SR Time Dilation in Rigid Structure Clocks

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Special relativity says that all clocks will show same time dilation, irrespective of clock mechanism. But Time period of a clock is a formula that must continue to hold even if time dilates. Let us look at a tuning fork clock. Here time period depends on the dimensions of the vibrating structure, and the density and elasticity of the rigid material. When T changes due to motion of clock, the T formula should continue to hold because the laws of physics on which the formula works have not changed; this would mean dimensions, density and/or elasticity changed. But how can motion possibly change these - would that not be in violation of known physics? And which of these would possibly change to match the new T value? Formula for frequency of tuning fork clock is below.

e2f72f7bf08be5c787201c0a3dfd3c338a4d01b3


where:

Note that for atomic clocks (moving electrons), light clock (moving photons), muons there is an inner mechanism where motion of inner particles is causing the T of these clocks, so there is an intuitive sense in T formula being affected. But in a rigid structure clock such as a tuning fork clock, do we not have a paradox of sorts?
 

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  • #2
PeroK
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Time dilation is a relative difference in time itself between two reference frames in relative motion, with respect to each other. This can be deduced from the postulates of SR and is not dependent, per se, on the mechanism of any particular clock. Although, it is often demonstrated by considering a "light clock".

If you have two identical clocks in relative motion, then each keeps accurate time in its own rest frame. Neither is affected by the relative motion. The formula you quote is valid in the rest frame of the tuning fork. But, if the fork is moving at relativistic speed relative to you, then quantities such as "length", "density" and "area" will not be the same for you as in the rest frame of the fork.
 
  • #3
Orodruin
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When T changes due to motion of clock, the T formula should continue to hold because the laws of physics on which the formula works have not changed; this would mean dimensions, density and/or elasticity changed. But how can motion possibly change these - would that not be in violation of known physics?
This assertion is false. What you are talking about here are properties as defined in an object’s rest frame. Typically they are non-relativistic properties that actually form part of a larger relativistic construction, it is just that at non-relativistic speeds you never notice the other parts in the same manner. For example, the stresses in an object are intimately tied to its energy (including mass) density and momentum. It is just that at the speeds you are used to, you likely would never make the connection.

Time dilation is also not due to a change in internal processes. It is a change in how space and time are perceived.
 
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But, if the fork is moving at relativistic speed relative to you, then quantities such as "length", "density" and "area" will not be the same for you as in the rest frame of the fork.


Typically they are non-relativistic properties that actually form part of a larger relativistic construction, it is just that at non-relativistic speeds you never notice the other parts in the same manner. For example, the stresses in an object are intimately tied to its energy (including mass) density and momentum. It is just that at the speeds you are used to, you likely would never make the connection.

Time dilation is also not due to a change in internal processes. It is a change in how space and time are perceived.

Fine, but my point is that you could make myriad clocks with all sorts of T formulas, and not have a quantitative explanation of what causes the required change in the quantities T depends on by its formula, so that their values in the other frame work out to match the T in the other frame. Length is a physical quantity that SR has a formula for in the other frame, but what if the T in other frame does not depend on Length in other frame in a matching manner.

In the tuning fork clock, we just assume that "somehow" the density and elasticity of the material in the other frame will change in such a way as to match the T in the other frame. There are no formulas in relativistic physics that connect such quantities to motion, yet they will "somehow" have exact needed changes.

So there must exist elasticity and density formulas between the frames. So "somehow" unknown formulas exist that make numerous diverse quantities have values in the other frame that match in an exact way T in the other frame -- in all sorts of clocks with T depending on all sorts of quantities. No one has listed these formulas between frames for these other physical properties?
 
  • #5
PeroK
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Fine, but my point is that you could make myriad clocks with all sorts of T formulas, and not have a quantitative explanation of what causes the required change in the quantities T depends on by its formula, so that their values in the other frame work out to match the T in the other frame. Length is a physical quantity that SR has a formula for in the other frame, but what if the T in other frame does not depend on Length in other frame in a matching manner.

In the tuning fork clock, we just assume that "somehow" the density and elasticity of the material in the other frame will change in such a way as to match the T in the other frame. There are no formulas in relativistic physics that connect such quantities to motion, yet they will "somehow" have exact needed changes.

So there must exist elasticity and density formulas between the frames. So "somehow" unknown formulas exist that make numerous diverse quantities have values in the other frame that match in an exact way T in the other frame -- in all sorts of clocks with T depending on all sorts of quantities. No one has listed these formulas between frames for these other physical properties?

