Science > Physics > Causation/Causality, Memory, and Convolution 17: Partial Circle Paradox
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
25 Feb 2006 05:23:51 PM |
| Object: |
Causation/Causality, Memory, and Convolution 17: Partial Circle Paradox |
From Osher Doctorow
The beta or 1/beta or gamma or 1/gamma factor in Special Relativity:
1) sqrt(1 - v^2/c^2)
is a circular coordinate in Cartesian coordinates. That is to say, let
x^2 = v^2/c^2 be the x-coordinate equation for an object, and let y be
defined by:
2) y = sqrt(1 - v^2/c^2) = sqrt(1 - x^2)
Then:
3) x^2 + y^2 = 1
is a unit circle. However, the y of (2) only runs through half of that
unit circle since it is not negative.
The "Partial Circle Paradox" is the incomplete specification of the
geometric object (circle) by (2). If we examine equations of physics
in Cartesian coordinates, there are sometimes nonnegative variables or
constants such as the 1 in (1) or (2) or the y in (2), but they almost
always represent an observable or real object in which the variable or
constant is a distance or length or area or volume or some other
quantity whose negative has no intuitive interpretation (not to mention
rigorous ones). However, y in (2), or in other words -sqrt(1 - x^2),
is only a coordinate, and negative coordinates have intuitive
interpretations as well as rigorous ones.
This does not prevent us from using (2) in a specified "phase" or
"scenario", but to claim that it is universal or represents a physical
law is arguably unjustified. The phase, of course, is the subluminal
or even luminal or luminal-subluminal phase if the claim that c is
finite is accepted (although it doesn't necessarily represent anything
other than an upper bound of visual perception as noted in another
posting of this thread). The remaining phase then is the superluminal
phase.
If c is not finite, then exceeding any alleged constant positive number
in speed is not prohibited by any "law of nature".
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Causation/Causality, Memory, and Convolution 17: Partial Circle Paradox |
25 Feb 2006 05:44:04 PM |
|
|
From Osher Doctorow
It might be argued that y = sqrt(1 - v^2/c^2) and x^2 = v^2/c^2 are
only "hypothetical" coordinates and that no "real object" corresponds
to them. However, a real object is alleged to correspond to v/c and
v^2/c^2, and incidentally or not, even an interpretation of -v vs +v as
a two-directional velocity exists with a speed that can be formally
written v. The same real object has speed v, it is allegedly bounded
by c, and so 1 - v^2/c^2 is arguably a property or characteristic of a
real object in a real scenario, and the same for sqrt(1 - v^2/c^2).
There is then no reason why we cannot define a y coordinate as +/-
sqrt(1 - v^2/c^2) = +/- sqrt(1 - x^2). It is true that if x = v/c,
then this is overtly dimensionless, but we can multiply it by a
dimensional constant with value (magnitude) 1 and dimension L (length).
Here x is not the quantity u such that v = du/dt in general.
Osher Doctorow
.
|
|
|
|
|
Related Articles |
|
|
