Speed of Light and the Causality Problem

[Zan]

Supposedly nothing can travel faster than the speed of light because this would violate causality and produce paradoxes. Someone on Planet Alpha will travel to Planet Beta at ten times the speed of light, so that to someone watching from planet Gamma it will LOOK like the person was on Beta before Alpha. But isn't this just a problem with APPEARANCES -- and no more strange than when a stick in water LOOKS like it is where it isn't? How is causality violated or a paradox produced -- just because some observer misinterprets the data before him, and confuses appearance with reality?

 

[cesiumfrog]

It isn't just appearances. See this old post for example.

 

[OOO]

Supposedly nothing can travel faster than the speed of light because this would violate causality and produce paradoxes. Someone on Planet Alpha will travel to Planet Beta at ten times the speed of light, so that to someone watching from planet Gamma it will LOOK like the person was on Beta before Alpha. But isn't this just a problem with APPEARANCES -- and no more strange than when a stick in water LOOKS like it is where it isn't?

Well, in a sense, it is just appearances. In one frame you may arrange a line of instantaneously flashing light bulbs. When your mate travels at some relativistic speed, he will report to you, that he has seen the flashes in succession and that the movement from one flash to the next was faster than the speed of light.

But, as an informed physicist, you know that there is no action traveling faster than light, and so you tell your mate that the causality between the flashes had only been apparent to him, and not real.

Thus, what is appearances, is not just the "velocity" of spacelike separated events (light flashes) but rather the causality between them, which your mate did insinuate implicitely and which can not exist.

How is causality violated or a paradox produced -- just because some observer misinterprets the data before him, and confuses appearance with reality?

That's right. No paradoxes arise if special relativity is interpreted correctly (no causal relationship along spacelike separation).

 

[cesiumfrog]

OOO, that's not right. The OP is asking about actual superluminal travel, not just shadows.

 

[JesseM]

Supposedly nothing can travel faster than the speed of light because this would violate causality and produce paradoxes. Someone on Planet Alpha will travel to Planet Beta at ten times the speed of light, so that to someone watching from planet Gamma it will LOOK like the person was on Beta before Alpha. But isn't this just a problem with APPEARANCES -- and no more strange than when a stick in water LOOKS like it is where it isn't? How is causality violated or a paradox produced -- just because some observer misinterprets the data before him, and confuses appearance with reality?

Do you understand about the relativity of simultaneity? If two events are simultaneous in one frame, then in another frame one happens before the other. In particular, if I send a signal "instantaneously" in my frame so that the event of my sending it and the event of your receiving it happen simultaneously in my frame, then if you are moving away from me, in your frame the event of your receiving the signal happened before the event of my sending it. And if you then send a reply which is "instantaneous" in your frame, so the event of your sending the reply happens simultaneously with the event of my receiving it, then in my frame I receive the reply before you sent it. The end result is that it's possible for me to receive your reply before I sent the first message, so if my first message was about the winning lottery numbers and I received your reply before that, I could know the numbers in advance. If you're familiar with how Minkowski diagrams work, there are some good ones illustrating this type of situation here.

Of course this only works if we assume that every frame is able to send signals instantaneously--if there's only one frame that can do this, you can have FTL without causality problems. But in that case you'd have a preferred reference frame, violating the first postulate of relativity which says the laws of physics work the same way in every inertial frame. So, of relativity, causality, and FTL, you can pick two, but all three can't be valid at once.

 

[OOO]

OOO, that's not right. The OP is asking about actual superluminal travel, not just shadows.

Sorry for my misunderstanding. I got the impression that he asked about the relativity of simultaneity and I thought superluminal travel just served as a tool for asking his question.

(but hopefully you didn't want to dispute the actual content of my post, did you.)

 

[meopemuk]

So, of relativity, causality, and FTL, you can pick two, but all three can't be valid at once.

It would be more precise to say that "Lorentz transformations + FTL" means violation of causality. The principle of relativity (even if you add the invariance of the speed of light to it) does not immediately imply Lorentz transformations.

Eugene.

 

[JesseM]

It would be more precise to say that "Lorentz transformations + FTL" means violation of causality. The principle of relativity (even if you add the invariance of the speed of light to it) does not immediately imply Lorentz transformations.

Eugene.

How do you mean? Normally the Lorentz transformations are derived from the two postulates--what are the loopholes that would allow both postulates to be true but not give you a Lorentz-invariant theory?

 

[Ich]

So, of relativity, causality, and FTL, you can pick two, but all three can't be valid at once.

Well said!

The principle of relativity (even if you add the invariance of the speed of light to it) does not immediately imply Lorentz transformations.

