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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...

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In summary, the question of why or how time slows as you accelerate a time tracking device is currently driving me nuts. I was hoping someone might have the answer as to why or how time slows as it measured during acceleration.f

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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...

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If you are looking for an answer within the theory, it would simply be that it is a logical consequence of the speed of light being constant for all observers and the special principle of relativity.

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Time Dilation has nothing to do with acceleration. It has only to do with the tick rate of a clock according to an Inertial Reference Frame (IRF) in which the clock is moving. The easiest way I know to show this is to start with a clock at rest in an IRF and then transform to an IRF moving with respect to the first one.

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...

Here is a spacetime diagram showing a clock at rest in an IRF. The dots indicate one-nanosecond ticks of the clock:

Now using the Lorentz Transformation process let's transform to an IRF moving at 60%c with respect to the above IRF:

At 60%c, the Time Dilation factor is 1.25. As you can see, the tick marks are expanded by a factor of 1.25 so that after 4 ticks, the Coordinate Time has covered 5 nanoseconds.

Pretty simple, don't you think?

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Hi welcome to physicsforums. :)

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...

It may be that you are asking for a model with which we can understand how to make sense of it, physically. If so, then you have half bad luck: special relativity doesn't have such a model (it only has a mathematical model, relating to observations) and while there are conceptual physical models, they have led to fruitless debates so that that "metaphysical" topic is now not appreciated on this forum - and that's the bad news.

The good news is that you can find some of those debates in the archives here, and in that way you can learn about two ways to make sense of it all. In a nutshell they correspond to the views of Lorentz versus the view of Minkowski, and you can find a summary of the debate here: https://www.physicsforums.com/threads/what-is-the-pfs-policy-on-lorentz-ether-theory-and-block-universe.772224/ [Broken]

Some explanations about those models can be found when you search this forum with the relevant key words.

By the way, a note to the Mentors: I find the relativity FAQ of this forum hard to find. Can it be made more visible? That will be useful for all, especially for newcomers.

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I understand the theory of special relativity and the mathematics which support it... 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...

Noting that the OP made clear that he agrees with the maths, I think the following explanation would answer his question.

According to the time dilation theorem, if the period of the clock is one nanosecond when described in a reference frame where the clock is at rest, then the period is 1.25 nanosecond when described in a reference frame where the radial velocity of the clock is 0.6c. Both options (and many more) are concurrently valid. These are two equivalent representations (/descriptions) for the period of a single clock. Nothing happens to the clock, its physical behaviour is the same in both cases. There is no action exerted on it, no acceleration has been applied to it... there is no physical change and therefore there is no rationale for invoking a physical cause.

Time dilation is not a physical effect, it does not reflect any change in the prevailing physical conditions.

Time dilation traces a change of the representation scheme within the mathematical formalism. Different numerical values (e.g. 1ns vs 1.25ns) must be used to describe the period of the clock depending on the inertial reference frame chosen for this representation, i.e. depending on whether the clock is represented at rest or in radial motion, this being an arbitrary decision by the theoretician (invoking “observers”, “observations” or “measurements” is simply irrelevant to explain the time dilation concept).

An analogy can be made with a change in the orientation of the coordinate system: it changes the space coordinates of any event. Different numerical values hold as the x,y,z coordinates of the event depending on the orientation of the space axes. There is no physical change, only a change of the mathematical representation scheme for the same physical event. Again, whether this event gets “observed” or not is irrelevant.

Please tell me if this explanation is correct.

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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...

Well, a simple thought-experiment that shows that something like time dilation must happen is a light clock. Imagine a train car on a track, with a pair of mirrors set up on top of the car, oriented parallel to the tracks (so perpendicular to the motion of the train). Imagine a pulse of light bouncing back and forth between those mirrors. You could use those bounces to measure time: When the train is at rest, then light will take a time [itex]T = \frac{2D}{c}[/itex] to travel back and forth between them, where [itex]c[/itex] is the speed of light and [itex]D[/itex] is the distance between the mirrors. Pure geometry shows that if you start the train car moving forward, then the light pulse will take longer to make the round trip: instead of [itex]T[/itex], it will take time [itex]T' = T/\sqrt{1-\frac{v^2}{c^2}}[/itex]. So as viewed from the rest frame of the train tracks, the clock aboard the moving train must run slow.

