Caveat: All u need iz a lil hi skool algebra for this one...
Sidd's comments (not in the original post from antimatter33@yahoo.com)
prefixed by Comment:
From antimatter33@yahoo.com Tue Jan 1 15:20:51 EST 2002
From: Danny Ross Lunsford
Newsgroups: sci.physics.research
Subject: Re: Unification of Electromagnetism and Gravity
Date: Sat, 22 Dec 2001 17:38:07 +0000 (UTC)
Lines: 128
Approved: spr@rosencrantz.stcloudstate.edu (sci.physics.research)
Comment: I have snipped
--
...In fact the framework of relativity is derives from a
group-theoretic analysis based on very basic and simple assumptions about
space, time, and motion, taken from experience. The conclusion is that the
allowed transformations depend on parameter with the dimensions of a
velocity that is either finite or not. In the real world, it turns out to
be finite. Of course, it is c. That light goes at this speed is incidental
to the analysis, which would still be correct if light went at some other
speed. Light happens to go at c because the photon is massless.
To give this derivation, we assume
1) The allowable transformations are linear between frames in uniform
relative motion, so that they depend on the relative (vectorial) velocity V
2) That time can enter into the transformations in an essential way, that
is we do not assume t' = t.
3) That space and time are isotropic, that is, time flows evenly and there
are no preferred directions.
Spatial isotropy eliminates the directional character of V and so we can
concentrate on the simple case of one spatial dimension.
Temporal isotropy implies that what is true of frame A in relation to frame
B for V, is true of frame B in relation to frame A for -V.
--
Comment:So what iz a reference frame, nyhoo?
i like to tink of it this way
frame A is that associated with an observer on a railway station platform
Frame B is that associated with an observer in a railway car moving with
velocity V with respect to the station platform.
x' and t' are the coordinates of an event as measured in reference
frame B, while x and t are the coordinates of an event as measured in reference
frame A.
So whats an event ?
lets say the observer in the railway car passes a line scribed in the station at
position x' and time t' as measured by himself in whatever coordinate system he
has set up
the observer on the platform also measures the position x and the time t
in his own coordinate system that the passenger passes the line
then , one can use assumptions 1-3 to write down the following equations
the notation a(V),b(V) etc. just means that the quantities a,b,c,d
depend on V.
'...what is true of frame A in relation to frame
B for V, is true of frame B in relation to frame A for -V' is rooted
in the fact that we dont really care if we call the coordinates of the
guy in the railway car x',t' and the guy on the platform x,t or the
other way round , as far as this argument is concerned u could also
have a stationary railway car and nudder one passing it
goin the udder vay with velocity -V ...
--
Now, writing
x' = a(V) x + b(V) t
t' = c(V) x + d(V) t
x = a(-V) x' + b(-V) t'
t = c(-V) x' + d(-V) t'
--
Comment: a simple way to see the following is
plug in the last 2 equations into the first two .. ie in
the first equation , replace x and t by the expressions
on the right hand side of the 3rd and 4th equations
and the same for the second equation....
so
x' = a(V) [a(-V) x' + b(-V) t'] + b(V) [c(-V) x' + d(-V) t']
= a(V) a(-V) x' + a(V) b(-V) t' + b(V) c(-V) x' + b(V) d(-V) t'
= [a(V) a(-V) + b(V) c(-V)] x' + [a(V) b(-V) + b(V) d(-V)] t'
and
t' = c(V) [a(-V) x' + b(-V) t'] + d(V) [c(-V) x' + d(-V) t']
= c(V) a(-V) x' + c(V) b(-V) t' + d(V) c(-V) x' + d(V) d(-V) t'
= [c(V) a(-V) + d(V) c(-V)] x' + [c(V) b(-V) + d(V) d(-V)] t'
or u could plug in the first set of eqns into the second set
for similar results
since we know that (cf. Aristotle et al) x'=x' and t'=t' we can
the terms in the equation for x' that involve t' and
the terms in the equation for t' that involve x' must have their
coefficients in the square brackets equal to zero
and the remaining bracketed terms in each equation must each add up to 1
und so weiter
what, u wanna me to do ALL the verk ...? fuggedaboutit !
--
so
a(V) a(-V) + b(-V) c(V) = 1
d(V) d(-V) + b(V) c(-V) = 1
a(-V) b(V) + b(-V) d(V) = 0
a(V) c(-V) + c(V) d(-V) = 0
and of course we can replace V -> -V and these still hold. So
a(V) b(-V) + b(V) d(-V) = 0
a(V) c(-V) + c(V) d(-V) = 0
so b(V) = c(V).
Now
a(V) a(-V) + b(V) b(-V) = 1
a(V) b(-V) + b(V) d(-V) = 0
thus
--
Comment: b(V)^2 means b(V) squared, sqrt means square root
--
b(V) [ b(-V)^2 - a(-V)d(-V) ] = b(-V)
and this also holds for V -> -V, which implies either
a(V) d(V) - b(V)^2 = 1, b(V) = -b(-V)
a(V) d(V) - b(V)^2 = -1, b(V) = b(-V)
The latter is ruled out by letting V=0 in which case
x = x'
t = t'
We can solve the equations now with
a(V) = d(V) = a(-V) = d(-V)
and so
a(V)^2 - b(V)^2 = 1
and
a(0) = 1, b(0) = 0
The origin x=0, which moves at speed V in the other frame, transforms as
x' = b(V) t
t' = a(V) t
so
--
Comment : I lied, said there was only algebra and a calculus equation follows
if it makes u feel better tink of it this way
velocity of the origin is x'/t' instead of dx'/dt'
--
dx'/dt' = V = b(V)/a(V)
so
a(V) = 1 / sqrt(1 - V^2)
b(V) = V / sqrt(1 - V^2)
Finally we dimensionalize time vs. space and replace
--
Comment: So far we have been treating x and t on the same footing:
but since we do not commonly measure time in meters or distance in seconds
(altho we could, and some of us doo...)
we make a transformation to new variables involving a parameter called C
this parameter has dimensions of velocity, meter/second
we dont need to have this velocity C be equal to that of light.
also note that we could have worked in terms of these new variables
all along and all our conclusions would be unchanged
--
V -> V/C
t -> Ct
and write
x' = 1/sqrt(1 - (V/C)^2) ( x + (V/C) Ct )
Ct' = 1/sqrt(1 - (V/C)^2) ( Ct + (V/C) x )
If we let C go to infinity,
--
Comment: letting C become infinite allows us to drop terms that have
C in the denominator
This is the Newtonian aproximation, leading to Galilean relativity
note that the time coordinate t is the same for all frames
no matter what their relative velocity
an Absolute Time if u vill ...
lookit assumption 2 , we now see that
the only case that vee can haf a
Universal or Absolute Time is if C is infinite
recall that C is not necessarily the speed of light in this discussion
--
x' = x + Vt
t' = t
So either C is finite, or not. Experience shows that it is finite.
--
Comment: and experience also shows that C is equal to the speed of light
there are also theoretical considerations that indicate why this should
b the case. but i digress
--
Of course, historically relativity emerged from the contradictions in
electron theory implied by the tacit, wrong assumptions about the nature of
simultaneity.
-drl
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