Galileo and Lorentz transformation

[Fredrik]

Regarding the first "postulate": I have said this many times before, and I guess I'll have to say it many times again. It isn't a well-defined statement that you can use as the starting point of a derivation! It's often stated in the form "The laws of physics are the same in all inertial frame". The most generous interpretation of this that we can make, is that this represents a set of well-defined statements, that has one member for each definition of "inertial frame", each definition of "law of physics" and each definition of what it means for two laws of physics to be "the same".

 

[meopemuk]

What you've been claiming is the equivalent of claiming that a single clock can display two different times at a single point in spacetime, in two different coordinate systems. This is extremely non-standard, so I find it strange that when I ask you about it, you're replying with a comment about trivial standard stuff.

I didn't say that. First, I have never referred to a "single point in spacetime". I even don't understand what this phrase means. I don't know which experimental measurements can tell two observers whether the events they see occur "at the same spacetime point" or at different spacetime points. I prefer not to use the "spacetime" language at all. Second, I don't like the term "coordinate system". It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events. I think that the terms "inertial observer" or "inertial laboratory" are more appropriate, because they do not exclude the possibility that different observers may actually see different events happening.

Eugene.

 

[meopemuk]

Regarding the first "postulate": I have said this many times before, and I guess I'll have to say it many times again. It isn't a well-defined statement that you can use as the starting point of a derivation! It's often stated in the form "The laws of physics are the same in all inertial frame". The most generous interpretation of this that we can make, is that this represents a set of well-defined statements, that has one member for each definition of "inertial frame", each definition of "law of physics" and each definition of what it means for two laws of physics to be "the same".

I disagree with that. In my opinion the first postulate is the most powerful, deep, and non-trivial statement in all physics. It just tells you that idential experiments in different laboratories produce identical results. It does not matter whether the laboratory is in Paris or in London. It does not matter whether the experiment was made today or a century ago. It does not matter whether the experiment was made in a spaceship standing still or in a spaceship moving with high velocity. Without this relativity principle, it would be impossible to compare results obtained by different researchers. Entire physics would be impossible.

Eugene.

 

[Al68]

Second, I don't like the term "coordinate system". It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events.

If their observations are contradictory, logic dictates that at least one of them is simply wrong.

 

[Dale]

Let's say we have a one-meter rod viewed by two different observers. The observer at rest with respect to the rod will find its length to be 1 meter exactly. The observer moving with respect to the rod will find its length to be shorter than 1 meter. This is called "length contraction". This is what I mean by saying that different observers disagree about measurement results.

This is standard SR length contraction, but even here every frame agrees about the outcome of all measurements. The moving observer agrees that the stationary observer measures 1 m, and the stationary observer agrees that the moving observer measures less than 1 m.

I am saying that, in addition to that, the two observers may disagree about more substantial things. ... like "the bomb does not explode today, while it will explode tomorrow" looks completely normal.

This is just standard relativity of simultaneity. Is there something more to your position or are you just trying to state standard SR in a provocative manner?

 

[meopemuk]

If their observations are contradictory, logic dictates that at least one of them is simply wrong.

Two inertial observers may see quite different events (not just the same events with different space-time labels, as usually postulated). This does not contradict any law of logic or physics.

Eugene.

 

[Dale]

For example?

 

[meopemuk]

I am saying that, in addition to that, the two observers may disagree about more substantial things. ... like "the bomb does not explode today, while it will explode tomorrow" looks completely normal.

This is just standard relativity of simultaneity. Is there something more to your position or are you just trying to state standard SR in a provocative manner?

This has nothing to do with the relativity of simultaneity.

In standard special relativity two moving observers may disagree about such "kinematical" properties as the length of an object or the duration of a time interval. However, they always agree about "dynamical" properties, like whether the bomb is exploded or not.

I am saying that these views must be generalized. Dynamical properties should be considered relative as well. I.e., different observers may disagree about them. It is possible that observer at rest does not see any explosion, while the moving observer (in the same location, at the same time) sees the explosion of the same object.

Eugene.

 

[Dale]

It is possible that observer at rest does not see any explosion, while the moving observer (in the same location, at the same time) sees the explosion of the same object.

