Galileo and Lorentz transformation

[yinfudan]

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

Many textbooks 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?

 

[Mentz114]

Though I believe I have understood some basic ideas, theories and mathematic formulas of SR ...

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.

 

[meopemuk]

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.

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.

 

[Al68]

Many textbooks 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.

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.

 

[meopemuk]

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

Many textbooks 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?

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.

 

[Cleonis]

Many textbooks 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.

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

 

[atyy]

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?

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.

 

[HallsofIvy]

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?

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.

 

[Mentz114]

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.

Eugene,

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

M

 

[A.T.]

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?

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.

 

[bcrowell]

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?

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 ) 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 , 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.)

 

[Al68]

It's certainly logically possible that motion would have an effect on clocks, but with a different effect on different types of clocks.

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.

 

[bcrowell]

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

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.

 

[bcrowell]

It's certainly logically possible that motion would have an effect on clocks, but with a different effect on different types of clocks.

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.

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.

 

[meopemuk]

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

How?

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.

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.

 

[meopemuk]

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.

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.

 

[A.T.]

Let's say we have a clock pair A,B at rest and another pair A',B' is moving.

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.

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.

Yes you can make this situation symmetrical. But you have just doubled the paradox above.

 

[meopemuk]

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.

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.

A moving observer cannot see (A>B) on timeout, and then after he stops moving relative to the clocks suddenly A=B.

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.

 

[bcrowell]

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.

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.

 

[Al68]

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."

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.

 

[bcrowell]

A.T. wrote:

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.

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.

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.

 

[bcrowell]

If I have two different types of clocks, both on the same spaceship, 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 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.

 

[meopemuk]

Hi bcrowell, thank you for the reference.

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.

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.

 

[meopemuk]

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.

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.

 

[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

 

[meopemuk]

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.

bcrowell,

I agree that "incidence relations" play a crucial role in the formulation of special relativity and the Minkowski space-time geometry. However, I find it very unfortunate that the role of these relations in SR is not sufficiently emphasised in textbooks (at least, in textbooks that I've read). I think it would be more beneficial to list the "incidence relations" as the third postulate of special relativity. Then readers would have a chance to question the plausibility of this postulate and to seek its independent experimental confirmation.

Personally, I don't find this postulate convincing. I think it is possible for worldlines of two interacting particles intersect in one frame and not intersect in another frame. This point of view is supported by the famous "no interaction" theorem, which tells that worldlines of interacting particles cannot transform by usual linear Lorentz formulas.

D. G. Currie, T. F. Jordan, E. C. G. Sudarshan, "Relativistic invariance and Hamiltonian theories of interacting particles", Rev. Mod. Phys., 35 (1963), 350.

Eugene.

 

[A.T.]

Be careful when you claim what observer would see "after he stops moving relative to the clocks".

He doesn't need to stop. He can pass the clocks very closely on timeout. And if A=B they explode and kill him. So according to the guy at rest to the clocks he's dead. But in his own frame he's fine because A>B.

This doesn't happen in SR because it is not a multiple universe theory.

You can call this the third postulate if you want.

 

[meopemuk]

He doesn't need to stop. He can pass the clocks very closely on timeout. And if A=B they explode and kill him. So according to the guy at rest to the clocks he's dead. But in his own frame he's fine because A>B.

This doesn't happen in SR because it is not a multiple universe theory.

You can call this the third postulate if you want.

Let me change the circumstances a bit to show why your logic does not work. Instead of two observers related by a boost (i.e., moving with respect to each other) I would like to consider two observers connected by a time translation (which is also a permissible inertial transformation). Let's say A is observer here "now" and B is observer here "tomorrow". Otherwise the two observers are completely equivalent. Suppose that A and B observe the same physical object, which is a time bomb, i.e., a piece of explosive connected to a clock. The mechanism is designed to explode tomorrow.

Now, if I follow your logic I must conclude (erroneously) that both A and B should see arms of the clock in the same position. If this was not so, then observer B might explode together with the bomb, while A does not experience any explosion.

