Re: starship-design: Vectors and Scalars

[kuo@bit.csc.lsu.edu (Isaac Kuo)]

Ken Wharton wrote:
>Zenon Writes:
>>> But for FTL, D<0. You also get D<0 for local time travel; set x=0 (you
>>> don't go anywhere) and set t<0 (you go back in time).

>>t^2 >= 0 always, also when t < 0.
>>In my version of the formula for D,
>>D < 0 always when x = 0 and t =/= 0.
>>Something is still wrong...

>You're right... according to what I said anyway.  But there's a subtle 
>sign issue here, and I'm pretty sure I am right that FTL is equivalent to 
>staying in the same place and travelling backwards in time.

What is this nonsense?  No, FTL is not equivalent to staying in
place and travelling backwards in time.

>Here's the thing:  D = t^2 - x^2  (let's use this notation) is not a 
>vector; it's a scalar.  All it is is a number that tells you how far 
>apart two events in spacetime are, in a way that is independant of ALL 
>reference frames.  Travelling from A to B will always give the same value 
>as travelling from B to A.  It's a distance, not a direction.

This number can still be negative or positive.  It isn't a "distance"
except in the loosest colloquial sense.

>The reason there's no direction built in to D, is that in STL travel all 
>journeys go in the same direction -- forward in time.

I don't think you understand what's going on here mathematically at
all.  The reason it's not a "vector" is that it is a scalar (which
is just a special case of a vector).  And this is because of how
it's defined mathematically.

It has absolutely nothing to do with STL, FTL, or time travel.
It's just a formula for a number.  Mathematically, it's not
even a metric (a mathematician's notion of distance in a metric
space), although it is a relation (a fancy name for a function
on two inputs).

>As I understand it, there are no imaginary numbers involved; when you 
>move from STL to FTL the sign of D changes, which merely signifies that 
>the Direction of travel shifts from forward-in-time to backwards-in-time.

You have got a really screwed up understanding.

>The crucial issue is thinking about FTL and STL in terms of invariant 
>parameters like D, parameters that don't change from frame to frame.  One 
>you start talking about frame-dependant parameters like time and space by 
>themselves, FTL without time travel might seem to make sense.  But when 
>viewed in terms of frame-independant parameters, the speed of light is no 
>longer some random large number, but rather the only possible number: an 
>infinite speed, a speed that can only be exceeded by sacrificing 
>causality.

What nonsense is this?  How do you propose defining speed in terms
of invariant parameters?  Since you talk about light having infinite
"speed", I presume you are using D as a notion of time
(speed=distance/time).  What do you propose using as a notion of
distance in an invariant way?

In order for use D in any way which makes sense here, you need to
take the square root of it.  For two events within each other's light
cones, this square root exists and can be either positive or negative.
For two events outside each other's light cones, this square root
doesn't exist, unless you want to use complex numbers.
-- 
    _____     Isaac Kuo kuo@bit.csc.lsu.edu http://www.csc.lsu.edu/~kuo
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