This is completely contrary to the first postulate. It is also not appropriate for this forum.

 

[Al68]

Two inertial observers may see quite different events (not just the same events with different space-time labels, as usually postulated). This does not contradict any law of logic or physics.

Of course they may see quite different events, as long as the observations don't contradict each other.

Logic only dictates that two mutually exclusive events don't both happen, so if both are observed, one of the observations is in error.

But this still has nothing to do with transforming the coordinates of a single event between reference frames.

 

[meopemuk]

For example?

I keep repeating the example of a bomb, where two observers disagree about whether the explosion has occurred or not. I use this dramatic and unrealistic example just to make the general point absolutely clear.

More realistic examples concern observations of decays of unstable particles. In the post #5 I've cited a few references in which decay laws of moving particles have been studied in a rigorous relativistic quantum-mechanical setting. It follows, for example, that the particle that is seen as yet undecayed by the observer at rest (at time 0) has a non-zero decay probability from the point of view of the moving observer (at the same time). This is a more realistic analog of the unexploded/exploded bomb discussed above.

Of course, for known unstable particles and realistic observer speeds the "boost induced decay probability" is extremely small and cannot be presently observed. So, the whole issue is rather academic, but I think it is important nevertheless.

Eugene.

 

[meopemuk]

This is completely contrary to the first postulate.

The first postulate tells that experiments in different laboratories yield the same results. This means that each laboratory studies its own copy of the physical system. The first postulate does not say how measurements performed by different observers on the same object are related to each other. In order to find these transformation laws one needs a full dynamical description of the system, i.e., the representation of the Poincare group in the Hilbert space of the system. In the instant form of Dirac's dynamics, this description demands the non-trivial dynamical character of boosts. See, for example

S. Weinberg "The quantum theory of fields" vol. 1. section 3.3.

It is also not appropriate for this forum.

I am ready to stop if you think so.

Eugene.

 

[Fredrik]

I didn't say that.

Actually you did. I said this:

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.

You replied:

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".

In SR, a person's entire existence is represented by a set of curves in Minkowski space. For our purposes, we can ignore the spatial separation between these curves and describe a person's existence approximately using only one curve. The endpoints of the curve represent the beginning and the end of the person's life. If a person gets shot and killed at the age of 20, then the endpoint of the curve that has the higher time coordinate (in all inertial frames) is the mathematical representation of his death in the real world. All of the other points on the curve are mathematical representations of events earlier in his life. Every one of those points represents an event where his age is 20 or less. And you said that there are points on this guy's world line at which his age is 125.

So you have clearly said (possibly without realizing it) that there's a point in Minkowski space where this particular "clock" (a person is a clock too) is displaying 125 years in one coordinate system and 20 or less in another.

First, I have never referred to a "single point in spacetime". I even don't understand what this phrase means.

How can you not? You must know that each point in Minkowski space is supposed to be a representation of an event in the real world, or rather in the universe described by the theory. (I prefer to think of a descriptive theory as SR as an exact description of a fictional universe that resembles our own, than as an approximate description of our universe).

I prefer not to use the "spacetime" language at all. Second, I don't like the term "coordinate system".

Special relativity is by definition a theory that uses a manifold called Minkowski spacetime to represent events. The definition of a manifold includes a bunch of stuff about coordinate systems, and a Lorentz transformation is a transition function between coordinate systems. So if you don't like those things, you must hate special relativity.

Edit: I have now read the posts where you talk about how the principle of relativity says that certain laboratories get the same results. I see what you mean now about laboratories vs. coordinate systems.

It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events. I think that the terms "inertial observer" or "inertial laboratory" are more appropriate, because they do not exclude the possibility that different observers may actually see different event happening.

I haven't completely ruled out that something like what you're suggesting might actually be valid, but the way you're talking about it is really strange. It's like you don't even see that what you're saying is something extremely different from anything that most of us have ever heard of in the context of SR. If it hadn't been you saying this (I've seen threads where you're the only one who gets it right), and if I hadn't read Leonard Susskind's claim that if you fall into a black hole, you pass through the horizon unharmed in one coordinate system and get incinerated by radiation in another, I would have dismissed it as crackpot nonsense right away.