I understand that this explanation raises more questions than it answers:

- Why does light have a characteristic speed, [itex]c[/itex], in the first place?
- Why should a clock that is not light-based, such as a wind-up clock, or an electric clock, experience the same sort of time dilation as a light clock?
- Why do we assume that the distance [itex]D[/itex] between mirrors is unchanged by the motion of the train?
- What if the light clock mirrors were oriented so that the light pulse traveled in the same direction as the tracks, instead of perpendicularly?
- What if the light clock is stationary, but the observers are moving? Do you get the same result? How is that possible?

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In ANY scientific theory the answer to any "why" or "how" question is the postulates/axioms of the theory. Those are the key concepts that explain everything else in the theory. In the case of SR, the traditional postulates are the principle of relativity and the invariance of c. So time dilation occurs because the laws of physics are the same in any inertial frame as is the speed of light.. The question I can not seem to find an answer to anywhere is the why or how time dilation occurs?

Of course, a theory never can explain its own postulates/axioms. Those can sometimes be derived from a more fundamental theory with its own postulates/axioms, but you always get to a point where there are unexplained postulates/axioms that are assumed because doing so fits the data better than any other known assumptions.

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If you are looking for an answer within the theory, it would simply be that it is a logical consequence of the speed of light being constant for all observers and the special principle of relativity.

If the answer is not in the domain of physics, why is SR a theory of physics?

The question is "how or why does time dilation occur?".

How does "a logical consequence of the speed of light being constant for all observers and the special principle of relativity" result in clocks running slower?

You don't explain anything by substituting a statement that asks another question.

harrylin #5

It may be that you are asking for a model with which we can understand how to make sense of it, physically. If so, then you have half bad luck: special relativity doesn't have such a model (it only has a mathematical model, relating to observations) and while there are conceptual physical models, they have led to fruitless debates so that that "metaphysical" topic is now not appreciated on this forum - and that's the bad news.

All models are conceptual, since that is all the mind can produce.

Mathematics is required for anything involving measurement.

The "light clock" is an effective model, demonstrating time dilation, using established physical phenomena; light propagation and object motion. Time dilation is an experimentally verified fact, and doesn't qualify as "metaphysical".

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If the answer is not in the domain of physics, why is SR a theory of physics?

I don't understand that question. A theory of physics describes what happens, it doesn't describe why something happens. (Except in the reductive sense that you can explain why something happens in terms of more fundamental properties. You can explain why some substance has certain chemical properties by showing how those properties follow from the behavior of the molecules making up that substance. You can explain why those molecules have that behavior in terms of the behavior of protons, neutrons, electrons that make up the molecules. You can explain why protons, neutrons and electrons behave the way they do in terms of quantum mechanics and the strong and electromagnetic forces. Maybe someday we will be able to explain why the strong and electromagnetic forces work the way they do in terms of some more fundamental theory of interactions. But at some point, explaining why in terms of more fundamental laws of physics has to stop. It has to stop with something that is just descriptive of HOW things work, not WHY they work that way.)

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Why can’t the question about physics, be answered in terms of known physics?

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Because it isWhy can’t the question about physics, be answered in terms of known physics?

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Why is used in the same context as how in this case. It's not why, as in motive or reason by some creative source, but process explaining why/how a clock rate is different when moving than at rest. No different than explaining why a car moves faster when the fuel flow increases, or a TV picture appears when you press a certain button. The clock is a physical object in motion. Why is that not about physics? Yes, physics describes how the universe works, by using abstract theories, whose elements should correspond to physical phenomena. Unless it's a postulate, each abstract element should be transformable to an observable behavior. If not, you could never verify a theory.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.

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Why is used in the same context as how in this case. It's not why, as in motive or reason by some creative source, but process explaining why/how a clock rate is different when moving than at rest. No different than explaining why a car moves faster when the fuel flow increases, or a TV picture appears when you press a certain button. The clock is a physical object in motion. Why is that not about physics? Yes, physics describes how the universe works, by using abstract theories, whose elements should correspond to physical phenomena. Unless it's a postulate, each abstract element should be transformable to an observable behavior. If not, you could never verify a theory.

