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starship-design: Spacetime, Shmacetime

Timothy correctly responds to my assertion:

>>Now, this "back-in-time" behavior will happen with or without FTL.  This
>>happens in the twin paradox when one twin turns around; the stationary 
>>"sees" the other twin travel backwards in time.
>No, I don't think so:
>[long explanation removed]
>So the stationary twin thinks his brother lived slower or equal all the 
>But the moving twin thinks his brother first lived slower, than started
>living hyperfast, and finally slower again. However his brother lived
>hyperfast more than he lived slower, so the net result is that his brother
>lived faster.

That's correct.  I had a sign error in my thinking; things only go 
backwards in time when you accelerate AWAY from them, not toward them as is 
done in the Twin Paradox.

The Twin Paradox can be easily modified so that (far away) objects DO seem 
to go backwards in time.  After the rocket-ship twin turns around toward 
Earth, have him turn around a second time, back in the direction that he 
was going in the first place.  During this process, time on Earth (instead 
of zipping forward very fast) zips BACKWARDS very fast.  It's a reversable 
process: you can accelerate one way to make time go forward, and the other 
way to make time go backwards.  No paradox, though, because by the time you 
get back to Earth time has always gone forwards.  Unless, of course, you 
have FTL.

This comes directly out of the Lorentz transformation of time: 

t' = gamma *(t - vx).  

If you're at t=0, and you're looking at a far-away object (say x=1), A 
space time event that seems like it's "now" (x=1, t=0) is transformed to a 
t' = gamma * (-v).  If v is negative (i.e. you're accelerating away from x) 
then t' is now a positive quantity.  But what does that mean?  It means 
that an event that was once "now", is now happening at a positive time 
coordinate; in the future!  So something that has "already" happened, now 
hasn't happened yet! (In your reference frame).  You've just "sent" the 
far-away object back in time.  Fortunately you can't do this for close-up 

Strange but true...