I don't understand Susskind's example either, but at least there's an event horizon in his example to make things more complicated

 

[Fredrik]

It is also not appropriate for this forum.

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

So if some moderator is thinking about closing the thread, please don't. At least not yet.

In my opinion the first postulate is the most powerful, deep, and non-trivial statement in all physics. It just tells you that idential experiments in different laboratories produce identical results. It does not matter whether the laboratory is in Paris or in London. It does not matter whether the experiment was made today or a century ago. It does not matter whether the experiment was made in a spaceship standing still or in a spaceship moving with high velocity. Without this relativity principle, it would be impossible to compare results obtained by different researchers. Entire physics would be impossible.

All of this is true, but it's still ill-defined. It doesn't unambiguously identify which laboratories produce the same results.

I'm going to get some sleep, so I won't read any answers for at least 8 hours.

 

[meopemuk]

Special relativity is by definition a theory that uses a manifold called Minkowski spacetime to represent events.

You and I are talking about two rather different approaches to relativity. They are based on different sets of postulates. Your (as well as Einstein's and lot of other people's) approach assumes the following postulates:

1. The principle of relativity (never mind that you don't like it, this principle remains true nevertheless).
2. The invariance of the speed of light.
3. The Minkowski space-time manifold in which all events are "embedded" (this postulate can be derived from 1. and 2. if you add the "coincidence condition" that we discussed earlier).

One problem with this logic is that it is not easily compatible with quantum mechanics. One example is the difficulty of defining the "time operator" (which, according to your approach, must exist, because space and time coordinates must be "interchangeable"). This difficulty was discussed in one of recent threads in the "Quantum physics" forum.


I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

In my approach I can also describe events with their space and time coordinates. However, these events are not regarded as points in the Minkowski space-time. The whole idea of the Minkowski space-time is just absent. Transformations of space and time labels of events between different observers can be calculated from quantum laws, and these transformations do not necessarily agree with Lorentz formulas, which are assumed exact and universal in your approach. Moreover, if one observer sees an event (e.g., an explosion or a collision of particles), another observer may not see it. The "coincidence condition" is not valid.

As far as I can tell, experimental consequences of the two approaches are pretty close. I couldn't find experiments, where predicted differences can be measured with modern tools. So, which approach is better should be decided on the basis of logic and consistency. You can make your own judgement.

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

So if some moderator is thinking about closing the thread, please don't. At least not yet.

Thank you, Fredrik. I appreciate that.

Eugene.

 

[Fredrik]

You and I are talking about two rather different approaches to relativity. They are based on different sets of postulates. Your (as well as Einstein's and lot of other people's) approach assumes the following postulates:

1. The principle of relativity (never mind that you don't like it, this principle remains true nevertheless).
2. The invariance of the speed of light.
3. The Minkowski space-time manifold in which all events are "embedded" (this postulate can be derived from 1. and 2. if you add the "coincidence condition" that we discussed earlier).

It's not that I don't like the principle of relativity. It's just that it's ill-defined, and therefore useless as a mathematical axiom. 3 can't be derived from 1 and 2. It can be guessed from 1 and 2. Alternatively, we can interpret 1 as a set of well-defined statements and then determine which members of the set are consistent with the assumptions a) that spacetime is the set $\mathbb R^4$ with the standard manifold and vector space (or affine space) structure, and b) that functions that represent a change between coordinate systems are smooth and take straight lines to straight lines. These assumptions can be weakened, but it's definitely never correct to start with an ill-defined axiom.

Both of the approaches I described lead to 3, but the steps that take us from 1 and 2 to 3 (a guess, or a derivation based on clarifying and supplementary axioms) are not a part of the definition of the theory. The theory is defined by axioms that tell us how to interpret the mathematics of Minkowski space as predictions about results of experiments. (But there's of course a rigorous version of 1 and 2 that's implied by the definition of Minkowski space).

One problem with this logic is that it is not easily compatible with quantum mechanics. One example is the difficulty of defining the "time operator" (which, according to your approach, must exist, because space and time coordinates must be "interchangeable").