In one sense you are never going to be happy with SR. Once you have analysed a tuning fork, and probably with some difficult relativistic calculations, demonstrated that a consistent mapping exists between the measurable quantities in each frame, then you merely construct a series of ever more elaborate clocks. Until the calcations become impossibly difficult.

It is a good idea, of course, to validate the results of SR where you can and check consistency.

However, a proof of time dilation is still a proof. Once you have sufficient experimental evidence that space and time are linked as predicted by SR, then you are not simply assuming that magically everything else falls into place but that all your classical formulas must be reanalysed before use in a relativistic context.

Finally, you should go back to the derivation of the formula for frequency of a tuning fork and look for the assumptions that fail if fork is moving at relativistic speed. Or, in fact, replace the quantities with their equivalent relativistic versions.
 
  • #6
PeroK
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PS note that even ##F = ma## is not a law of physics post SR. All the classical formulas, even the most fundamental, must be reanalysed in a relativistic context.
 
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PS note that even ##F = ma## is not a law of physics post SR. All the classical formulas, even the most fundamental, must be reanalysed in a relativistic context.
I had tried but does not work out. Density = mass/volume. We have relativistic mass, and for relativistic volume we use the cuboid prong of the tuning fork being contracted. We also have l, A and I which are affected by Length contraction. For Elasticity value between frames there is a suggested formula between frames (in advanced papers) but no nice formula of the length or time type. It would seem that this last fact alone would be sufficient to ensure that in the other frame the T would not match what one calculates from the values of these quantities in the other frame.

(Also, if we turned the tuning fork 90 degrees then the other side of the rectangle would contract but T would still has to have the same value. That independently creates problems.)
 
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Orodruin
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I had tried but does not work out. Density = mass/volume. We have relativistic mass, and for relativistic volume we use the cuboid prong of the tuning fork being contracted. We also have l, A and I which are affected by Length contraction. For Elasticity value between frames there is a suggested formula between frames (in advanced papers) but no nice formula of the length or time type. It would seem that this last fact alone would be sufficient to ensure that in the other frame the T would not match what one calculates from the values of these quantities in the other frame.
This approach simply does not work. You cannot do relativity in this way. The better way to understand it is in terms of the actual relativistically covariant expressions. You also seem to be suffering from the very common misconception that properties of an object somehow actually change when they are moving. This is incorrect and the objects appear perfectly normal in their rest frame. What does change between frames is how objects are described in each frame, very similar to how components of a vector change when you change the coordinate system. Regarding relativistic mass, my suggestion is to forget you ever heard the term.
 
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  • #9
PeroK
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I had tried but does not work out. Density = mass/volume. We have relativistic mass, and for relativistic volume we use the cuboid prong of the tuning fork being contracted. We also have l, A and I which are affected by Length contraction. For Elasticity value between frames there is a suggested formula between frames (in advanced papers) but no nice formula of the length or time type. It would seem that this last fact alone would be sufficient to ensure that in the other frame the T would not match what one calculates from the values of these quantities in the other frame.

(Also, if we turned the tuning fork 90 degrees then the other side of the rectangle would contract but T would still has to have the same value. That independently creates problems.)

A simpler example would be a mass oscillating on a spring perpendicular to its motion. The period only depends on the mass and the spring constant. Therefore, the spring constant cannot be frame invariant.

An assumption that the spring constant must be invariant would lead to problems.

In your case the assumption that elasticity is frame invariant is likewise false.
 
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This approach simply does not work. You cannot do relativity in this way. The better way to understand it is in terms of the actual relativistically covariant expressions.
You say: The better way to understand it is in terms of the actual relativistically covariant expressions.
Can someone kindly work it out using these expressions? They are not working out when I do them, as detailed in my post. What are these expessions for density and elasticity?

In your case the assumption that elasticity is frame invariant is likewise false.
I did not make this assumption. What I had said is below:
For Elasticity value between frames there is a suggested formula between frames (in advanced papers) but no nice formula of the length or time type. It would seem that this last fact alone would be sufficient to ensure that in the other frame the T would not match what one calculates from the values of these quantities in the other frame.

(Also, if we turned the tuning fork 90 degrees then the other side of the rectangle would contract but T would still has to have the same value. That independently creates problems.)
 
