Speed Limit

[Anvoice]

So simultaneity breaks down as we move to relativistic speeds. For example, if we have frame S' moving relative to S with velocity v (positive x-direction), two events separated by SPACE that are simultaneous in frame S' will not be simultaneous in S. Using Lorentz transformations, we can find that if we put an observer in S at x = 0 and t = 0, then time is S' will be different for that observer everywhere except for at x' = 0, where t' = t = 0. Everywhere with

x' < 0 will be "ahead" in time of the point in S (0,0), and every point with x' > 0 would be "behind". If a ship with proper length in S' suddenly disappears in that frame (a single event), in S it would be disappearing in a "wave", from back to nose. NOTE that I am talking about actual occurrence of the event, imagine that the observer at S (0,0) could "know about every occurrence instantaneously, disregarding light speed.

Basically, that observer would advance through time and notice that while when he was at S (0,0), he already saw the ship as having disappearing up to the point where x' = 0 (x' = x = 0, t = t' = 0). Next, the ship will continue disappearing as the observer progresses in time through frame S. Eventually the nose will be gone.

When v is low, the time speed of the "wave" is huge since the time "shift" is very small.

However, what if something could travel faster than the speed of light? If it was fast enough, imagine that as nose of the ship disappears, it sends a faster-than-light particle towards the point S (0,0). Then it is possible for that particle (assuming that speed is possible in our space) should technically be able to surpass the speed of the "wave" and get to point S (0,t) BEFORE the event of the nose disappearing occurs in frame S. If everything that has been said above is correct, the observer at S would receive information from an event that has not occurred in his space. Technically, if he could "act" at a high enough speed, he could interfere with the very occurrence of the event in his space. I think this should be impossible since an event cannot occur in one frame and not occur in another. Does this mean that speed CANNOT be limitless?

Also as v gets close to c, the speed of the wave as derived from the space-time transformation equations becomes smaller and smaller. At v = 99% of c, it works out to about 60 million meters/second. Meaning that if light still traveled at c = 300 million meters/second in that frame, a photon could become the above-mentioned "particle from the future". Does that mean that the speed of light cannot be constant in for all reference frames?

 

If anyone can understand what I have written and answer, I would really appreciate it. I am a bit confused by this but don't really have anyone to ask.

 

Also keep in mind that my experience in quantum mechanics is EXTREMELY small (beginning of a college course in "Nonclassical Physics"), so if you see something horribly wrong with what I have written or if it is easily explained please tell me.

Thank you.

[timo]

Ok, a few notes about what I understood so far. I might or might not elaborate later, depending on mood/time/ability to understand what you said/... .

[... everything before that sentence ...] Does this mean that speed CANNOT be limitless?

At first glance, your example looks like the causality problem (that's not the official name, but "causality" itself is). The idea is pretty much the same as the one you (if I understood you correctly) tried to sketch: You run into paradoxa with time ordering as soon as you allow FTL. The idea boils down to this: Under orthochronous Lorentz transformations (i.e. excluding tricks like t -> -t), the time order of two events is preserved if their connection line is time-like or light-like. So if you have a cause at spacetime point A which ... err ... causes an effect at B, the order of cause and effect will always be the same if cause and effect have time-like or light-like relations (e.g. by being mediated via a particle travelling with v<c or v=c). As soon as you allow them to have space-like relations (which would then require a messenger particle with v>c), the order of cause and effect is not preserved under changes of coordinate systems. That is believed to be paradoxical.

 

 

Also as v gets close to c, the speed of the wave as derived from the space-time transformation equations becomes smaller and smaller. At v = 99% of c, it works out to about 60 million meters/second. Meaning that if light still traveled at c = 300 million meters/second in that frame, a photon could become the above-mentioned "particle from the future". Does that mean that the speed of light cannot be constant in for all reference frames?

I'd have to do some calculations to understand what you did, where you got your numbers from and where a potential error might lie.

