Galileo and Lorentz transformation

[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).

 

[meopemuk]

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 if the Poincare group really imply.

This is not how I see the essence of relativity. In my opinion, a relativistic theory must provide transformation rules, which allow one to calculate states of the system in all reference frames as soon as one knows the state in one particular frame. What I am suggesting is not different. The only difference is that the rules are slightly more complicated than usual length contraction and dilation of time intervals.

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).

Apparently, the property that you call "fate" does not belong to the list of relativistic invariants. This is true in both quantum and classical physics. The many worlds interpretation has nothing to do with it.

Eugene.

 

[Al68]

This is not how I see the essence of relativity. In my opinion, a relativistic theory must provide transformation rules, which allow one to calculate states of the system in all reference frames as soon as one knows the state in one particular frame. What I am suggesting is not different. The only difference is that the rules are slightly more complicated than usual length contraction and dilation of time intervals.

The term "reference frame" is normally used to describe a system to assign space and time coordinates to an event, not a system to determine whether or not the event occurred. That's more than "slightly more complicated". That's a whole new conceptual definition of reference frame, a definition that simply isn't the one used by everyone else.

How can the coordinates of an event be transformed from one reference frame into a different reference frame unless it's the same event?

 

[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.

As I have argued many times before in this forum, there are no logical steps that can accomplish this, for the simple reason that the "postulates" are ill-defined. What you can do is to assume that inertial frames, whatever they will eventually be defined as, must be such that the functions representing a change of coordinates between two frames, take straight lines to straight lines, and form a group. This is only consistent with the Galilei and Poincaré groups. The second postulate is then interpreted as "Let's go with Poincaré".

The postulates are clearly insufficient to define a theory. They are just loosely stated guidelines that can help us guess what theory we should be using. It turns out that the natural choice is to define the theory by a set of axioms that tells us how to interpret the mathematics of Minkowski space as predictions about results of experiments, axioms like "A clock measures the proper time of the curve in Minkowski space that represents its motion".

Are you actually suggesting that this definition of SR is wrong? What then, is the proper definition of SR?

D'oh, I have more to say, but I have to go. Later.

 

[meopemuk]

The term "reference frame" is normally used to describe a system to assign space and time coordinates to an event, not a system to determine whether or not the event occurred. That's more than "slightly more complicated". That's a whole new conceptual definition of reference frame, a definition that simply isn't the one used by everyone else.

How can the coordinates of an event be transformed from one reference frame into a different reference frame unless it's the same event?

You are right, I have a different definition of "reference frame". In my opinion, a better term is "observer" or "laboratory". In my definition, a reference frame is a laboratory fully equipped with measuring devices for all basic observables, which include position, momentum, energy, mass, spin, etc. Time is also measured by the laboratory's clock. Assigning space and time labels to events is just a small part of functions performed by the observer/laboratory. There are many other observables that determine the state of the observed system. It is not a big deal if different observers see different events. A viable relativistic theory must provide transformation laws for connecting system's description between different observers. If these laws are more complicated than relabeling of space-time coordinates, then so be it.

Eugene.

 

[hamster143]

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

Would you agree that the difference of clock readings is a valid observable? I.e. time shown by clock B when clock A reads 12:00 PM. This difference has to transform as a representation of Lorentz group under boosts and rotations. The principle of relativity tells us that there can't be any preferred reference frames, which greatly constrains the choice of representations.

 

[meopemuk]

Are you actually suggesting that this definition of SR is wrong? What then, is the proper definition of SR?

I think that in order to have a fully operational relativistic theory it is sufficient to adopt three postulates:

1. All inertial frames are equivalent (the principle of relativity).
2. Transformations between inertial frames form the Poincare group.
3. Postulates of quantum mechanics.

The most relevant references are

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

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

Eugene.

 

[A.T.]

But in classical relativity there is only one bomb, that explodes or not. And every frame has to agree on that.

Apparently, the property that you call "fate" does not belong to the list of relativistic invariants.

Yes it does. The 'fate' of the bomb is determined by the proper-times of the two clocks, which are frame invariant.

You are right, I have a different definition of "reference frame". In my opinion, a better term is "observer" or "laboratory"

Or a "parallel universe", where things happen, that don't happen in other frames.

 

[meopemuk]

Would you agree that the difference of clock readings is a valid observable? I.e. time shown by clock B when clock A reads 12:00 PM. This difference has to transform as a representation of Lorentz group under boosts and rotations. The principle of relativity tells us that there can't be any preferred reference frames, which greatly constrains the choice of representations.

I think we can agree that positions of clock arms \mathbf{r}_A and \mathbf{r}_B are good observables. Wigner-Dirac relativistic quantum theory allows us to find these positions in the moving reference frame


\mathbf{r}_A (\theta) = e^{-iK^A_x \theta} \mathbf{r}_A e^{iK^A_x \theta}
\mathbf{r}_B (\theta)= e^{-iK^B_x \theta} \mathbf{r}_B e^{iK^B_x \theta}

where K^A_x and K^B_x are total boost operators characteristic for the two clocks A and B, and \theta is the rapidity of the boost. Since two clocks are complex interacting systems, operators K^A_x and K^B_x depend not only on observables of the clocks' arms, but also on observables of other clocks' parts. Therefore, above transformations are complex functions, which cannot be written explicitly without the detailed knowledge of interactions acting inside the two clocks. So, it is not possible to conclude (as you suggest) that from \mathbf{r}_A =\mathbf{r}_B it must follow that \mathbf{r}_A(\theta) =\mathbf{r}_B (\theta).

Eugene.

 

[Fredrik]

1. All inertial frames are equivalent (the principle of relativity).
2. Transformations between inertial frames form the Poincare group.
3. Postulates of quantum mechanics.

In what way are the frames equivalent when they (according to you) don't even agree about measurement results?

How do you define classical SR? #2 is enough to imply Minkowski space, so what does #1 add? And didn't you just argue against Minkowski space?

 

[Dale]

Meopemuk, I think you may be confusing some concepts here, or I am misunderstanding your point. You seem to be erroneously applying a many-worlds interpretation to special relativity where every inertial reference frame is a separate world which can disagree about the existence of physical events.

In MWI if an event happens in this world then it will happen in all reference frames in this world. There may be another world where it does not happen, and in that world it does not happen in any frame. There is no world where it happens in one frame but not another.

If I am misunderstanding your position could you please clarify?