirrors and therefore results in slower clock ticks. For any specified path distance between two points in a rest frame, the same path ALWAYS has a different distance when this system is placed in uniform motion. Greater path distances is exactly why clocks slow in inertial frames...period!

For a more comprehensive explanation, see; <<link deleted>>

 

[Sugdub]

Looking at the responses in 71, 72, 73, and 74 it appears that there is complete consensus about the meaning of "proper time". Your doubts seem to be based on a simple misunderstanding on your part.
That is not particularly surprising, all of us are trying to express the math in English, and many things get garbled in the translation. However, the meaning is clear in the math. Please review post 78. To see the complete link between your post 76 and stevendaryl's post 78 please note that $ds^2=g_{\mu\nu} dx^{\mu} dx^{\nu}$.
There is no ambiguity, proper time is an invariant measure of the spacetime interval along a timelike path. That is what clocks measure.

I thank you and stevendaryl for your inputs. Your confirmation that my proposal is fully backed-up by the most generic mathematical formalism of the SR and GR theories is a major step.
Due to the logical precedence of the “space-time” concept over its time and space coordinate components, I felt it was no longer possible to consider that what gets measured by a clock is of a “time” nature. Therefore I criticised the semantics of the language used by physicists, in particular wordings such as “proper time” and “elapsed time” since their intuitive meaning refers to amounts of a “time” nature, whereas the mathematical concept corresponding to what gets actually measured deals with amounts of space-time, i.e. the measure of a space-time interval.
It is amazing to read that the misunderstanding is only on my part. Where has it been explained that a clock measures amounts of the S quantity, i.e. amounts of “space-time”? I can't remember any presentation of SR, any lecture heading in this direction. For me the wording “proper time” is not only highly misleading, it is symptomatic of a misconception about what gets actually measured: why did physicists introduce this wording whereas “space-time” was already available and fully appropriate? The “major step” I refer to above precisely consists in getting rid of this misconception.
But never mind. Let's try and secure our common understanding. I referred to “a pair of time-like physical events”. This is certainly not correct: “time-like” relates to the pair, not to individual events. So a better wording would be “a time-like pair of physical events”, with “time-like” indicating that there exists an inertial frame of reference in which the physical events at stake are represented as being co-located. Hopefully this is what you call a “timelike path” and the following statement will be backed-up: “ A clock delivers an invariant measure of the space-time interval along a path connecting a time-like pair of physical events”. Please let me know.
Let's now come back to my post #53 and consider the “ageing” of the twins along their respective journeys. There is absolutely no doubt that the word “ageing” has been chosen because it designates what we usually consider being an increase in the age of the twins, hence a “time” interval. Now we must acknowledge that “ageing” actually designates an increase in S, an amount of space-time, the measure of a space-time interval. Subtracting S' from S is certainly possible, but this so-called “difference in ageing” can in no way be considered as a difference in the age of the twins. How could that lead to a statement whereby one of the twins comes back “younger” than the other? The only way would be to isolate the “time” components of S and S' respectively and to subtract one from the other... but first, one would need to ascertain that it is physically meaningful to breakdown S and S' onto the same base of the same manifold (I've never seen any consideration about this) and second, one would have to ascertain that the difference between both “time” components is frame-invariant (which I believe is not true: only the difference between S and S' is frame-invariant).
Both twins have a different life history life history. A comparison can certainly be drawn, but no objective qualification of the difference can be made in terms of a frame-invariant time interval. So a statement whereby one of the twins comes back “younger” or “ages less” propagates an erroneous conclusion.
The same goes for clocks “slowing down” or “being late”. Of its own, the experiment cannot sort out whether the non-inertial clock changes behaviour or whether it follows a different path in space-time. However your inputs show that semantic characterisations accounting for a “time” interval are necessarily frame-variant. Therefore I think such expressions should be firmly rejected.

 

[PeterDonis]

the mathematical concept corresponding to what gets actually measured deals with amounts of space-time, i.e. the measure of a space-time interval.

It is amazing to read that the misunderstanding is only on my part. Where has it been explained that a clock measures amounts of the S quantity, i.e. amounts of “space-time”?

You mean no presentation of SR that you have read has used the term "spacetime interval"? That seems hard to believe. In the text I learned SR from (Taylor & Wheeler's Spacetime Physics), the term "spacetime interval" is all over the place.

