Speed of Light and the Causality Problem

[DrGreg]

We can probably agree that special relativity can be based on three postulates:

1. All inertial reference frames are equivalent.

2. The speed of light does not depend on the velocity of the source or the observer.

3. If space-time coordinates (x,t) of two events N and I coincide in one reference frame, then they coincide in all other reference frames.

Then postulates 1. and 2. are sufficient to prove Lorentz transformations for some simple events (e.g., intersections of light pulses), and postulate 3. allows to extend these transformations to arbitrary events with interacting particles.

Can we agree about that?

Eugene.

When physicists and mathematicians use the word "event" in the context of spacetime, they use it in a specific technical sense. An event in spacetime is the four-dimensional equivalent of a "point" in 3D space. It is something that occupies zero volume in space and whose duration in time is zero. Furthermore if two events have identical coordinates, then there are not two events at all, there is just one event (although there might be more than one way to describe it).

Events are hypothetical mathematical constructs. In the real universe, any phenomenon occupies a non-zero volume of space and persists for a non-zero duration of time. So any real-universe collision is not single event but a whole continuum of events.

Now can you rephrase your objections while using the word "event" in only its correct technical sense?

 

[meopemuk]

When physicists and mathematicians use the word "event" in the context of spacetime, they use it in a specific technical sense. An event in spacetime is the four-dimensional equivalent of a "point" in 3D space. It is something that occupies zero volume in space and whose duration in time is zero. Furthermore if two events have identical coordinates, then there are not two events at all, there is just one event (although there might be more than one way to describe it).

Events are hypothetical mathematical constructs. In the real universe, any phenomenon occupies a non-zero volume of space and persists for a non-zero duration of time. So any real-universe collision is not single event but a whole continuum of events.

I use word "event" with a different meaning. For me "event" is a physical process that occurs in a small region of space during short time interval. For example - collision of two billiard balls, or blink of a lightbulb.

Abstract points in the 4-dimensional Minkowski space-time cannot be observed by any experiment (if there is no real physical system, there is nothing to observe). So, I would prefer to avoid using these abstract points as representatives of events.

Eugene.

 

[cesiumfrog]

Perhaps the OP would benefit from [..] a concrete example of how superluminal velocity can directly lead to causality paradoxes

nota bene: the very first reply gave just that.

A quantum relativistic analysis of this problem was performed in

E. V. Stefanovich, [..] Int. J. Theor. Phys. [..]1996[..]

It was concluded that[..]

You cite yourself in third person?

I don't reject the possibility that one observer does see the explosion while the other observer doesn't.
[..]
we know that if observer O sees an explosion, then observer O' translated or rotated with respect to O sees exactly the same explosion. That's why space translations and rotations are called "kinematical". Now take time translations. Let's say observer O' makes observation 1 day later than O. It is quite possible that O finds the bomb unexploded and O' (one day later) sees that the bomb has exploded. [..] Now, the question is whether boosts are "kinematical" like space translations and rotations (so that observers related by a boost would always agree on the state of the bomb) or boosts are "dynamical" like time translations (so that it is possible that one observer sees the bomb exploded, while the other sees it unexploded)?

I disagree with your explanation: if one observer sees the needle in the "pointing left" state, then your "kinematically" related observer will disagree. In exactly the same superficial way, the "dynamically" related observer will disagree on when (but not whether) the explosion occurred.

Moreover, while different observers may not agree on the time and location (in their respective coordinate systems) of the explosion, they will all agree on what blew up (whether it was a rock, a nearby muon, or the clock, and whether the technician survived).

Eugene, do you accept that all observers will agree on the reality of what blew up? Or are you perhaps arguing that boosts can steer the observer into a particular trouser-leg of the quantum many-worlds?

 

[meopemuk]

I disagree with your explanation: if one observer sees the needle in the "pointing left" state, then your "kinematically" related observer will disagree. In exactly the same superficial way, the "dynamically" related observer will disagree on when (but not whether) the explosion occurred.

I think we are disagreeing in part because we are using different definitions of "observer". In my definition observer is "instantaneous". This observer cannot see time evolution. Time evolution is an inertial transformation between two "instantaneous" observers displaced in time. For example "I today" and "I tomorrow" are two different observers connected by a time displacement transformation. This is important, because with this definition all ten types of inertial transformations of observers (space and time translations, rotations and boosts) can be treated on equal footing and combined into one Poincare group.

With my definition of "instantaneous" observers the non-trivial dynamical character of time translations becomes obvious: the observer "I today" does not see the explosion; the observer "I tomorrow" does see it.

In the quoted paper I demonstrated that the effect of boosts on particle decays is not trivial as well. Let us consider an observer O which is at rest with respect to the muon and which sees 100% muon without any probability of decay products. Time translated observers "O time t later" and "O time t earlier" will see the muon as partly decayed. All this is well-known.