Not true. The only operators that must exist because of what I've said are the ones that can be constructed from the Poincaré algebra. The fact that Minkowski space has a non-trivial group of isometries make it obvious (once you understand the math, as I think you already do) that we should change the axioms of quantum theory to include the axiom that there's a group homomorphism from that group (the Poincaré group) into the group of probability preserving bijections on the set of unit rays. Wigner's theorem takes care of the rest.

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

It's hard to see a difference between this and my approach, other than that you choose not to mention Minkowski space. I think the proper way to do that is to use the algebraic approach to QM. As I said in #42, according to this article (which I have only skimmed...I intend to return to it later), it's actually possible to reconstruct Minkowski spacetime from the Poincaré algebra (and the axioms of the algebraic approach).

 

[Dale]

The first postulate does not say how measurements performed by different observers on the same object are related to each other.

Yes, it does, all of the experimental results must be the same regardless of which coordinate system is used.

The bomb either explodes or does not explode based on the action of some set of physical laws from some set of boundary conditions. The coordinate system that you use to express those laws and those boundary conditions must lead to the same experimental outcomes in all cases. Otherwise you have violated the first postulate.

 

[Dale]

More realistic examples concern observations of decays of unstable particles. In the post #5 I've cited a few references in which decay laws of moving particles have been studied in a rigorous relativistic quantum-mechanical setting. It follows, for example, that the particle that is seen as yet undecayed by the observer at rest (at time 0) has a non-zero decay probability from the point of view of the moving observer (at the same time).

As long as the rest observer also predicts that the moving observer measures a non-zero probability and the moving observer also predicts that the rest observer measures a zero probability then this is standard fare for SR.

 

[Dale]

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

I also have a nagging suspicion that this is a large miscommunication. Usually I am better at understanding what is being proposed than in this thread.

 

[atyy]

Yes, eq. (60) is an approximation. However, this is a pretty good approximation, as discussed in the beginning of section 11. I believe that the error associated with this approximation is much less than the magnitude of the obtained effect.

Yes, it seems like a good approximation, but why do you think the associated error is less than the obtained effect?

 

[meopemuk]

...it's actually possible to reconstruct Minkowski spacetime from the Poincaré algebra (and the axioms of the algebraic approach).

That's exactly the point where we disagree. Poincare group plus quantum mechanics does not imply (and does not need) the Minkowski spacetime.

Eugene.

 

[meopemuk]

As long as the rest observer also predicts that the moving observer measures a non-zero probability and the moving observer also predicts that the rest observer measures a zero probability then this is standard fare for SR.

Yes, in my approach (which is also the approach used by Wigner, Dirac, and Weinberg, though, unlike me, they never questioned the usefulness of the Minkowski spacetime) there are well-defined rules that connect system's descriptions by the two observers. However, in contrast to standard SR, these rules are more complicated than simple linear Lorentz transformation formulas (x,t) -> (x',t'). The exact boost transformation rules are different for different physical systems, they depend on interactions acting in the system and on the system's state. If F is operator of observable in the reference frame at rest, then operator of the same observable in the moving frame is obtained by formula

$$F(\theta) = e^{-iK_x \theta} F e^{iK_x \theta}$$

where $$K_x$$ is the total boost operator which (similarly to the total Hamiltonian) contains interaction-dependent terms. These terms cannot be avoided in any relativistic interacting theory. The presence of these terms is responsible for the difference between exact transformation laws and (approximate) Lorentz formulas.

Eugene.

 

[meopemuk]

Yes, eq. (60) is an approximation. However, this is a pretty good approximation, as discussed in the beginning of section 11. I believe that the error associated with this approximation is much less than the magnitude of the obtained effect.

Yes, it seems like a good approximation, but why do you think the associated error is less than the obtained effect?

The reason for my belief is that different parameters control the accuracy of formula (60) and the size of the effect (the violation of the Einstein's time dilation formula).

The approximation used in the derivation of (60) is given in eq. (58). In words this means that the position-space wave function of the unstable particle is localized better than the distance passed by light during the particle's lifetime.

On the other hand, the size of the effect predicted by (60) is controlled by the ratio $$\Gamma/m$$, where $$Gamma$$ is the width of the mass distribution and m is the particle's mass.