That said, this "S quantity" you refer to is not well-defined as it stands. What does "amount of spacetime" mean? Spacetime is a 4-dimensional geometric object. Does the "amount" of it refer to the "size" of a 4-dimensional subset? Of a 3-dimensional hypersurface? A 2-dimensional surface? A 1-dimensional curve? All of these are conceptually distinct, so you can't just use one term, "amount of spacetime", to refer to all of them. That's why physicists have different terms for these different things. See below.

For me the wording “proper time” is not only highly misleading, it is symptomatic of a misconception about what gets actually measured: why did physicists introduce this wording whereas “space-time” was already available and fully appropriate?

Because "spacetime" describes the 4-dimensional geometric object, and "proper time" describes the arc length along a one-dimensional timelike curve within this 4-dimensional geometric object. They're different things, so it's entirely appropriate to have different terms for them.

“ A clock delivers an invariant measure of the space-time interval along a path connecting a time-like pair of physical events”.

Yes (with the appropriate definition of "a time-like pair of events", which you give earlier in the same paragraph. Note that this "space-time interval" is an arc-length along a 1-dimensional curve, as above.

There is absolutely no doubt that the word “ageing” has been chosen because it designates what we usually consider being an increase in the age of the twins, hence a “time” interval. Now we must acknowledge that “ageing” actually designates an increase in S, an amount of space-time, the measure of a space-time interval.

They are the same thing; the term "time interval" is just shorthand for "spacetime interval along a timelike curve". Why do you think we call such curves "timelike"? Because you measure arc length along them with a clock, not a ruler.

Subtracting S' from S is certainly possible, but this so-called “difference in ageing” can in no way be considered as a difference in the age of the twins.

Yes, it is. You have an incorrect understanding of what the term "time interval" means; see above. With the correct understanding, as given above, S and S' are indeed "time intervals", and subtracting them does give a difference in aging. You don't need to separate out "time components"; in fact you don't even need to define coordinates at all. The difference between S and S' is an invariant physical difference: the physical manifestation of this difference is the difference in age of the twins (as recorded on their clocks, in their biological processes, their experienced time, etc.) when they come back together.

The same goes for clocks “slowing down” or “being late”. Of its own, the experiment cannot sort out whether the non-inertial clock changes behaviour or whether it follows a different path in space-time.

Yes, they can; following a different path in spacetime can be measured. The fact that the two twins in the twin paradox follow different paths in spacetime is an invariant physical fact, just like the difference in arc length along those different paths. The measurement is simple: do the two objects (twins, clocks, whatever) stay co-located all the time (i.e., do they pass through exactly the same set of events)? If not, they are following different paths through spacetime.

You appear to be getting hung up on superficial features of the words we use to describe SR in English, instead of looking at the underlying concepts. If you look at the actual math, there is no ambiguity at all; and if you look at how the math gets translated into predictions about physical observables, there is no ambiguity there either.

 

[nitsuj]

Due to the logical precedence of the “space-time” concept over its time and space coordinate components, I felt it was no longer possible to consider that what gets measured by a clock is of a “time” nature. Therefore I criticised the semantics of the language used by physicists, in particular wordings such as “proper time” and “elapsed time” since their intuitive ...whether it follows a different path in space-time. However your inputs show that semantic characterisations accounting for a “time” interval are necessarily frame-variant. Therefore I think such expressions should be firmly rejected.

Does the above post mean that clocks measure spacetime, not just time?

 

[Sugdub]

You mean no presentation of SR that you have read has used the term "spacetime interval"? That seems hard to believe. In the text I learned SR from (Taylor & Wheeler's Spacetime Physics), the term "spacetime interval" is all over the place.

No, I mean that I've never seen a statement whereby the measure S of a space-time interval varies depending on the path followed end-to-end, I've never seen a statement whereby what a clock measures is nothing else than S along a definite path. These are things you don't find in presentations of SR easily accessible by non-physicists, or by those having a limited background in maths. When I suggested in a previous post that a clock actually measures S instead of a time quantity, I was really “fishing”.

That said, this "S quantity" you refer to is not well-defined as it stands. What does "amount of spacetime" mean? Spacetime is a 4-dimensional geometric object. Does the "amount" of it refer to the "size" of a 4-dimensional subset? Of a 3-dimensional hypersurface? A 2-dimensional surface? A 1-dimensional curve? All of these are conceptually distinct, so you can't just use one term, "amount of spacetime", to refer to all of them. That's why physicists have different terms for these different things. See below.

This is an excellent comment. Thanks.

