# Galileo and Lorentz transformation

1. Dec 27, 2009

### yinfudan

Though I believe I have understood some basic ideas, theories and mathematic formulas of SR, I still have a pretty fundamental question:

Many text books start SR with a light clock consisting of two mirrors and a light blip bouncing in between, claiming that when the light clock moves, the light blip travels longer distance per bouncing, resulting in time dilation. Then it claims that other physics phenomena will also slow down - even a person ages slower.

But it does not explain why if the light clock ticks slower, other physics phenomena also slow down. Is it possible that lorentz transformation only applies to electromagnetism while galileo transfer still applies to mechanics, even at high speed? As a result, the light clock will slows down but a mechanic clock (for example, spring based clock) will not slow down?

2. Dec 27, 2009

### Mentz114

I'd say you still have some thinking to do.

Your question about clocks has a self-evident answer. If different clocks behaved differently ( when viewed by uniformly moving observers ) then different observers would disagree about what they saw, which is a paradox. It is not clocks that are affected, it is time itself. Any process undergoing change would be equally affected.

Clocks running slower for moving observers is an illusion in any case and is observer dependent, which means it has no true physical significance. SR is based on the invariance of the proper-interval, i.e. all observers will agree on the elapsed time on a clock, when they coincide spatially with the clock.

3. Dec 28, 2009

### meopemuk

How is it a paradox? Let's say two different clocks A and B go at the same rate when they are at rest. And their rates are (slightly) different when they are moving (with the same speed). Does this contradict the principle of relativity?

Eugene.

4. Dec 28, 2009

### Al68

The textbooks should probably start with the postulate that light has a constant velocity in any inertial reference frame. Then it follows that since the light travels a longer distance in one frame than another, with the same velocity in each, time must pass slower in one frame than the other.

The claim that a person "ages slower" is more accurately a claim that that person experienced less elapsed time, and aged normally during that time.

5. Dec 28, 2009

### meopemuk

yinfudan,

Einstein's principle of relativity establishes that *all* physical processes are invariant with respect to the Poincare group (=Lorentz group plus translations in space and time). From this principle and from the invariance of the light speed it is not difficult to conclude that the rate of the moving light clock slows down exactly $$\gamma$$ times. However, you are absolutely right that one cannot prove that all other physical processes should slow down exactly by the factor $$\gamma$$ as well. In fact, it is possible to show that behavior of moving clocks can be more complicated than this universal slowdown. There are recent works, which analyze the decay rate of moving unstable particles within relativistic quantum mechanics. They predict very small (but fundamentally important) corrections to the Einstein's "time dilation" law.

E. V. Stefanovich, "Quantum effects in relativistic decays", Int. J. Theor. Phys., 35 (1996), 2539.

L. A. Khalfin, "Quantum theory of unstable particles and relativity", (1997), Preprint of Steklov Mathematical Institute, St. Petersburg Department, PDMI-6/1997
http://www.pdmi.ras.ru/preprint/1997/97-06.html

M. I. Shirokov, "Decay law of moving unstable particle", Int. J. Theor. Phys., 43 (2004), 1541.

M. I. Shirokov, "Evolution in time of moving unstable systems", Concepts of Physics, 3 (2006), 193. http://www.arxiv.org/abs/quant-ph/0508087

E. V. Stefanovich, "Violations of Einstein's time dilation formula in particle decays", http://www.arxiv.org/abs/physics/0603043

Eugene.

6. Dec 28, 2009

### Cleonis

That is the problem that is staring every textbook writer in the face. The student demands to know why.

In physics, what the student comes to expect is that when he is wondering why does this happen, physics can show him. When the atmosphere is filled with water droplets, why do we see a rainbow? Why has the second rainbow, the fainter one, the colors of the spectrum in inverted order? How can that be? Our physics gives the answers; the physics of light reflecting and refracting in and out of water droplets accounts for observing rainbows, in terms of readily understandable, intuitive principles.

Ironically, physics is trapped by its success: when it comes to introducing special relativity the student expects that any moment the curtains will be drawn aside, and that the apparently self-contradicting picture will be shown to be readily understandable in terms of intuitive principles.

And that is just not going to happen.
Teachers can present the principles of special relativity, they can demonstrate that mathematically no self-inconsistency arises, but the counter-intuitive nature cannot be lifted.

Special relativity isn't newtonian; it cannot be reduced to the familiar, intuitive newtonian principles. That is the problem that is staring every textbook writer in the face.

What is the textbook writer to do? It's understandable that the textbook writer decides to introduce special relativity step-by-step, rather than throwing in everything at once. And yeah, those first steps go really against intuition. But it's not productive to go into skeptical mode right away. You need to give yourself time to acquire the overall picture.