The "why" of time dilation isn't observable, but the fact of time dilation certainly is. If you take a clock, and fly it around the world on a plane, the amount of time elapsed will be different than that of a clock that remains at one spot on the Earth the whole time. If you take a clock to the top of a mountain, or up to a geosynchronous satellite, and leave it for a year, then bring it back down, it will show a different amount of elapsed time than one that is at sea level the whole time.

You're completely right, that physics has to make contact with observations, and relativity (special and general) makes plenty of testable predictions. But you're wrong to think that the "why" of time dilation is relevant to observation.

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Time Dilation is a mathematical coordinate effect having nothing to do with any physical attribute. The physical effect that can be proven is whether or not physical laws are invariant under the Lorentz Transformation. If they are, then Time Dilation is a useful mathematical process that falls out of doing Lorentz Transformations on the coordinates of a diagram. Time Dilation is no more physical than the coordinates, the scales, the directions of the axes or the origins of the diagrams.

Note that the OP specifically asked about Time Dilation of a clock during acceleration which is a subject that cannot be answered or addressed by Special Relativity. Time Dilation cannot predict what happens to a clock before, during, after and as a result of, acceleration. But if you can specify how a clock behaves during this process in one Inertial Reference Frame it can establish how it behaves in another IRF.

Note that the OP specifically asked about Time Dilation of a clock during acceleration which is a subject that cannot be answered or addressed by Special Relativity. Time Dilation cannot predict what happens to a clock before, during, after and as a result of, acceleration. But if you can specify how a clock behaves during this process in one Inertial Reference Frame it can establish how it behaves in another IRF.

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Time Dilation is a mathematical coordinate effect having nothing to do with any physical attribute. The physical effect that can be proven is whether or not physical laws are invariant under the Lorentz Transformation. If they are, then Time Dilation is a useful mathematical process that falls out of doing Lorentz Transformations on the coordinates of a diagram. Time Dilation is no more physical than the coordinates, the scales, the directions of the axes or the origins of the diagrams.

It depends on what you mean by "time dilation". The prediction that the elapsed time on a clock is given by [itex]\tau = \int \sqrt{1-\frac{v^2}{c^2}} dt[/itex], where [itex]v[/itex] and [itex]t[/itex] are both measured in an inertial Cartesian coordinate system, is a physical prediction.

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Time Dilation is a mathematical coordinate effect having nothing to do with any physical attribute... Time Dilation is no more physical than the coordinates, the scales, the directions of the axes or the origins of the diagrams.

It's good that you acknowledge that time dilation is not a physical effect. It is obvious that the Lorentz transformation deals with the coordinates of events and therefore the time dilation formula deals with the elapsed time separating the time coordinates of two events: so the SR formalism itself establishes that the “period” of a clock is no longer an intrinsic property of this object as it was the case in the Newtonian mechanics. The same happens for the “length” of an object which also becomes a coordinate-like quantity in SR, since its numerical value is IRF-dependent.

Any metaphysical statements such as "the clock slows down" or "time slows" should be firmly rejected.

...But if you can specify how a clock behaves during this process in one Inertial Reference Frame it can establish how it behaves in another IRF.

This statement could lead to misunderstandings insofar a change of the IRF has no bearing at all to a change of the (physical behaviour) of the clock, as you rightly pointed out above. Time dilation deals with a change of description of the clock, namely a change of its coordinate-like parameters, which include the so-called “period” of the clock: following a change of the velocity of the clock in respect to the IRF, a different numerical value must be assigned to this coordinate-like parameter. This is a change in the formal representation (/ description), with no bearing to a change of (physical) behaviour.

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What makes such statements "metaphysical"? Which branch of metaphysics do they deal with?Any metaphysical statements such as "the clock slows down" or "time slows" should be firmly rejected.

I would call them "coordinate dependent" or "frame variant".

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What makes such statements "metaphysical"? ...

It's a matter of consistency with other coordinate quantities. If the theoretician decides to move the origin on the time axis, all dates change value but we don't conclude that we have jumped into the future or in the past. Only the formal representation of history has changed.

In the same way, reversing the direction of the time axis is not going to change the fact that our future is unknown, not our past. One would not conclude that the time flows backwards.

Should the theoretician decide to move the origin or the orientation of the spaces axes, all positions will be affected; but we don't conclude that physical objects have suddenly moved away. The world (more precisely our simulated world) is still the same. Changes of the coordinate system deal with parameters of the representation process, not with what gets represented.

