# An honest question about time dilation in SR

confutatis

## Main Question or Discussion Point

Hello everyone. I just kicked myself out of the philosophy section of the forums, and came here looking for some intelligent conversation at last. I'd like to ask a question about relativity that has been puzzling me for a long time.

By looking at these threads it's hard (for me anyway) to be sure whether all physicists agree on the nature of time dilation in SR. That is, whether moving clocks really tick slower or just appear to do so. I used to think the clocks actually ticked slower, but I have seen physicists argue that it's only appearances. So what is it really? What is the official position regarding time dilation in SR?

If time dilation is real, in a physical sense, I'd also like to understand what is the physical cause for the slower rate of moving clocks. Given that movement is relative and dependent on another frame of reference, how can real phenomena arise out of the act of observation? In other words, how can my clock tick slower just because somebody else is in uniform motion relative to me?

(I think the apparent contradiction in that question may imply that time dilation for bodies in uniform motion is not real, but I'm not an expert on the matter)

As a bit of background on myself, I'm an electrical engineer with an amateurish interest in physics. Since I do not work in the field, some issues about relativity can be hard to understand. I hope you experts will be able to help me.

Related Special and General Relativity News on Phys.org
jcsd
Gold Member
The nature of time dialiton is agreed; it's a an actually 'happens' and it's not just 'appearnces'.

Take the well-know example of the light clock:

Imagine a photon bouncing inbetween two mirrors:

Code:
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|<--->|
|     |
As the photons speed is constant we can use this to measure time, we could make the distance between the clocks such that the time it takes the photon to travel from one mirror to the other is 1 second, so that essentially eveytime it hits a mirror is a 'tick' of our clock.

Now imagine the clock viewd from someone moving relative (paralell) to the mirrors at some constant velocity, this is what they'll see (for two ticks of our clock):

Code:
|\   |
| \  |
|  \ |
|   \|
|   /|
|  / |
| /  |
|/   |
It's clear that the path the light takes in the second diagram is longer than the path taken in the first. One of the postulates of relativty is that light travels at the same speed in all reference frames therfore one tick measured by someone moving relative to the mirrors is no longer 1 second, the clock has slowed down! You might argue as some of the less knowledable members of this board have (mentioning no names) that we can still work out the rate that the rate that the clock 'should' be ticking and hence ime dialtion is not something real but soemthing to do with 'appearnces', but imagine if the moving observer has his own light clock which is at rest to him, there is no way that we can say that one clock is 'right' and the other clock is 'wrong'. Hence time dialtion is a fundamentally physically real phenomenon.

Now you could argue that the fact that the clocks don't agree is a pecularity of the particular method we have choosen to measure time, but that is not so; consider the other fundamental postulate of relatvity that the laws of physics are the same in all rest frames. If we choose a different method to measure time it must agree with the light clock in our rest frame, if that were not the case and it agreed with the light in some other rest frame it would mean that that the laws of physics differed from rest frame to rest frame, this is not the case.

confutatis
jcsd said:
The nature of time dialiton is agreed; it's a an actually 'happens' and it's not just 'appearnces'.
I'll take your word for it.

Imagine a photon bouncing inbetween two mirrors...
As the photons speed is constant we can use this to measure time, we could make the distance between the clocks such that the time it takes the photon to travel from one mirror to the other is 1 second, so that essentially eveytime it hits a mirror is a 'tick' of our clock.
Now imagine the clock viewd from someone moving relative (paralell) to the mirrors at some constant velocity, this is what they'll see (for two ticks of our clock) ...
It's clear that the path the light takes in the second diagram is longer than the path taken in the first.
I don't understand how this addresses the issue. You are obviously not saying the distance between the two mirrors really increases just because someone happens to be looking at it from a moving frame of reference; that would be nonsense. I take it that what you are saying is that moving observers do not agree as to their measurements of length anymore than they agree as to their measurements of time. But to me that seems to imply that time dilation is caused by length dilation, and length dilation is a function of measurement, not a real phenomenon.

I'm not arguing for it, just trying to understand.

Perhaps I should clarify what I mean by "physical sense". If time dilation is a distortion caused by movement, then it can't possibly affect the rates at which clocks tick. In that case, a moving clock will still show the same time if somehow reunited with a clock that had been stationary. The problem is that such a scenario is difficult in SR: you can't reunite the clocks without having one of them go through acceleration. But there is a scenario that seems possible to me.

