Is Proper Time Contraction Valid for Accelerated Watches?

[facenian]

Le K_0 and K_1 be two inertial frames moving relative to each other with velocity v,
from Lorentz transfromation we have that a watch(t') at the origin of K_1 is slow(proper time contraction) acording to dt'=dt/gamma=(1-(v/c)^2)^{1/2}dt where v is the velocity of the watch.
My question relates to fact that when the watch is moving with an arbitrary time dependent velocity then physicists still use the former formula arguing that since the watch is at each instant t at rest relative to an inertial frame moving with the same instant velocity of K_1 then the accelerated whatch and the one in the inertial frame at rest with respect to K_1 show the same dt.
It seems to me that this reasonig does not follow logically from any postulate and must be stated as a new one.

 

[JesseM]

I haven't thought about it too carefully, but it does seem logically possible that you could have a universe with Lorentz-symmetric fundamental laws where a clock's rate of ticking was influenced by its past history of accelerations, so that even if two clocks are currently at rest relative to each other they might tick at different rates if one had been traveling at a different velocity in the past while the other hadn't been. It would probably be difficult if not impossible to construct a theory like this that would match observations up until 1905, but if it's logically possible at all, then you'd be correct that this needs to be added as a new postulate if we want to calculate proper time for a non-inertial clock.

 

[Jerbearrrrrr]

Seems to be a calculus question.

We approximate the worldline as a series of tangents, then consider physical values (cf 'length of curve') in the appropriate limits.

 

[JesseM]

Seems to be a calculus question.

We approximate the worldline as a series of tangents, then consider physical values (cf 'length of curve') in the appropriate limits.

Yes, but that only works if you assume that a clock whose worldline is a series of constant-velocity segments joined by accelerations will, during each constant-velocity segment, tick at the same rate as a purely inertial clock next to it that has been moving at constant velocity for all eternity. This is a pretty natural assumption I think, but it seems logically possible that even if two clocks are current at rest relative to each other and currently moving inertially, they might still tick at different rates if one accelerated in the past and the other didn't.

 

[Jerbearrrrrr]

Oh right. As in, in the worldline that is composed of a sequence of straight lines, we don't know if instantaneously "turning" to go along the next segment affects the clock permanently.

The whole clock thing is just a way of explaining what's meant by a frame though.

But surely - the definition of proper time is (insert integral here), and that is the end of that? (whether or not it has physical significance can be tested with clocks)

[edit]
Oh my bad. The OP was asking whether we could derive the physical significance of this integral, I think.

 

[atyy]

Yes. http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html

 

[bcrowell]

There is a good discussion of this at p. 9 of Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf . Dieks' point of view, which I find convincing, is that it doesn't need to be a separate postulate. This is the opposite of the opinion expressed by Baez in the link at #6.

 

[JesseM]

There is a good discussion of this at p. 9 of Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf . Dieks' point of view, which I find convincing, is that it doesn't need to be a separate postulate.

Interesting, I hadn't considered the idea of using a light clock specifically. That might be enough to derive it from the basic postulates without adding a new one.

 

[atyy]

There is a good discussion of this at p. 9 of Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf . Dieks' point of view, which I find convincing, is that it doesn't need to be a separate postulate. This is the opposite of the opinion expressed by Baez in the link at #6.

I'm not sure it's opposite - maybe it's complementary? The clock postulate defines an ideal accelerated clock. Dieks's considerations show that given special relativity and the definition of an accelerated ideal clock, that the accelerated ideal clock exists.

Maybe it'd be more accurate to say the "clock definition" rather than the "clock postulate"?

 

[bcrowell]

I'm not sure it's opposite - maybe it's complementary? The clock postulate defines an ideal accelerated clock. Dieks's considerations show that given special relativity and the definition of an accelerated ideal clock, that the accelerated ideal clock exists.

Maybe it'd be more accurate to say the "clock definition" rather than the "clock postulate"?

If experts disagree on whether it should be a postulate, my feeling is that that's just an example of the fact that physical theories aren't formal mathematical systems in the way that Euclidean geometry or the real number system are. Another good example is that experts disagree about whether Newton's first law is logically independent of Newton's second law.

 

[atyy]

If experts disagree on whether it should be a postulate, my feeling is that that's just an example of the fact that physical theories aren't formal mathematical systems in the way that Euclidean geometry or the real number system are. Another good example is that experts disagree about whether Newton's first law is logically independent of Newton's second law.