I think we must behave consistently in respect to changes of the IRF, because they also deal with coordinate-like quantities. We must keep in mind that the choice of an IRF belongs to the theoretician, not to the operator of an experiment. A change of the IRF does not affect in any way the conditions of the experiment, and neither (of course) its outcome. If a clock gets represented as being in motion in a given IRF, its frequency must be lowered: the numerical value of this coordinate-like quantity decreases but that does not mean that “the clock is slow”. The so-called “period” of the clock is no longer an intrinsic property of the clock. SR does not deal with intrinsic properties of physical objects, it constrains how coordinate-like quantities must evolve depending on the representation scheme selected by the theoretician. But indeed it is difficult to "digest" the fact that durations and lengths are no longer intrinsic properties.

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None of that seems to justify the term "metaphysical". The term "metaphysical" means that something is part of the philosophical discipline of metaphysics. I have never seen anything that indicated that coordinate charts and coordinate dependent quantities are part of that philosophical discipline (although I only took a single course on the topic).

In fact, quite to the contrary you speak at length of theoreticians. Theoretical physicsts are still practicing physics, not metaphysics. Theory is an important part of science.

I think you are using the term as a means of deprecating the concept of time dilation. I am fine with that deprecation, but I think that "metaphysical" is an inaccurate description. I would simply call them "coordinate dependent" or "frame variant", which I believe is more accurate.

In fact, quite to the contrary you speak at length of theoreticians. Theoretical physicsts are still practicing physics, not metaphysics. Theory is an important part of science.

I think you are using the term as a means of deprecating the concept of time dilation. I am fine with that deprecation, but I think that "metaphysical" is an inaccurate description. I would simply call them "coordinate dependent" or "frame variant", which I believe is more accurate.

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It's a matter of consistency with other coordinate quantities. If the theoretician decides to move the origin on the time axis, all dates change value but we don't conclude that we have jumped into the future or in the past. Only the formal representation of history has changed.

In the same way, reversing the direction of the time axis is not going to change the fact that our future is unknown, not our past. One would not conclude that the time flows backwards.

Should the theoretician decide to move the origin or the orientation of the spaces axes, all positions will be affected; but we don't conclude that physical objects have suddenly moved away. The world (more precisely our simulated world) is still the same. Changes of the coordinate system deal with parameters of the representation process, not with what gets represented.

Well, that's a reason that many people feel that the only physically meaningful effects are ones that can be expressed in coordinate-free language. Many of the predictions of SR can be recast in that form. A clock that accelerates away, turns around, and accelerates back will show less elapsed time than a clock with the same starting and ending point but which travels inertially the whole time. That's a prediction of SR that can be written in a coordinate-free (or coordinate covariant) way.

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None of that seems to justify the term "metaphysical". The term "metaphysical" means that something is part of the philosophical discipline of metaphysics. I have never seen anything that indicated that coordinate charts and coordinate dependent quantities are part of that philosophical discipline (although I only took a single course on the topic)...

I did not state, as you seem to believe, that a change of coordinate dependent quantities is relevant to metaphysics. It seems that some misunderstanding is developing here. I qualified as “metaphysical” such statements like “we have jumped into the future or in the past”, “the time flows backwards”, “physical objects have suddenly moved away”, and as well “the clock is slow” or “time is slowing”. For me such statements do not belong to physics, and this is why I consider they are “metaphysical”. I think it is the appropriate word but are welcome to propose a better one. The first three statements were examples of my own illustrating potential mis-interpretations of the consequences of changing the space and time coordinate system (origin and orientation of axes). Nobody makes such mistakes. Conversely the last two expressions are very common in debates about time dilation, however they play exactly the same role as the previous examples in case of a change of the inertial frame of reference: they are mis-interpretations of what the SR formalism is about. The SR formalism, including the time dilation formula, deals with coordinate-like quantities, e.g. the gap between the time coordinates of two events, and therefore should not be interpreted as dealing with objective qualifications of physical objects, e.g. the “period of the clock”. Indeed a change occurs in the SR mathematical formalism in response to a change of the IRF, but claiming that this indicates a change of a physical attribute, an intrinsic property or the physical behaviour of the clock goes far beyond what can be reasonably expected from a mathematical formalism dealing with the coordinates of events.

