Galileo and Lorentz transformation

[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.

 

[Al68]

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.

Sure, but that's not analogous to transforming an event from one frame to another.

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.

Yes, it does. It says that object b (and a) will obey the same laws of physics in A as it does in B.

A single object observed by different frames obeys the same laws of physics in both. That's the whole subject of this thread, and the context of the first postulate in SR.

 

[meopemuk]

A single object observed by different frames obeys the same laws of physics in both. That's the whole subject of this thread, and the context of the first postulate in SR.

It is impossible to argue with that. Of course, all objects in all reference frames obey the same laws of physics. The interesting question is "what these laws of physics are?"

For example, how can we find properties of the system 'a' observed from the reference frame B? If the only information given to us are the results of measurements of 'a' performed in the reference frame A. In other words, we need transformation laws, which connect observables in A with observables in B. The principle of relativity alone is not sufficient to obtain these transformation laws. Even the knowledge that inertial transformations connecting frames A, B, C,... form the Poincare group is not sufficient to determine the transformation laws for observables. This problem can be solved if we know the representation of the Poincare group in the Hilbert space of the system 'a'. In classical mechanics, the same task is fulfiled by constructing the appropriate representation of the Poincare group by canonical transformations in the phase space of the system 'a'.

Eugene.

 

[Al68]

It is impossible to argue with that. Of course, all objects in all reference frames obey the same laws of physics. The interesting question is "what these laws of physics are?"

For example, how can we find properties of the system 'a' observed from the reference frame B? If the only information given to us are the results of measurements of 'a' performed in the reference frame A. In other words, we need transformation laws, which connect observables in A with observables in B. The principle of relativity alone is not sufficient to obtain these transformation laws.

I agree with all of that, but I don't see the point.

In the context of an explosion, if the explosion is a result of the laws of physics, then it occurs in all reference frames according to the first postulate. Of course the laws of physics don't tell us how to label the coordinates. Neither does the first postulate.

In fact I can arbitrarily define a coordinate system any way I choose, with arbitrary transformation laws, without violating the laws of physics or the first postulate. But whether or not the explosion occurs or not is completely independent of my choice.

 

[Fredrik]

Eugene, I agree that there's nothing logically inconsistent about your interpretation of the principle of relativity, but don't you see how different it is from standard SR? Just about every calculation in SR is based on the axiom that physical events are (coordinate independent) points in Minkowski space.

Consider e.g. the method we use to prove that two laboratories that don't have the same velocity will measure different lengths of an object that's co-moving with one of the laboratories. The co-moving laboratory measures the proper distance between the endpoints at two events that are assigned the same time coordinate by the coordinate system that's associated with its motion. The other laboratory also measures a proper distance, but between two different events. To understand what the result will be, the first thing we have to do is to figure out which two events that is. We can e.g. draw a spacetime diagram that shows all the relevant events mapped to \mathbb R^2 by the coordinate system associated with the motion of the co-moving laboratory. The world lines are vertical in this diagram, and we can prove that events that the other laboratory considers simultaneous are on a line with slope v, and from that we can figure out which two events the other laboratory will consider, and what the result will be.

When we use your way of looking at things, that object may not even exist to the second laboratory. That may not be logically inconsistent, or even in contradiction with every possible interpretation of the principle of relativity, but it certainly isn't special relativity.

 

[Al68]

When we use your way of looking at things, that object may not even exist to the second laboratory. That may not be logically inconsistent, or even in contradiction with every possible interpretation of the principle of relativity, but it certainly isn't special relativity.

It would be pretty difficult to observe an object violate the laws of physics if the object just isn't observed at all. Surely that's not what meopemuk is talking about?

 

[Dale]

I would like to reiterate my suggestion that this is inappropriate for this forum. We have given meopemuk more than enough time to say something remotely related to standard SR, which he has not. The present discussion does not belong here. Particularly the very non-mainstream interpretation of the first postulate.

Of course, for known unstable particles and realistic observer speeds the "boost induced decay probability" is extremely small and cannot be presently observed.

In meopemuk's own words, it is at this time a speculative theory with differences from SR that are so small as to live within the errors of all current theories. Once there is some reputable experimental evidence supporting this theory then it would be appropriate to discuss it in another portion of PF, but even then it will not belong in the relativity sub-forums.

 

[meopemuk]

Eugene, I agree that there's nothing logically inconsistent about your interpretation of the principle of relativity, but don't you see how different it is from standard SR? Just about every calculation in SR is based on the axiom that physical events are (coordinate independent) points in Minkowski space.

Consider e.g. the method we use to prove that two laboratories that don't have the same velocity will measure different lengths of an object that's co-moving with one of the laboratories. The co-moving laboratory measures the proper distance between the endpoints at two events that are assigned the same time coordinate by the coordinate system that's associated with its motion. The other laboratory also measures a proper distance, but between two different events. To understand what the result will be, the first thing we have to do is to figure out which two events that is. We can e.g. draw a spacetime diagram that shows all the relevant events mapped to \mathbb R^2 by the coordinate system associated with the motion of the co-moving laboratory. The world lines are vertical in this diagram, and we can prove that events that the other laboratory considers simultaneous are on a line with slope v, and from that we can figure out which two events the other laboratory will consider, and what the result will be.