Now that I read Don Koks's http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html more carefully, I see he explicitly says it is not a definition. I believe he is wrong. Rindler http://books.google.com/books?id=MuuaG5HXOGEC&dq=rindler+relativity&source=gbs_navlinks_s p 65 states the accelerated ideal clock as a definition and deduces its existence from the absoluteness of acceleration, in line with Dieks.

 

[Ich]

That might be enough to derive it from the basic postulates without adding a new one.

Interesting point.
But IMO there is a hidden assumption, that could count as a postulate in itself. It is conceivable that there exist whole classes of clocks with different - but independent of the construction details - bevaviour under acceleration. It is not evident why one should then gauge such clocks against light clocks. The light clock might be the exception, not the norm.

 

[bcrowell]

Interesting point.
But IMO there is a hidden assumption, that could count as a postulate in itself. It is conceivable that there exist whole classes of clocks with different - but independent of the construction details - bevaviour under acceleration. It is not evident why one should then gauge such clocks against light clocks. The light clock might be the exception, not the norm.

I think the real issue you're getting into is whether relativity is to be interpreted in terms of distortion of time itself (Einstein), or in terms of dynamical effects on clocks, such as aether drag (Lorentz). If time dilation was clock-dependent, then you could have a pair of clocks A and B such that in frame K, A and B run at the same rate, while in frame K' they run at different rates. A pair of clocks like this would be a very handy navigational device. You could use it to determine your state of absolute motion. But every axiomatization of SR that I'm aware of includes a postulate that says that all inertial frames are equivalent.

 

[Ich]

If time dilation was clock-dependent, then you could have a pair of clocks A and B such that in frame K, A and B run at the same rate, while in frame K' they run at different rates.

I don't think so, if time dilation were a function of proper acceleration. You'd still have complete Poincaré symmetry.

 

[facenian]

even if two clocks are current at rest relative to each other and currently moving inertially, they might still tick at different rates if one accelerated in the past and the other didn't.

three points about your above statement:
1)two clocks are current at rest relative to each other -> yes
2)currently moving inertially-> NO, one of them is not moving inertially on the contrary
it is accelerated and that's the whole issue of my question
3)they might still tick at different rates if one accelerated in the past and the other didn't-> I don't think that the history is impotant here because we are concerned with instantaneous non inertial clock rate tick and I think it would be too far fechted to assumed it is dependent on it's past history

 

[bcrowell]

I don't think so, if time dilation were a function of proper acceleration. You'd still have complete Poincaré symmetry.

I see. I think you're imagining what would happen if all clocks the same v-dependence in their rates, but that only light clocks have an a-dependence that vanishes to terms of a certain order in the size of the clock. But let's think for a minute about what the clock postulate/clock theorem/clock statement really says. It doesn't say that every real clock's rate depends only on v and not a, because that's false. It's false for a pendulum clock, and it's false for a light clock. What it really says is that for an idealized clock, the a-dependence vanishes. One way of getting an idealized clock is to take a limit, e.g., make your light clock smaller and smaller, as described by Dieks. Another way is to take a real clock and apply certain corrections to it. Because the acceleration (unlike the velocity) is detectable to instruments moving along with the clock, we can always calibrate it away. Dieks has a good discussion of this on pp. 9-10.

 

[facenian]

[edit]
Oh my bad. The OP was asking whether we could derive the physical significance of this integral, I think.

My question is because the physical significance of that integral is the time measured by a non inertial frame and I think that if this is so it can not be simply derived from the ussual principles of SR however it can be tested and that's why I think is not a definition.

 

[atyy]

I see. I think you're imagining what would happen if all clocks the same v-dependence in their rates, but that only light clocks have an a-dependence that vanishes to terms of a certain order in the size of the clock. But let's think for a minute about what the clock postulate/clock theorem/clock statement really says. It doesn't say that every real clock's rate depends only on v and not a, because that's false. It's false for a pendulum clock, and it's false for a light clock. What it really says is that for an idealized clock, the a-dependence vanishes. One way of getting an idealized clock is to take a limit, e.g., make your light clock smaller and smaller, as described by Dieks. Another way is to take a real clock and apply certain corrections to it. Because the acceleration (unlike the velocity) is detectable to instruments moving along with the clock, we can always calibrate it away. Dieks has a good discussion of this on pp. 9-10.

But why would we calibrate it away? Is the proper time the unique geometric invariant that can be integrated on a worldline?