The true power of the principle of relativity and the Poincare group is that all 10 inertial transformations (3 space translations, 3 rotations, 3 boosts, and 1 time translation) can be treated on the same footing. Then, according to Dirac, if you deny the possibility of non-trivial dynamical boost transformations you must also deny the possibility of non-trivial dynamical time evolution, which is absurd.

Eugene.

 

[bcrowell]

Personally, I don't find this postulate convincing. I think it is possible for worldlines of two interacting particles intersect in one frame and not intersect in another frame. This point of view is supported by the famous "no interaction" theorem, which tells that worldlines of interacting particles cannot transform by usual linear Lorentz formulas.

That's interesting. Thanks for pointing me to that. It's something I'd never heard of before.

The paper is very long, and I wanted to try to understand the significance of the result without having to wade through all 26 pages of it. I found the following talk by Gordon Fleming at the Perimeter Institute, which seemed to do a nice job of explaining the physical ideas behind it, and putting it in historical context: http://streamer.perimeterinstitute.ca/Flash/1a7787fa-5478-49ca-82c2-4b7a342117c8/index.html

One place where I really got stuck, though, was on understanding what is meant by the "invariant world-line condition," which seems to be a fundamental issue in this kind of thing. Can you enlighten me at all?

If I'm understanding correctly, this is all related to attempts to formulate relativistic theories of quantum mechanics in which point particles interact without mediation by a field, i.e., by instantaneous action at a distance. Fleming describes this as motivated by the fact that at the time, nobody knew how to do field theory for any force other than electromagnetism. The whole idea seems kind of odd to me. Why would you try to formulate a relativistic theory based on instantaneous action at a distance?

 

[meopemuk]

The paper is very long, and I wanted to try to understand the significance of the result without having to wade through all 26 pages of it. I found the following talk by Gordon Fleming at the Perimeter Institute, which seemed to do a nice job of explaining the physical ideas behind it, and putting it in historical context: http://streamer.perimeterinstitute.ca/Flash/1a7787fa-5478-49ca-82c2-4b7a342117c8/index.html

Yes, this is a nice talk, though I don't agree with some of Fleming's points.

One place where I really got stuck, though, was on understanding what is meant by the "invariant world-line condition," which seems to be a fundamental issue in this kind of thing. Can you enlighten me at all?

The "invariant world-line condition" basically means that world-lines of particles seen from two reference frames are related by the usual Lorentz formulas (Lorentz transformations for x,y,z,t). The Currie-Jordan-Sudarshan theorem says that if world-lines are transformed by Lorentz formulas then the system of particles must be non-interacting. This theorem is formulated for interactions described in terms of Hamiltonian dynamics (i.e., the unitary representation of the Poincare group that I've mentioned earlier). The usual reaction to this theorem is that Hamiltonian formalism is not applicable in relativistic physics. Other people say that we should abandon the notion of particles altogether. In my opinion, the solution of this paradox is different. I think that boost transformations of particle world-lines must be dynamical (interaction-dependent), so that universal linear Lorentz transformation formulas are not accurate.

If I'm understanding correctly, this is all related to attempts to formulate relativistic theories of quantum mechanics in which point particles interact without mediation by a field, i.e., by instantaneous action at a distance. Fleming describes this as motivated by the fact that at the time, nobody knew how to do field theory for any force other than electromagnetism. The whole idea seems kind of odd to me. Why would you try to formulate a relativistic theory based on instantaneous action at a distance?

Quantum field theories (such as QED) are great tools, but their applications are limited to calculations of such things as S-matrix and energies of bound states. Renormalized QFT do not have well-defined Hamiltonians (their Hamiltonians must contain divergent counterterms) therefore they can't be applied to calculations of the time evolution of states and observables. In particular, it is impossible to calculate trajectories (world-lines) of interacting systems of particles and how these world-lines transform with respect to boosts. This problem can be solved by a reformulation of QFT known as the "dressed particle" approach

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim., 8 (1958), 378.

E. V. Stefanovich, "Relativistic quantum dynamics", http://www.arxiv.org/abs/physics/0504062

It appears that in this approach "dressed" or "physical" particles interact with each other via instantaneous action-at-a-distance potentials. Usually, such an action-at-a-distance is considered a bad thing. However, I don't think there is anything wrong with it.

