Re: Why relativists don't understand Einstein's 1905 mathematics.

On Aug 24, 2:31 pm, "Androcles" <Headmas...@xxxxxxxxxxxxxxxx> wrote:

If x' = x-vt, (Cartesian graph at
http://www.androcles01.pwp.blueyonder.co.uk/function.GIF
under the words "Composition of functions")
and xi = x'/sqrt(1-u^2/c^2), what is the value of u?

You are asking for the value of u in the equation
xi = x'/sqrt(1-u^2/c^2); which – other than the symbols in it,
is identical to xi = (x-vt)/sqrt(1-u^2/c^2).
I am glad you asked; because it is right here that you
misunderstand Einstein's equations. The answer is: v = u, and may have
any numerical value at all.
The place where you (and many calculus "experts") go wrong
is in thinking that x' is moving at v relative to the K-frame
(x,y,z,t), thus is a member of a different frame of reference
than his "'stationary' system K."
I will try to explain it to you, in the same civil spirit you
seem to have recently adopted.
Co-ordinate system (cs) K (x,y,z, with the clock-time t) and
cs k (xi,eta,zeta, with the clock-time tau) are initially at rest
in Einstein's "'stationary' space" in which "the speed of light
is a universal constant, c".
"Now to the origin of [system k] let a constant velocity be
imparted" along the X axis of cs K. "To any system of values
x,y,z,t, which completely define the place and time of an event
in the stationary system [K], there belongs a system of values
xi,eta,zeta,tau, determining that event relatively to the system k,
and our task is to find the system of [transformation] equations
connecting these quantities."
That means this: If an event happens at x = y = 2, z = 3, at
t = 1, we want to find the equations that will give us the values
xi, eta, zeta, tau that system k, moving at v on X of cs K, would
have plotted for that identical event at that identical moment.
"… it is clear that [a phrase E often used when introducing
something that often turned out to be false] the equations must be
linear [thus algebraic will do] on account of the properties of
homogeneity which we attribute to space and time."
In his GR paper, he realized that his space (matter itself)is NOT
homogeneous anywhere at all, and that due to the variable density of
this luminiferous matter (now called "dark matter" by physicists) the
rate at which an event transpires in any local place may be different
than if it happened somewhere else. Therefore his GR math substituted
curved lines with no origin at all, in place of an orthogonal system
of co-ordinates with an origin and axes; and he let the degree of
curvature be a function of zippidy dooda, whatever that means.
(Physics thinks it is a function of the structure of empty four-to-21-
dimensional space-time; or something equally incomprehensible to
Mankind.) Actually, E's GR u,v,w lines MAP the local density of space-
filling natter and his "time" (the indications of the hands of a local
clock) is a function of that density also. But that's another story
altogether, unrelated to his 1895 paper to here.
Anyway, let an event be at x = y = z = 1 at t = 1 as plotted by cs
K, i. e. a bee is there at that instant. Where will the bee be on cs
k, at that same instant?
Since X and Xi coincide and k moves at v on X, if we use a Galilean
transformation it will be at eta = zeta = 1. If the origins
of K and k coincided at t = tau = 0 (as marked by their coinciding
clocks at that instant) then xi = 0 will be at x = 0 + vt and any
point xi will have moved vt units to the right on X.
If we let v=.6c the bee at x=1 will be at xi = x - vt = .4 at t=1..
Where would the bee be on k at a later time t? That depends
on what the bee is doing, i.e. is it alive and moving around or is
it dead and pinned to x = y = z = 1, and if the bee is alive, then
where it will be depends on how fast and in which direction it
is moving. It also depends on the value of t, i.e. when K plots it.
If we assume that the bee is permanently at x = y = z = 1, then
where it would be on Xi of k depends on the value of v and t!
(We might therefore say that xi is a function of v and t.)
Example: If t = 1, then as shown above, xi_0 is at x = vt = .6.
The bee at x = 1 would therefore be at xi = x – vt = .4. If t = 2,
it would be at xi = x – vt = 1 – 1.2 = -.2.
In order to avoid this problem, Einstein said, "If we place x'=x-vt,
it is clear that [clear to me, though evidently to very few others] a
point at rest in the system k must have a system of values x', y, z
[AS PLOTTED BY K], independent of time."
Where, then, is x'? NOWHERE!
It isn't a point at rest of cs k. It isn't a point at rest on cs K
either. It is an abstraction, a value to be substituted for x
anywhere x appears in an equation.
Its purpose was exactly what E said it was: To be able to transform x
into xi REGARDLESS of the value(s) of t.

So, Androcles, there isn't any k-frame x',y.z',t'; nor is there any
K-frame x',y,z,t; nor is the value of x' a function of x per se.
The numerical values of x and x' are co-functions of v and t; and
so is the value of xi. (Although the wording may be wrong, the idea is
correct. )

glird
.

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