 

[matheinste]

Because the acceleration (unlike the velocity) is detectable to instruments moving along with the clock, we can always calibrate it away. Dieks has a good discussion of this on pp. 9-10.

I was unable to access the Dieks link where this query may be addressed. When in a previous thread I quoted Rindler saying the same thing an objection was, not unreasonably, made to the effect that if the clock had extension in space then the actual acceleration type or method would complicate or render the calibration impossible because different locations within the clock could have different accelerations.

Matheinste.

 

[bcrowell]

I haven't thought about it too carefully, but it does seem logically possible that you could have a universe with Lorentz-symmetric fundamental laws where a clock's rate of ticking was influenced by its past history of accelerations, so that even if two clocks are currently at rest relative to each other they might tick at different rates if one had been traveling at a different velocity in the past while the other hadn't been.

I'm assuming you don't mean that the clock's current rate depends on its past absolute velocities, but only on its past accelerations. Even then, I think there are two cases:
(1) The clock's history-dependence shows up in a way that can be determined by physical inspection of the clock. For instance, the parts of a mechanical clock might get stretched by accelerations. You can measure the parts with calipers, and say that the clock must be running 0.01% fast because of these distortions.
(2) The history-dependence can't be predicted by inspection of the clock.

I think case #1 is uninteresting for the reasons described in my #16, while case #2 creates problems because the theory would lack predictive value -- initial-value problems posed in this theory would not have unique solutions.

You would also have problems with the equivalence principle and with the geometrical interpretations of SR and GR. For instance, suppose a planet is orbiting a star. The period of its orbit would depend on the history of the solar system's accelerations as it orbited around the galaxy. If the effect is sensitive to the history of gravitational accelerations, then it violates the equivalence principle, because according to an observer in a frame that's always been co-moving with the solar system, the solar system never underwent any accelerations. Even if the effect is only sensitive to nongravitational accelerations, it still causes similar problems. The planet doesn't move along some geodesic that is determinable from knowledge of the metric in its own patch of spacetime; it moves along a trajectory that is different from that of some other object that has had a different history. This would cause a non-null result in Eotvos experiments, interpreted as a violation of the equivalence principle. E.g., a pendulum clock with a bob made from the planet's own materials would run at a different rate than one with a bob made from a meteor.

It would probably be difficult if not impossible to construct a theory like this that would match observations up until 1905, but if it's logically possible at all, then you'd be correct that this needs to be added as a new postulate if we want to calculate proper time for a non-inertial clock.

An actual theory that had along these lines was the Weyl gauge theory, which was a classical unified theory meant to unify gravity and electromagnetism. Some sources of information on this kind of thing:
- http://en.wikipedia.org/wiki/Classical_unified_field_theories
- Hubert F. M. Goenner, "On the History of Unified Field Theories," http://www.livingreviews.org/lrr-2004-2
- Eddington, "Space, time and gravitation: an outline of the general relativity theory," http://books.google.com/books?id=uU1WAAAAMAAJ&pg=PA167
The Eddington book gives a pretty complete and readable layman's description of the theory. In this theory atomic emission frequencies depend on the history of the atom (the electromagnetic fields that it has moved through in the past). The problem with lack of predictive power hadn't occurred to me before, but as far as I can tell it would be a serious problem with the theory.

 

[facenian]

Yes. http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html

Thank you Guys, I think atyy's reference answers my question completely although there seems to be some controversy

 

[atyy]

Thank you Guys, I think atyy's reference answers my question completely although there seems to be some controversy

And it turns out I don't agree completely with the reference either. I prefer Rindler's discussion. But essentially, while we're counting references here, Dieks is outnumbered. Both Koks and Rindler agree that something new is required. However, it is possible to count Rindler as sideing with Dieks. Anyway, this is not literary interpretation , it's democracy - I cast my vote with Rindler.

 

[bcrowell]

I was unable to access the Dieks link where this query may be addressed.

Strange. It works for me right now. Maybe their server was just momentarily busy.

When in a previous thread I quoted Rindler saying the same thing an objection was, not unreasonably, made to the effect that if the clock had extension in space then the actual acceleration type or method would complicate or render the calibration impossible because different locations within the clock could have different accelerations.

Dieks defines an idealized clock in terms of the limiting behavior of a light clock that is made small. Taking this limit is necessary precisely because of the ambiguity you're referring to. The light clock isn't immune to acceleration effects. Its just that those acceleration effects are calculable, and vanish at a certain known rate as the clock is made smaller.