So, if we choose a particle with a sharply localized wave packet, small mass, and wide mass distribution (=short lifetime), we will minimize the error in formula (60) and we will maximize the effect of violation of the time dilation formula. So, it is possible to have situations in which the magnitude of the effect exceeds the error.

Eugene.

 

[Fredrik]

That's exactly the point where we disagree. Poincare group plus quantum mechanics does not imply (and does not need) the Minkowski spacetime.

I checked the article I mentioned again, and it seems that I remembered it wrong. What they claim to be able to do is something more complicated, and I neither have the time nor the knowledge to fully understand what their doing at this time, so let's drop that part of the discussion.

 

[atyy]

The reason for my belief is that different parameters control the accuracy of formula (60) and the size of the effect (the violation of the Einstein's time dilation formula).

The approximation used in the derivation of (60) is given in eq. (58). In words this means that the position-space wave function of the unstable particle is localized better than the distance passed by light during the particle's lifetime.

On the other hand, the size of the effect predicted by (60) is controlled by the ratio $$\Gamma/m$$, where $$Gamma$$ is the width of the mass distribution and m is the particle's mass.

So, if we choose a particle with a sharply localized wave packet, small mass, and wide mass distribution (=short lifetime), we will minimize the error in formula (60) and we will maximize the effect of violation of the time dilation formula. So, it is possible to have situations in which the magnitude of the effect exceeds the error.

Eugene.

If the time dilation formula is not exact, then does that mean the speed of light is not the exactly the same in all inertial frames?

Edit: I guess the speed of light being constant is usually given by the dispersion relation in free space, ie. no interaction. But your point is that interaction modifies stuff? Also, even in classical SR there are processes where the time dilation formula doesn't apply just because they are not localized in any frame (I think), is the decay process analagous or not in your view? I know I'm being somewhat dense here, thanks for taking the time to answer questions!

 

[meopemuk]

If the time dilation formula is not exact, then does that mean the speed of light is not the exactly the same in all inertial frames?

No, these are totally unrelated issues. The speed of light is always c and this value is observer-independent. To prove that the speed of light is c, I note that light particles - photons - are massless, therefore their energy E is related to their momentum P as E=Pc. From the definition of relativistic speed I then obtain

$$V=Pc^2/E = c$$

The frame independence of this value can be proven by applying the unitary operator of boost transformation to V. For simplicity I consider the case in which the photon is moving along the x-axis, and the boost is apllied along the x-axis as well

$$V_x(\theta)=e^{-iK_x \theta} \frac{P_xc^2}{E} e^{-iK_x \theta}= \frac{(P_x \cosh \theta - E/c \sinh \theta)c^2}{E\cosh \theta -Pc \sinh \theta} = \frac{(P_x c^2(\cosh \theta -\sinh \theta)}{E(\cosh \theta - \sinh \theta)} =\frac{P_x c^2}{E} = c$$

Eugene.

 

[Al68]

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

If that's the case, then it is inappropriate for the SR/GR forum, since SR/GR does not use those postulates.

But I think it should be obvious that some of your claims are clearly incompatible with #1, which says effectively that every reference frame agrees on what does or doesn't physically happen.

 

[atyy]

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

These are absolutely standard, aren't they? So if there is a mistake, it's not at this point.

 

[meopemuk]

But your point is that interaction modifies stuff?

That's exactly my point. Formulas of special relativity are perfectly OK for systems not involving interactions, e.g, in the time clock where a free photon is bouncing between two mirrors. However, if interactions are present (as in the case of unstable particles), then Lorentz transformations and other SR formulas (such as the time dilation law) must be modified to take this interaction into account.

Also, even in classical SR there are processes where the time dilation formula doesn't apply just because they are not localized in any frame (I think), is the decay process analagous or not in your view?

I am not sure what you are talking about? I thought that the time dilation formula is always valid in SR independent on localization.

Eugene.

 

[meopemuk]

But I think it should be obvious that some of your claims are clearly incompatible with #1 [the principle of relativity], which says effectively that every reference frame agrees on what does or doesn't physically happen.