Why not?

 

[meopemuk]

How do you mean? Normally the Lorentz transformations are derived from the two postulates--what are the loopholes that would allow both postulates to be true but not give you a Lorentz-invariant theory?

All proofs that I am aware of use some additional assumptions. For example, they often assume that events (whose times and positions in different frames are considered) are associated with non-interacting systems (such as colliding free particles or intersecting light rays). The insufficiency of the two postulates is clear already from the fact that the second postulate (the invariance of the speed of light) has relevance only to specific kinds of systems - light pulses and has nothing to say about behavior of other systems, such as interacting systems of massive particles.

If you like, we can take apart your favorite "proof" of Lorentz transformations together.

Eugene.

 

[JesseM]

All proofs that I am aware of use some additional assumptions. For example, they often assume that events (whose times and positions in different frames are considered) are associated with non-interacting systems (such as colliding free particles or intersecting light rays). The insufficiency of the two postulates is clear already from the fact that the second postulate (the invariance of the speed of light) has relevance only to specific kinds of systems - light pulses and has nothing to say about behavior of other systems, such as interacting systems of massive particles.

Well, of course it's true that the second postulate deals only with light, but wouldn't the dynamical laws governing any other system (such as the equations that would allow us to predict the dynamics of any system of interacting massive particles, given their initial conditions) be covered by the first postulate?

I suppose since the postulates deal with inertial systems, you do have to assume that inertial coordinate systems are possible, which wouldn't be true globally in curved spacetime, for example. I'm not sure how to best define an inertial coordinate system, perhaps just a coordinate system where anything with a constant coordinate position will experience no G-forces, and where as Einstein says in his 1905 paper "the equations of Newtonian mechanics hold good" (in the appropriate limits of low relative velocities and scales where quantum effects are unimportant). One could also define them in terms of measurements on a physical grid of rigid rulers that were not experiencing G-forces, with clocks attached to each point on the rulers and synchronized using the Einstein synchronization convention. However we define them, would you say that if we assume inertial coordinate systems are possible, then that assumption plus the two postulates is enough to derive the Lorentz transformations, or would you say this is still not enough?

 

[meopemuk]

...would you say that if we assume inertial coordinate systems are possible, then that assumption plus the two postulates is enough to derive the Lorentz transformations, or would you say this is still not enough?

Of course, inertial frames of reference are possible, and I don't have any objections against the first and second Einstein's postulate. However, this is not sufficient to prove Lorentz transformations. If you have a proof that does not involve any additional assumptions (most importantly, it shouldn't tacitly assume that coordinates (x,t) refer to events in non-interacting systems) I would be very interested to see it.


Eugene.

 

[JesseM]

Of course, inertial frames of reference are possible, and I don't have any objections against the first and second Einstein's postulate. However, this is not sufficient to prove Lorentz transformations. If you have a proof that does not involve any additional assumptions (most importantly, it shouldn't tacitly assume that coordinates (x,t) refer to events in non-interacting systems) I would be very interested to see it.

Well, can we imagine as a thought-experiment that we physically instantiate the coordinate system using rigid rulers that have clocks set at each point along the rulers, where the clocks are synchronized using light as in the Einstein clock synchronization convention? In this case, to say an event happened at (x,t) would simply mean that it happened in the immediate locality of the mark "x" on a ruler that lies on or parallel to the x-axis, and that the clock at that marking read "t" at the moment the event occurred...it wouldn't matter what the event actually was (whether it referred to an event in an interacting system or a non-interacting one, for example), just what ruler-marking and clock-tick it happened next to (and for the purposes of a thought experiment 'next to' can mean something like 'infinitesimally close to'). Is this acceptable as the basis for a derivation or do you think it involves additional assumptions beyond "inertial frames of reference are possible"?

 

[meopemuk]

...it wouldn't matter what the event actually was (whether it referred to an event in an interacting system or a non-interacting one, for example),

That's the unjustified assumption I was talking about.

Let us consider, for definiteness, two kinds of events. Events of the first kind are collisions of billiard balls. Each collision is characterized by coordinates (x,t) in the reference frame O and (x',t') in the moving reference frame O'. (let us also assume that the balls have very small radius and that measurement uncertainties are irrelevant). The balls move freely before and after the collisions. In this case, I can agree that (x,t) and (x't') are related to each other by usual Lorentz transformation formulas.

For events of the second kind let us put some (positive) electric charge on each of the balls. The charges are not too high, so the balls can still collide with each other, but now there is some repulsion between them that makes their trajectories curvilinear. My claim is that in this case coordinates (x,t) and (x',t') of balls' collisions cannot be related by Lorentz formulas. There should be some corrections that take into account the interaction between balls.