Imagine two radio signals with the same frequency (relative to their source), coming from two sources in relative uniform motion. Let's call those two sources A and B. According to my understanding of SR, A's measurement of B's frequency will be lower than A's measurement of its own frequency, and vice-versa for B. That can be understood and fully accounted by your explanation of parallel mirrors. However, if from A's perspective time really runs slower for B, then B's clocks must be ticking just as slower as B's signal generator is. And because B's clocks are running slower, B's measurement of A's frequency must yield a higher value - when you have less seconds, you have more cycles per second.

I don't know if this can be easily understood.

You might argue as some of the less knowledable members of this board have (mentioning no names) that we can still work out the rate that the rate that the clock 'should' be ticking...
In no way I mean to imply I found a flaw in SR. I'm just interested in understanding it as well as the experts do.

DW
confutatis said:
Hello everyone. I just kicked myself out of the philosophy section of the forums, and came here looking for some intelligent conversation at last. I'd like to ask a question about relativity that has been puzzling me for a long time.

By looking at these threads it's hard (for me anyway) to be sure whether all physicists agree on the nature of time dilation in SR. That is, whether moving clocks really tick slower or just appear to do so. I used to think the clocks actually ticked slower, but I have seen physicists argue that it's only appearances. So what is it really? What is the official position regarding time dilation in SR?

If time dilation is real, in a physical sense, I'd also like to understand what is the physical cause for the slower rate of moving clocks. Given that movement is relative and dependent on another frame of reference, how can real phenomena arise out of the act of observation? In other words, how can my clock tick slower just because somebody else is in uniform motion relative to me?

(I think the apparent contradiction in that question may imply that time dilation for bodies in uniform motion is not real, but I'm not an expert on the matter)

As a bit of background on myself, I'm an electrical engineer with an amateurish interest in physics. Since I do not work in the field, some issues about relativity can be hard to understand. I hope you experts will be able to help me.

Any physicists here are sure to know that experiments have been done where clocks were flown around brought back together and times compared the end results of which were different times consistent with the predictions of relativity. You can't get much more real than that.

confutatis
DW said:
Any physicists here are sure to know that experiments have been done where clocks were flown around brought back together and times compared the end results of which were different times consistent with the predictions of relativity. You can't get much more real than that.
Certainly not, but a clock flying around the earth's gravitational field can hardly be said to be in uniform relative motion. Have there been experiments in which time dilation is shown to be a real effect in an SR scenario?

The essential difficulty I have to understand SR is that it seems to imply that time dilation is symmetrical, which is at odds with the notion that it can be a real phenomenon. For if my clock ticks slower than yours, and yours ticks slower than mine, it seems to me one effect must necessarily cancel the other.

jcsd
Gold Member
confutatis said:
I'll take your word for it.

I don't understand how this addresses the issue. You are obviously not saying the distance between the two mirrors really increases just because someone happens to be looking at it from a moving frame of reference; that would be nonsense. I take it that what you are saying is that moving observers do not agree as to their measurements of length anymore than they agree as to their measurements of time. But to me that seems to imply that time dilation is caused by length dilation, and length dilation is a function of measurement, not a real phenomenon.
Time dialtion and Lorentz contraction go hand in hand, but I wouldn't worry about the lengths too much, just worry about the differnce in time as we're just trying to illustrate. Lorentz contraction is a real phenomenon, when two observers measure a length or a time and ytthey get diffrent results, there is no objective way of saying who is right and who is wrong, so the only conclusion MUST be that they are both right and length and time are functions of the relative velocities of the observers.

I'm not arguing for it, just trying to understand.

Perhaps I should clarify what I mean by "physical sense". If time dilation is a distortion caused by movement, then it can't possibly affect the rates at which clocks tick. In that case, a moving clock will still show the same time if somehow reunited with a clock that had been stationary. The problem is that such a scenario is difficult in SR: you can't reunite the clocks without having one of them go through acceleration. But there is a scenario that seems possible to me.
Think about it: a clock must agree with the light clock in it's rest frame, if it doesn't it isn't measuring time. It is possible to preserve the symmetry of the situation and have both clocks undergo acceleration, but if you don't presevre the symmetry you do not get measuremnts that are symmetric.

