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.