 

EDIT: If I define "the speed of the wave" the change in length (as soon as part of the ship started to disappear) with time (i.e. dL/dt), then I get [math]\frac{dL}{dt} = \left(\frac{1}{\beta} - \beta \right) c > c[/math], i.e. always a "vanishing speed" greater than the speed of light. I either misunderstood what you meant with "speed of the wave", or you got some numbers or the calculation wrong. You should possibly present your calculation if you don't find the error.

 

Also keep in mind that my experience in quantum mechanics is EXTREMELY small (beginning of a college course in "Nonclassical Physics"), so if you see something horribly wrong with what I have written or if it is easily explained please tell me.

I am pretty convinced that there is no QM necessary in this thread at all (and also almost convinced that someone will mention it to complicate things a bit, though).

[Anvoice]

Ok, a few notes about what I understood so far. I might or might not elaborate later, depending on mood/time/ability to understand what you said/... .

 

At first glance, your example looks like the causality problem (that's not the official name, but "causality" itself is). The idea is pretty much the same as the one you (if I understood you correctly) tried to sketch: You run into paradoxa with time ordering as soon as you allow FTL. The idea boils down to this: Under orthochronous Lorentz transformations (i.e. excluding tricks like t -> -t), the time order of two events is preserved if their connection line is time-like or light-like. So if you have a cause at spacetime point A which ... err ... causes an effect at B, the order of cause and effect will always be the same if cause and effect have time-like or light-like relations (e.g. by being mediated via a particle travelling with v<c or v=c). As soon as you allow them to have space-like relations (which would then require a messenger particle with v>c), the order of cause and effect is not preserved under changes of coordinate systems. That is believed to be paradoxical.

 

 

I'd have to do some calculations to understand what you did, where you got your numbers from and where a potential error might lie.

 

EDIT: If I define "the speed of the wave" the change in length (as soon as part of the ship started to disappear) with time (i.e. dL/dt), then I get [math]\frac{dL}{dt} = \left(\frac{1}{\beta} - \beta \right) c > c[/math], i.e. always a "vanishing speed" greater than the speed of light. I either misunderstood what you meant with "speed of the wave", or you got some numbers or the calculation wrong. You should possibly present your calculation if you don't find the error.

 

I am pretty convinced that there is no QM necessary in this thread at all (and also almost convinced that someone will mention it to complicate things a bit, though).

 

So as I understand, you DO agree that exceeding the speed of light (maybe to a certain extent), causality could be reversed? In general, this means that events that already occurred could be affected by events that did not occur yet, crudely put "changing the future" and that is a paradox, since it would allow an event to occur in one IF and not in another.

 

As for near-light speeds, your equation [math]\frac{dL}{dt} = \left(\frac{1}{\beta} - \beta \right) c > c[/math]

since beta = v/c

would yield:

(1/(v/c) - v/c)c = (c/v - v/c)c = ((c^2 - v^2)/vc)c = (c^2 - V^2)/v

That is definitely NOT > c at close-to-light speeds, so I don't know why you say "speed of the wave is always > c" if you used this equation.

I did not bother to simplify it, I simply used the Lorentz transformations (simple time dilation can do as well because I measured time intervals from the point of frame synchronization) to get the time SHIFT delta t' for a displacement delta x: the de-synchronization that occurs between time frames. Then I found what that time would translate to in S (again, Lorentz unnecessary because interval begins at t = 0) by multiplying by gamma. Then I simply divide delta x by delta t to find the disappearance SPEED as it looks (as it IS in frame S) to the observer in frame S.

 

If that calculation still makes no sense, I will post the actual math I did (right now I'm a bit low on time).

[ydoaPs]

So as I understand, you DO agree that exceeding the speed of light (maybe to a certain extent), causality could be reversed? In general, this means that events that already occurred could be affected by events that did not occur yet, crudely put "changing the future" and that is a paradox, since it would allow an event to occur in one IF and not in another.