I disagree. The principle of relativity says that two different observers get exactly the same results for experiments with systems confined to their respective laboratories. The principle of relativity says absolutely nothing about how views of different observers on the *same* system are related. For example, it is not possible to derive the length contraction formula from the principle of relativity alone. You need an additional postulate. Usually, the invariance-of-the-spped-of-light postulate is chosen.

Eugene.

 

[Al68]

I disagree. The principle of relativity says that two different observers get exactly the same results for experiments with systems confined to their respective laboratories.

If an observation is made, it was by definition part of their "laboratory". If an event was "outside" their laboratory, it isn't observed at all.

 

[meopemuk]

These are absolutely standard, aren't they? So if there is a mistake, it's not at this point.

Yes, this is a textbook stuff. The best textbook taking this point of view is S. Weinberg "The quantum theory of fields", vol. 1.

The mistake is pretty obvious if you know where to find it. Take any SR textbook and find a place where Lorentz transformations are derived from the two Einstein's postulates. Note that the physical system used in this derivation does not involve interactions. Usually, the derivation involves light pulses or photon bunches (otherwise, the 2nd postulate cannot be applied). It is a mistake to generalize these transfrormation laws to interacting physical systems. This generalization (and subsequent introduction of the Minkowski spacetime) is never properly justified in textbooks.

Eugene.

 

[atyy]

I am not sure what you are talking about? I thought that the time dilation formula is always valid in SR independent on localization.

t'=g.(t-v.x)

t2'-t1'=g.[(t2-v.x2)-(t1-v.x1)]

If x2=x1, then t2'-t1'=g[t2-t1], so the two events must be at the same location in one frame.

I understand your main intuitive arguments are
(i) the usual derivation assumes no interactions
(ii) the Hamiltonian is generates time translations, so if there are interactions, then things are different.

And these do seem quite intuitive to me, it's just that I've not come across your result before, so am being skeptical before I accept it for myself, I suppose like those learning classical SR whom JesseM always helps out - I used to do those detailed calculations from all the different points of view when learning SR, but now having done them in my distant past, I'm happy to accept them to the point where I would rather not calculate that way, since I usually get confused all over again.

I don't even know how to define event if there is no concept of intersecting worldlines, as I think is true in relativistic quantum field theory, and as you point out in the introduction of one of your papers. So I'm wondering if this is why you get a modification to the usual time dilation.

Another way which the your result could make sense to me if it's something like the dispersion relation of light being changed when passing through a material (interaction!)?

 

[meopemuk]

(ii) the Hamiltonian is generates time translations, so if there are interactions, then things are different.

The most important thing that is usually missed in relativity textbooks is that the generator of boosts also must contain interactions (just as the Hamiltonian of any interacting system does). This is inevitable in any theory (either quantum or classical) invariant with respect to the Poincare group. Dirac was first who realized this important point

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

Then it follows that boost transformations of dynamical variables must be different in different interacting systems. They cannot be the same as universal Lorentz transformations of special relativity.

Eugene.

 

[meopemuk]

If an observation is made, it was by definition part of their "laboratory". If an event was "outside" their laboratory, it isn't observed at all.

Let me try to make it more clear. Suppose we have an inertial laboratory A, which observes object a. Suppose also that we have another laboratory B which observes object b. The experimental setups A+a and B+b are exactly the same. The only difference is that they are moving with respect to each other. Then the principle of relativity tells us that all results of measurements in A+a and in B+b are the same.

The principle of relativity does not tell us anything about what observer A will find by making measurements on the object b, or what will be measurement results in B+a. To answer these questions we need to have a full dynamical theory describing the observed system.

For example, suppose that we want to find results of measurements in the pair B+a assuming that results in the pair A+a are known. In quantum mechanics the solution of this problem requires following steps.

1. Construct the Hilbert space H_a describing the physical system a.
2. Define an uinitary representation of the Poincare group in H_a, which is consistent with interactions acting in a.
3. Find the Poincare group element (the inertial transformation) which connects reference frames A and B.
4. Find the unitary operator U in H_a, which corresponds to the inertial transformation in 3.
5. If F is operator of observable measured in the setup A+a, then the same observable measured in the setup B+a should be obtained by formula

$$F' = U FU^{-1}$$

Eugene.