Now we take the point of view of observer O', which moves with respect to O. It can be shown that O' will not find the muon with 100% probability. Moreover neither observer "O' time t later" nor observer "O' time t earlier" will not find the muon with 100% probability. All of them will see a non-zero decay probability for all values of t. So, the whole group of "instantaneous" observers related to O' by time translations would agree that the muon has decayed (exploded).

Eugene.

 

[cesiumfrog]

I think we are disagreeing in part because we are using different definitions of "observer". In my definition observer is "instantaneous".

Let's call your concept an "observation".

There are some facts that two observations must agree on if the transformation that relates them is a spatial rotation, but will disagree on if the transformation is a space and/or time translation. An example of such a fact might be "this observation is of an exploding apple" (it's conceptually cleaner to consider macroscopic facts). In this sense SR would include space-time rotations (i.e. boosts) together with other (purely spatial) rotations, whereas you would place boosts with translations, right?

Now, a major difficulty is how we can formally even say this much if you deny the very space-time itself: how do you intend to specify that two observations are "at the same point" (hence purely boosted)? In SR we would consider an "event" (corresponding to one of the facts described above) and define that all observations of that event are "at the same point (in space-time)"; the events are presumed to represent elements of objective reality. Are you proposing that such defining events be restricted to facts like "this observation is of both billiard balls meeting" and not ".. both electrically repulsive pith balls meeting"?

 

[meopemuk]

Let's call your concept an "observation".

There are some facts that two observations must agree on if the transformation that relates them is a spatial rotation, but will disagree on if the transformation is a space and/or time translation. An example of such a fact might be "this observation is of an exploding apple" (it's conceptually cleaner to consider macroscopic facts). In this sense SR would include space-time rotations (i.e. boosts) together with other (purely spatial) rotations, whereas you would place boosts with translations, right?

Let's take this "exploding apple" as an example. Let us denote "observation" O which is at rest with respect to the apple and is made exactly at the time when the apple explodes. Now using inertial transformations we can obtain a few other "observations". For example, we can translate O 1 meter to the North. This new "observation" O-translated will also see the apple exploding, exactly as O. The only difference is that the point of explosion will have different coordinates with respect to the O-translated.

Another example: we can rotate "observation" O around its axis and obtain O-rotated. Obviously, from the point of view of O-rotated there is exactly the same explosion. Simply it is seen from a different direction. These examples show that space translations and rotations of "observations" do not have any significant effect on what "observations" are seeing. The effect is purely geometrical: translated and rotated observers view the same thing from different distances and angles. I will call these transformations "dynamical".

Now, let us consider time translations of "observations". Suppose that we displaced our "observation" O 1 year back in time. We are simply asking what happened to the apple 1 year ago. Apparently, the result of such a transformation is far from being "geometrical" or trivial. One year ago the apple might not even exist. If we displaced O 1 year forward in time, the O-time-displaced "observation" would not see anything but rotten debris from the explosion. This is very different from what O sees. This means that time translations are "dynamical". Their results depend very much on interaction acting in the observed system (e.g., on the type of explosive inside the apple).

The next question is about boosts. What will an O-boosted "observation" see? Will it see exactly the same exploding apple as O? Surely, the apple seen by the O-boosted will have a non-zero velocity. It will be also affected by a relativistic length contraction. But can we be sure that there will be no other effects? For example, the explosion seen by the O-boosted may change its properties. Or, perhaps, O-boosted may not see any explosion at all (if the velocity of the boost is high enough).

If I understand you correctly, you firmly believe that boost transformations must be purely kinematical (i.e., change of velocity, length contraction,...) and independent on interactions that control apple's dynamics. You believe that somewhat similar to space translations and rotations, the effect of boosts is a simple change of space-time coordinates of events without any effect on the inner composition of the system (i.e., exploded versus un-exploded). You seem to be so convinced about this, that you are ready to use the kinematical interaction-independent character of boosts as the third postulate of relativity. If this postulate plays such an important role, you should be pretty sure that it is correct. What is the basis for your belief?

Eugene.

 

[cesiumfrog]

[..spatially]translated will also see the apple exploding, exactly as O. The only difference is that the point of explosion will have different coordinates [..But a one year ]time-displaced "observation" would not see anything but rotten debris from the explosion. This is very different

No, if you spatially translate your observation by one (light-)year, you will certainly not see the explosion.

It's a bit biased to construct some complicated network of clocks and rulers to (after waiting for distant messages to arrive) describe the explosion as being included in the spatially-translated observation, if you then refuse for that same network to also be used to describe the explosion as included in the time-translated observation.

What is the basis for your belief?