Because "spacetime" describes the 4-dimensional geometric object, and "proper time" describes the arc length along a one-dimensional timelike curve within this 4-dimensional geometric object. They're different things, so it's entirely appropriate to have different terms for them.

Yes, I agree to the need for different terms for different things, however the terms must be chosen in a meaningful way. In the Newtonian context, there is no doubt that what a clock measures relates to an interval alongside the time axis. In the SR context, it is no longer the case. Contrary to “time dilation” which describes the variation of an interval alongside the time axis, the curved line which gets measured by a clock is not, in general, alongside the time coordinate axis: hence the importance of using appropriate terms. The common misconception whereby “time dilation” reflects what a clock measures (that is what this thread was initially dealing with) is a recurrent thematic which cannot be eliminated as long as physicists claim that a clock actually measures a time interval.

Yes (with the appropriate definition of "a time-like pair of events", which you give earlier in the same paragraph. Note that this "space-time interval" is an arc-length along a 1-dimensional curve, as above.

Yes, well-done. Please don't forget to mention that in general, this curved line does coincide with the time axis. This is essential.

They are the same thing; the term "time interval" is just shorthand for "spacetime interval along a timelike curve".

No. the curved line is not alongside the time axis. So it is not appropriate to call this a "time interval". I've no problem with the maths definition, but I strongly disagree with the naming.

Why do you think we call such curves "timelike"? Because you measure arc length along them with a clock, not a ruler.

I think my wording whereby there exists a reference frame in which both limiting events are represented as being co-located is better than “because we use a clock”.

Yes, it is. You have an incorrect understanding of what the term "time interval" means; see above.

No, see above. The semantics of “time interval” points to an interval alongside the time axis, whereas the mathematical definition of S which you refer to as “time interval” actually deals with a space-time interval, i.e. a curved line which does not coincide with the time axis. Your mathematical derivation is perfect, but the english language you display is not in accordance with it. We have had this problem all along this thread: I criticize the language used by physicists and they answer that their maths formalism is perfect. It is perfect indeed, but this is not the point. It is clear we won't agree until this gets sorted out. At least I get the feeling that I finally succeeded to identify the root cause of all these discrepancies. Hopefully there is a margin for progressing toward a common understanding. Thanks a lot for your efforts.

 

[Sugdub]

Does the above post mean that clocks measure spacetime, not just time?

I think the last post by stevendaryl was very helpful insofar it becomes clear that a clock delivers an invariant measure of a space-time interval along a one-dimensional curve. Since this curve does not, in general, coincide with the time coordinate axis, I think it is misleading to claim that a clock measures a time interval. I'm the only one defending this, so far.

 

[Orodruin]

cannot be eliminated as long as physicists claim that a clock actually measures a time interval.

And it should not, because that is by definition what a clock does. That you in SR chose to define a coordinate $t$ and call it a "time coordinate" is unrelated to this fact and in my opinion more symptomatic of the coordinate actually behaving as the time for an observer following a world line where only this coordinate changes.

No. the curved line is not alongside the time axis. So it is not appropriate to call this a "time interval". I've no problem with the maths definition, but I strongly disagree with the naming.

Again you have it backwards, the "fault" here if there is one is not in calling what the clock measures "time", but calling the time axis "time".

The semantics of “time interval” points to an interval alongside the time axis

Physicists, in particular the ones fairly familiar with GR will disagree with you. What you call your coordinates is utterly irrelevant to the physics.

 

[stevendaryl]

I think the last post by stevendaryl was very helpful insofar it becomes clear that a clock delivers an invariant measure of a space-time interval along a one-dimensional curve. Since this curve does not, in general, coincide with the time coordinate axis, I think it is misleading to claim that a clock measures a time interval. I'm the only one defending this, so far.

Well, whether we are talking about Newtonian physics or special relativity or general relativity, there are two different notions of "time". One is a coordinate, which depends on a coordinate system, and the second is a measurable quantity, which only depends on having a clock. When someone says to bake the cookies for 15 minutes, she is talking about time as a measurable quantity. When someone says to meet her at the park at 12:30 am, she is talking about time as a coordinate. Of course, given Newtonian physics, the two are simply related: the measurable quantity, elapsed time, is just the difference between two coordinate times.

In SR and GR, the relationship between the two notions of time becomes more complicated. So people use "proper time" and "coordinate time" to indicate which notion is meant. But they are both rooted in the different notions of time from Newtonian/Galilean physics.