Eugene.

 

[Al68]

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.

Sure, but if the clocks stay next to each other, if they are out of synch, they are out of synch in every reference frame.

 

[Al68]

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.

By "logically impossible", I simply mean that different observers would disagree about what did or didn't happen. For example, you could rig the clocks so that they would explode if they ever went out of synch by a minute. Since they stay right next to each other, the light travel time between them will be insignificant. It's logically impossible for one observer to watch the clocks explode when one reads noon and the other reads 12:01, while a different observer watches both clocks tick way past noon without exploding.

It's analogous to having one observer see Earth get hit by an asteroid while a different observer sees the same asteroid miss Earth and hit venus.

 

[meopemuk]

It's logically impossible for one observer to watch the clocks explode when one reads noon and the other reads 12:01, while a different observer watches both clocks tick way past noon without exploding.

It's analogous to having one observer see Earth get hit by an asteroid while a different observer sees the same asteroid miss Earth and hit venus.

I agree that your examples are rather odd, but I don't think they are logically impossible.

Clearly, two observers displaced in time with respect to each other have very different views on the same physical system. I don't see why views of two relatively moving observers cannot be just as different. As I explained above, the structure of the Poincare group demands that both time translations and boosts of the reference frame must induce non-trivial dynamical changes in the observed physical system.

Eugene.

 

[A.T.]

Let me change the circumstances a bit to show why your logic does not work. Instead of two observers related by a boost (i.e., moving with respect to each other) I would like to consider two observers connected by a time translation

You cannot solve the paradox by presenting a different situation without a paradox. A physical theory has to work in any situation.

(which is also a permissible inertial transformation).

You have introduced a temporal separation. In my scenario there is neither spatial nor temporal separation, between the clocks and both observers on A-timeout. In a single-world theory they have to agree if the clocks explode or not.

The true power of the principle of relativity and the Poincare group is that all 10 inertial transformations (3 space translations, 3 rotations, 3 boosts, and 1 time translation) can be treated on the same footing.

For me the only point of the principle of relativity is being able to predict if the clocks explode from any frame, with the same result. If that is not possible anymore, the principle of relativity has lost its usefulness to physics.

Then, according to Dirac, if you deny the possibility of non-trivial dynamical boost transformations you must also deny the possibility of non-trivial dynamical time evolution, which is absurd.

It is not the problem of physics, if some abstract mathematical concepts lead to paradoxical results. It just means they are useless to physics, not that nature has to obey the math which some humans thought up.

 

[bcrowell]

Eugene -- Thanks for taking the time to explain those points about CJS in such detail. Much appreciated.

The usual reaction to this theorem is that Hamiltonian formalism is not applicable in relativistic physics.

If this is the usual interpretation, what is your reason for not liking it?

 

[meopemuk]

The usual reaction to this theorem is that Hamiltonian formalism is not applicable in relativistic physics.

If this is the usual interpretation, what is your reason for not liking it?

There are few postulates in physics, which are so simple, powerful and well-verified that they just cannot be wrong. These are (i) the principle of relativity, (ii) the idea that transformations between inertial frames form the Poincare group, (iii) postulates of quantum mechanics. I think E.P. Wigner was first to realize that it follows immediately from these postulates that there exists a unitary representation of the Poincare group in the Hilbert space of any physical system.

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math.,40 (1939), 149.

The 10 Hermitian generators of this representation coincide with total observables in the system - total energy, total momentum, total angular momentum, and total boost operator (=center of mass). The commutators between these operators follow directly from the Poincare group structure. In this theory the time evolution is generated by the operator of total energy - the Hamiltonian. So, the Hamiltonian formalism is an inevitable consequence of the most basic postulates in physics. Any non-Hamiltonian approach to particle dynamics sacrifices one or more postulates. This is unacceptable, in my opinion.

Eugene.