I hope this clarifies my previous input.

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I propose "coordinate dependent". A statement like "the clock is slow" can be assigned a well-defined meaning in physics. Whether or not that statement is true of a given clock depends on the coordinate system. So the statement is "coordinate dependent", but it does belong to physics.I qualified as “metaphysical” such statements like “we have jumped into the future or in the past”, “the time flows backwards”, “physical objects have suddenly moved away”, and as well “the clock is slow” or “time is slowing”. For me such statements do not belong to physics, and this is why I consider they are “metaphysical”. I think it is the appropriate word but are welcome to propose a better one.

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I did not state, as you seem to believe, that a change of coordinate dependent quantities is relevant to metaphysics. It seems that some misunderstanding is developing here. I qualified as “metaphysical” such statements like “we have jumped into the future or in the past”, “the time flows backwards”, “physical objects have suddenly moved away”, and as well “the clock is slow” or “time is slowing”. For me such statements do not belong to physics, and this is why I consider they are “metaphysical”.

To expand on what Dale said, I disagree with that characterization, completely. I don't consider such statements as "metaphysical" at all. They are simply coordinate-dependent statements. If someone asks me which way to Joe's restaurant, and I answer him by saying: "Go to the left 1/4 of a mile", have I said something "metaphysical"? No, I said something very mundane, but its interpretation is dependent on context. Whether the restaurant is to the left of to the right depends on which way you are facing. So it's not an absolute fact about Joe's restaurant.

Similarly, in Special Relativity, anything that you might say about the timing of distant events, or about the lengths of moving objects is similarly context-dependent. Those statements are not absolute about the objects, or about the objects, but are facts relative to a particular coordinate system. That doesn't make them "metaphysical". I would say it is the farthest you can get to metaphysical. Metaphysics is a branch of philosophy having to do with "explaining the fundamental nature of being and the world that encompasses it". It's an investigation into what is real and what is fundamental about the world. A coordinate-dependent statement is almost exactly the opposite. There is nothing fundamental about a description in terms of coordinates.

Metaphysics doesn't just mean "not physics" or "not physical".

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...Similarly, in Special Relativity, anything that you might say about the timing of distant events, or about the lengths of moving objects is similarly context-dependent. Those statements are not absolute about the objects, or about the objects, but are facts relative to a particular coordinate system. That doesn't make them "metaphysical". I would say it is the farthest you can get to metaphysical. Metaphysics is a branch of philosophy having to do with "explaining the fundamental nature of being and the world that encompasses it". It's an investigation into what is real and what is fundamental about the world. A coordinate-dependent statement is almost exactly the opposite. There is nothing fundamental about a description in terms of coordinates...

To be honest I'm not sure we disagree . We are debating on the meaning of “time dilation”. My claim is that “time dilation” relates to a change of the inertial frame of reference, and that an IRF sets a space-and-time framework into which the relative motion between physical objects gets formally represented. According to SR, all IRFs are equally valid for such a representation, and SR caters for transformation rules to enforce this equivalence. Replacing a representation in a given IRF with another representation based on another IRF does not entail any change in WHAT gets represented, it only deals with HOW it gets represented. This means that the Lorentz transformation formula (dealing with the coordinates of events), the time dilation formula (dealing with the time separation between events) and the length contraction formula (dealing with the space separation between events) should not be interpreted as tracing changes in the world. Conversely they deal with different but equivalent representations of the same realm, whatever that is. Therefore “time dilation” has no bearing to any kind of “physical effect” and there is no rationale for seeking a “physical cause” for this non-effect.

This is in essence my answer to the OP. I'm not going to expand on whether some statements are “metaphysical” or not until I get convinced that we agree on the meaning of “time dilation”. Let's first thrive to offer a conclusive answer to the OP, if possible.

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...This means that the Lorentz transformation formula (dealing with the coordinates of events), the time dilation formula (dealing with the time separation between events) and the length contraction formula (dealing with the space separation between events) should not be interpreted as tracing changes in the world. Conversely they deal with different but equivalent representations of the same realm, whatever that is. Therefore “time dilation” has no bearing to any kind of “physical effect” and there is no rationale for seeking a “physical cause” for this non-effect.