Imagine two radio signals with the same frequency (relative to their source), coming from two sources in relative uniform motion. Let's call those two sources A and B. According to my understanding of SR, A's measurement of B's frequency will be lower than A's measurement of its own frequency, and vice-versa for B. That can be understood and fully accounted by your explanation of parallel mirrors. However, if from A's perspective time really runs slower for B, then B's clocks must be ticking just as slower as B's signal generator is. And because B's clocks are running slower, B's measurement of A's frequency must yield a higher value - when you have less seconds, you have more cycles per second.

I don't know if this can be easily understood.

In no way I mean to imply I found a flaw in SR. I'm just interested in understanding it as well as the experts do.
What you sya ids tur BOTH observers observe each others clocs running slower as situation is symmetric, but there is no paradox here as the explanation is self-consistent. When you break that symmetry you find that more time would of passed for one observer than another when they meet up again.

reilly
Radioactive particles "live longer" than ones at rest -- this is crucial for scattering experiments wih pi mesons, for example. Further, the change in lifetime is exactly explained by the time-dilation effect. this is one of the syrongest experimentl proofs of the effect -- works for both linear and circular accelerators, and for cosmic rays.
Further, if the speed of light is to be invariant across all inertial frames, then time dilation must be a realiy -- which it is.

Read Einstein's little book, Relativity, which has little mathematics in it, just great physically based explanations.
Regards,
Reilly Atkinson

confutatis
I don't think anyone understood my question so far. I'm tired of knowing that time dilation has been verified in experiments; what I do not know is whether it has been verified in experiments involving uniform relative motion. I suspect not, because such an experiment would be extremely difficult to carry out. So all those arguments that time dilation has been proved real in GR do not address my original question, which was about SR.

If time dilation is real in an exclusively-SR scenario, then it can't possibly be symmetrical. If my clock has slowed down to you, in a real sense, then I must necessarily see your clock running faster, not slower. And if the effect is symmetrical, then it can't possibly be real, but simply an artifact of measurement. As far as I can tell anyway.

I'd appreciate if people understood that I'm fully aware that time dilation is both real and asymmetrical in a GR scenario. That has nothing to do with my question, which is exclusively about SR.

Regards

Staff Emeritus
Gold Member
Dearly Missed
The cosmic rays and the particles in linear colliders move in straight lines, and over the span of an experiment they move at substantially constant speeds. And they do obey the Lorentz transformations, including the symmetry of the transformations between two inertial observers.

Trying to wiggle out of it by insisting on more and more finicky experimental proof is not going to save you. You just have to get used to the fact that if you and I are inertial observers, and you find my clock slowing down relative to yours, then you can infer that I observe your clock slowing down relative to mine. And we both measure c to have the same value. And speeds don't add linearly but in the particular formula that's been shown here a lot of times already.

confutatis
The cosmic rays and the particles in linear colliders move in straight lines, and over the span of an experiment they move at substantially constant speeds. And they do obey the Lorentz transformations, including the symmetry of the transformations between two inertial observers.
That's interesting. May I ask you, again in the spirit of honest inquiry, how physicists know that the time dilation effects in those particular experiments are real, in the sense that comparing two out-of-sync clocks would be real? I understand such a comparison would not be possible with subatomic particles, but again I'm not an expert. I'd be very interested to learn how they do it.

Trying to wiggle out of it by insisting on more and more finicky experimental proof is not going to save you.
I do not understand what that particular comment is supposed to mean. Is there something wrong in having a desire to understand what one knows to be a fact? I do not deny the facts, but I do confess I do not properly understand what makes them facts.

You just have to get used to the fact that if you and I are inertial observers, and you find my clock slowing down relative to yours, then you can infer that I observe your clock slowing down relative to mine.
I'm fully aware of that, but I can't reconcile it with the fact that there's only one reality. I perfectly understand how two clocks may appear to be running slower from each other's perspective, but I have trouble understanding how both clocks can run slower by exactly the same amount, and still be found to show different time measurements when compared.

And we both measure c to have the same value.
That couldn't possibly be otherwise when we use clocks in our own reference frame to measure c. I have no problem at all understanding this.