One way of getting an idealized clock is to take a limit, e.g., make your light clock smaller and smaller, as described by Dieks. Another way is to take a real clock and apply certain corrections to it. Because the acceleration (unlike the velocity) is detectable to instruments moving along with the clock, we can always calibrate it away. Dieks has a good discussion of this on pp. 9-10.

But why would we calibrate it away?

We calibrate it away for the same reason that we would calibrate away a clock's dependence on temperature or air pressure.

Is the proper time the unique geometric invariant that can be integrated on a worldline?

Hmm...I could be wrong, but I'll take a stab at this. In SR, it's pretty clear that the spacetime interval between two events is a unique geometric invariant. If you want to generalize to GR, where particles might accelerate due to gravity, then you carry out the generalization by applying the equivalence principle. One way of stating the equivalence principle is that GR is locally equivalent to SR. So it seems to me that the proper time does have a unique significance as a geometric invariant in both SR and GR. The local nature of the equivalence principle shows up in Dieks' treatment because the light clock gets closer and closer to measuring proper time as you make the clock smaller and smaller.

 

[Fredrik]

In both SR and GR, proper time is a coordinate-independent property of a curve in spacetime. The statement "A clock measures the proper time of the curve that represents its motion" is an axiom of both theories. To get what Ich is talking about, i.e. Poincaré invariance along with clocks that are affected by acceleration, we need Minkowski spacetime (i.e. the same mathematical structure that we use in SR) and a different axiom about what clocks measure. So it would be a different theory.

Would that theory be consistent with Einstein's postulates? I haven't thought about it, and I don't really care. People care far too much about those "postulates" anyway.

 

[Dale]

People care far too much about those "postulates" anyway.

I agree. With any theory it is always possible to choose more than one set of statements and call them "axioms" and then derive the rest. The important part is whether or not the theory as a whole agrees with experiment. If you find a conflict between experiment and the theory then regardless of how you have defined your axioms at least one is false.

 

[Ich]

But let's think for a minute about what the clock postulate/clock theorem/clock statement really says. It doesn't say that every real clock's rate depends only on v and not a, because that's false. It's false for a pendulum clock, and it's false for a light clock. What it really says is that for an idealized clock, the a-dependence vanishes.

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?

 

[bcrowell]

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?

This is case (2) in my #20. If experiments showed that this happened, there would be no way to decide which clock to use. However, we'd also have to give up (i) the equivalence principle, (ii) the geometrical interpretations of SR and GR, and (iii) the ability to make predictions in initial-value problems. If you like, you can have your choice of any of four postulates to add to SR that would rule out this case: i, ii, iii, or the "clock postulate." To my taste, the "clock postulate" would be a poor choice, compared to, say, a postulate asserting iii. Nobody has any interest in creating a classical field theory that violates iii. Also, if we had a theory that obeyed the clock postulate but disobeyed iii, it would clearly be a non-viable theory, so the clock postulate is too weak. To my taste, it is also not worth trying to formalize these extra axioms, because relativity is not a formal mathematical system, so it's futile to try to come up with a logically complete set of axioms. It's also possible that GR disobeys iii (Earman's "green slime and lost socks" coming out of singularities), in which case GR arguably *isn't* a viable classical field theory.

 

[Ich]

This is case (2) in my #20.

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. 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?

 

[Fredrik]

I agree. With any theory it is always possible to choose more than one set of statements and call them "axioms" and then derive the rest. The important part is whether or not the theory as a whole agrees with experiment. If you find a conflict between experiment and the theory then regardless of how you have defined your axioms at least one is false.

That's actually not what I meant. I just don't think we should be referring to the postulates all the time when we can refer to the actual theory that the postulates were used to find a century ago.

 

[Al68]

Le K_0 and K_1 be two inertial frames moving relative to each other with velocity v,
from Lorentz transfromation we have that a watch(t') at the origin of K_1 is slow(proper time contraction) acording to dt'=dt/gamma=(1-(v/c)^2)^{1/2}dt where v is the velocity of the watch.
My question relates to fact that when the watch is moving with an arbitrary time dependent velocity then physicists still use the former formula arguing that since the watch is at each instant t at rest relative to an inertial frame moving with the same instant velocity of K_1 then the accelerated whatch and the one in the inertial frame at rest with respect to K_1 show the same dt.
It seems to me that this reasonig does not follow logically from any postulate and must be stated as a new one.

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.