 

[stevendaryl]

Of course, given Newtonian physics, the two are simply related: the measurable quantity, elapsed time, is just the difference between two coordinate times.

Actually, even ignoring relativity, the relationship between coordinate time and elapsed time can be a lot more complicated. If I tell you that one event takes place in Paris on June 23, 1987 at 12:45 pm, and a second event takes place in New York on December 3, 2015 at 10:30 am, it's actually pretty complicated to convert those coordinates to an elapsed time. You have to take into account time zones and leap years and conversions between minutes, hours, days and years. But in Newtonian physics, it's possible (though people don't do it) to have a coordinate time that is simply a real number, no matter what your location, and elapsed time is always the difference between two coordinate times.

 

[PeterDonis]

I mean that I've never seen a statement whereby the measure S of a space-time interval varies depending on the path followed end-to-end, I've never seen a statement whereby what a clock measures is nothing else than S along a definite path. These are things you don't find in presentations of SR easily accessible by non-physicists.

Ah, I see. Yes, pop science or layman's presentations of SR (like pop science or layman's presentations of science in general) don't take the same care as textbooks or scientific papers do. That's why you shouldn't try to learn a science from pop science or layman's presentations. Textbooks on SR, at least the ones I'm familiar with, do address these points.

In the Newtonian context, there is no doubt that what a clock measures relates to an interval alongside the time axis.

No, this is not correct. In Newtonian physics, clocks measure absolute time, which is not linked to the "time axis" of any inertial frame. (It can't be, because the "time axes" of different inertial frames are different, but absolute time is the same in all of them.) It appears that your confusion about terminology in SR arises from a confusion about terminology in Newtonian physics.

The semantics of “time interval” points to an interval alongside the time axis

No, it doesn't. See above.

As a further point, the semantics of "time interval", or any such term involving "time", has to change when you go from Newtonian physics to SR, because SR does not have absolute time. So your general argument that we should adjust the semantics of terms like "time interval" so they match the Newtonian semantics is not valid, because it can't be done.

 

[Sugdub]

And it should not, because that is by definition what a clock does. ... Again you have it backwards, the "fault" here if there is one is not in calling what the clock measures "time", but calling the time axis "time".

Excellent. I'm ready to follow whichever representation system provided it is internally consistent. According to your suggestion, clocks will be said measuring “time” intervals and thus the twins will effectively "age" differently. Clocks will follow geodesic lines in a 4-dimensions manifold, its geometrical structure (curvature) being dependent on the prevailing physical conditions, and this will in turn determine the coordinate axes (curved lines).

However “time dilation” will need re-naming since it won't deal any longer with a “time” interval and the coordinates of today's “space-time” which will no longer be called “space” and “time”. That's fine with me. For what concerns this specific thread, I found today's language being inconsistent because the terms "time dilation" and "proper time" can't both refer to a time interval. Also the path followed by the clock did not coincide with the "time" coordinate axis. Again the maths are correct but the language used by physicists was so far inconsistent.
Should the above be acceptable by physicists, I think it positively resolves my long-lasting discrepancy. Thanks a lot.

 

[PeterDonis]

Clocks will follow geodesic lines in a 4-dimensions manifold

No. Clocks can follow any timelike worldline. There is no requirement that it be a geodesic. The traveling twin's clock in the twin paradox follows a non-geodesic worldline.

geometrical structure (curvature) being dependent on the prevailing physical conditions

Yes, if by "prevailing physical conditions" you mean "the stress-energy tensor".

this will in turn determine the coordinate axes

No. Coordinates are an arbitrary choice; there is no requirement that a particular set of coordinate axes must be chosen.

“time dilation” will need re-naming since it won't deal any longer with a “time” interval and the coordinates of today's “space-time” which will no longer be called “space” and “time”.

I don't understand how you're getting any of this out of what we've been saying. Your understanding of how the terms "time" and "space" are used is incorrect, and we've been telling you so for many posts now. In the particular passage you quoted, Orodruin was not saying that the "time axis" of a coordinate chart is not a "time" coordinate; he was saying that coordinates are an arbitrary choice, so you should stop fixating on the "time axis" of an arbitrarily chosen coordinate chart, and start thinking about actual observables, like what a clock reads as it follows a particular timelike curve.

I found today's language being inconsistent because the terms "time dilation" and "proper time" can't both refer to a time interval.

No, you found it inconsistent because your understanding of it is incorrect.

the path followed by the clock did not coincide with the "time" coordinate axis.

