# B Time Dilation Derivation

#### benorin

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Ok so I've got a question after walking through the time dilation derivation that used 'light clocks' (think a beam of light bouncing back and forth between mirrors) to derive $\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}$. So my Q is could you derive the same equation if you had used atomic clocks instead? Don't actually do so here, just wanted to know if it would have lead to the same relation. Thanks for your time!

Edit, my tex tags seem to not work?! I'm sure that was how...

<mentor edit latex>

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#### Janus

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The whole idea behind the light clock example is that someone at rest with respect to the light clock would measure the light pulse as traveling straight up and down at c, and thus would measure the time it takes to make the round trip as being shorter than the time the person which measures the light clock as moving does ( for which the light pulse travels a longer path at c). If he used an atomic clock, and measured the trip time, it would tick off N sec per round trip. The observer that measures the clock as moving would also see the moving atomic clock tick off N sec per round trip of the light pulse. Thus he would measure the atomic clock as ticking slower by the same rate as he measures the light clock as ticking.

#### Ibix

So my Q is could you derive the same equation if you had used atomic clocks instead?
The maths is harder if you do it from scratch. However, it's easy to get from the light clock result to "all clocks must be affected the same".

Set up a light clock with a movable mask in front of the lower mirror. Drive the mask with another clock, synchronised to the light clock so that the mask is out of the way when the light pulse needs to bounce off the lower mirror and in the way when the ligjt needs to bounce off the upper mirror. This does not affect the light clock, since the mask is always out of the way when the light pulse needs to pass through.

Now observe this system using a frame where it is moving. Either the other clock is time dilated the same as the light clock or the mask gets out of sync and the light clock stops, contradicting our analysis in the frame where the clock is not moving. I've made no assumption about the nature of the other clock, so all clocks must be affected like a light clock.

#### PeroK

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Ok so I've got a question after walking through the time dilation derivation that used 'light clocks' (think a beam of light bouncing back and forth between mirrors) to derive $\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}$. So my Q is could you derive the same equation if you had used atomic clocks instead? Don't actually do so here, just wanted to know if it would have lead to the same relation. Thanks for your time!

Edit, my tex tags seem to not work?! I'm sure that was how...
In terms of the thought experiment, imagine that there are two clocks in each IRF: a light clock and an atomic clock. In each IRF, they keep time with each other, otherwise something is badly wrong. In other words, given that every other type of clock stays synchronised with the light clock in your IRF, then every type of clock must run slow in a frame where they are moving.

The reason you use a light clock to deduce this is that the invariance of the speed of light is a postulate of SR. Starting with this postulate, you show that light clocks show time dilation and the above argument shows that all other clocks must do too.

The next step, of course, is to infer that as clocks measure time it is time itself that is dilated.

#### Ibix

Edit, my tex tags seem to not work?! I'm sure that was how...
You missed a close brace. Adding that:
$\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}$
It would look better with dollars instead of hashes:$$\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$

#### stevendaryl

Staff Emeritus
Ok so I've got a question after walking through the time dilation derivation that used 'light clocks' (think a beam of light bouncing back and forth between mirrors) to derive $\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}$. So my Q is could you derive the same equation if you had used atomic clocks instead? Don't actually do so here, just wanted to know if it would have lead to the same relation.
The principle of relativity says that the laws of physics have the same form in every inertial reference frame. If atomic clocks are not affected by time dilation, then that would imply an observable difference between frames: there would be a preferred frame in which the light clock and atomic clock keep in synch. That would violate relativity.

And in fact, tests of relativity all give the same results, regardless of the type of clock (within reason---obviously a pendulum clock or a sundial won't keep correct time if you put them on a plane).

As far as proving that the atomic clock experiences time dilation, the implications go the other way around: From experiment, we know that the relativity principle is correct, and that motivates physicists to derive relativistically invariant theories to describe particles. So yes, you can prove that our current theories predict atomic clocks experience time dilation, but that's because our current theories were specifically constructed with relativity in mind.

#### Mister T

Gold Member
OSo my Q is could you derive the same equation if you had used atomic clocks instead?
If you had an atomic clock you could set it next to the light clock and see that they are in sync. Then, just by passing by the apparatus, you cannot make them go out of sync. Thus, if the light clock is observed to run slow, so is the atomic clock.

The example involving the light clock is a lesson about the way time behaves, it's not just a lesson about how light clocks behave.