Special relativity makes an assumption that Lorentz formulas are exactly valid in both cases. Where this assumption comes from? How is it justified?

Eugene.

 

[Integral]

All proofs that I am aware of use some additional assumptions. For example, they often assume that events (whose times and positions in different frames are considered) are associated with non-interacting systems (such as colliding free particles or intersecting light rays). The insufficiency of the two postulates is clear already from the fact that the second postulate (the invariance of the speed of light) has relevance only to specific kinds of systems - light pulses and has nothing to say about behavior of other systems, such as interacting systems of massive particles.

If you like, we can take apart your favorite "proof" of Lorentz transformations together.

Eugene.

So you reject Einstein's 1905 paper?

 

[meopemuk]

So you reject Einstein's 1905 paper?

"Improve" is a better word.

Eugene.

 

[JesseM]

For events of the second kind let us put some (positive) electric charge on each of the balls. The charges are not too high, so the balls can still collide with each other, but now there is some repulsion between them that makes their trajectories curvilinear. My claim is that in this case coordinates (x,t) and (x',t') of balls' collisions cannot be related by Lorentz formulas. There should be some corrections that take into account the interaction between balls.

I don't get it. Were you willing to grant my suggestion that we could assign coordinates based on local readings on an actual ruler/clock system like the one I described? Do you agree that when events happen "next to" each other (in the idealized sense of being infinitesimally close) this property is transitive, so if A happens next to B and B happens next to C, then A happens next to C? If so, just consider some event E that happens at coordinates (x,t). What this means is that event E happens next to the event of the clock at position x on the ruler showing a time t. Likewise, if we say that (x,t) transforms to (x',t') in another coordinate system, this means that the event (clock which is at position x on the ruler shows a time t) on the first ruler/clock system happens next to the event (clock which is at position x' on the ruler shows a time t') on the second ruler/clock system which is in motion relative to the first one. So, if events being next to one another is transitive, then this automatically means E happens next to the event (clock which is at position x' on the ruler shows a time t') on the second ruler/clock system, so it must have coordinates (x',t'). All you really have to worry about is which clock-readings and ruler-markings on one system happen next to which clock-readings and ruler-markings on the other, the event E is actually irrelevant.

 

[meopemuk]

All you really have to worry about is which clock-readings and ruler-markings on one system happen next to which clock-readings and ruler-markings on the other, the event E is actually irrelevant.

Let me formulate your statement in a slightly different way. Suppose that N and I are two events that occur next to each other (they have the same coordinates (x,t)) in the reference frame O. N is an event (collision) with non-interacting balls, and I is an event (collision) with interacting balls. Then your assumption is that these two events will be seen next to each other in all other reference frames O', i.e. (x', t')_N = (x', t')_I. Is it what you are saying?

I agree that if this assumption is made, then we can forget about the nature of events N and I and interactions in corresponding physical systems. Then Lorentz transformations acquire universal interaction-independent status, and they can be regarded as "geometrical" transformations of global space-time coordinates. Then entire formalism of special relativity follows.

But is this assumption obvious? Not for me. Could you please explain why do you believe that this assumption is true?

Eugene.

 

[JesseM]

Let me formulate your statement in a slightly different way. Suppose that N and I are two events that occur next to each other (they have the same coordinates (x,t)) in the reference frame O. N is an event (collision) with non-interacting balls, and I is an event (collision) with interacting balls. Then your assumption is that these two events will be seen next to each other in all other reference frames O', i.e. (x', t')_N = (x', t')_I. Is it what you are saying?

I agree that if this assumption is made, then we can forget about the nature of events N and I and interactions in corresponding physical systems. Then Lorentz transformations acquire universal interaction-independent status, and they can be regarded as "geometrical" transformations of global space-time coordinates. Then entire formalism of special relativity follows.

But is this assumption obvious? Not for me. Could you please explain why do you believe that this assumption is true?

Eugene.

Well, we're idealizing "next to" to mean infinitesimally close...are you saying two events can be infinitesimally close in one coordinate system but not another? This would seem tantamount to saying that physical facts about things like collisions can be different in different coordinate systems--for example, suppose we have two very small physical clocks and N represents the event of one clock showing some time t and I represents the event of the other clock showing some time t', and each clock will stop when an object collides with it, mustn't there be single frame independent truth about what time each shows when they collide and therefore what times they'll show after stopping?