Cleonis

7. Dec 28, 2009

### atyy

It's not obviously impossible (to me). If such a mathematically consistent picture can be constructed, it would violate the Principle of Relativity (only Galilean and Lorentz transformations are consistent with the Principle). The Principle of Relativity and the validity of the Lorentz transformations are an experimental fact.

8. Dec 28, 2009

### HallsofIvy

That was, in fact, Lorentz's explanation of the null result of the Michaelson-Morley experiment when he derived the Lorentz transforms. That theory, however, would require that only physical objects contract with motion, not the space between them while Einstein's theory requires that space itself contract and that all motion, not just electromagnetic, slow down. A version of the Michaelson-Morely experiment, called, I think, the "Kennedy experiment" showed that Einstein's theory was right and Lorentz's was wrong.

9. Dec 28, 2009

### Mentz114

Eugene,

that's not what I'm saying. I'm refering to the case postulated by the OP where
a mechanical clock might behave differently from a chemical clock, when viewed
from another frame.

M

10. Dec 28, 2009

### A.T.

Yes it does, because this would allow to determine the absolute rest of that clock pair.

It is paradoxical as well. Imagine each clock stops itself and the other clock after reaching 1 min. Different observers would disagree if the clocks stopped at the same mark, but the setup can have only one correct final physical state, that can examined by any observer in his frame after both clocks stopped.

11. Dec 28, 2009

### bcrowell

Staff Emeritus
It's certainly logically possible that motion would have an effect on clocks, but with a different effect on different types of clocks. We simply have to do experiments to find out whether the effect is different or the same. Here are two such experiments. The Hafele-Keating experiment ( http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html#Section1.1 [Broken] ) shows that we get a certain amount of time dilation with a certain type of atomic clock, the amount being consistent with Einstein's $\gamma=(1-v^2/c^2)^{-1/2}$. A 1974 experiment at CERN ( http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html#Section1.2 [Broken] , see example 1 and figures p and q ) shows that we get a certain amount of time dilation with a different type of "clock," this one being a beam of muons undergoing radioative decay. Again, the amount is consistent with the standard formula for gamma. There are many such experiments, and they are all consistent with the standard gamma factor for time dilation. Because all these experiments show a consistent result, we have support for Einstein's theory of relativity, and specifically for the standard interpretation it as a theory of the geometry of spacetime (not as a theory involving some kind of dynamical effect like aether drag).

You've suggested a particular variation on relativity, based on a combination of Lorentz transformations for some effects and Galilean transformations for others. Most likely this particular idea is not logically self-consistent; it's quite difficult to come up with self-consistent theories of this type. For example, you're going to run into problems trying to separate mechanics cleanly from electromagnetism. Mechanical bodies are made of atoms, and atoms are objects that interact electromagnetically. As an example of a check on the self-consistency of standard relativity, W.F.G. Swann did an explicit QED calculation in 1941 that showed the relativistic length contraction should occur for solid meter-sticks in relative motion with respect to each other. Under the standard interpretation of relativity, you could say that such a calculation was a waste of time; but it is certainly reassuring to know that it does come out consistent when you do an explicit check.

(Note added later: The description of the Swann paper above is misleading. See later discussion below.)

Last edited by a moderator: May 4, 2017
12. Dec 28, 2009

### Al68

Certainly, but then it's logically impossible that both are keeping proper time. The clock hypothesis says that a clock will keep proper time (equal to a light clock) regardless of its relative motion or acceleration. A clock that fails this test is not a valid clock in SR/GR.

13. Dec 28, 2009

### bcrowell

Staff Emeritus
The issue, as pointed out by atyy, is whether or not we're talking about holding on to the principle of relativity. If we abandon it, then there can be a preferred rest frame, and we can say that clocks of different types agree with one another only if they're at rest with respect to the preferred rest frame. The validity of the principle of relativity can only be determined by experiment.

14. Dec 28, 2009

### bcrowell

Staff Emeritus
I think we're all in agreement here that if different types of clocks disagree, then SR is falsified. It is, however, logically possible that SR is false. The OP was explicitly stating that this was all under the assumption that SR was false, since he talked about creating a hybrid of SR and Galilean relativity. What is less clear is whether the OP understood that his hybrid theory was incompatible with the more generic idea that all inertial frames are equivalent. Both SR and Galilean relativity are theories in which all inertial frames are equivalent; in his hybrid theory, this is not the case.

15. Dec 28, 2009

### meopemuk

How?

Let's say we have a clock pair A,B at rest and another pair A',B' is moving. The observer at rest finds that clocks A,B have stopped at the same mark (A=B), while in the moving pair clock A' showed later time (A'>B') when both clocks have stopped. From the point of view of observer co-moving with the pair A',B' the situation is reverse: He finds that A'=B' and A>B. There is no contradiction with the principle of relativity: Both observers are equivalent. Both of them agree that two clocks at rest stop at the same time, and two moving clocks stop at different times.

