Causality Violation: Exploring Group Velocity & Its Reversal Effects

[ZealScience]

Exceeding the speed of light would necessarily violate causality to some reference frames. I think the derivation is related to group velocity. How does it show that if c is exceeded, cause and results could be reversed to some reference frames? And why it is so significant?

 

[bcrowell]

http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html See the discussion of figure i. It's not related to group velocity.

And why it is so significant?

Typically I think people find it interesting because they usually believe strongly that causality is necessary, and therefore they take this as a demonstration that FTL is impossible. It's also a useful argument for explaining why SR's prohibition of FTL is not just a prohibition of FTL motion for particles, but a much more general prohibition of any kind of FTL transmission of cause and effect (e.g., transmission of information). This highlights the fact that relativity is fundamentally a theory of causality.

 

[Saw]

I think people find it interesting because they usually believe strongly that causality is necessary, and therefore they take this as a demonstration that FTL is impossible. It's also a useful argument for explaining why SR's prohibition of FTL is not just a prohibition of FTL motion for particles, but a much more general prohibition of any kind of FTL transmission of cause and effect (e.g., transmission of information). This highlights the fact that relativity is fundamentally a theory of causality.

I am a little lost with this sort of comments you make from time to time. Let me tell you my view and you might yourself comment if you disagreed. Causality cannot be violated because it is not a theory, it is a fact. A conceptual interpretation may be subject to exceptions. But if I scratch my nose too much and due to that cause I suffer the effect that my nose aches..., that is it, no way out, no exception. As to FLT, I believe it is impossible for physical reasons. But if by chance one day a means for FLT were discovered, causality would not in the least suffer for that reason. And if some conceptual interpretation, some understanding of SR held that causality should in that case suffer, that would only mean that such interpretation is in that respect flawed, but my nose should keep aching, as judged from any reference frame...

 

[bobc2]

I am a little lost with this sort of comments you make from time to time. Let me tell you my view and you might yourself comment if you disagreed. Causality cannot be violated because it is not a theory, it is a fact. A conceptual interpretation may be subject to exceptions. But if I scratch my nose too much and due to that cause I suffer the effect that my nose aches..., that is it, no way out, no exception. As to FLT, I believe it is impossible for physical reasons. But if by chance one day a means for FLT were discovered, causality would not in the least suffer for that reason. And if some conceptual interpretation, some understanding of SR held that causality should in that case suffer, that would only mean that such interpretation is in that respect flawed, but my nose should keep aching, as judged from any reference frame...

Hi Saw,

"...but my nose should keep aching, as judged from any reference frame..."

What if I found many reference frames for which the nose ache occurred before the scratch?

 

[Superstring]

Look at the Lorentz Transformation for the time separating two events:

$$\Delta t'=\gamma (\Delta t-v\Delta x/c^2)$$

The primed frame sees the events reversed in time (i.e. $\Delta t'$ is negative) when $v\Delta x/c^2 >\Delta t$.

Restated:

$\Delta t'<0$ when $v\Delta x/c^2 >\Delta t$.

Rearranging the condition gives:

$$\frac{v}{c} > \frac{c\Delta t}{\Delta x}$$$c\Delta t$ can be though of as the distance that light travels in the time separating the events. $v$ is the velocity of the primed frame with respect to the unprimed frame, which must always be less than c (or you get imaginary numbers out of the LT). The left side of the above inequality must always, therefore, be less than one. The only way for the primed frame to see the events reversed is therefore when $\Delta x > c \Delta t$. In other words, the distance in space separating events must be large enough so that light does not have enough time to travel from the first event to the second event. It is therefore concluded that if light does not have enough time to travel from event A to event B, then event A could not have possibly caused event B, and therefore no signals can travel faster than light. Why?

Say, for example, that you send out a faster-than-light signal and a short time afterwards I shoot and kill you. Another person who is at a distance and moving at a velocity such that $v/c > c\Delta t/ \Delta x$ receives your signal. But who sent it? From his frame, your death preceded your transmission of the signal.

 

[Saw]

therefore no signals can travel faster than light. Why?

Say, for example, that you send out a faster-than-light signal and a short time afterwards I shoot and kill you. Another person who is at a distance and moving at a velocity such that $v/c > c\Delta t/ \Delta x$ receives your signal. But who sent it? From his frame, your death preceded your transmission of the signal.

Are you implying that FLT is impossible because if it were possible causality would be violated in some frames? If so, I would disagree. FLT is most probably impossible for some physical reason. But if FLT were not impossible, causality would still be preserved. If a method for FLT existed, then we would use it for all purposes. We would measure time and distances with it and we would plug those measurements into the equations, whose speed limit would not be the speed of light any more but the speed of the said FLT method. Thus the events in question would not be spacelike any more and all frames would agree on their order in the timeline.

 

[edgepflow]

Look at the Lorentz Transformation for the time separating two events:

$$\Delta t'=\gamma (\Delta t-v\Delta x/c^2)$$

The primed frame sees the events reversed in time (i.e. $\Delta t'$ is negative) when $v\Delta x/c^2 >\Delta t$.

Restated:

$\Delta t'<0$ when $v\Delta x/c^2 >\Delta t$.

Rearranging the condition gives:

$$\frac{v}{c} > \frac{c\Delta t}{\Delta x}$$

$c\Delta t$ can be though of as the distance that light travels in the time separating the events. $v$ is the velocity of the primed frame with respect to the unprimed frame, which must always be less than c (or you get imaginary numbers out of the LT). The left side of the above inequality must always, therefore, be less than one. The only way for the primed frame to see the events reversed is therefore when $\Delta x > c \Delta t$. In other words, the distance in space separating events must be large enough so that light does not have enough time to travel from the first event to the second event. It is therefore concluded that if light does not have enough time to travel from event A to event B, then event A could not have possibly caused event B, and therefore no signals can travel faster than light. Why?

Say, for example, that you send out a faster-than-light signal and a short time afterwards I shoot and kill you. Another person who is at a distance and moving at a velocity such that $v/c > c\Delta t/ \Delta x$ receives your signal. But who sent it? From his frame, your death preceded your transmission of the signal.

Good treatment; I followed your equations but not the text discussion after. There is a similar discussion on:

http://en.wikipedia.org/wiki/Tachyonic_antitelephone

I had to review Lorentz Transformations and study this derivation for hours to get the idea. In this model, the gamma factor cancels. Remarkable.

Seems the math of SR can support FTL travel, at least with some hand waving. That is, FTL based on starting out faster than c (Tachyons), not accelerating something that starts below c to above c.

I am not sure about avoiding causality violation. Maybe nature prohibits Tachyons, or they are unable to provide the type of interaction that leads to antitelephones.

 

[bcrowell]

I am a little lost with this sort of comments you make from time to time. Let me tell you my view and you might yourself comment if you disagreed. Causality cannot be violated because it is not a theory, it is a fact.

I guess it depends on what you mean by causality. If causality means no closed timelike curves, then there is quite a large literature that takes this as not being obvious at all, e.g.:

Friedman, 1990, Cauchy problem in spacetimes with closed timelike curves, http://authors.library.caltech.edu/3737/

Echeverria, 1991, Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory, http://authors.library.caltech.edu/6469/