I have absolutely no basis for my belief that there exists an objective reality (e.g., that you exist outside of my imagination), but all of my existing experience supports the presumption that changing my velocity does not change what else is real. Admittedly there are a few postulates of modern physics which do contradict my everyday experience, but I also have verification of many consequences of those postulates.

 

[meopemuk]

No, if you spatially translate your observation by one (light-)year, you will certainly not see the explosion.

I agree about that, but I regard this fact as mere technical annoyance rather than something fundamental.

but all of my existing experience supports the presumption that changing my velocity does not change what else is real.

I also agree that nobody has seen directly any dynamical effect of boosts. So, apparently these effects are rather weak (at accessible velocities). However, there is an argument, which, in my opinion, demonstrates conclusively that these effects should be real.

In relativistic quantum theory we must construct an unitary representation of the Poincare group U_g in the Hilbert space of observed system. This representation tells us how to connect operators of observables in two different reference frames

F' = U_g F U_g^{-1}...(1)

In particular, when g is time translation

U_g = \exp(\frac{i}{\hbar} Ht)

where

H = H_0 +V .....(2)

is the interacting Hamiltonian. Then eq. (1) tells us how observable F evolves in time. The fact that the Hamiltonian contains interaction V explains non-trivial interacting dynamics, such as reactions, decays of apples, etc. Without this interaction term the dynamics of particles would be trivial and boring (all particles move with uniform velocities along straight lines independent on the presence of other particles)

When g is boost with rapidity \vec{\theta}, then

U_g = \exp(\frac{i}{\hbar} \mathbf{K} \vec{\theta})

and eq. (1) tells us how observable F is seen from different moving frames of reference. In particular, eq. (1) should describe for us Lorentz transformations (for time-position, energy-momentum, etc.). It is not difficult to demonstrate that if the operator of boost is non-interacting

\mathbf{K} = \mathbf{K}_0... (3)

then eq. (1) would lead to usual Lorentz transformation formulas. These formulas would be linear and universal (i.e., independent on the type of observed physical system and interactions acting there).

However, if the boost operator contains interaction term

\mathbf{K} = \mathbf{K}_0 + \mathbf{W}...(4)

then boost transformations of observables computed by formula (1) become non-trivial and interaction-dependent.

The punchline is this: It is known from the theory of Poincare group representations (You can find this discussion, for example, in Weinberg's "The quantum theory of fields", vol. 1)that if the Hamiltonian contains interaction dependence (2), then the boost operator cannot remain interaction-independent as in (3). It must have interaction terms, like in (4). So, necessarily, boost transformations of observables must depend on interactions, i.e., they must be "dynamical".

Eugene.

 

[cesiumfrog]

I regard this fact [that real observations are specific to a time and place, not only a time] as mere technical annoyance rather than something fundamental.

So far I don't follow your full argument, but since you treat time and space on such a different footing right from the outset (despite that no actual observation seems to exhibit this difference), I'm not sure I even find it surprising or interesting that the difference carries through to boosts.

If you're actually proposing something that is different from mainstream, can you describe a simple (hypothetical) experiment to demonstrate the point?

 

[meopemuk]

... you treat time and space on such a different footing right from the outset (despite that no actual observation seems to exhibit this difference)...

Space (distance) is measured with rulers. Time is measured with clocks. What can be more different than that?

If you're actually proposing something that is different from mainstream, can you describe a simple (hypothetical) experiment to demonstrate the point?

One possible experiment is described in the references that I gave earlier. Their idea is that the decay law of fast moving particles should deviate from the usual expression given by Einstein's time dilation formula. For known unstable particles these deviations are several orders of magnitude smaller than the precision of modern instruments. So, special relativity remains a very good approximation for all practical purposes.

Eugene.

 

[cesiumfrog]

Does that mean you think there is an absolute frame of reference?

 

[meopemuk]

Does that mean you think there is an absolute frame of reference?

No, it doesn't. I accept both Einstein's postulates (the equivalence of all inertial reference frames and the independence of the speed of light on the velocity of the source and/or observer). So, there is no absolute frame. However, I do not accept the third tacit postulate of special relativity that we discussed above (two events coinciding in one reference frame must coincide in all other frames).

Eugene.

 

[cesiumfrog]

I think what you've shown using QM is that the result of a measurement can depend on the relative velocities of the state preparing system and the measurement apparatus.

It seems like a bit of a jump to deny classical macroscopic reality to the extent of claiming different inertial reference frames will give different answers to such a question as "do these two billiard balls ever collide".

 

[meopemuk]

It seems like a bit of a jump to deny classical macroscopic reality to the extent of claiming different inertial reference frames will give different answers to such a question as "do these two billiard balls ever collide".

