starship-design: Vectors and Scalars

[wharton@physics.ucla.edu (Ken Wharton)]

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.

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.

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.  You're always 
travelling forward in time, so there's no need to consider going 
backwards.  But now, since we Are considering travel in both directions, 
we need to create some new notation.  If travel from A to B is forward in 
time, travel from B to A is backward in time: back in time simply means 
travel in the opposite direction, so we can define a positive D to be 
travel forward in time, (the usual way), and a negative D to correspond 
to backwards in time travel.  We're turning the scalar D into a vector, 
by adding a sign that correspons with the direction of travel.

SO - when you start going FTL and D changes signs, that signifies that 
you're now travelling the opposite direction in time than you were.  The 
idea that you take the square root of a negative number and get an 
imaginary proper time is wrong, I believe.  The negative sign only 
signifies that you're now travelling in the opposite direction that is 
normally assumed when you're travelling STL.  

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.

This works the other way around, too.  If there's an object that's 
(somehow) travelling backwards in time while going STL, if that object 
goes FTL it will then go forwards in time again; two negative signs make 
a plus.  


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.

Ken