 

[Fredrik]

There are few postulates in physics, which are so simple, powerful and well-verified that they just cannot be wrong. These are (i) the principle of relativity, (ii) the idea that transformations between inertial frames form the Poincare group, (iii) postulates of quantum mechanics.

I agree with that, but isn't the idea that the result of an experiment obtained at a specific point p in spacetime must be independent of the coordinates x(p) that we assign to it even more obviously correct than any of the ideas you mentioned? You seem to be saying that a person can get shot and killed at age 20 in one coordinate system and die of old age at the age of 125 in another.

The Currie-Jordan-Sudarshan theorem sounds interesting, but their proof looks very complicated, so it would take too long to examine it. It seems that others have though. I found http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000019000004000780000001&idtype=cvips&gifs=yes&ref=no for example. I haven't read the article, but the abstract seems to be saying that the theorem is irrelevant unless the Lagrangian contains terms at least of order c^-6.

 

[meopemuk]

You seem to be saying that a person can get shot and killed at age 20 in one coordinate system and die of old age at the age of 125 in another.

Your example is rather extreme. Calculations show that the dynamical effects of boost are rather weak (there are no experiments capable of seeing these effects today). However, as a matter of principle, I would answer "yes". A moving observer can see things very differently than the observer at rest. The difference can go beyond the usual kinematical effects of length contraction and time dilation.

The Currie-Jordan-Sudarshan theorem sounds interesting, but their proof looks very complicated, so it would take too long to examine it.

Yes, the full proof is not so easy to follow. However, the idea behind the proof is rather simple. From Dirac's theory of relativistic interactions

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

it follows that the generator of time translations (the Hamiltonian) and generators of boosts must both contain interaction-dependent terms (here we are talking about the "instant form" of dynamics in which space translations and rotations are interaction-free). This means that boost transformations of particle observables cannot be universal and must depend on interactions acting between particles. Lorentz transformations of special relativity do not have this property. They are assumed to be universal and independent on interactions. This assumption can be realized only in particle systems without interactions. This is the contradiction pointed out by the CJS theorem.

As you can see, conditions of the theorem are very general. It applies to any relativistic interacting theory based on Hamiltonian dynamics and Poincare group. In particular, quantum field theories also belong to this class (see S. Weinberg's "The quantum theory of fields" vol. 1).

Eugene.

 

[meopemuk]

You cannot solve the paradox by presenting a different situation without a paradox. A physical theory has to work in any situation.

The logic of my example was this: If we agree about the Poincare group properties of transformations between inertial observers, then all 10 transformations (time translations, space translations, rotations, and boosts) should be treated on the same footing. Whatever is said about one transformation can apply to other transformations as well. It is obvious that dynamical effects are characteristic to time translations (a bomb seen unexploded by one observer may be seen exploded by a time-translated observer). Therefore, in principle, similar dynamical effects can occur with other interial transformations. I dismiss as unphysical the possibility of dynamical effects of space translations and rotations (I don't think that a bomb seen unexploded by myself can be seen exploded by an observer on the other side of the street). This means that in physics we are dealing with Dirac's "instant form" of dynamics, in which time translations and boosts lead to "dynamical" effects, while space translations and rotations are only "kinematical". It then follows that we should seriously consider the possibility of non-trivial dynamical effects of boosts. For example, the bomb seen unexploded by me may be perceived as exploded by a moving observer.

It is important to note that it is impossible to have a relativistic theory in which dynamical effects are associated only with time translations, while space translations, rotations, and boosts are "kinematical". Such a theory would violate group properties of the Poincare group.

Eugene.

 

[meopemuk]

... but isn't the idea that the result of an experiment obtained at a specific point p in spacetime must be independent of the coordinates x(p) that we assign to it even more obviously correct than any of the ideas you mentioned?

The whole idea of the Minkowski 4-dimensional spacetime and the idea that boost transformations are simply pseudo-rotations in this space-time, i.e., mere changes of coordinate labels, is based on the assumption that Lorentz transformations are exact universal formulas that are equally applicable to all kinds of events independent on their physical nature and on interactions controlling these events.

That's exactly the point of view I am arguing against.

