Time dilation why or how, Special Relativity causes

[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.