From this I conclude that the principle of relativity itself does not forbid different moving clocks to have different rates. However, Einstein's theory of special relativity does forbid such an effect. This means that Einstein's special relativity is not limited to two famous postulates (the principle of relativity and the constancy of the speed of light). There should be another important postulate which is rarely spelled out explicitly. It goes something like this: "clocks of different type slow down by exactly the same amount; rods made of different materials shorten by exactly the same amount."

Only if this (third) postulate is true, we can say that rate slowdowns and length contractions are universal for all objects. Then it would be natural to say that these effects are just manifestations of the global time dilation and space contraction. Then it would be logical to introduce the 4-dimensional Minkowski space-time picture, in which Lorentz transformations are represented as geometrical pseudo-rotations.

It is true that all our present experiments confirm that the "third relativity postulate" is valid. However, there is no guarantee that a more precise experiments in the future will not show some small deviations (see papers about particle decays cited above). Then the 4D geometrical formulation of special relativity will be in geopardy.

Eugene.

16. Dec 28, 2009

### meopemuk

Could you give a more precise reference? I doubt very much that such a calculation was possible in 1941, even before renormalization was invented by Tomonaga, Schwinger, and Feynman.

Eugene.

17. Dec 28, 2009

### A.T.

I was talking of just two clocks A,B at rest to each other. If one observer sees them stop at the same mark (A=B), every observer does so as well. That is not a third postulate of SR, but simple consistency. A moving observer cannot see (A>B) on timeout, and then after he stops moving relative to the clocks suddenly A=B. The mark at which the clocks stop is frame invariant, and so is their rate-ratio.
Yes you can make this situation symmetrical. But you have just doubled the paradox above.

18. Dec 28, 2009

### meopemuk

I do not accept your statements (e.g., A=B in all reference frames) as self-evident "simple consistency". They are not evident to me. Moreover, I've studied concrete examples of relativistic quantum systems (unstable particles) in which your statements are not realized (the decay law of a moving particle does not experience simple uniform dilation). So, in my opinion your statements must be formulated as a separate postulate of special relativity and subjected to careful analysis.

Be careful when you claim what observer would see "after he stops moving relative to the clocks". Special relativity is designed to talk about inertial observers only. The "twin paradox" is a good example how you can get wrong conclusions by "extending" special relativity to accelerated observers.

Eugene.

19. Dec 28, 2009

### bcrowell

Staff Emeritus
Thanks, Eugene, for calling me on this one :-) My description of the Swann paper was, as you suspected, second-hand and inaccurate.

W.F.G. Swann, "Relativity, the Fitzgerald-Lorentz Contraction, and Quantum Theory," Rev. Mod. Phys., 13, 197 (1941).

http://prola.aps.org/abstract/RMP/v13/i3/p197_1

I was basing my description on what Ohanian says in "Einstein's Mistakes," p. 283. He describes it as a calculation in the "context of relativistic quantum mechanics." Now that I've looked up the original article, it's clear that it's not really a QED calculation. It's got quantum mechanics in it, and it's got relativity in it, but it doesn't use the full machinery of QED, which, as you point out, hadn't been invented yet in 1941.

Ohanian's description actually seems somewhat misleading to me: "It was not until 1941 that the American physicist W.F.G. Swann revisited Lorentz's arguments in the context of relativistic quantum mechanics and showed that, indeed, the length contraction emerges from a quantum-theoretical calculation of the length of a solid body when the length of a moving solid body is compared with the length of a similar body at rest."

What the Swann actually does is this. He describes the process of accelerating a measuring rod from an initial state of rest in the lab frame. He considers the problem that it may be difficult to distinguish between two possibilities: (1) the rod becomes Lorentz-contracted, and (2) the rod suffers a mechanical contraction because of the stress imposed by accelerating it. He claims (and I think this is correct) that if all you know is the Lorentz transformation, you can't tell whether the result of the experiment actually verifies the Lorentz transformation (#1) or not (#2); you need some specific physical theory that's capable of describing the structure and dynamics of solid rods. He hypothesizes a Lorentz-invariant theory of quantum mechanics, which didn't actually exist at the time. What he does know, based on the state of the art at the time, is that quantum-mechanical systems have ground states. Then he argues that after you're done accelerating the rod, it will settle back down into its ground state (assuming you accelerate it gently enough). Thus by picking a specific physical theory (quantum mechanics) to lay on top of the foundation of the bare Lorentz transformation, you gain the ability to distinguish between interpretations 1 and 2. Basically it's an argument that SR by itself has predictive value (e.g., it predicts a null result for the Michelson-Morley experiment), but it doesn't have full explanatory value unless you augment it with some dynamical theory that describes how particles interact.

