The following text was written by Wayne Throop:
Wayne Throop throopw%sheol.uucp@dg-rtp.dg.com
throop@aur.alcatel.com
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: Could you explain more where Special Relativity rules out "instantaneous"
Ok. I'll do so with no mathematics to speak of (just a bit of
trivial arithmetic), and I'll try to minimize confusion between
space or time *intervals* vs space or time *coordinates* by discussing
how coordinate systems are set up based on intervals. So hopefully,
this will be a fresh approach that will be clear to some folks to whom
other approaches have not.
This'll start *very* simple-mindedly, but the roots of people's
(mis/non)understandings of relativity are (I think) very deep.
Bear with me.
Let's start with something Everybody Knows about relativity. When you
"go fast" your "time slows down". Now, what does that mean, really.
What is meant by "go fast" and "time slows down"? Well, to do these
sorts of measurements and calculations, we have to have a coordinate system.
You need to have a way of organizing events, and saying things like
"these two events happened at the same place, but at different times",
or "these two events happened at the same time, but in different places".
So, if a metronome is sitting somewhere ticking, each tick of that
metronome "paces off" a regular interval of time. Each tick is then
said to happen "at different times" but "in the same place".
It's a bit trickier to try to define what we might mean by two things
happening at the same time in diferent places, but we can basically say
that if we "see them" happen at the same time, that means essentially
that light (or whatever) took the same amount of time to get to where
you are from each event. So, by having a few observers at different
places, and not moving each relative to the others, they can bounce
light pulses around and reach agreement about "happening at the same time".
You'll note this isn't rigorous, because I've appealed to a great deal
of intuition... such as "how do we know the metronome is regular" and
"how do we know the observers aren't in relative motion". But I think
the important concepts are there: regular periodicity defines a
time-axis, and a regular tiling of space defines space axes.
Now, people noticed that when two such coordinate systems were in motion
relative to each other, observers using either system measured the same
speed for light, and in general couldn't distinguish "which one was
moving" and "which was was still" compared to the light. From this odd
property of light (no matter what grid of metronomes you use to measure
from, you get the same value for its speed) various consequences follow,
having to do with figuring out what some critter using a different
system of metronomes means when that critter says "same time as" or
"same place as".
Which brings us back to the something "everybody knows" about SR. That
"clocks slow down". But now we know a *bit* better what this means. It
means that if you are sitting still tending your metronome, and another
observer swoops past you at some constant speed tending *another*
metronome (and we'll assume all metronomes tick at "the same rate" when
they aren't moving past each other, right?), then as your metronome
ticks ten times, then the moving one will tick some number N<10 times.
Let's put some simple numbers to this. This will be tedious, but
stick through it, because it is central to why it is that
FTL <--> acausality.
First, we need to define what's going on. Let's say that your metronome
and the swooping one tick just as they pass. (They pass close enough
together to ignore the difference.) Then, when your metronome ticks
again, the swooping metronome is some distance away, but hasn't ticked
again yet. (Note that the exact same thing appears to be true to the
person holding the swooping metronome. To that person, *your* metronome
is swooping, and *your* metronome doesn't tick yet by the time that
person's does. So which of the two is "right"? SR says, you can't tell
which is right, they are *both* right. We'll see what this means in
numbers in just a minute.)
Now, how fast. Let's say you and the swooping person are moving at 0.87
lightspeed each relative to the other. That means that for every ten
ticks your metronome makes, the swooping person's metronome makes 5 (and
again, bizarrely enough, vice versa).
As your metronome makes its fourth tick since the close flyby, according
toy your metronome grid, it is 4 ticks since the flyby (obviously) and
the swooping person is 3.84 light-metronome-ticks distant in space.
We'll call that coordinate 4,3.84 (or y(4,3.84) to be explicit that
that's accordin to your metronome grid). The swooping person, however,
assigns a completely different coordinate to that same point in space.
Namely, s(2,0). That is, "2 metronome ticks after the flyby, at the
same point in space".