E. V. Stefanovich, "Is Minkowski space-time compatible with quantum mechanics?", Found. Phys., 32 (2002), 673.

Eugene.

 

[Al68]

I agree that your examples are rather odd, but I don't think they are logically impossible.

Clearly, two observers displaced in time with respect to each other have very different views on the same physical system. I don't see why views of two relatively moving observers cannot be just as different. As I explained above, the structure of the Poincare group demands that both time translations and boosts of the reference frame must induce non-trivial dynamical changes in the observed physical system.

Eugene.

I suppose we can disagree about what the word "impossible" means, but if two clocks are local, and stay local, and they stay in synch in one frame, but go out of synch in another, I'd say at least one of the clocks must display a different reading to different observers for the same time in the clock's rest frame.

I would call a clock that does that "two clocks" disguised as a single clock. Logically possible, yes. But that's cheating! :!)

 

[Fredrik]

This all sounds very strange to me. The reason why we can define non-interacting relativistic QM by postulating that there's a symmetry (probability preserving bijection on the set of unit rays) for each restricted Poincaré transformation, is that restricted Poincaré transformations are isometries of spacetime. They therefore define equivalent but different ways to represent operationally defined events as points in a manifold, and for each of those ways there's an equivalent but different way to represent operationally defined states as unit rays on a Hilbert space.

So if we drop Minkowski space, we also loose our reason to keep the Poincaré algebra, and that kills the definition of non-interacting relativistic QM. Besides, I've read that Minkowski space can actually be reconstructed from the Poincaré algebra in the C*-algebraic approach to QM. It was mentioned in this article, which I have only skimmed. (I'm not sure what the significance of that is, but it seems like it should be significant in some way).

I know that things aren't always described the same way in different coordinate systems. For example, the vacuum, which is empty when examined from any inertial frame, is filled with particles when examined from an accelerating frame. (The Unruh effect). But we always assume that everyone will agree that the inertial dectector will not click and that the accelerating detector will. And inertial frames are so special (because they correspond to the isometries of the metric) that it's very hard to believe that things could be described differently from different inertial frames. It also seems to be a huge contradiction of the principle of relativity, which you insist on keeping, even though you're dismissing something that seems to be both much more fundamental and the best reason to believe in the principle of relativity.

I didn't find your article online. I could only find the abstract.

 

[meopemuk]

This all sounds very strange to me. The reason why we can define non-interacting relativistic QM by postulating that there's a symmetry (probability preserving bijection on the set of unit rays) for each restricted Poincaré transformation, is that restricted Poincaré transformations are isometries of spacetime. They therefore define equivalent but different ways to represent operationally defined events as points in a manifold, and for each of those ways there's an equivalent but different way to represent operationally defined states as unit rays on a Hilbert space.

So if we drop Minkowski space, we also loose our reason to keep the Poincaré algebra, and that kills the definition of non-interacting relativistic QM.

Fredrik,

There is a long (logical) distance between two postulates of special relativity (the equivalence of all inertial frames and the invariance of the speed of light) and the idea of Minkowski space-time. To fill this gap one needs to make a few logical steps. The two postulates work quite well when one derives Lorentz transformations for simple systems, like "light clocks", in which interactions do not play any significant role. Then textbooks need to justify why the same transformations can be applied to all other physical systems, where interactions are important. In the worst case the universality of Lorentz transformations and the Minkowski space-time picture is simply postulated without much discussion. In the best case, the (dubious) idea of "incidence relations" is mentioned to make this "logical" step.

I think that the assumption of "incidence relations" (which say that if two events coincide in one frame then they coincide in all other frames) is not supported by anything except wishful thinking. So, I prefer different logic. In this logic relativistic physics does not require introduction of the Minkowski spacetime. The Poincare group is introduced as a group of transformations (space and time translations, rotations and boosts) between different intertial observers. According to Wigner and Dirac, this idea can be combined with quantum mechanics by building a unitary representation of the Poincare group in the Hilbert space of the system. If this representation is known we can find how various observables (e.g., positions of particles) transform with respect to time translations and boosts. So, we can see whether these transformations are the same Lorentz transformations postulated in special relativity or they are different. The answer is that for systems of non-interacting particles Lorentz transformations are reproduced exactly. For interacting particles, there are small (but important) corrections to Lorentz formulas.