So what? Coordinates are an arbitrary choice.

the maths are correct but the language used by physicists was so far inconsistent.

No, your understanding of the language is incorrect.

Should the above be acceptable by physicists

Not likely since it's based on an incorrect understanding of the issues involved.

 

[nitsuj]

... I think it is misleading to claim that a clock measures a time interval. I'm the only one defending this, so far.

In turn do you find it misleading to claim a ruler measures length? In other words do you find your perspective makes sense when seen from the perspective of length?

 

[Dale]

Where has it been explained that a clock measures amounts of the S quantity, i.e. amounts of “space-time”? I can't remember any presentation of SR, any lecture heading in this direction.

It is in any mainstream SR or GR textbook, it is even in the Wikipedia entry on proper time.

For me the wording “proper time” is not only highly misleading, it is symptomatic of a misconception about what gets actually measured: why did physicists introduce this wording whereas “space-time” was already available and fully appropriate?

All proper times are spacetime intervals, but not all spacetime intervals are proper time. Proper time is the spacetime interval along a purely timelike world line. There are spacelike, null, and mixed worldlines also.

Complaining about bad terminology is fruitless. Even widespread disagreement with a term can be insufficient, for example"relativistic mass". If all the top scientists can't get rid of "relativistic mass" then you are not going to be able to get rid of "proper time". Besides, it is a useful term, with a more specific meaning than just spacetime interval.

Subtracting S' from S is certainly possible, but this so-called “difference in ageing” can in no way be considered as a difference in the age of the twins.

Of course it can. The human body can be used as a clock. Not a very accurate clock, but a clock nonetheless.

 

[Dale]

I referred to “a pair of time-like physical events”. This is certainly not correct: “time-like” relates to the pair, not to individual events. So a better wording would be “a time-like pair of physical events”, with “time-like” indicating that there exists an inertial frame of reference in which the physical events at stake are represented as being co-located. Hopefully this is what you call a “timelike path” and the following statement will be backed-up: “ A clock delivers an invariant measure of the space-time interval along a path connecting a time-like pair of physical events”. Please let me know.

The correct wording is "a pair of timelike separated events", but I understood what you meant to begin with so I didn't make a big deal of it. It is the separation that is timelike, the events are just events. Timelike separated events have a timelike path which connects them, and a timelike path is a path whose tangent vector is timelike at all events along the path.

I would not use the description in terms of inertial frames since there may not be a global inertial frame at all if you are dealing with GR. However, if you are in flat spacetime and if you have a pair of timelike separated events using the general definition, then your definition follows.

Yes, a clock measures the spacetime interval along its worldline, and that is invariant.

Let's now come back to my post #53 and consider the “ageing” of the twins along their respective journeys. There is absolutely no doubt that the word “ageing” has been chosen because it designates what we usually consider being an increase in the age of the twins, hence a “time” interval. Now we must acknowledge that “ageing” actually designates an increase in S, an amount of space-time, the measure of a space-time interval.

Yes.

Subtracting S' from S is certainly possible, but this so-called “difference in ageing” can in no way be considered as a difference in the age of the twins.

Sure it can. Both S' and S are invariant numbers, true in any frame, they both represent ages, they have the same units and so forth, so subtracting them is a well-defined operation. If I am 40 and my wife is 37 then everyone I know would consider the difference in our age to be 3 years.

How could that lead to a statement whereby one of the twins comes back “younger” than the other?

The word "younger" means less age.

The only way would be to isolate the “time” components of S and S' respectively and to subtract one from the other

Why would you need to do that. Subtracting the spacetime interval, or proper time is sufficient. No need to take an invariant and break it into components.

... but first, one would need to ascertain that it is physically meaningful to breakdown S and S' onto the same base of the same manifold (I've never seen any consideration about this) and second, one would have to ascertain that the difference between both “time” components is frame-invariant (which I believe is not true: only the difference between S and S' is frame-invariant).

I see no benefit it breaking it into components

Both twins have a different life history life history. A comparison can certainly be drawn, but no objective qualification of the difference can be made in terms of a frame-invariant time interval. So a statement whereby one of the twins comes back “younger” or “ages less” propagates an erroneous conclusion.

What is erroneous? Seems fully justfied to me

However your inputs show that semantic characterisations accounting for a “time” interval are necessarily frame-variant. Therefore I think such expressions should be firmly rejected.

I don't follow either the "semnatic characterisation" or the resulting "firmly rejected"