I am not sure I am willing to go as far as claiming that two balls colliding in one frame never collide in another moving frame. However, it seems perfectly reasonable to me that if the balls are charged (i.e., interacting) then the space-time coordinates of their collision may not transform by Lorentz formulas between different moving frames.

Actually, this fact (the impossibility of Lorentz transformations of worldlines of interacting particles) is a well-know fact. There is a theorem that says that if worldlines (or trajectories) in a system of particles transform exactly by Lorentz formulas, then the particles must be non-interacting:

D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, "Relativistic Invariance and Hamiltonian Theories of Interacting Particles" Rev. Mod. Phys. 35, 350 - 375 (1963)

Eugene.

 

[Dale]

This argument is becoming absurd. If "observation" O of "events" A and B with spacetime coordinates a and b determines that they are coincident then, by definition a - b = 0 so a = b. Then if "observation" O' is given by a' = f(a) and b' = f(b) then a' = f(a) = f(b) = b' so a' - b' = 0.

The only way for your 3rd postulate to be false is for there to be no such transformation function (implying a preferred reference frame) or for logic to not hold.

 

[meopemuk]

This argument is becoming absurd. If "observation" O of "events" A and B with spacetime coordinates a and b determines that they are coincident then, by definition a - b = 0 so a = b. Then if "observation" O' is given by a' = f(a) and b' = f(b) then a' = f(a) = f(b) = b' so a' - b' = 0.

The only way for your 3rd postulate to be false is for there to be no such transformation function (implying a preferred reference frame) or for logic to not hold.

This is exactly the point I am trying to make. I am suggesting that there is no single universal function f (a.k.a. Lorentz transformation) that can be applied for all events. The transformation of the (space-time coordinates of the) event a may be given by a function f_a, and transformation of the (space-time coordinates of the) event b may be given by function f_b \neq f_a. The transformation functions f_a and f_b may depend on the type of event they are acting on, in particular, on whether the event occurs in an interacting system or in a non-interacting system.

It is not so unusual to have different transformation functions f_b \neq f_a for different events. If we consider the transformation of time translation (instead of the boost discussed above), then it would become obvious that there is no universal function f that would tell us how all physical events develop in time. The time evolution of any system depends on interactions in the system.

Of course, there are also examples to the contrary: i.e., the effect of rotations and space translations is independent on the nature of the event and on interactions, so the equality f_b = f_a is perfectly valid. I am trying to say that boosts may have properties more similar to time translations than to rotations.

Eugene.

 

[Dale]

This is exactly the point I am trying to make.

If you honestly think that you and I are trying to make the same point then that goes pretty far in explaining the absurdity of this discussion.

If a = b then f_b = f_a by substitution. QED. It ain't rocket science, it ain't physics, it ain't even advanced math.

Your 3rd postulate is just silly, if it doesn't hold then the universe cannot be described by math, therefore the universe is fundamentally illogical and there is really no point to doing science since there are no laws and no rules anyway.

 

[pervect]

This is exactly the point I am trying to make. I am suggesting that there is no single universal function ... (a.k.a. Lorentz transformation) that can be applied for all events.

I'm unclear in my own mind if the idea is self consistent (I suspect it is probably not, but I'm not positive).

Is there any literature on this idea, i.e. any peer-reviewed papers that discuss it?

 

[meopemuk]

Is there any literature on this idea, i.e. any peer-reviewed papers that discuss it?

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

Eugene.

 

[meopemuk]

If "observation" O of "events" A and B with spacetime coordinates a and b determines that they are coincident...

If a = b then f_b = f_a by substitution. QED.

Hi DaleSpam,

I am sorry for using a sloppy notation. I actually meant to consider transformation functions f_A and f_B (rather than f_a and f_b). So, these functions depend on the physical nature of events A and B (rather than on their space-time positions a and b). For example, event A can be a "collision of two neutral billiard balls" and event B can be a "collision of two charged billiard balls". Then, it is possible that boost transformation formulas for these different kinds of events are different f_A \neq f_B . So, if the two events happen to have the same space-time coordinates in the reference frame O (a=b), they do not necessarily have the same space-time coordinates in the moving reference frame O' f_A(a) \neq f_B(a). I hope I made it clear now.

Eugene.

 

[Dale]

Yes, that is clear. It is not as trivial as I had thought at first glance, but it is still too poorly defined to discuss reasonably. Once you figure out what f_A and f_B are and can generate some testable predictions then you will have something worth discussing further.

My guess is that you will find any such transformation incompatible with the relativity postulate.

 

[meopemuk]

My guess is that you will find any such transformation incompatible with the relativity postulate.

As I mentioned earlier, two Einstein's postulates remain exactly valid in this approach. If you decide to check these claims for yourself, you are welcome to look at already quoted (peer reviewed) references:

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

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

Eugene.