  • #11
Orodruin
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You say: The better way to understand it is in terms of the actual relativistically covariant expressions.
Can someone kindly work it out using these expressions? They are not working out when I do them, as detailed in my post. What are these expessions for density and elasticity?
No, you are fighting a red herring. Also, you did not work anything out in your post.
For Elasticity value between frames there is a suggested formula between frames (in advanced papers) but no nice formula of the length or time type.
It is completely unclear what you mean by this or even what "advanced papers" you are referring to. Please provide references.
 
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  • #12
stevendaryl
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Fine, but my point is that you could make myriad clocks with all sorts of T formulas, and not have a quantitative explanation of what causes the required change in the quantities T depends on by its formula, so that their values in the other frame work out to match the T in the other frame. Length is a physical quantity that SR has a formula for in the other frame, but what if the T in other frame does not depend on Length in other frame in a matching manner.

Let me make an analogy: We know that energy is conserved, so you are not going to invent a perpetual motion machine that produces endless amounts of energy. But if you give me a complicated design for a machine and claim it is a perpetual motion machine, it might be very difficult (or impossible) for me to point out exactly where your mistake lies. In such cases, you have to faith in mathematical proofs: If the laws of physics predict conservation of energy, then you aren't going to violate conservation of energy using those laws.

In the case of time dilation, we can reason as follows: If the laws of physics are relativistically-invariant (the same laws apply in every inertial reference frame), then any clock constructed using those laws will experience time dilation. Your clock is going to be made out of atoms, held together using electromagnetic forces. The theory of atoms and the theory of electromagnetic forces are relativistically invariant. Therefore, any clock you make out of atoms is going to experience the same time dilation.

Note: There's a fine-print on the claim that any clock will experience time dilation. It has to be a self-contained clock, where the timing mechanism is completely within the clock. Obviously, if your "clock" is using a sun dial, or satellite signals to keep time, then you aren't going to experience time dilation in those clocks unless you get the whole solar system moving--not just the clock.
 
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  • #13
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No, you are fighting a red herring. Also, you did not work anything out in your post.

It is completely unclear what you mean by this or even what "advanced papers" you are referring to. Please provide references.

It would seem the formulas for E in "advanced papers" involve general relativity. So there is no accepted special relativity formula for E between frames that I can find.

e2f72f7bf08be5c787201c0a3dfd3c338a4d01b3


What I had noted was working out the other quantities in the formula. We also have l, A and I which are affected by Length contraction and for these we can do an exact calculation. E, Elasticity and ρ, density remain. For E - no nice formula. ρ, density = mass/volume, we could have used relativistic mass, and for relativistic volume we could use the cuboid prong of the tuning fork being contracted. But is that correct for ρ under special relativity?

E and ρ remain. So by putting in all the rest, we have a nice formula connecting E and ρ for relativistic speeds. Why would this not an acceptable result?

So by choosing clocks with different formulas for period T, we can find new unpublished formulas connecting quantities in special relativity! Is there not something wrong with such methodology?

My original point was that for time dilation to be true across myriad clocks, "somehow" (or magically) unknown formulas must exist that make numerous diverse quantities in the T formula for the clock work together in the other frame to match in an exact way the T in this other frame. This has to happen even if these relationships between quantities make no intuitive physical sense.
 
  • #14
stevendaryl
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My original point was that for time dilation to be true across myriad clocks, "somehow" (or magically) unknown formulas must exist that make numerous diverse quantities in the T formula for the clock work together in the other frame to match in an exact way the T in this other frame. This has to happen even if these relationships between quantities make no intuitive physical sense.

It's not "magical". If the laws of physics are relativistically invariant, then it follows that any clock you make is going to either experience time dilation, or else disprove the laws of physics. If you have a theoretically derived formula for the clock rate, then that formula was derived using the laws of physics, and those laws are relativistically invariant. You can't violate relativity using relativistically invariant laws.

Now, if the derivation of the formula is complicated enough, then it might be difficult to derive how the formula would change in response to the clock being set into motion. That's sort of the point of investigating the fundamental laws. You can prove properties about those laws (such as conservation of energy, or relativistic invariance) once and for all. The proof at the level of fundamental forces is simpler that proving conservation of energy or relativistic invariance for some complicated system.
 