#### PeroK

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If you had an atomic clock you could set it next to the light clock and see that they are in sync. Then, just by passing by the apparatus, you cannot make them go out of sync. Thus, if the light clock is observed to run slow, so is the atomic clock.

The example involving the light clock is a lesson about the way time behaves, it's not just a lesson about how light clocks behave.
Which is more succinctly expressed than I managed in post #4!

#### pervect

Staff Emeritus
If you had an atomic clock you could set it next to the light clock and see that they are in sync. Then, just by passing by the apparatus, you cannot make them go out of sync. Thus, if the light clock is observed to run slow, so is the atomic clock.

The example involving the light clock is a lesson about the way time behaves, it's not just a lesson about how light clocks behave.
I would describe this as being a consequence of the principle of relativity, the one that says that all inertial frames are equivalent. If all inertial frames were not equivalent, one might have some special frame where light clocks agreed with atomic clocks, and other frames where they didn't necessarily agree. For instance, classical physics prior to Special relativity suggested that there was such a sepcial frame, usually called the "ether". It's mainly of historical interest now.

Assuming that the laws of physics are the same in all frames of reference leads one to believe that if a light clock agrees with an atomic clock in one frame of reference, it agrees with it in all frames of reference.

In the end, it's experiment that decides if all frames of reference, no matter their state of motion, have the same laws of physics. There's been a lot of such experiments of various degrees of subtlety done to see if we can detect any differences, and none of them have ever found a difference.

#### benorin

Homework Helper
Ok, so I get it now, to be honest I expected someone to point out that from experiments we know atomic clocks experience relativistic time dilation so of course we can derive it from principal. I'm really heartened by the outpouring of posts by all you physicists considering this was a noob question, thanks! So when I asked it seemed that the time dilation relation might have been dependant on that particular light clock model but then as I knew experiment agreed with the result something more than luck was involved. So no reference frame is preferred, that's an answer!

Does anyone happen to know which physical law would throw out the speed of light constant if we were to try to derive the atomic clock version of this? Not really remembering that part of physics (college was 10+ years ago).

Also, could you please recommend a good text on relativity? I think I could handle SR and have an eye on GR for the future when I finally conquer tensor calculus (even as a math major that course gives me dreadful fits of anxiety, I exaggerate but only to tell the truth more clearly).

#### meopemuk

If you had an atomic clock you could set it next to the light clock and see that they are in sync. Then, just by passing by the apparatus, you cannot make them go out of sync. Thus, if the light clock is observed to run slow, so is the atomic clock.
I would like to disagree with your logic.

Suppose that we have two inertial observers Alice and Bob, which are moving with respect to each other. Suppose that both of them have their own pair of synchronized clocks: a light clock and an atomic clock. According to the principle of relativity, Alice and Bob are equivalent. This means that Alice's pair of clocks is always in sync from her point of view, and Bob's pair of clocks is always in sync from his point of view. However, the principle of relativity does not tell us how Alice sees Bob's clocks and how Bob sees Alice's clocks.

In order to answer this question, we can use the postulate about the invariance of the speed of light. This postulate can be applied to the light clocks only. So, we can say for sure that Bob's light clock slows down from the point of view of Alice and that Alice's light clock slows down from the point of view of Bob. However, this postulate says nothing about atomic clocks, because they are much more complicated than a single light pulse bouncing between two mirrors.

So, it might happen that Alice will see Bob's pair of clocks as out of sync. And Bob may notice the same behavior of Alice's clocks. This (seemingly paradoxical) situation does not contradict Einstein's postulates. If you insist that clocks of different design are always in sync for all observers, you'll have to introduce this statement as a separate (third) postulate of special relativity: "boost transformations are kinematical, i.e., universal and independent on interactions acting in the observed system". Only then you can make the next step and claim that the clock design doesn't matter, and that the time itself is slowing down in the moving frame.

Eugene.

#### Ibix

Does anyone happen to know which physical law would throw out the speed of light constant if we were to try to derive the atomic clock version of this?
As someone noted above, you'd need to use quantum field theories, which are built on top of relativity. So you might regard it as cheating. That said, it's Maxwell's equations (our first relativistic theory) that caused Einstein to start thinking about the constancy of the speed of light in the first place, so maybe it isn't cheating.

Interesting fact - you do not need to assume anything about the speed of light to derive relativity. You can show that there are only two ways you can respect the principle of relativity - Newtonian physics or Einsteinian physics. Then experiment tells you we aren't in a Newtonian universe and we haven't caught any violation of the principle of relativity yet, so...

