Is Proper Time Contraction Valid for Accelerated Watches?

[bcrowell]

Then I didn't understand. You're suggesting that a clock's state is somehow unmeasurably/unpredictably dependent on its acceleration history?
I'm suggesting that it's in a well defined state depending on momentary proper acceleration.

The case I thought we were referring to was the one in which a clock's rate depends in a measurable, predictable way on its acceleration history. But because it depends on the *history*, not just the boundary conditions on a spacelike surface, it means that we can't predict the future evolution of a system based on knowledge of the boundary conditions. That is, the standard way of stating initial conditions in a classical field theory would be to give the coordinates and their first derivatives on some timelike surface. In the case where the clock's behavior depends on its history, you would need more information that in the standard statement of the initial value problem. Given a system in a known state, without knowledge of its history, you would be unable to predict its future evolution.

I see that in such a theory there's some argument about how time should be defined, and that it's not a simple geometric theory (ii). But why should (i) or (iii) follow?

That's based on the argument I gave in #20.

 

[Al68]

I was talking about idealized clocks, too. Suppose you had muon decay dependent on acceleration. That's a very idealized clock. Or suppose you had atomic clocks converge to a rate that differs from the expected one after all plausible corrections are done. Suppose these two (muon and atomic) were equal, but different from the light clock. How would you justify taking the light clock as master?

Because it's more convenient.

If we were to instead use as a standard convention (take as master) a clock with a rate that depended on its acceleration, such a convention would be very cumbersome indeed.

It wouldn't violate any laws of physics to take such a clock "as master". Just a big pain in the rear.

The clock postulate is a convention, not a law of physics, but a very useful convention.

 

[facenian]

It follows logically from the clock postulate,

Yes, is yust that I did not know about the clock postulate because the books a I have been reading don't explain anything about it and simply derive the formula as if it followed logically from first potulates.
So this forum is usefull to show some ligtht thank to you guys.

 

[atyy]

It follows logically from the clock postulate, since the formula is only valid for clocks meeting the clock postulate, and the clock postulate states that a clock's time will match that formula regardless of acceleration.

Obviously, clocks could be (and are) constructed that don't meet the clock postulate, and the above formula isn't valid for such clocks.

Edit: It would seem a simple matter to purposely construct a clock that would meet the prediction of any arbitrary equation, such as dt'=(1-(v/c)^2)^{1/2}dt*F, where F is a function of acceleration. An hourglass would seem to be a good example, as well as an electronic clock which used an accelerometer as input.

I share the sentiment that the clock postulate is just a definition or a convention. I was trying to come up with clocks that depend on acceleration, but stuff like the hourglass didn't make it, since I wanted something like dt=(1-(v/c)^2)^{1/2}dt*(1+F), ie. it should behave like a normal time dilating clock when moving inertially in flat spacetime. The hourglass needs gravity, and so it wouldn't run when moving inertially in flat spacetime. I guess the clock with an accelerometer attached would work, but it seems a bit klugey.

 

[Ich]

The case I thought we were referring to was the one in which a clock's rate depends in a measurable, predictable way on its acceleration history.

No. That was JesseM.

 

[facenian]

I share the sentiment that the clock postulate is just a definition or a convention.

I'm not sure if this is correct. To my understanding the clock postulate states that if you build a clock that is ticking independantly of gravitation or aceleration then it will show a proper time contraction according to the "proposed formula" and this assertion can be tested that's why think is more than a simple convention.

 

[Fredrik]

The only way I can make sense of the claim that it's only a definition/convention is that we could define the word "clock" in a different way than we do now. A "clock" would then measure some other quantity instead of the proper time of the curve in spacetime that represents its motion, and the axioms of SR and GR would have to be updated to reflect that.

 

[Dale]

IMO (and this is nothing more than an opinion) any scientific theory consists of three things:
1. A mathematical framework
2. A set of experimental data
3. A way to interpret the data in terms of the framework

The clock postulate seems to belong in 3., a statement linking the measurements of devices called "ideal clocks" with the theoretical construct called "proper time".

 

[atyy]

I'm not sure if this is correct. To my understanding the clock postulate states that if you build a clock that is ticking independantly of gravitation or aceleration then it will show a proper time contraction according to the "proposed formula" and this assertion can be tested that's why think is more than a simple convention.

I would guess so. In my understanding, the clock postulate first defines an ideal clock as one that reads the integrated proper time, and secondly asserts the existence of such a device. The definition part is uncontroversially an "additional thing". The existence part is not an "additional thing", since one can use the absoluteness of acceleration etc in special relativity to assert that those can always be corrected for, ie. that the proper time along a timelike worldline can always be determined. The existence part can be tested, in the sense that special relativity (Poincare invariance of the dynamical laws) can be tested.

 

[Al68]

I'm not sure if this is correct. To my understanding the clock postulate states that if you build a clock that is ticking independantly of gravitation or aceleration then it will show a proper time contraction according to the "proposed formula" and this assertion can be tested that's why think is more than a simple convention.

No, that's not exactly what the clock postulate says. It says that an ideal clock by definition is unaffected by acceleration, ie it will show the proper time predicted by SR regardless of physical conditions (acceleration), not that an ideal clock must exist. Such a test would only test a particular clock to see if that clock meets the clock postulate's standard within a specified margin of error. The standard itself exists whether any existing physical clock meets it or not.