Fredrik,

I am fully aware that my approach is different from standard SR. It suggests a different solution for the length contraction problem, which does not involve construction of space-time diagrams.

Let me choose two particles on the opposite ends of the stick and denote their position operators (or their position dynamical variables in the classical case) in the frame at rest by x_1 and x_2. I use 1-dimensional case for simplicity. Then the length of the stick in the rest frame is L = |x_1-x_2|. Then I switch to the moving frame description. I denote the boost operator by K_x, and obtain the length of the stick in the moving frame by usual quantum-mechanical formula

L' = e^{-iK_x \theta} L e^{iK_x \theta}......(1)

If interactions between atoms in the stick are weak, then this result will not be different from the usual SR length contraction formula

L' = L/\cosh \theta..........(2)

However, for very strong interactions, results (1) and (2) will be different.

When we use your way of looking at things, that object may not even exist to the second laboratory. That may not be logically inconsistent, or even in contradiction with every possible interpretation of the principle of relativity, but it certainly isn't special relativity.

Yes, this is not the standard special relativity. For example, the operator K_x in (1) may contain interaction terms that lead to decays of particles 1 and 2. Then, in the moving frame even the particle content of the stick can be altered. So, strictly speaking, the notion of the "length of the stick" will be altered as well.

I would like to reiterate my suggestion that this is inappropriate for this forum. We have given meopemuk more than enough time to say something remotely related to standard SR, which he has not. The present discussion does not belong here. Particularly the very non-mainstream interpretation of the first postulate.

Even if you don't buy my arguments, I think in our discussion we touch some basic and interesting issues regarding the logical structure of special relativity. A critical discussion of SR postulates could be benefitial in learning this theory by everyone, IMHO.

On the other hand, I understand that I over-used your hospitality on this forum. So, if you decided to lock this thread, I will not be offended.

Eugene.

 

[meopemuk]

In the context of an explosion, if the explosion is a result of the laws of physics, then it occurs in all reference frames according to the first postulate.

The first postulate tells us that (in the notation I've used above) if an explosion is seen in the setup A+a, then an explosion must be seen also in the setup B+b. However, the principle of relativity does not tell us what we should see in the experimental setups A+b and B+a. It might be true that in these combinations Observer+object explosions are not observed. I admit that such a possibility is odd, but it does not violate the principle of relativity at all.

Eugene.

 

[Fredrik]

However, for very strong interactions, results (1) and (2) will be different.

I don't see how the strength of the interaction enters into the calculation. These are the things I think I do see:

The Hilbert space of two non-interacting systems is the tensor product of the two component systems. Each of those admits a representation of the Poincaré group (or its covering group), and we can use those to construct a representation on the tensor product space. For example, the Hamiltonian is defined as H=H_1\otimes H_2, and the other generators are defined the same way. An interaction between these systems is defined as a modification of the Hamiltonian H → H+V that entangles the two systems. (What I mean by this is that the time evolution operator applied to an unentangled state gives us an entangled state). When we do this, we also have to modify the boost generator, K → K+W, to ensure that the commutation relations are still satisfied.

You have defined a length operator L, and you're using the modified boost generator to transform it. I'm OK with that. What I don't see is how to verify the statement I quoted from your post.

I don't think any of this looks like an argument against Minkowski spacetime. We're adding that term to the boost generator to preserve the commutation relations that we get from the assumption that spacetime is Minkowski space. So it looks more like an argument for Minkowski spacetime than against it.

Yes, this is not the standard special relativity. For example, the operator K_x in (1) may contain interaction terms that lead to decays of particles 1 and 2. Then, in the moving frame even the particle content of the stick can be altered. So, strictly speaking, the notion of the "length of the stick" will be altered as well.

As long as we're assuming the symmetries of Minkowski spacetime are symmetries of the quantum theory, I'd say that it is standard SR, or rather standard special relativistic quantum mechanics. If the particles can decay in one frame while not doing so in the other, then the whole object could disintegrate before the experiment is over. If this is in fact what we get from simply combining SR and QM, I'd say it's a pretty significant result. But I strongly doubt that we can get anything that extreme from SRQM.

 

[Fredrik]

I would like to reiterate my suggestion that this is inappropriate for this forum.
...
In meopemuk's own words, it is at this time a speculative theory with differences from SR that are so small as to live within the errors of all current theories. Once there is some reputable experimental evidence supporting this theory then it would be appropriate to discuss it in another portion of PF, but even then it will not belong in the relativity sub-forums.