I didn't find your article online. I could only find the abstract.

Try http://www.springerlink.com/content/3k48wj87g310v042/fulltext.pdf
but the most complete and up-to-date presentation is in

E. V. Stefanovich, "Relativistic quantum dynamics", http://www.arxiv.org/abs/physics/0504062

 

[atyy]

Try http://www.springerlink.com/content/3k48wj87g310v042/fulltext.pdf
but the most complete and up-to-date presentation is in

E. V. Stefanovich, "Relativistic quantum dynamics", http://www.arxiv.org/abs/physics/0504062

So all the S-matrix quantities are the same, but not the non-S-matrix stuff? What sort of experimental outcome goes beyond the S-matrix?

 

[meopemuk]

So all the S-matrix quantities are the same, but not the non-S-matrix stuff? What sort of experimental outcome goes beyond the S-matrix?

Everything that involves observation of time dependence. Unfortunately, there are not many HEP experiments where time dependence can be seen. The notable exceptions are oscillations (of kaons, neutrinos, etc.) and decays. Decay laws of moving particles have been studied by a number of authors (see references in my post #5). The conclusion is that these decay laws are different from predictions of the standard Einstein's time dilation formula. However, the deviations are several orders of magnitude smaller than the precision of modern experiments. So, standard special relativity remains a pretty good approximation (but still an approximation!).

Eugene.

 

[hamster143]

I think that the assumption of "incidence relations" (which say that if two events coincide in one frame then they coincide in all other frames) is not supported by anything except wishful thinking.

Coincidence is a scalar property and it can only transform under the trivial representation of the Poincare group. Which is the same as saying that it is the same for all observers.

 

[meopemuk]

Coincidence is a scalar property and it can only transform under the trivial representation of the Poincare group. Which is the same as saying that it is the same for all observers.

I haven't heard about such an observable as "coincidence" and I can't say what is the corresponding transformation law.

I am more familiar with particle positions that are described by Newton-Wigner operators in quantum mechanics. Using unitary representations of the Poincare group one can calculate trajectories of interacting particles in different reference frames. In general case, it is *not* true that (as you suggested) if two trajectories intersect in one frame, then they must intersect in all other frames.

Eugene.

 

[A.T.]

It is important to note that it is impossible to have a relativistic theory in which dynamical effects are associated only with time translations, while space translations, rotations, and boosts are "kinematical". Such a theory would violate group properties of the Poincare group.

So? Who cares? All that matters for a physical theory is if its predictions match the observation. And not if the math of the theory can be generalized in an elegant way.

 

[meopemuk]

So? Who cares? All that matters for a physical theory is if its predictions match the observation. And not if the math of the theory can be generalized in an elegant way.

The group property of inertial transformations is not just elegant math. It is a physical necessity. The composition of two transformations is another transformation. The associativity property is also fairly obvious. So, there is no way around groups.

There is no much freedom in choosing the group structure too. The number of more or less suitable 10-parameter Lie groups is very limited (I know about Galilei, Poincare and de Sitter groups). The Poincare group is the best candidate for relativistic physics. All predictions obtained from combining the Poincare group with quantum mechanics are in perfect agreement with observations.

Eugene.

 

[A.T.]

The Poincare group is the best candidate for relativistic physics.

If it really implies what you described here (the same bomb explodes in one frame, but doesn't in other frames) then it contradicts the principle of relativity, which allows you to pick any frame for calculation and yet arrive at the same conclusion about the bombs fate. Therefore I have doubts if that is what the properties of the Poincare group really imply.

All predictions obtained from combining the Poincare group with quantum mechanics are in perfect agreement with observations.

Quantum physics has many-worlds-interpretations. But in classical relativity there is only one bomb, that explodes or not. And every frame has to agree on that. Contrary to what the name "Relativity" suggests, the key of this theory are the absolute (frame invariant) quantities. Actually Einstein preferred the name "Invariantentheorie" (theory of invariants).