And speeds don't add linearly but in the particular formula that's been shown here a lot of times already.
Does that imply then that the phenomenon is not really symmetrical? That would make sense to me, but it doesn't seem to be what most physicists say.

confutatis,

The time dilation measured in cosmic-ray meson experiments can be accounted for with SR. This is essentially a uniform motion experiment (the mesons travel from the top of a mountain to the bottom). The half life of these moving mesons has been measured to be greater than the half life for stationary mesons.

So if I have two stationary, synchronized clocks, A and B, separated by some distance and a third clock passes by, the time it takes for the moving clock to go from A to B as measured by the two synchronized clocks will be greater than the time as measured on the moving clock. This difference is real, but as you say, it's not symmetric. A and B are synchronized in their frame of reference but not in the frame of the moving clock. Simultaneity is not absolute. That's the key, and it's the key in most (so called) paradoxes in SR.

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confutatis
jdavel said:
The time dilation measured in cosmic-ray meson experiments can be accounted for with SR. This is essentially a uniform motion experiment (the mesons travel from the top of a mountain to the bottom). The half life of these moving mesons has been measured to be greater than the half life for stationary mesons.
I have heard of that experiment before but I didn't know all the details. Thanks for the information.

So if I have two stationary, synchronized clocks, A and B, separated by some distance and a third clock passes by, the time it takes for the moving clock to go from A to B as measured by the two synchronized clocks will be greater than the time as measured on the moving clock. This difference is real, but as you say, it's not symmetric. A and B are synchronized in their frame of reference but not in the frame of the moving clock. Simultaneity is not absolute. That's the key, and it's the key in most (so called) paradoxes in SR.
I'm happy someone finally understood what my difficulty really was. So time dilation is not symmetric - I'm relieved to have that confirmed from a knowledgeable person. Now would you be so kind as to elucidate another issue for me? If the falling clock in your example really ticks slower compared to A and B, wouldn't it be correct to say that an observer moving together with it would perceive A and B as running faster than the falling clock? From what you explained above, I gather the answer is yes, so here is the bit I don't understand: how exactly does one get that effect from the Lorentz transformation? Can you provide a mathematical example?

I do not understand this light clock, the bouncing photon, example about time dilation.

If I am approaching an intersection and a car is coming on my right, I do not suppose that in some sense the car has been following me to the intersection. Anyway, that would make sense only for one time, the bouncing photon is not going to follow me as I move further and further from the intersection, bouncing over and over again as I pulling away.

I can tell the direction of a car coming on my right at the intersectin, as for a photon, it would move so fast nothing would be seen, at least by me.

Could the example be specious? Several readers have said they could make a correction for this, which suggests that they don't agree with the example either. Who would take such an observation at face value?

Doc Al
Mentor
time dilation is symmetric

confutatis said:
I'm happy someone finally understood what my difficulty really was. So time dilation is not symmetric - I'm relieved to have that confirmed from a knowledgeable person.
While I agree completely with jdavel's analysis, I think it is a mistake to think of this as showing time dilation is not symmetric in SR . All SR effects (in their general statement) are perfectly symmetric. What is not symmetric (and key to understanding the meson experiment) is the number of clocks involved in the measurements made in each reference frame. The "moving" frame uses but one clock; the stationary frame, two.
Now would you be so kind as to elucidate another issue for me? If the falling clock in your example really ticks slower compared to A and B, wouldn't it be correct to say that an observer moving together with it would perceive A and B as running faster than the falling clock?
Both frames observe that moving clocks run slow as measured by their own clocks. As jdavel points out, the key to understanding how the SR effects can be symmetric but consistent is to realize that simultaneity is frame dependent.
From what you explained above, I gather the answer is yes, so here is the bit I don't understand: how exactly does one get that effect from the Lorentz transformation? Can you provide a mathematical example?
The answer to that one is no. Let the two events be the passing of the moving clock by stationary clocks A and B. The Lorentz transformation gives the time between those events according to the stationary (unprimed) clocks in terms of the measurements made in the moving (primed) frame as:
$$\Delta t = \gamma (\Delta t' + v\Delta x'/c^2)$$
In this example, Δx' = 0 -- the moving clock doesn't move in its own frame. So:
$$\Delta t = \gamma (\Delta t')$$
which is the usual expression for the slowing down of moving clocks.