Also, isn't rejecting the notion that "infinitesimally close" is frame-independent tantamount to rejecting that spacetime can be treated as a manifold with a metric on it? Metrics are supposed to provide some frame-independent notion of "distance" between points on a manifold (or paths between points), no?

 

[Dale]

Causality is frame invariant, so there is no problem.

 

[meopemuk]

...are you saying two events can be infinitesimally close in one coordinate system but not another? This would seem tantamount to saying that physical facts about things like collisions can be different in different coordinate systems-

Yes, that's what I am saying. Isn't it what relativity is about? "Physical facts about things can be different in different reference frames". (Note that I don't use words "coordinate systems", because they imply simple relabeling of coordinates by different observers, which is something we haven't proved yet.) Special relativity assumes that the difference is purely kinematical, i.e., different observers can measure different time intervals, angles, distances, etc. However, it may be true that different observers view also dynamical effects differently. For example, where observer O sees a neutron the observer O' may see its decay products (proton + electron + antineutrino). Why not?

-for example, suppose we have two very small physical clocks and N represents the event of one clock showing some time t and I represents the event of the other clock showing some time t', and each clock will stop when an object collides with it, mustn't there be single frame independent truth about what time each shows when they collide and therefore what times they'll show after stopping?

Clocks you are talking about are interacting systems. So, according to my views, their transformations to the moving reference frame can be non-trivial and interaction-dependent. So, I can't tell what different moving observers would see without first examined interactions involved in clock's design.

Also, isn't rejecting the notion that "infinitesimally close" is frame-independent tantamount to rejecting that spacetime can be treated as a manifold with a metric on it? Metrics are supposed to provide some frame-independent notion of "distance" between points on a manifold (or paths between points), no?

Spacetime manifold, metric, and other stuff of special relativity can be introduced only if we decided that transformations to the moving frame of reference are purely kinematical and do not depend on interactions acting in the observed physical system. If this assumption is made, then it follows immediately that Lorentz transformations are universal for all systems (interacting and non-interacting) and can be described in 4D geometrical terms. I am questioning this assumption, so I am not convinced that 4D "geometry" is the exact way to approach relativity.

Eugene.

 

[cesiumfrog]

The principle of relativity (even if you add the invariance of the speed of light to it) does not immediately imply Lorentz transformations.

I am not convinced that 4D "geometry" is the exact way to approach relativity.

Eugene, you're entitled to this view, but if you want to express it then can you see it would be more appropriate to start a separate thread?

Unfounded theories are statistically less likely individually to be fruitful compared to the single theory that has already passed expert scrutiny, and this isn't a research journal. There would be too much noise for anyone here to learn anything, if every request to explain mainstream physics was met with interruptions from every person who has an unproven hunch (and everyone does on at least some small topic) disagreeing someway from the accepted mainstream. Remember, understanding mainstream theories is a goal that is prerequisite to effectively furthering science.

 

[meopemuk]

Remember, understanding mainstream theories is a goal that is prerequisite to effectively furthering science.

"Understanding mainstream theories" was exactly the goal of my posts here. Mainstream special relativity claims that Lorentz transformations can be proved from two Einstein's postulates. I examined many such "proofs" and found that all of them use other tacit assumptions, which I find doubtful. I am asking your help in justifying these assumptions. Are you saying that such inquiries are inappropriate for this Forum? Or that they are not helping "furthering science"?

Eugene.

 

[cesiumfrog]

I was saying counter-mainstream comments are off-topic unless specified otherwise by an OP.

SR was originally obtained not from the dynamics of ideal classical (your "non-interacting") hard spheres, but from electromagnetic interactions. This immediately calls into question your claim that it applies less to particles undergoing such interactions. Also, the fact that nuclear decay (governed by the colour and weak interactions) exhibits the same time dilation is a demonstration that SR is not exclusive to EM phenomena. The successes of GR further validate the whole approach. Nobody doubts that it may take a conceptual revolution to combine GR with QM, but in the classical domain there is no evidence whatsoever to support your criticism of SR.

And how can you even consider that a stopped clock will display different digits to different observers? You're denying classical reality?

 

[meopemuk]

And how can you even consider that a stopped clock will display different digits to different observers? You're denying classical reality?

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.

 

[cesiumfrog]

Sure. But since you postulate 3 explicitly, I wonder what reason (or preferably, evidence) you have have to question it more than the countless other implicit assumptions (eg. 4: validity of logic, and so on)?

 

[meopemuk]

Sure. But since you postulate 3 explicitly, I wonder what reason (or preferably, evidence) you have have to question it more than the countless other implicit assumptions (eg. 4: validity of logic, and so on)?