As I said in another post recently, the claim that one clock experiences time dilation is in some sense, not physical, because it's a statement about the relationship between two different coordinate systems.

On the other hand, the claim that a clock that moves away from Earth at a high speed, turns around, and comes back at high speed will show less elapsed time than one that stays put the whole time (at least if we ignore gravitational time dilation) is a physical claim that is independent of coordinates.

Those two claims are related, in the sense that you can use the coordinate-dependent time dilation to derive the difference in elapsed times on the two clocks.

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Yes. Being a logical consequence of replacing an IRF with another IRF, time dilation deals with comparing two representations, in two different inertial frames, of a single clock. A unique physical pattern gets described twice. No physical change is involved. Also the reciprocity / symmetry of time dilation between two frames confirms that it is not a physical effect.As I said in another post recently, the claim that one clock experiences time dilation is in some sense, not physical, because it's a statement about the relationship between two different coordinate systems.

Yes, but... The twins experiment consists in comparing the physical behaviour of two identical clocks, one of them being subject to an acceleration for part or all of its journey. Since “being subject to an acceleration” is an absolute physical determination, the prevailing physical conditions are different for the “inertial” clock and the “accelerated” clock respectively. This accelerated motion is the (objective) physical cause for the (objective) difference in their physical behaviour. Here we have a genuine physical effect (a gap between the elapsed times respectively predicted and measured on each clock), which can be traced back to a genuine physical cause (one of the clocks is subject to an acceleration). The asymmetry of the effect can be traced back to the asymmetry of the cause.On the other hand, the claim that a clock that moves away from Earth at a high speed, turns around, and comes back at high speed will show less elapsed time than one that stays put the whole time (at least if we ignore gravitational time dilation) is a physical claim that is independent of coordinates.

Unfortunately many people seem to confuse the time dilation paradigm (one clock, a unique physical pattern described twice in two different IRFs) and the twins experiment paradigm (two clocks, two different physical patterns, both described in the same IRF in view of their comparison).

Indeed, because in order to predict the difference in their behaviour, SR must represent both clocks in the same inertial frame. Hence the role played by the time dilation formula in deriving the predicted gap between both clocks: a delta-time quantity must be converted from one frame to another frame in order to compute the elapsed time of both clocks in the same IRF. However, it is the acceleration pattern applied to one of the clocks (combined with the initial relative velocity of both clocks) which determines how far the motion of the “accelerated” clock “departs” from an inertial motion. The magnitude of the predicted time gap (physical effect) is directly linked to the magnitude of this “departure”, which is obviously non-symmetrical.Those two claims are related, in the sense that you can use the coordinate-dependent time dilation to derive the difference in elapsed times on the two clocks.

Let's hope that the above convinces you that the outcome of the twins experiment cannot be explained on the sole basis of considerations about time dilation, which are symmetrical in essence. Only the (objective) non-symmetrical acceleration pattern can be set as the physical cause for the (objective) non-symmetrical outcome of the twins experiment. A more detailed discussion of this derivation should certainly be ported to a dedicated thread.

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The twins experiment consists in comparing the physical behaviour of two identical clocks, one of them being subject to an acceleration for part or all of its journey. Since “being subject to an acceleration” is an absolute physical determination, the prevailing physical conditions are different for the “inertial” clock and the “accelerated” clock respectively. This accelerated motion is the (objective) physical cause for the (objective) difference in their physical behaviour.

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

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

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Let's hope that the above convinces you that the outcome of the twins experiment cannot be explained on the sole basis of considerations about time dilation, which are symmetrical in essence. Only the (objective) non-symmetrical acceleration pattern can be set as the physical cause for the (objective) non-symmetrical outcome of the twins experiment. A more detailed discussion of this derivation should certainly be ported to a dedicated thread.

I do not agree with that conclusion. I think it helps to look at the analogous case of ordinary lengths in Euclidean geometry. Suppose you are traveling down a straight highway, and the highway has road markers every 10 meters (say). Another road crosses yours at a slope (tangent of the angle) of [itex]m[/itex]. As you travel down your road, counting the roadmarkers as you pass, you can look perpendicularly toward the other road, and see how many roadmarkers you've passed of the other road. There is a formula relating the two: If [itex]N[/itex] is the number of roadmarkers you've passed on your road since the roads crossed, and [itex]N'[/itex] is the number you've passed on the other road, then

[itex]N' = \sqrt{1+m^2} N[/itex]

So, you conclude that [itex]N'[/itex] is increasing faster than [itex]N[/itex]. If the other road ever curves and rejoins your road, you know that [itex]N'[/itex] will be greater than [itex]N[/itex]. To curve back, [itex]m[/itex] for the other road has to change, but since [itex]m^2 \geq 0[/itex], you know that the factor [itex] \sqrt{1+m^2}[/itex] is greater than 1.