In the other direction, the inverse Lorentz transformation gives:
$$\Delta t' = \gamma (\Delta t - v\Delta x/c^2)$$
In the stationary frame, the moving clock moves (D'oh!) so Δx is not zero: that's where the asymmetry comes in.

However, if the moving frame observes a clock in the stationary frame, then Δx = 0, and the Lorentz transformation gives:
$$\Delta t' = \gamma (\Delta t)$$
Symmetry restored!

confutatis - welcome aboard - a topic that I frequently post on myself - and get a lot of flack. In fact janus and i have an ungoing discussion about exactly the same thing - is time dilation real, and if so, how can it be symmetric, and if it is real, what is the physical cause?

There are some good articles on the net regarding whether Einstein was true to his original trajectory that you had to break the symmetry by noting which twin felt the acceration etc. The way I approach the subject is this - taking for example a one way twin excursion - and whether he will experience a different time when he reaches the turn around point. --- when one states that one twin moves past the other at velocity v, the situation is symmetric - then when you add the next statement, that twin 2 travels 10 light years to a turn around point your have immediately created a non symmetrical situation - because the 10 light year distance is a proper distance in the earth frame and since you know the velocity v, you can calculate the proper time in the earth frame to reach the turn around point (time measured in the earth frame) - this means that you now know both the proper time and proper distance in the earth frame - so since the spacetime intervals are always equal - you can calculated the proper time for the traveler, and from that the effective distance he measures for the journey in order that the two spacetime intervals be equal (but the temporal and spactial parts will be different)

Adding to the above - if we substitute a high speed particle for the traveling twin, and we specify that it decays in 2 usec in the lab, but it travels very near c to a distance planet 10 light years from earth before it decays - according to the interval transforms, the proper time registered by a clock accompanying the muon will be 2 usec. The proper time registed by an earth clock will be 10 light years (+ a little more) - in other words, here we have specified 3 factors (the proper time in the earth frame, the proper distance in the earth frame, and the proper time in the muon frame.
The 4th factor (the apparent distance as determined by the time and velocity (approx c) is ct where t is 2 usec. This appears to imply that time dilation is real, and length contraction apparent.

russ_watters
Mentor
yogi said:
This appears to imply that time dilation is real, and length contraction apparent.
I don't see it. Here's another one: to someone in a spacecraft traveling just under C, how far away is Alpha Centuari? If a person on earth says its 4.5 light years and the person on the spaceship says it took 1 year to get there, can we conclude that he thinks he was traveling at just under C but was actually traveling at 4.5C? I don't think so. Then again, unless we can be sure that the Earth is at rest in an absolute reference frame, how can we even know what the "real" distance to Alpha Centuari is?

Its sometimes a hard pill to swallow, but reality is as we observe it to be.

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The geomentric argument for time dilation

This argument on time dilation shows up in popularizations, but I have never seen it in a text. This geometric "proof" about something that is internal to the system confuses me. Firstly if you are talking of one photon, how am I supposed to see that? Secondly if the photon travels with the speed of light, the tick would be one second before the tock at which time, if we are coming to the apex of the triangle, the photon would be one second closer to the parallel traveler and thus the tick and tock would both arrive at the same time!!!! I don't get it. What is carrying the message to the parallel traveler?

jcsd said:
The nature of time dialiton is agreed; it's a an actually 'happens' and it's not just 'appearnces'.

Take the well-know example of the light clock:

Imagine a photon bouncing inbetween two mirrors:

Code:
|     |
|<--->|
|     |
As the photons speed is constant we can use this to measure time, we could make the distance between the clocks such that the time it takes the photon to travel from one mirror to the other is 1 second, so that essentially eveytime it hits a mirror is a 'tick' of our clock.

Now imagine the clock viewd from someone moving relative (paralell) to the mirrors at some constant velocity, this is what they'll see (for two ticks of our clock):

Code:
|\   |
| \  |
|  \ |
|   \|
|   /|
|  / |
| /  |
|/   |
It's clear that the path the light takes in the second diagram is longer than the path taken in the first. One of the postulates of relativty is that light travels at the same speed in all reference frames therfore one tick measured by someone moving relative to the mirrors is no longer 1 second, the clock has slowed down! You might argue as some of the less knowledable members of this board have (mentioning no names) that we can still work out the rate that the rate that the clock 'should' be ticking and hence ime dialtion is not something real but soemthing to do with 'appearnces', but imagine if the moving observer has his own light clock which is at rest to him, there is no way that we can say that one clock is 'right' and the other clock is 'wrong'. Hence time dialtion is a fundamentally physically real phenomenon.