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It's not "magical". If the laws of physics are relativistically invariant, then it follows that any clock you make is going to either experience time dilation, or else disprove the laws of physics. If you have a theoretically derived formula for the clock rate, then that formula was derived using the laws of physics, and those laws are relativistically invariant. You can't violate relativity using relativistically invariant laws.

Now, if the derivation of the formula is complicated enough, then it might be difficult to derive how the formula would change in response to the clock being set into motion. That's sort of the point of investigating the fundamental laws. You can prove properties about those laws (such as conservation of energy, or relativistic invariance) once and for all. The proof at the level of fundamental forces is simpler that proving conservation of energy or relativistic invariance for some complicated system.
Agreeing with everything you said ... then, as I pointed out, using the tuning fork clock we can work out a formula connecting E and ρ for relativistic speeds. And other formulas using other clocks.
 
  • #16
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Agreeing with everything you said ... then, as I pointed out, using the tuning fork clock we can work out a formula connecting E and ρ for relativistic speeds. And other formulas using other clocks.

No, I don't think so. I don't know the exact derivation of the formula you quote, but my guess is that it is only valid for clocks at rest. So you can't use it together with time dilation to predict how ##E## and ##\rho## change with velocity.

It's like the formula for the speed of sound. I don't remember what it is, but it involves the temperature and density of air. That formula is only valid if the air is at rest. In a reference frame in which the air is moving, sound travels differently, and its speed is not described by that formula.
 
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No, I don't think so. I don't know the exact derivation of the formula you quote, but my guess is that it is only valid for clocks at rest. So you can't use it together with time dilation to predict how ##E## and ##\rho## change with velocity.

It's like the formula for the speed of sound. I don't remember what it is, but it involves the temperature and density of air. That formula is only valid if the air is at rest. In a reference frame in which the air is moving, sound travels differently, and its speed is not described by that formula.

You say the formula could be "only valid for clocks at rest." But should Time period of a clock and its relation to various quantities not continue to hold even in the other frame? If the T formula changes in the other frame then does that not mean that we have a violation of the laws of physics being the same in all frames?
 
  • #18
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If the T formula changes in the other frame then does that not mean that we have a violation of the laws of physics being the same in all frames?
No, because a formula for a clock at rest holds in all inertial frames, for that clock at rest in that frame.
 
  • #19
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Malvia, you seem to be approaching this from an anti-relativity direction. Another direction would be more helpful.

I think stevendaryl's point is a good starting point. You can't pick an equation out of a book and say "Aha! It's not compatible with SR!" and learn anything useful. In most cases, it''s a low energy approximation. In some cases, the equation is valid, but the individual terms transform in non-trivial ways.
 
  • #20
Mister T
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Special relativity says that all clocks will show same time dilation, irrespective of clock mechanism.

It is the goal of the clock designer to make clocks that show the same time. Yes, special relativity and the other theories of physics can describe how such clocks behave, but making clocks that show the same time is a goal that metrologists strive for. What special relativity tells us is that if a pair of clocks show the same time when they are at rest relative to you, they will also show the same time dilation when you move relative to them.

But Time period of a clock is a formula that must continue to hold even if time dilates. Let us look at a tuning fork clock. Here time period depends on the dimensions of the vibrating structure, and the density and elasticity of the rigid material.

Yes, but not only those properties. There are other properties that it depends on, and relative motion of the clock is one of those properties!

What you're describing is proper time, and it is a relativistic invariant, meaning that all observers will agree on its value, regardless of their motion relative to the clock.
 
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  • #21
russ_watters
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Fine, but my point is that you could make myriad clocks with all sorts of T formulas, and not have a quantitative explanation of what causes the required change in the quantities T depends on by its formula, so that their values in the other frame work out to match the T in the other frame. Length is a physical quantity that SR has a formula for in the other frame, but what if the T in other frame does not depend on Length in other frame in a matching manner.

In the tuning fork clock, we just assume that "somehow" the density and elasticity of the material in the other frame will change in such a way as to match the T in the other frame. There are no formulas in relativistic physics that connect such quantities to motion, yet they will "somehow" have exact needed changes.

So there must exist elasticity and density formulas between the frames. So "somehow" unknown formulas exist that make numerous diverse quantities have values in the other frame that match in an exact way T in the other frame -- in all sorts of clocks with T depending on all sorts of quantities. No one has listed these formulas between frames for these other physical properties?
Density most certainly *does* depend on length: density is mass divided by volume and length is part of the calculation for volume.