How do you know causality would be reversed? What if the present(and indeed much of the past) is explicitly the result of all future time traveling into what is our past? From what I can tell, your "paradox" is no more of a paradox than that of the Grandfather "Paradox".

[Anvoice]

How do you know causality would be reversed? What if the present(and indeed much of the past) is explicitly the result of all future time traveling into what is our past? From what I can tell, your "paradox" is no more of a paradox than that of the Grandfather "Paradox".

 

I don't think you bothered to read the question I asked. It is not about the time shift in general. What I mean is: does going FTL mean the possibility of "outrunning" the time shift wave from the future into the past? If so, that should be proof of the impossibility of FTL travel. That is what I asked (+ another similar question pertaining to near-light speed). The grandfather paradox does not have much to do with my question.

 

Although you should know that the grandfather paradox is not completely explained, only theories have been put forth regarding the possible explanations. Because if I, with my free will, actually do go back in time into the same exact world where I was born, I DO have the option of killing myself unless I disappear from that world completely, in which case I can still kill myself. Then there are additional mass/energy consequences. In any case, that is a complicated and separate issue.

[timo]

So as I understand, you DO agree that exceeding the speed of light (maybe to a certain extent), causality could be reversed?

Yes. Not only do I agree, but that's a widely spread argument against faster than light phenomena, meaning the idea is not really mine.

 

 

As for near-light speeds, your equation [math]\frac{dL}{dt} = \left(\frac{1}{\beta} - \beta \right) c > c[/math]

since beta = v/c

would yield:

(1/(v/c) - v/c)c = (c/v - v/c)c = ((c^2 - v^2)/vc)c = (c^2 - V^2)/v

That is definitely NOT > c at close-to-light speeds, so I don't know why you say "speed of the wave is always > c" if you used this equation.

Because I screwed up that statement.

 

I did not bother to simplify it, I simply used the Lorentz transformations (simple time dilation can do as well because I measured time intervals from the point of frame synchronization) to get the time SHIFT delta t' for a displacement delta x: the de-synchronization that occurs between time frames. Then I found what that time would translate to in S (again, Lorentz unnecessary because interval begins at t = 0) by multiplying by gamma. Then I simply divide delta x by delta t to find the disappearance SPEED as it looks (as it IS in frame S) to the observer in frame S.

 

If that calculation still makes no sense, I will post the actual math I did (right now I'm a bit low on time).

Not completely sure if I understand; I wonder where you left length contraction. Anyways, to get some common point. Assume the ship exists during some time interval [math][0,\tau][/math] (where the zero could as well be -infinity, but I want to get a rectangle in the spacetime diagram to make life possibly easier) in a space interval [0,L] (meaning it has a length of L). So in its frame, the edges of the rectangle it forms in spacetime-coordinates (x,t) are (0,0), (0,L), [math](\tau,L), (\tau, 0)[/math].

 

After a boost to another frame, these coordinates of the edges are (0,0), [math](\beta \gamma L, \gamma L), (\gamma \tau + \beta \gamma L, \beta \gamma \tau + \gamma L), (\gamma \tau, \beta \gamma \tau)[/math]. You could get the length at any time t' from that as I did, but I already wondered which information that should carry. The real question (if I got your original post correctly) is: From which point within this quad to which other point do you want to send a signal?

 

An interesting idea would be sending along or faster than the upper line , the one where parts of the ship disappear (meaning a vanishing part would communicate something to a not-vanished part). This line obeys the equation [math](\gamma \tau, \gamma \beta \tau) + \alpha (\gamma \beta, \gamma)[/math] (with alpha just being the line parameter) and now this gradient is faster than lightspeed (because beta <1).

 

As a sidenote why the vanishing speed might not suffice: The speed with which one of the sides disappears is not equal to the vanishing speed, but equal to the vanishing speed plus the speed with which the ship is already moving itself. Iow, to get the speed with which the "place of vanishing" moves away, you'd have to add another v/c to my equation about the vanishing speed, leaving you with c²/v, which then is indeed >1.