The thread has drifted far from the original topic, and isn't even about classical relativity anymore. I agree that this discussion doesn't belong in the relativity forum. But I think it would be OK in the quantum physics forum. Meopemuk has been describing this as a speculative non-standard theory, but most of it is just the standard way to combine SR and QM. Now he's suggesting that this (very mainstream) theory has some implications that we have so far been unaware of (some of them pretty extreme), and even though I think some of his ideas are based on misunderstandings, I still think this is a meaningful discussion.

If some moderator would like to take action because this stuff doesn't belong in the relativity forum, then I would rather have it cut out and put into a new thread in the quantum physics forum, than to have the thread locked or posts deleted.

 

[meopemuk]

I don't think any of this looks like an argument against Minkowski spacetime. We're adding that term to the boost generator to preserve the commutation relations that we get from the assumption that spacetime is Minkowski space. So it looks more like an argument for Minkowski spacetime than against it.

The best argument against Minkowski spacetime is the Currie-Jordan-Sudarshan theorem that I've mentioned earlier

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

This theorem says that in any relativistic theory (where both the Hamiltonian and the boost operator contain interaction terms) world-lines (or trajectories) of particles do not transform by usual Lorentz formulas. So, the Minkowski space-time picture is not applicable.

Eugene.

 

[Al68]

The first postulate tells us that (in the notation I've used above) if an explosion is seen in the setup A+a, then an explosion must be seen also in the setup B+b. However, the principle of relativity does not tell us what we should see in the experimental setups A+b and B+a.

Of course it does. It tells us that b will obey the same laws of physics in A as it does in B.

It might be true that in these combinations Observer+object explosions are not observed. I admit that such a possibility is odd, but it does not violate the principle of relativity at all.

Yes, it does. Even in Newtonian physics, the object b obeys F=ma in both A and B. So does object a.

This is the whole reason for the first postulate in SR. That a single object obeys the same laws of physics in different reference frames.

 

[meopemuk]

The first postulate tells us that (in the notation I've used above) if an explosion is seen in the setup A+a, then an explosion must be seen also in the setup B+b. However, the principle of relativity does not tell us what we should see in the experimental setups A+b and B+a.

Of course it does. It tells us that b will obey the same laws of physics in A as it does in B.

If explosion occurs in the setup A+a, then there is no guarantee that the same explosion is seen in the setup B+a. Let me give you an example.

Suppose that object 'a' is a time bomb. Suppose also that two inertial observers A and B are related to each other by a time translation transformation. For example, observer B makes his observations 1 hour earlier than A. Now, we have agreed that A sees an explosion of the time bomb 'a'. One hour before this observation the bomb was intact. Therefore, no explosion is seen in the setup B+a.

If you agree that my example with time translations is correct, then there is only a little step to change this example by replacing the time translation with a boost. My claim is that if observers A and B are related to each other by a boost, we can get a similar situation: explosion is observed in the setup A+a and not observed in the setup B+a.

Eugene.

 

[Al68]

If explosion occurs in the setup A+a, then there is no guarantee that the same explosion is seen in the setup B+a. Let me give you an example.

Suppose that object 'a' is a time bomb. Suppose also that two inertial observers A and B are related to each other by a time translation transformation. For example, observer B makes his observations 1 hour earlier than A. Now, we have agreed that A sees an explosion of the time bomb 'a'. One hour before this observation the bomb was intact. Therefore, no explosion is seen in the setup B+a.

If you agree that my example with time translations is correct, then there is only a little step to change this example by replacing the time translation with a boost. My claim is that if observers A and B are related to each other by a boost, we can get a similar situation: explosion is observed in the setup A+a and not observed in the setup B+a.

Eugene.

Your example has nothing to do with whether or not the explosion happens. It either happens or not.

The only thing that varies is the time and space coordinate that is assigned to it. Which means if the explosion is assigned the time noon in A, and 1 pm in B, then it's true that the explosion already happened at 12:30 in A, but had not happened at 12:30 in B. But the difference is only with the time coordinate of the explosion assigned by each frame.

But that's equivalent to saying that observers on the east and west coast disagree about whether or not the Superbowl kickoff happened, since at 5 pm on the west coast it has happened, but at 5 pm on the east coast it hasn't happened.

Does that mean that whether or not the Superbowl kickoff happens or not depends on which time zone you're in?

What about someone who's TV is broke? Does whether or not the kickoff happened depend on when/whether it is actually observed?

 

[A.T.]

For example, observer B makes his observations 1 hour earlier than A.

You mean he opens his eyes for a second, says "Ahh no explosion yet", and keeps sitting on the bomb that will blow him to pieces 1h later?

Now, we have agreed that A sees an explosion of the time bomb 'a'.

Let also say that A sees B sitting on that bomb while it explodes. It is a possible scenario.

One hour before this observation the bomb was intact. Therefore, no explosion is seen in the setup B+a.

Yeah B will not see much with his eyes closed, but he will still notice the bomb blast under his butt.

Bottom line is: You claiming that events don't happen in some frames, just because the observer 'wasn't making his observation' at that particular time. This might be consistent with your private definition of frames of reference and observers, but is rather a useless concept from practial standpoint. And it is definitely not what Relativity says.