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sage

发表文章数: 1125
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Causality Tutorial

Let me begin with a quote from Xing Kong

>为了避免因果悖论，信号是不能超光速的，这是共识，至于它如何会导致a close time curve,
>我不清楚.利用是否产生close time curve来分析因果律问题，属于相对论专家级水平，对于
>非相对论专业的，只有设计悖论来判断。

>如果sage兄发射一颗子弹打死拉登，那子弹是超光速的，那么另一个观察者可能看到sage
>兄还没开枪时，拉登已经中弹身亡。这时候这个观察者赶紧拿一块板子挡在拉登面前，
>sage兄发出的子弹终于来到，但是被板子挡住了。这样就有了一个悖论：
>拉登到底该不该死？

You might have heard similar claims elsewhere.

It is probably one of the most wide spread misconception that
propagating outside of the light-cone means violation of
causality. Let's recall why we often see comments like this.

In special relativity, nothing can propagate outside the
light-cone. Therefore, there is no causal relation between space-like
events outside of the light-cone because there is no way we can
propagate outside of light-cone. Two things should not be causally
connected if there no way to propagate signal bewteen them. This is true.

However, let's suppose we can propagate signal outside of the
lightcone (obviously we are breaking Lorentz invariance). This shows
that space-like (defined still by light-speed) events could be
causally connected. However, does this necessarily implies violation
of causality? Definitely not.

Technically, causality means whether A could propagate a signal,
send it to B. B should not be able to send it back before A send
it. Sending a signal outside of the lightcone is very far away from
achieving that.

It should be very easy to understand this point by drawing graphs with
light-cones and worldlines. However, I cannot do it here. I could only
give instructions on how to draw them and interpret them. Since you
are all experts on relativity, you should be able to do it on a piece
of paper and understand.

Now, let's only have 1 space dimension. In the frame of A, x
(horizontal) is some distance. Suppose A is at x=0. A's world-line,
vertical axis, defines the time coordinate for A. Let's say B is at
x=x0. Now, A send a signal to B. Let's call it s. s travels faster than
light in the frame of A. the worldline for s is a light with a slope
less than 45 degrees.

Let call the event A send out signal a and put point a at the origin. Let's
call the event B receives signal point b. a and b are space-like and
now causally connected. however, would B be able to send the signal
back to A before A send it? the best B can do is send s back right
away. s will go though a worldline in -x direction with a the slope
defined by speed of s. That world line will intersect the world line
of A at some positive value t. that is, A will receive the signal back
AFTER send it out.

Now. If there is a C moving respect to A, what is this episode
appearing to C?

There is a way of drawing the reference frame of C on the same
plot. C's worldline, with slope being inverse of the velocity of C
relative to A, defines the time axis, t', for the reference frame of C. x'
axis for C is also tilted upward with a slope determined by Lorentz
contraction.

now suppose x' of C is titled in such a way that point b is below
x'. Now, project b to t' axis by drawing a line from b parallel to
x'. you will see that in the frame of C, b happen at some negative
t'. This means, in the frame of C, b has a time coordinate earlier
than a (which is at the origin).

this is nothing but the statement that Lorentz boost could change the
time order of two space-like event. In special relativity, this is not
a problem because a and b are not causally connected, their time
labels mean nothing. However, in our case, a and b are causally
connected by signal s. is this a violation of causality?

It is not.

Let assume the role of observer C. Now, how should C know the event b
happened? The best way to do it is for B to send back a signal s to
C. The s is again a world line in -x direct with a slope of the speed
of s. if you draw it correctly, you will see it intersects t' axis of
frame C at positive t' value. This means C only learned about event b
after the knowledge of event a.

A good physicist C will conclude the following

1) C sees event a at t'=0, and then event b at t'>0, after event
a. There is nothing C can do to prevent a after C knows b
happened. There is no paradox can be constructed here. No problem
with causality.

2) C will measure the speed of s, and then extrapolate backwards and
conclude that b corresponds to t'<0 in his coordinates. He will
then conclude, based on this observation, that b is the reason for
a.

3) Therefore, if C derive physics laws based on this observation, it
will be backward in comparison to A.

4) Physics laws will depend on the frame (A or C) in this case. This
is what one expect anyway when one violates Lorentz invariance!

It will be a strange world. However, it is still a world with
causality, a reasonable world.

发表时间：2005-07-13, 22:53:09