 

General remark: In case you didn't do it already, drawing a spacetime diagram x vs. ct (so that the speed of light is a line with a slope of 1) is often, and imho in this case, very helpful. I personally see little to nothing from the numbers alone.

[Anvoice]

Yes. Not only do I agree, but that's a widely spread argument against faster than light phenomena, meaning the idea is not really mine.

 

 

Not completely sure if I understand; I wonder where you left length contraction. Anyways, to get some common point. Assume the ship exists during some time interval [math][0,\tau][/math] (where the zero could as well be -infinity, but I want to get a rectangle in the spacetime diagram to make life possibly easier) in a space interval [0,L] (meaning it has a length of L). So in its frame, the edges of the rectangle it forms in spacetime-coordinates (x,t) are (0,0), (0,L), [math](\tau,L), (\tau, 0)[/math].

 

After a boost to another frame, these coordinates of the edges are (0,0), [math](\beta \gamma L, \gamma L), (\gamma \tau + \beta \gamma L, \beta \gamma \tau + \gamma L), (\gamma \tau, \beta \gamma \tau)[/math]. You could get the length at any time t' from that as I did, but I already wondered which information that should carry. The real question (if I got your original post correctly) is: From which point within this quad to which other point do you want to send a signal?

 

An interesting idea would be sending along or faster than the upper line , the one where parts of the ship disappear (meaning a vanishing part would communicate something to a not-vanished part). This line obeys the equation [math](\gamma \tau, \gamma \beta \tau) + \alpha (\gamma \beta, \gamma)[/math] (with alpha just being the line parameter) and now this gradient is faster than lightspeed (because beta <1).

 

As a sidenote why the vanishing speed might not suffice: The speed with which one of the sides disappears is not equal to the vanishing speed, but equal to the vanishing speed plus the speed with which the ship is already moving itself. Iow, to get the speed with which the "place of vanishing" moves away, you'd have to add another v/c to my equation about the vanishing speed, leaving you with c²/v, which then is indeed >1.

 

General remark: In case you didn't do it already, drawing a spacetime diagram x vs. ct (so that the speed of light is a line with a slope of 1) is often, and imho in this case, very helpful. I personally see little to nothing from the numbers alone.

 

Thanks for the hints.

 

However, the Lorentz transformations take into account the speed at which the ship is moving. In fact, the ship is only an example: what I am really talking about is the real time-shift of another frame. If you look at point S (x = 0, t = 0) and find the time t' = gamma (t - vx/c^2) for t = 0 x = some length x, you will, you will find the time shift dt'. However, the observed time interval it takes to get from S(0,0) to the point where x = 1 and t' gamma (t - vx/c^2) is dilated by gamma. Basically, the observed time (t - 0) in S it takes for the vanishing wave to travel from S (0,0) to S(x,t) is gamma^2 (t - vx/c^2). Since the velocity of the wave in frame S is the differential displacement / differential time interval in THAT frame, it becomes (unless I made a mistake):

 

speed(wave) = x / (gamma^2 (t - vx/c^2))

 

Now, that is my equation. If it is done incorrectly, then you will hopefully see where.

 

 

All that is left is to plug in the values. Since t at the moment we are concerned with is 0, x does not matter. Let's imagine v = .99c. Then plugging that into this equation, I got

speed(wave) = 6030303.03 m/s (actually it was negative, but that is because I forgot to reverse the sign for the negative time interval in the equation).

 

Once again, ONLY if I made no mistakes (that is what I want to know, if I am wrong HOW am I wrong), speed(wave) is considerably less than the speed of light. If a light beam still traveled at c relative to this frame, then it would easily exceed the speed of the wave.

 

Feel free to destroy this if you see the error, this is driving me crazy.