Now you could argue that the fact that the clocks don't agree is a pecularity of the particular method we have choosen to measure time, but that is not so; consider the other fundamental postulate of relatvity that the laws of physics are the same in all rest frames. If we choose a different method to measure time it must agree with the light clock in our rest frame, if that were not the case and it agreed with the light in some other rest frame it would mean that that the laws of physics differed from rest frame to rest frame, this is not the case.

The equations posted by Doc Al flow directly from the LT - and when one considers different locations in the non proper frame, they reveal the symmetry that leads to the statement that "each guy sees the other clock running slow" But when one makes the pass to the interval transformations we are only left with 4 factors, the proper space interval in each frame and the proper temporal interval in each frame - and since the composite spacetime interval is invarient as between two relatively moving systems - we arrive at the bizaar results that clocks attached to their respective frames read different real times because each is measuring proper time for events that take place in the frame to which it is resident. Strictly speaking, to measure proper time in the earth frame, only one clock can be used. But this can be obviated if two or more clocks are properly sync(ed) in the same frame - so when we say that alpha centuri is 4.5 light years - we are tacitly assuming the distance between the earth and the star is fixed for the purposes of the experiment - that is, the 4.5 years is measured in the earth-alpha frame - and it is a proper distance because it represents a length at rest in the earth-alpha frame (according to SR this would be true even if the earth alpha frame were moving relative to the universe, because it is again - a measurement in the earth alpha frame). When measured by the traveler, it is not a proper distance, and therefore any calculation the traveler makes must be an apparent length. We can place a clock on alpha and if properly sync(ed) with an earth clock, it will read earth time when the traveler arrives. Of course, if there is a real time difference as recorded by the earth alpha frame vs the travelers frame, the notion of symmetry as an absolute concept becomes would need further review).

Doc Al
Mentor
light clock questions

robert Ihnot said:
This argument on time dilation shows up in popularizations, but I have never seen it in a text.
Funny, I don't think I've seen an elementary text that didn't have the "light clock" example.
This geometric "proof" about something that is internal to the system confuses me. Firstly if you are talking of one photon, how am I supposed to see that?
Think of the light clock as emitting a flash--observable by everyone--every time the photon bounces back to the first mirror. Everyone can see those periodic flashes and use them as a clock.
Secondly if the photon travels with the speed of light, the tick would be one second before the tock at which time, if we are coming to the apex of the triangle, the photon would be one second closer to the parallel traveler and thus the tick and tock would both arrive at the same time!!!!
I don't see what you are saying here. The photon merely bounces back and forth. The apparent path of the photon depends on who is observing it. Just like dropping a ball on a moving train: in the train, the ball moves straight down a distance h. But from the view of someone on the stationary platform, the ball moves sideways as well: a different total distance.

The "tick" and "tock" are symmetric.
I don't get it. What is carrying the message to the parallel traveler?
Imagine there to be an observable flash everytime the photon hits the first mirror.

jcsd
Gold Member
Robert the light clock example is well-known thought exepriment, the point is you don't have to worry about the practicalities of seeing a single photon (but it only takes a few adhustments to turn it into a physically viable expetriment).

You can even show that it is consistent with the formula for time dialtion as the lengths (in the frame moving relative to the mirrors) of the sides of the triangle formed by the bouncing mirror are:

Code:
     /|
ct  / | ut
/  |
/___|
ct'
so by the Pythagorean theorum:

$$c^2t'^2 = c^2t^2 - u^2t^2$$

therefore:
$$t'^2 = t^2(1-\frac{u^2}{c^2})$$

therefore:

$$t' = \frac{t}{\gamma}$$

(where t is measured from the rest frame of the observer moving relative to the mirrors).

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DW
confutatis said:
Certainly not, but a clock flying around the earth's gravitational field can hardly be said to be in uniform relative motion. ...
So what? I said that the results were what relativity predicts not what "only gravitational effects in relativity predict". In other words SR time dilation assosiated with motion is a part of the amount. If it was truely an honest question, not loaded with a hidden presumption you would accept that. I have to wonder now.

confutatis
DW said:
So what? I said that the results were what relativity predicts not what "only gravitational effects in relativity predict". In other words SR time dilation assosiated with motion is a part of the amount. If it was truely an honest question, not loaded with a hidden presumption you would accept that. I have to wonder now.
I do not have "hidden presumptions". It would do a lot of good if you stopped judging my character; I'm not judging yours.