Books - I recommend Taylor and Wheeler Spacetime Physics. If you want something free you might look up former PF Mentor Ben Crowell's book on www.lightandmatter.com/books. I haven't read his SR one but his GR one is good.

#### Ibix

If you insist that clocks of different design are always in sync for all observers, you'll have to introduce this statement as a separate (third) postulate of special relativity: "boost transformations are kinematical, i.e., universal and independent on interactions acting in the observed system". Only then you can make the next step and claim that the clock design doesn't matter, and that the time itself is slowing down in the moving frame.
Not really. You just require consistency (i.e. that there's an underlying reality that is being described differently by different observers, which is pretty much a requirement for science to work at all). See my post #3 for a trivial example of inconsistency if different clocks behave differently under boost.

#### meopemuk

Not really. You just require consistency (i.e. that there's an underlying reality that is being described differently by different observers, which is pretty much a requirement for science to work at all). See my post #3 for a trivial example of inconsistency if different clocks behave differently under boost.
I don't see any inconsistency in your example. No laws of physics are violated if different inertial observers (Alice and Bob) disagree on whether the clock is working or not. Different observers see world differently. For some reason we like to think that this difference can be only of kinematical (universal) nature. For example, we are pretty sure that under a space translation of the observer all atoms in the universe shift exactly by the same distance, independent on how these atoms interact with other atoms etc. Likewise, we are convinced that under rotation all atoms in the universe rotate through the same angle.

How certain are we that the same rule holds for boosts, i.e., that under a boost all atoms and bodies behave exactly the same: they get the same additional speed, they experience a universal contraction, and the rate of all processes slow down by exactly the same factor? What if actual boost transformations are more complicated than these universal kinematical effects?

Actually, there is an example of an inertial transformation, which is neither kinematical nor universal. This is time translation. I hope, we agree that two reference frames separated by a time interval are equivalent. (Many people even like to combine all 10 types of inertial transformations - 3 space translations, 3 rotations, 3 boosts and 1 time translation - into one Poincare group.) However, observations of the same physical system (e.g., a clock) from time-shifted frames may be drastically different. For example, the clock was working yesterday, but it is not working today, because the battery is dead.

So, returning to boosts, the question is: are boosts like space translations and rotations (simple, universal, kinematical) or they are like time translations (complex, system-dependent, interaction-dependent)? Special relativity insists that boosts are kinematical. But my point is that this is not required by any consistency arguments. This is just an additional tacit postulate of the theory.

Eugene.

#### PeroK

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So, it might happen that Alice will see Bob's pair of clocks as out of sync. And Bob may notice the same behavior of Alice's clocks. This (seemingly paradoxical) situation does not contradict Einstein's postulates. If you insist that clocks of different design are always in sync for all observers, you'll have to introduce this statement as a separate (third) postulate of special relativity: "boost transformations are kinematical, i.e., universal and independent on interactions acting in the observed system". Only then you can make the next step and claim that the clock design doesn't matter, and that the time itself is slowing down in the moving frame.

Eugene.
Technically you do, of course, have to add a little more to @Mister T 's analysis to make it watertight, but you do not need a new postulate. For example, if you have the two clocks stacked perpendicular to the direction of relative motion. The two pairs of clocks can exchange information over an arbitrarily short distance and hence compare times.

If Alice measures Bob's atomic clock to be ahead, then Bob must measure Alice's atomic clock to be ahead and that would be a contradiction, as they all have the same four numbers.

In fact, a simpler case would be for Alice simply to make two permanent time stamps as she passes Bob. If these numbers are the same for Alice, they must be the same for Bob. That would constitute a valid measurement of Alice's clocks in Bob's frame.

#### stevendaryl

Staff Emeritus
Suppose that we have two inertial observers Alice and Bob, which are moving with respect to each other. Suppose that both of them have their own pair of synchronized clocks: a light clock and an atomic clock. According to the principle of relativity, Alice and Bob are equivalent. This means that Alice's pair of clocks is always in sync from her point of view, and Bob's pair of clocks is always in sync from his point of view. However, the principle of relativity does not tell us how Alice sees Bob's clocks and how Bob sees Alice's clocks.
There is a consistency requirement, which is that events taking place at the same time and same place must happen at the same time and place for all observers. So if Alice has two clocks, let $e_1$ be the event where her first clock shows time $1:00$, and let $e_2$ be the event where her second clock shows time $1:00$. If those events are simultaneous and (approximately) at the same location for Alice, then those events must be simultaneous and approximately at the same location for Bob.