But note that the slope [itex]m[/itex] is relative! From the point of view of the other highway, it is YOUR highway that has a nonzero value for [itex]m[/itex]. A traveler on that road could use the same formula to conclude that when you get back together, [itex]N \geq N'[/itex] instead of the other way around.

The resolution to the paradox is that the formula [itex]N' = \sqrt{1+m^2} N[/itex] is only valid for comparing [itex]N'[/itex] to the number [itex]N[/itex] for a STRAIGHT road. The formula can't be used if [itex]N[/itex] is counting the number of roadmarkers along a nonstraight road. Every time the road makes a curve, there is a change in the notion of what point along the other road is in correspondence.

- #31

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Lets suppose that Alice is staying put and Bob is traveling away at [itex]v = 0.866 c[/itex] (giving a time dilation factor of 2). So Bob's position as a function of time, using Alice's coordinates is given by: [itex]x=vt[/itex].

Let [itex]e_0[/itex] the the event at which Alice and Bob depart (coordinates [itex]x_0=0, t_0=0[/itex]. Let [itex]e_1[/itex] be the event with coordinates [itex]x_1=0, t_1=10[/itex] in Alice's coordinate system (with t measured in years). Let [itex]e_2[/itex] be the event with coordinates [itex]x_2 = v \cdot 10, t_2 = 10[/itex]. Alice can use her coordinate system, using the relativistic time-dilation formula to compute:

- At event [itex]e_1[/itex], Alice is 10 years older than at event [itex]e_0[/itex]
- At event [itex]e_2[/itex], Bob is only 5 years older than at event [itex]e_0[/itex]

The only thing that different coordinate systems disagree about is whether events [itex]e_1[/itex] and [itex]e_2[/itex] are SIMULTANEOUS, or not. According to Alice's coordinate system, the two events are simultaneous, so she concludes that she is 5 years older than Bob at time [itex]t_1 = 10[/itex]. According to Bob's coordinate system, the two events are NOT simultaneous, and as a matter of fact, event [itex]e_1[/itex] takes place 15 years AFTER event [itex]e_2[/itex].

Given those facts, I think it is wrong to dismiss the time dilation formula as "merely a coordinate effect". It absolutely gives the correct answer for Alice's or Bob's ages at any event along their two spacetime paths.

- #32

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Indeed, because in order to predict the difference in their behaviour, SR must represent both clocks in the same inertial frame.

Not really, taking the space-time interval approach, the particular coordinate(s) used to represent the clocks are immaterial. There isn't any such requirement in SR, though it is probably convenient to use a single coordinate system for both.

However, it is the acceleration pattern applied to one of the clocks (combined with the initial relative velocity of both clocks) which determines how far the motion of the “accelerated” clock “departs” from an inertial motion. The magnitude of the predicted time gap (physical effect) is directly linked to the magnitude of this “departure”, which is obviously non-symmetrical.

The magnitude of the predicted time gap is well-approximated by the angle between worldlines on the space-time diagram, in the same sense that the magnitude of the "distance gap" when you add two sides of a triangle and compare it to the hypotenuse depends on the included angle - the angle between the two sides.

The angle between the wolrdlies on a space-time diagram is another way of describing the velocity between the physical objects represented by the worldlines.

Thinking that focuses on the acceleration tends to cause confusion - the actual formula for time dilation and the time gap uses relative velocities, you won't find the acceleration in the calculation at all. From the standpoint of learning about SR as it is taught in textbooks and understood by professionals, the long philosophical arguments that time dilation and/or the time gap "should" depend on acceleration are just a distraction, when one looks at the actual formula written in the textbook, one does not see the acceleration referred to anywhere.