Since you understand this far better than I do, I've always been curious as to how they have determined the SR component of time dilation in that experiment with planes. Being an engineer, I consider that an extremely difficult thing to do, so I always assumed the experiment was more about verifying the reality of time dilation than the particular value measured. It seems to me it's very hard to take into account the acceleration of the aircraft, all the turbulence throughout the whole trip, all the variations in the gravitational field in different places of the planet...

Anyway, I'm glad to hear physicists were able to calculate the exact portion of time dilation that was due to unaccelerated movement, and the exact portion that was due to gravity and acceleration. Would you be so kind as to share your knowledge of how they achieved that?

Thanks so much.

DW
confutatis said:
Would you be so kind as to share your knowledge of how they achieved that?

Thanks so much.
By modeling the Earths external spacetime geometry with Schwarzschild geometry, the proper time of a clock $$\tau$$, which is undergoing arbitrary motion around the earth can be related to Schwarzschild coordinates $$(t, r, \theta , \phi)$$ by
$$d\tau = \sqrt{\frac{1}{\gamma ^{2}} - \frac{2GM}{rc^{2}}\frac{(\frac{dr}{dct})^2 + 1 - \frac{2GM}{rc^{2}}}{1 - 2\frac{GM}{rc^{2}}}}dt$$ where
$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$
and
$$v^{2} = (\frac{dr}{dt})^{2} + r^{2}(\frac{d\theta }{dt})^{2} + r^{2}sin^{2}(\theta )(\frac{d\phi }{dt})^{2}$$
One simply uses a spreadsheet to time step integrate this for whatever motion was undertaken. The $$\gamma$$ piece is the special relativistic contribution. One applies this to the net motion of whatever clocks are carried around to see if the predictions for the comparisons matches the experimental comparisons. They do and only do so when the special relativistic term is included in that equation.
(In reality one doesn't need an exact calculation, just an approximation which is of the level of significance of the experimental equiptment precision. So one really does an approximation for this formula to predict "the difference" between the times by time steps rather than calculate the total times by time steps. This is because experimental precision is down around nanoseconds and time stepping that for the entire experiment is not feasible, but the predicted difference between the times for feasible time steps is a matter of nanoseconds or less per time step and so that is what is actually calculated.)

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jcsd
Gold Member

Imagine two observers at rest relative to each other; then one observer accelerates with constant acceleration a (as measured in the rest frame of the staitionary observer) until he reaches velocity u, he then deccelerates with constant accelartion -a until he is at rest relative to the other observer again.

In my last post you can see how the relative times of the observers are related and we can substitue the simple kinematic equation

$$u = u_i + at$$

into it to obtain (if we adjust for the relative displacement of the two observers):

$$t' = t\sqrt{1-\frac{(u + at)^2}{c^2}}$$

We can then obtain expressions for how long each observer measures the journey to take.

For the stationary observer we can work that out from the simple kinematic formual above:

$$t=\frac{2u}{a}$$

Unfortunatley it's not so stariaghtforward for the accelarting obsever but still just using the above two formulas we can obtain:

$$t' = 2\int^{0}_{\frac{-u}{a}} \sqrt{1-\frac{a^2t^2}{c^2}}dt$$

Note the factor of 2. this is as there are two parts to the journey, the accelartion from 0 to u and the decccelartion from u to 0. Even though there is no longer symmetry between the two observers there is still symmetry between the two legs of the journey.

So lets put a few values in to the formula; if we set a as 60g (g = 9.81 m/s^2) we obtain the following (approximate values) values:

Code:
u/c        t(secs)        t'(secs)
0          0              0
0.001      1.019E-3       1.019E-3
0.1        1.019E5        1.018E5
0.5        5.097E5        4.876E5
0.9        9.174E5        7.707E5
0.99       1.009E6        7.997E5
0.999      1.018E6        8.006E5
Therefore you can see, though the two observers start and finish at rest to each other, the observer who accelerated as experinced less time than the observer who remained in the orginal rest frame.

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