So you *could* do the calculations you are alludingc to, it's just not as useful as you think it is, so scientists don't bother.

Take modulus of elasticity, for example. I can look up the elastic modulus of steel in a table. You could create a "relativistic modulus of elasticity", but there could be no table for it, since - as you point out - it is different between different frames.

My understanding is that this is why the concept of "relativistic mass" was discarded: it is unnecessary and causes confusion since "mass" is supposed to be a fixed property of matter.
 
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  • #22
Dale
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When T changes due to motion of clock, the T formula should continue to hold because the laws of physics on which the formula works have not changed
You have to be careful here. There are laws of physics, like the relativistic velocity addition formula, and there are approximations to the laws of physics which are only valid at low velocities, like the Galilean velocity addition formula. The T formula that you cite is not actually a law of physics, but a low velocity approximation. In fact, the contradiction that you identify is precisely the manner of proving that it is not a law of physics.
 
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You have to be careful here. There are laws of physics, like the relativistic velocity addition formula, and there are approximations to the laws of physics which are only valid at low velocities, like the Galilean velocity addition formula. The T formula that you cite is not actually a law of physics, but a low velocity approximation. In fact, the contradiction that you identify is precisely the manner of proving that it is not a law of physics.

Yes, the Galilean velocity addition formula is approximate and has relativistic velocity addition formula as the exact. Most importantly, we are able follow the reason for that.

For myriad clocks, are the known time period formulas only approximate and valid at low velocities?

How do we reason our way to the exact formula for clocks such as the tuning fork clock? I was suggesting a path where a high velocity formula connecting two quantities would have solved the dilemma. But that was clearly ad hoc and not the solution that seems reasonable.
 
  • #24
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Yes, the Galilean velocity addition formula is approximate and has relativistic velocity addition formula as the exact. Most importantly, we are able follow the reason for that.

For myriad clocks, are the known time period formulas only approximate and valid at low velocities?

How do we reason our way to the exact formula for clocks such as the tuning fork clock? I was suggesting a path where a high velocity formula connecting two quantities would have solved the dilemma. But that was clearly ad hoc and not the solution that seems reasonable.

There's no dilemma if you break the problem down into: describe the motion in the rest frame of the clock, where the parts are moving at non relativistic velocities. Then use the Lorentz Transformation to map the motion to another frame.

There's no reason to doubt this as a valid approach.
 
  • #25
Dale
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For myriad clocks, are the known time period formulas only approximate and valid at low velocities?
Yes, as you have shown.

How do we reason our way to the exact formula for clocks such as the tuning fork clock? I was suggesting a path where a high velocity formula connecting two quantities would have solved the dilemma. But that was clearly ad hoc and not the solution that seems reasonable.
The approach I would take is to use the relativistic Lagrangian for the system. In the case of the simple harmonic oscillator ##L=-mc^2\sqrt{1-\dot{x}^2/c^2}-(1/2)kx^2##. Contrast this with the classical Lagrangian ##L=(1/2) m \dot{x}^2 - (1/2)kx^2##. Note, this is the oscillator where the mass oscillates at relativistic velocities wrt the center of displacement. I don't know the Lagrangian for the situation where the whole oscillator is travelling at relativistic velocities, but that is the approach I would use.
 
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Yes, as you have shown.

The approach I would take is to use the relativistic Lagrangian for the system. In the case of the simple harmonic oscillator ##L=-mc^2\sqrt{1-\dot{x}^2/c^2}-(1/2)kx^2##. Contrast this with the classical Lagrangian ##L=(1/2) m \dot{x}^2 - (1/2)kx^2##. Note, this is the oscillator where the mass oscillates at relativistic velocities wrt the center of displacement. I don't know the Lagrangian for the situation where the whole oscillator is travelling at relativistic velocities, but that is the approach I would use.
Do you know of any examples of T-classical changed to T-high-speed that have been achieved by such approaches? Of course, only the latter would be the actual correct formula.

Also, how can you have a classical equation that describe the motions at low speed, which you then modify by such procedure, if that classical formula for the clock mechanism could be substantially wrong in unexpected ways, to begin with? Because perhaps a drastic change is needed. For example -- assume that two quantities in a formula need to be related at high speed, somewhat along the lines of the made-up modification I suggested, and at low speeds this extra relationship was insignificant.

It is not clear to me that there is a procedure that applies and gets the result.
 