The clock postulate itself, that an "ideal" clock is unaffected by acceleration, is valid even if no such ideal clock exists.

The only test result that would affect the clock postulate would be one that showed that its underlying assumption is incorrect, ie that the SR equations for proper time are invalid. But experimental evidence so far has only confirmed that assumption.

 

[facenian]

, ie it will show the proper time predicted by SR regardless of physical conditions (acceleration), not that an ideal clock must exist.

The problem with what you are saying here is that SR has nothing to say about accelarated clocks,ie proper time of an arbitrary motion, and in this sense I agree with a previous reference given by Atyy in post #6

 

[atyy]

The problem with what you are saying here is that SR has nothing to say about accelarated clocks,ie proper time of an arbitrary motion, and in this sense I agree with a previous reference given by Atyy in post #6

I don't agree with the link I gave in #6 where it says "It also won't do to simply define a clock to be a device whose timing is unaffected by its acceleration, because it's not clear what such a device has got to do with the real world: that is, how well it approximates the thing we wear on our wrist." It's easy to know that the watch on your wrist will be broken by very high accelerations, after which it will fail to read the proper time. Thus if the clock postulate is about the watch on your wrist, then it is false.

Hmmm, on reading the link I gave in #6, I see Don Koks's view is really quite different. He says "But still it can be shown to be a reasonable thing to assume, because it leads to something that has been tested in other ways, as we'll see in the next section." ie, Don Koks doesn't assume Minkowski spacetime to be part of SR minus the clock postulate. He says that SR without Minkowski spacetime plus the clock postulate gives Minkowski spacetime (the spacetime interval)! Wow! Interesting. Would love to hear comments from others on whether this alternate way of axiomatizing SR works!

 

[Al68]

, ie it will show the proper time predicted by SR regardless of physical conditions (acceleration), not that an ideal clock must exist.

The problem with what you are saying here is that SR has nothing to say about accelarated clocks...

Of course it does. Even Einstein's original 1905 paper uses an accelerated clock as an example. SR says the proper elapsed time of an accelerated (or unaccelerated) clock will be dt'=(1-(v/c)^2)^{1/2}dt, where t is the elapsed time in an inertial reference frame and t' is the proper elapsed time on the clock's worldline.

The clock postulate simply assumes SR to be correct and says that an ideal clock will read t' regardless of physical conditions (acceleration).

 

[facenian]

Of course it does. Even Einstein's original 1905 paper uses an accelerated clock as an example.

Well, it seems that not everybody agrees with this. It seems that SR is based on two postulates 1) Physical equivalence of inertial frames 2) Non existence of instantaneous interactions. And from this two Lorentz transformation can be derived and apply only to inertial frames from which we properly calculate proper time contraction of an inertial watch
The whole issue is : Does an accelated observer experience the same ageing process as does an inertial observer intantaneously at rest with him? I think physics know from experinece that the answer is yes but our question is : Do this follow logically from the above two principles or has to be considered as an independent assumption validated by experience?

 

[Dale]

Well, it seems that not everybody agrees with this.

That is true, not everybody is correct like Al68. SR does fine with accelerating observers as described by inertial frames.

 

[atyy]

The whole issue is : Does an accelated observer experience the same ageing process as does an inertial observer intantaneously at rest with him?

Well, probably we disagree because we are talking about different things. For any particular accelerated "pointlike thing" (eg. muon decay), the postulate that ageing follows proper time does require an additional dynamical postulate, which would be the laws of muon decay.

If muon decay did not follow this dynamical law, then it would not be an ideal clock by definition.

But for us to determine that it did not follow this dynamical law, we must be able to determine proper time along its worldline, which assumes that ideal clocks exist, which is the statement that the proper time can be determined, which pertains to the verifiability of special relativity (existence of Minkowski spacetime).

 

[yuiop]

I share the sentiment that the clock postulate is just a definition or a convention. I was trying to come up with clocks that depend on acceleration, but stuff like the hourglass didn't make it, since I wanted something like dt=(1-(v/c)^2)^{1/2}dt*(1+F), ie. it should behave like a normal time dilating clock when moving inertially in flat spacetime. The hourglass needs gravity, and so it wouldn't run when moving inertially in flat spacetime. I guess the clock with an accelerometer attached would work, but it seems a bit klugey.

The hour glass is probably not a bad example. I imagine that an hour glass that is calibrated to run at the same rate as an atomic clock on the surface of the Earth, would run noticeably slower than the reference atomic clock when they are both side by side on the surface of the moon for example. However the existence of a clock that does not match the requirements of an ideal clock in the clock postulate, does not deny the the possibility of such an ideal clock aproximated by the atomic clock in this example.

 

[atyy]

The hour glass is probably not a bad example. I imagine that an hour glass that is calibrated to run at the same rate as an atomic clock on the surface of the Earth, would run noticeably slower than the reference atomic clock when they are both side by side on the surface of the moon for example.

I was trying to keep to flat spacetime, to stay within special relativity.

However the existence of a clock that does not match the requirements of an ideal clock in the clock postulate, does not deny the the possibility of such an ideal clock aproximated by the atomic clock in this example.

Yes. I was trying to find an "elegant" non-ideal clock to highlight that the ideal clock is a definition.