X-Message-Number: 19232
From: 
Date: Sat, 8 Jun 2002 08:24:43 EDT
Subject: Re: CryoNet #19226Spacetime

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Bob said:

> Yvan Bozzonetti says that time is not a dimension in the same sense as the 
> dimensions of space, and I agree--it is profoundly different. Yet 
> Einstein's 
> theory--supported by countless experiments--showed that space and time are 
> intimately related and not independent.
> 
> 

Yes, in special Relativity,time is taken as a dimension. The angle between 
two space dimension is imaginary ( exp ix) this is an "ordinary angle". On 
the other hand, angle between any space dimension and time is real (exp w) 
this is the so-called hyperbolic angle. By definition: exp w = ch w + sh w  
where ch stands for cosinus hyperbolic and sh for sinus hyperbolic. We can 
define  tangent hyperbolic, th = sh/ch and for small values, th w is near 
beta = velocity/celetrty of light.  So, what we perceive as velocity is in 
fact th w, a quantity near w itself for velocity far under  light speed.

So, don't said you can't see hyperbolic angles, it is simply speed. Well, you 
can work with a time dimension if you agree that its square is negative and 
when you add it to space, the resulting Lorentz space is not metric (you 
can't travel in it).

If you want, you can extend that idea and take for example mass as a 
dimension beyond time. The maths work but I am not convinced and I assume Bob 
Ettinger with me) that such spaces with such dimensions are similar to our 
space. I think such spaces must be said to have n dimensional parameters, or 
free coordinates or parameters,... but not simply "dimensions".

In the same way, for me integrating the product of a complex function by its 
conjugate is not a dimension even if it is taken as such in all quantum 
mechanics books.

General Relativity is another story: in its fundamental form, it use four 
dimensions, all with the same properties, it is only in the so called 3+1 
theory that time is taken as the fourth dimension.

Yvan Bozzonetti.

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