- #33

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"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. I've only seen rambling comments like the following:To be honest I'm not sure we disagree . We are debating on the meaning of “time dilation”. My claim is that “time dilation” relates to a change of the inertial frame of reference, and that an IRF sets a space-and-time framework into which the relative motion between physical objects gets formally represented.

Could you please give what you consider to be the best definition of "Time Dilation" and then we'll see if we agree.According to SR, all IRFs are equally valid for such a representation, and SR caters for transformation rules to enforce this equivalence. Replacing a representation in a given IRF with another representation based on another IRF does not entail any change in WHAT gets represented, it only deals with HOW it gets represented. This means that the Lorentz transformation formula (dealing with the coordinates of events), the time dilation formula (dealing with the time separation between events) and the length contraction formula (dealing with the space separation between events) should not be interpreted as tracing changes in the world. Conversely they deal with different but equivalent representations of the same realm, whatever that is. Therefore “time dilation” has no bearing to any kind of “physical effect” and there is no rationale for seeking a “physical cause” for this non-effect.

This is in essence my answer to the OP. I'm not going to expand on whether some statements are “metaphysical” or not until I get convinced that we agree on the meaning of “time dilation”. Let's first thrive to offer a conclusive answer to the OP, if possible.

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.

- #34

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You could have stated your scenario in a different way: Two events on Bob's inertial worldline are separated by 5 years of his Proper Time. In a coordinate system in which he is traveling at 0.866c, the Time Dilation factor is 2 which means that there is 10 years of Coordinate Time separation between those two events. Any observer/object/clock at rest in that coordinate system will accumulate 10 years of Proper Time between two events on their own worldlines that are simultaneous with Bob's two events no matter where they are located with respect to Bob. Everything in this scenario is a coordinate effect except the specification of Bob's Proper Time of 5 years.

Lets suppose that Alice is staying put and Bob is traveling away at [itex]v = 0.866 c[/itex] (giving a time dilation factor of 2). So Bob's position as a function of time, using Alice's coordinates is given by: [itex]x=vt[/itex].

Let [itex]e_0[/itex] the the event at which Alice and Bob depart (coordinates [itex]x_0=0, t_0=0[/itex]. Let [itex]e_1[/itex] be the event with coordinates [itex]x_1=0, t_1=10[/itex] in Alice's coordinate system (with t measured in years). Let [itex]e_2[/itex] be the event with coordinates [itex]x_2 = v \cdot 10, t_2 = 10[/itex]. Alice can use her coordinate system, using the relativistic time-dilation formula to compute:

Far from being a coordinate-effect, those two facts are true, no matter WHAT coordinate system you use to compute Alice's and Bob's ages. So it's not completely true to dismiss time dilation as not physical. The elapsed times computed using the time dilation formula are objectively correct.

- At event [itex]e_1[/itex], Alice is 10 years older than at event [itex]e_0[/itex]
- At event [itex]e_2[/itex], Bob is only 5 years older than at event [itex]e_0[/itex]

The only thing that different coordinate systems disagree about is whether events [itex]e_1[/itex] and [itex]e_2[/itex] are SIMULTANEOUS, or not. According to Alice's coordinate system, the two events are simultaneous, so she concludes that she is 5 years older than Bob at time [itex]t_1 = 10[/itex]. According to Bob's coordinate system, the two events are NOT simultaneous, and as a matter of fact, event [itex]e_1[/itex] takes place 15 years AFTER event [itex]e_2[/itex].

Given those facts, I think it is wrong to dismiss the time dilation formula as "merely a coordinate effect". It absolutely gives the correct answer for Alice's or Bob's ages at any event along their two spacetime paths.

Stating that two events along a worldline have the same Proper Time separation in all coordinate systems doesn't prove your point since Proper Time is invariant. You are merely capitalizing on the fact that a Proper Time separation between two events on a worldline at rest in a coordinate system is the same as the Coordinate Time and it was the Coordinate Time separation that is a coordinate effect. Coordinates and their effects are not physical.

If you want to use the Time Dilation formula in a useful way, you could say that Bob separated from Alice at 0.866c and after 5 years of his Proper Time, he turned around and came back at the same speed. Then ask, how much time did Alice, who remained inertial, accumulate during Bob's absence?

- #35

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Stating that two events along a worldline have the same Proper Time separation in all coordinate systems doesn't prove your point since Proper Time is invariant.

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

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