It is good that we have this postulate spelled out. Now we can look at it and decide whether we like it or not. I hope you would agree that there are no experiments which directly test this postulate (unlike the first and second postulate, which have overwhelming experimental support). So, I find it intriguing to consider how the theory of relativity will look like without this postulate? Shall we encounter any obvious logical or experimental contradictions?

Eugene.

 

[JesseM]

Yes, that's what I am saying. Isn't it what relativity is about? "Physical facts about things can be different in different reference frames". (Note that I don't use words "coordinate systems", because they imply simple relabeling of coordinates by different observers, which is something we haven't proved yet.)

But relativity also has a notion of local facts such as facts about what two clocks were reading at the moment they collided (with this being truly 'local' only in the limit as the size of the clocks approaches zero), which are supposed to be the same in every frame. You seem to be denying this.

Clocks you are talking about are interacting systems. So, according to my views, their transformations to the moving reference frame can be non-trivial and interaction-dependent. So, I can't tell what different moving observers would see without first examined interactions involved in clock's design.

But you think it's logically possible that after the collision when the clocks stopped ticking, one observer would see the clocks stuck at "5 seconds" and "3 seconds" while another observer would see the clocks stuck at "7 seconds" and "2 seconds"? This seems tantamount to believing that different reference frames are like parallel universes...suppose we have a person sitting on one of the clocks, and that clock is programmed to explode (killing the person) if the clock reaches 6 seconds...if different frames disagreed about whether the clock stopped at 5 seconds or 7 seconds in the first version of the experiment where there were no explosives, then in this version would one frame say the clocked stopped at 5 seconds and the person was fine, while another frame says the clock reached 6 seconds before colliding with the other clock, so it exploded and killed the person?

If you don't believe different frames can be like parallel universes in this way, can you specify what it is you think cannot be different in different frames? Like I said, in relativity the answer would be "pairs of facts about the same local region" but you seem to disagree with this answer.

 

[DaveC426913]

Perhaps the OP would benefit from, rather than an academic discussion about the theory, a concrete example of how superluminal velocity can directly lead to causality paradoxes (paradoxen?)

 

[meopemuk]

But you think it's logically possible that after the collision when the clocks stopped ticking, one observer would see the clocks stuck at "5 seconds" and "3 seconds" while another observer would see the clocks stuck at "7 seconds" and "2 seconds"? This seems tantamount to believing that different reference frames are like parallel universes...suppose we have a person sitting on one of the clocks, and that clock is programmed to explode (killing the person) if the clock reaches 6 seconds...if different frames disagreed about whether the clock stopped at 5 seconds or 7 seconds in the first version of the experiment where there were no explosives, then in this version would one frame say the clocked stopped at 5 seconds and the person was fine, while another frame says the clock reached 6 seconds before colliding with the other clock, so it exploded and killed the person?

Hi JesseM,

Yes, this may seem weird, but I don't reject the possibility that one observer does see the explosion while the other observer doesn't.

Actually, this possibility is not as weird as it looks. There are analogs of this situation, which don't look unacceptable. In our discussion we were trying to compare observations of two observers related to each other by the inertial transformation of boost. Boosts are rather unusual transformations, and we don't have much knowledge about observers moving with very high velocities. So, let us first examine more familiar inertial transformations, such as space and time translations and rotations. These transformations form the famous Poincare group together with boosts, so we may expect that they share some common properties.

Let us first look at space translations and rotations. From our experience 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. We say that these drastic differences in observations of O and O' are related to interactions (occurring inside the bomb). We can also say that time translations are "dynamical" inertial transformations.

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)? Logically, you cannot exclude the latter possibility. Actually, there are pretty strong arguments that this latter possibility is more likely than the former one.

It would be difficult to explain these arguments using clocks, bombs, etc as examples, because these are rather complex interacting systems whose precise theoretical treatment is not possible at this moment. However, there is a simple class of interacting physical systems which are analogous to bombs in some respect and which permit rather rigorous mathematical description. I am talking about unstable particles. The particle (e.g., a muon) can exist in two states - undecayed (unexploded) and in the form of its decay products (electron + electron antineutrino + muon neutrino). So, we can investigate how the state of the muon is seen by different inertial observers, including those moving with high velocities. A quantum relativistic analysis of this problem was performed in

E. V. Stefanovich, "Quantum effects in relativistic decays", Int. J. Theor. Phys., 35, (1996), 2539. (see also http://www.arxiv.org/abs/physics/0603043)

It was concluded that, indeed, the muon may look as undecayed for the observer at rest and decayed for the moving observer.

Eugene.

 

[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.