  • #27
Dale
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Do you know of any examples of T-classical changed to T-high-speed that have been achieved by such approaches?
No, I only know of the opposite way, the relativistic Lagrangian comes first, and then you can check that it correctly reduces to the classical approximation in the low-speed limit.

However, it is true that knowing the low-speed approximate Lagrangian can often help a physicist make a good guess about the corresponding relativistic Lagrangian. Such a process is so common, in fact, that it is given the fancy German name "ansatz" which sounds better (at least in English) and more scientific than "educated guess".

It is not clear to me that there is a procedure that applies and gets the result.
I agree, I also do not think such a procedure exists. The issue is that there is often more than one way to generalize a specialized formula. So it is hard to see how a general procedure could always find the correct generalization out of all possible generalizations.

The other direction is straight forward.
 
  • #28
PeroK
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The other direction is straight forward.

@Malvia

A simple and good example of this is that the classical KE can easily be obtained from the relativistic expression by:

##KE = (\gamma - 1) mc^2 = \frac12 mv^2 + \dots##

But, if you start with ##\frac12 mv^2## there is no logical way to get to the relativistic formula.
 
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  • #29
pervect
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Special relativity says that all clocks will show same time dilation, irrespective of clock mechanism.
Yes.
But [the] Time period of a clock is a formula that must continue to hold even if time dilates.

I'm not sure quite sure what this means. Perhaps you could rephrase this? The time period of a clock is something physical that we measure, not a formula. The proper time period of a clock (the time period of a clock in a frame of reference in which it is at rest) is independent of the observer, as it should be. The proper time of a clock is the basis of the SI definition of the second.

NIST said:
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

Malvia said:
Let us look at a tuning fork clock. Here time period depends on the dimensions of the vibrating structure, and the density and elasticity of the rigid material. When T changes due to motion of clock, the T formula should continue to hold because the laws of physics on which the formula works have not changed; this would mean dimensions, density and/or elasticity changed. But how can motion possibly change these - would that not be in violation of known physics? And which of these would possibly change to match the new T value? Formula for frequency of tuning fork clock is below.

e2f72f7bf08be5c787201c0a3dfd3c338a4d01b3


where:

Note that for atomic clocks (moving electrons), light clock (moving photons), muons there is an inner mechanism where motion of inner particles is causing the T of these clocks, so there is an intuitive sense in T formula being affected. But in a rigid structure clock such as a tuning fork clock, do we not have a paradox of sorts?

It's not a paradox to point out that the classical laws of elasticity need modification to make them compatible with special relativity. What we can usefully say at the I level is that your equations work just fine in the frame of the tuning fork, but don't give correct predictions in relativistically moving frames. This isn't a huge practical obstacle, one simply applies the equations in the regime where they work, recognizing that they are not universal.

There is, of course, some work that has been done on how to change the theory of elasticity to make it fully compatible with special relativity, which among other things would imply that they would then work in any frame of reference, no matter how rapidly it was moving. However, I'm not intimately familiar with the literature on the topic, though I've dabbled a bit.. https://arxiv.org/abs/gr-qc/0605025 appears to have a good summary. Greg Egan has a good (IMO), but non-peer reviewed website which touches on the topic as well, though his focus is different than yours. Egan is interested in very strong materials, not in formulae that will correctly predict the operation of a tuning fork properly in a relativistically moving frame of reference.

For a short quote from the reference I cite above:

Attempts to formulate a relativistic theory of elasticity can be traced back as early as 1911 for special relativity ([26]) and 1916 for general relativity ([43]). The promptness of these results (being published on ly a short time after the advent of special (general) relativity) indicates the importance of elasticity to the scientific community back then.

If the non-relativisti theories we had for elasticity worked in the relativistic realm, we wouldn't need to consider how the theory had to be modified to be compatible with relativity. But, as the above quote points out, we do.
 
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My original point was that for time dilation to be true across myriad clocks, "somehow" (or magically) unknown formulas must exist that make numerous diverse quantities in the T formula for the clock work together in the other frame to match in an exact way the T in this other frame. This has to happen even if these relationships between quantities make no intuitive physical sense.

You may be able to extend your point to length measurement also. Materials of measures, e.g. steel wood with condition of temperature, pressure etc. could be investigated to make them work in coordinated ways. Laser beam measurement could be added.
Same way of coordination will be applicable in all the IFRs. This is a principle of SR.
 

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