You see, y and s have carved up space and time differently. Both are
talking about exactly the same event (that is, the swooping person's
metronome ticks for the second time since the flyby). But they give
different spacetime coordinates for it.
Now each person can tell what coordinates the other person will assign.
Let's go through this case one step further, because it illustrates the
most important point. Again, tedious, but bear with it.
At 4 ticks after flyby, you assign the coordinate y(4,0) to your own
position, and (4,3.84) to the swooping person's position. You can
figure out that the swooping person assigns coordinate s(2,0) for their
own posision at that instant. And you can also figure out that the
swooping person assigns your coordinate at that instant as s(2,-1.74).
Yes, it's confusing, but we're almost there. Finally, you can figure
out that the swooping person can figure out that your version of
s(2,-1.74) is y(1,0). (Which is NOT y(4,0) where we started!!!!)
We are so used to thinking of one, universal standard for "same time as"
that we just naturally think something's wonkey-jawed here. But we
never established that we were talking about the same event all the
time. We used "same time as" in both the y() coordinate system (er...
"metronome grid"), *and* in the s() grid. But these have two
different meanings.
Consider: think of "same place as" instead of "same time as". We are
unsurprised that y() and s() disagree about events separated in time
being at "the same place" in space or not. Consider what occurs if
s() flips a coin. The event of tossing the coin and the event of
catching it occur at the same place, just a tick or two apart in time,
according to s(). But clearly, according to y(), the toss and catch
are quite far apart in space: s() moved in the meanwhile!
Just so the other way around. We should not be surprised that y() and
s() disagree about events separated in space being "at the same time"
or not. Karl said (remember Karl? The fellow who didn't realize
I'd ramble on for so long...):
teleportation where you are at one point at one instant and
discontinuously at another point the next instant
Let's consider that. "The next instant" according to whom? Remember,
accordin to y(), the events y(4,0) and s(2,0) happen at just the
same instant, but according to s, the events s(2,0) and y(1,0)
happen at just the same instant. Just as the they can't agree
whether the coin toss and catch occured "at the same place", poor y()
and s() can't agree whether the teleport send and teleport receive
occur "at the same time".
In fact it's worse: y() and s() can't even necessarily agree whether the
teleport send happened *before* the receive, or long afterwards. There
could be a period of time where the teleportee existed at neither source
nor destination, or even a period of time where the teleportee existed
at *both* source *and* destination.
And here we've finally gotten to the acausal stuff. Remember the
lightcone Karl (remember Karl?) drew:
^
| \ /
t \ / can not be affected
i \ /
m------.------------space------------
e / \
| / \ can not affect
| / \
(the diagonals are equal to the speed of light)
If the event at the '.' *can* affect the events in the region labled
"can not [be] affected", there can be no universal agreement over which
event affected the other. Different observers draw that "space" axis
through the "." event *at* *an* *angle* to others, so whether you are
above it, and in the "can not be affected" category, or below it and
"can not affect" category is *observer* *dependent*, and not an absolute
fact about the universe.
So if you have two events, and they are linked causally, then the ambiguity
of which is cause and which effect never arises if and ONLY if each
event is in the lightcone of the other. If each is *outside* the
lightcone of the other, it is impossible to say which is cause and which
is effect.
Implications this has for the two forms of FTL that one might think
rightly ought to exist:
: Motion whose angle exceeds the angle at which light moves, and
: teleportation where you are at one point at one instant and
: discontinuously at another point the next instant.
are 1) the two forms are actually equivalent, because for any slope
greater than lightspeed-angle, there exists an observer to whom
that line connects all events happening at the same instant of time,
and 2) both forms lead to acausality, because given an event and that event's
lightcone, it is impossible to unambiguously tell whether an event
outside the lightcone preceeds the given event, or follows it.
This has been awfully long. I could do better if I had a whitebord
and some markers, because then I could draw in y()s coordinate axes
in s()s system, and vice-versa, and show how the symmetry of their
situation hold up. But maybe this will suffice?