#### A.T.

I don't see any inconsistency in your example.
Then attach the two different clocks to a bomb, that goes off when they go out of synch.

Different observers see world differently.
But it's the same world, where the bomb either goes off or doesn't. Since all frames must agree on whether the bomb goes off, they all must agree on whether the clocks stay in synch.

#### PeroK

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Then attach the two different clocks to a bomb, that goes off when they go out of synch.

But it's the same world, where the bomb either goes off or doesn't. Since all frames must agree on whether the bomb goes off, they all must agree on whether the clocks stay in synch.
That doesn't quite work, though, if the clocks are separated in the direction of motion.

#### A.T.

That doesn't quite work, though, if the clocks are separated in the direction of motion.
The clocks are locally attached to the bomb. And "clocks are in synch" practically means that their signals arrive in sync at the fuse.

#### PeroK

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The clocks are locally attached to the bomb. And "clocks are in synch" practically means that their signals arrive in sync at the fuse.
But, if the clocks are attached to the bomb and separated in the direct of motion, then synchronisation of the clocks and their signals reaching the bomb at the same time are incompatible. In a frame in which this apparatus is moving (in the direction of the separation of the clocks), the signals will only reach the bomb at the same time if the clocks are out of synch in that frame.

You could appeal to your being able to put the clocks arbitrarily close together, but I suggest it's better to separate the clocks perpendicular to the direction of motion and additionally argue that if they are in sync in their rest frame, they are also in sync in the frame where they are moving, as in post #15.

PS In general, I think it's quite an important point in the development of SR to show that there can be no length contraction or "simultaneity" issues in a direction perpendicular to the motion. This clears the way for the arguments we all want to make in this thread.

#### A.T.

I suggest it's better to separate the clocks perpendicular to the direction of motion.
I would rather change the trigger from "not in synch" to "have different tick rates". The point here is time dilation applying equally to all types of clocks.

#### PeroK

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I would rather change the trigger from "not in synch" to "have different tick rates". The point here is time dilation applying equally to all types of clocks.
Yes, and in fact that leads to a simple argument that time dilation cannot be dependent on position. @meopemuk introduced a red herring with the issue of syncronisation. We don't care whether the clocks are in sync, only that they tick at the same rate.

#### Ibix

I don't see any inconsistency in your example. No laws of physics are violated if different inertial observers (Alice and Bob) disagree on whether the clock is working or not.
As A.T. points out, it's trivial to extend the example. Put a light sensor on the moving mask that will trigger a bomb if illuminated. If the two clocks move out of synch (where synchronised is taken to be the motion of the mask compared to that of the light pulse) then there's an explosion.

That leaves you with three options. One - the bomb does not go off. If anyone observes the clocks ticking out of synch then they have no explanation for why it did not trigger. Two - the bomb does go off. If anyone observes the clocks ticking in synch then they have no explanation for why it triggered. Three - the bomb both exploded and didn't, and I am free to choose which one (and switch between results) by the purely mental exercise of selecting a reference frame. All of these require some violation of physical law - primarily the principle of relativity.

As @PeroK points out, it is possible for a frame change to produce a constant offset between clocks as long as they are spatially separated. I specifically constructed my example to avoid this issue by making the comparator mechanism inside one of the clocks - that kind of synchronisation issue must fall out in whatever mechanism is used to drive the shutter.

#### Ibix

For example, the clock was working yesterday, but it is not working today, because the battery is dead.
This raises no consistency issues of the kind raised by clocks ticking both at the same rate and different rates does. Sure you can change "now" from being t=0 to t=1 to t=-1, but if you work out any direct observable consequences to that switch you will find there are none. There is no way to set up an experiment such that a person can say "a bomb has gone off, now it hasn't, now it has again". If clocks tick at different rates then there can be direct observable consequences that vary when you switch frames - and you allow this kind of exploded-not-exploded-exploded-again behaviour.

#### meopemuk

There is a consistency requirement, which is that events taking place at the same time and same place must happen at the same time and place for all observers.
This is your version of the "third postulate" of special relativity I was talking about. I agree that this postulate implies the kinematical character of boosts and ultimately the 4D geometry of space-time. However, my point is that one can imagine a world where this "consistency requirement" does not hold, but all other laws of physics are still valid.

Eugene.

"Time Dilation Derivation"

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