Last edited: Dec 28, 2009
20. Dec 28, 2009

### Al68

If I have two different types of clocks, both together on the same spaceship moving fast relative to earth, and they run at the same rate in their own rest frame, it's logically impossible that the two clocks run at two different rates in a different reference frame because they are "different types".

If the ship's crew start and later stop both clocks and they have the same reading when stopped, it's just logically impossible that they have two different readings in a different frame, like earth's rest frame.

Last edited by a moderator: Dec 28, 2009
21. Dec 28, 2009

### bcrowell

Staff Emeritus
A.T. wrote:
Eugene, I'm perfectly willing to admit that you may be right here. However, I, like A.T., am having a hard time understanding the point of view you're advocating. A common way of handling the issue of consistency in SR and GR is to treat it using what a geometer would call "incidence relations." In Euclidean geometry, an incidence relation is a relationship that says two geometrical objects have a point in common. E.g., two points that are incident on one another are the same point. Incident lines are those that are either the same or not parallel. A point can be incident with a line, etc. Generally all observers in SR and GR agree on incidence relations, and this is considered a requirement of consistency. Either the bullet's world-line intersected the target's world-line, or it didn't. Incidence relations are preserved under general-relativistic coordinate transformations, since such transformations are required to be smooth and one-to-one.

Now I think that the statement A=B is in a form that can be stated as an incidence relation, assuming that the clocks follow the same world-line. E.g., we can talk about not just the incidence of the two clocks' centers of mass, but also about the incidence of the tips of their minute-hands, etc. So it seems to me that if one observer says A=B, everyone else must say the same.

22. Dec 28, 2009

### bcrowell

Staff Emeritus
If the principle of relativity is violated, then we can have a situation in which the clocks are synchronized in the universal rest frame (say the frame of the aether), but they go out of sync because the spaceship is in motion relative to the universal rest frame.

23. Dec 28, 2009

### meopemuk

Hi bcrowell, thank you for the reference.

Yes, in order to conclude how moving observer sees an interacting system (solid rod or clock) we need a dynamical theory of interactions in this system. The important progress in description of relativistic interactions was made by Dirac in 1949:

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys., 21 (1949), 392.

His basic point (explained also earlier by E . P. Wigner) is that in order to build a relativistic description of a quantum system one needs to build a unitary representation of the Poincare group in the Hilbert space of the system. (The same idea applies in classical mechanics, where the Poincare group must be represented by canonical transformations in the phase space of the system). The knowledge of ten generators of this representation allows us to answer any question about the behavior of the system seen from different reference frames. For example, the generator of time translations is the Hamiltonian H, and if we want to calculate the value of observable F at time t from its (known) value at time 0, we can use formula

$$F(t) = e^{iHt}F(0)e^{-iHt}$$.....................................(1)

Similarly, if we know the value of F in the reference frame at rest, then we can find its value in the moving reference frame by applying the generator of boosts $$K_x$$ (along the x-axis)

$$F(\theta) = e^{-iK_x \theta}F(0)e^{i K_x \theta}$$.......(2)

where $$\theta$$ is the rapidity of the boost, that is related to the boost velocity v by formula $$v = c \tanh \theta$$.

We all know that in interacting systems the Hamiltonian H contains non-trivial interaction terms, which lead to rather non-trivial dynamical effect (explosions, decays, etc.) happening in the course of time evolution (1). The important (and still not fully appreciated) point of the Dirac's paper is that in relativistic interacting systems the generator of boosts $$K_x$$ also must contain non-trivial interaction-dependent terms. It then follows that boost transformations (2) also must lead to non-trivial dynamical effects, like explosions and decays.

This result of rigorous relativistic quantum theory contradicts the traditional statement of Einstein's special relativity that moving observers see only (rather trivial) kinematic changes in the observed system. Special relativity treats boost transformations as "geometrical" pseudo-rotations in the 4D Minkowski space-time. So, it only allows such simple effects as length contraction and clock rate dilation. Wigner-Dirac theory suggests that more significant (dynamical) effects of boosts are not only possible, but necessary.

Eugene.

24. Dec 28, 2009

### meopemuk

You said twice "logically impossible", but which law of logic has been violated? In my opinion, the situation you've described does not violate any established physical law. I agree, that the situation in which one observer sees one thing (two clocks show the same time) and another observer sees another thing (two clocks show different times) is rather unusual. But, in my opinion, it is not more unusual than "relativity of simultaneity" or the "twin paradox". If no physical law has been violated, we should consider it as a possibility.

Eugene.

25. Dec 28, 2009

### matheinste

Perhaps I am missing the point here but isn't a stopped clock no longer a clock but just an unchanging object indicating the time at which it stopped and so the same for all observers (except for physical dimensions if it has any).

Matheinste