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Einstein (1905) AbsurditiesSection - 8. The data scale degradation absurdity.

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Previous Document: 7. Simultaneity and Measurement Prologue.
Next Document: 9. The absolute simultaneity SR transforms.
```The SR transforms and the Galilean transforms both
convert good, ratio scale data to inferior interval
scale data. The effect is corrected, allowed for,
when the transforms are conducted on the generalized
coordinate forms specified by analytic geometry - and
vector algebra, for that matter - but SR refuses to
do it right. The consequence is the appearance that
simultaneity does not hold across inertial frames,
and the consequence of that is the Twins Paradox
absurdity.

Both sets of transforms are 'translations' - lateral
movements of an axis, increasing over time in these
caes - but with the SR transform also containing a
rescaling. It is the translation term, -vt in the x
transform to x', and -xv/cc in the t transform to t',
that degrades the ratio scale data to interval scale
data.

SR likes to consider its transforms just rotations,
however, and in the case of 'good' rotations, ratio
scale data quality is indeed preserved, but SR violates
the conditions of good rotations; they are not rigid
rotations and they don't appropriately rescale all
the axes that must be rescaled to preserve compati-
bility.

The proof is in the pudding, and the pudding is the
combination of simple tests of the transformations.
We can tell if the transformed data are ratio scale
or interval.

Ratio scale data are like absolute Kelvin. A measure-
ment of zero means there is zero quantity of the
stuff being measured. Ratio scale data support add-
ition, subtraction, multiplication, and division.

The test of a ratio scale is that if one measure
looks like twice as much as another, the stuff
being measured is actually twice as much. With
absolute Kelvin, 100 degrees really is twice the
heat as 50 degrees. 200 degrees really is twice
as much as 100.

Interval scale data are like relative Celsius, which
is why your science teacher wouldn't let you use it
in gas law problems.  There is only one mathematical
operation interval scales support, and that has to
be between two measures on the same scale: subtraction.

100 degrees relative (household) Celsius is not twice
as much as 50; we have to convert the data to absolute
Kelvin to tell us what the real ratio of termperatures
is.

However, whether we use absolute Kelvin or relative
Celsius, the difference in the two temperature readings
is the same: 50 degrees.

Thus, if we know the real quantities of the 'stuff'
being measured, we can tell if two measures are on
a ratio scale by seeing if the ratio of the two
measures is the same as the ratio of the known quant-
ities.

If a scale passes the ratio test, the interval scale test
is automatically a pass.

If the scale fails the ratio test, the interval scale
test becomes the next in line.

It isn't just the bare differences on an interval
scale that provides the test, however. Differences
in two interval scale measures are ratio scale, so
it is ratios of two differences that tell the tale.

Let's do some testing, and remember as we do that our
concern is for whether or not the data are messed up,
not with 'reasons', excuses, or avoidance.
------------------------------------------------------

Are we going to take a transformed length and see
whether that length fits ratio or interval scale
definitions?

Of course, not. Interval scale data are ratio after
one measure is subtracted from another. That is the
major reason the SR transforms can be used in science.

Let there be three rods, A, B, C, of length 10, 20, 40,
respectively.  These lengths are on a known ratio scale,
our original x-axis, with one end of each rod at the
origin, where x=0, and the other end at the coordinate
that tells us the correct lengths.

Note that these x-values are ratio scale only because
one end of each rod is at x=0. That may remind you of
the correct way to use a ruler or yard/meter-stick:
put the zero end at one end of the thing you are
measuring. Put the one mark there instead of the zero,
and you have interval scale measures.

Let a,b,c    be x' at v=.7071c, t=0.
Let A',B',C' be x' at v=.7071c, t=10.
g=sqrt(1-(.7071)^2)=.7071.

A   B   C         a      b      c        A'     B'     C'
----------------  --------------------   ---------------------
10  20  40        14.14  28.28  56.57    4.14   18.28  46.57
----------------  --------------------   ---------------------
B/A = 2           b/a = 2                B'/A' =  4.42
C/A = 4           c/a = 4                C'/A' = 11.25
C/B = 2           c/b = 2                C'/B' =  2.55

C-A = 10          b-a = 14.14            B'-A' = 14.14
C-A = 30          c-a = 32.52            C'-A' = 42.42
C-B = 20          c-b = 28.28            C'-B' = 28.28

(C-A)/(B-A) = 3   (c-a)/(b-a) = 3        (C'-A')/(B'-A') = 3
(C-B)/(B-A) = 2   (c-b)/(b-a) = 2        (C'-B')/(B'-A') = 2.

The results show that the primed data (a,b,c)
are ratio scale as we'd expect since the vt term
is zero.

The ratios b/a, etc, are the same as the known
ratio scale ratios, B/A, etc.

When vt=0 the data are still ratio scale, but
the rescaling is why the differences (b-a, etc)
are not the same as before transform. The simple
ratios prove the data still ratio, and the ratios
of differences [(c-a)/(b-a), etc] just support
that finding.

When vt<>0, the data (A',B',C') are no longer
ratio scale, which is why the simple ratios now
differ from both the original and vt=0 data.

However, the ratios of differences show us that
the data do satisfy the one mathematical operation
of subtraction, the differences thus being shown
to be ratio scale.

If you do not understand that the above data table
proves that the SR transforms did indeed degrade
the ratio scale to interval scale, please study it
until you understand.
---------------------------------------------------
If we remember that the only effect of gamma=g
is to rescale the data, we realize that the
above results and conclusions also apply to the
galilean transform.

As we said in the introduction of this Subject,
use of the generalized cartesian coordinate form
corrects the interval scale problem. Using this
form for the galilean transformation upgrades the
of laws/equations up to invariant (so to speak)
invariance.

To test the results of the use of the generalized
cartesian coordinate form, with (x-x0) instead of
just (x), we can again let the SR version stand
in for both the galilean and SR results.

Here, our unprimed data were with x0=0.

Let a,b,c    be x' at v=.7071c, t=0.
Let A',B',C' be x' at v=.7071c, t=10.
g=sqrt(1-(.7071)^2)=.7071.

a'=      b'=      c'=
A   B   C    x0     (A-x0)'  (B-x0)'  (C-x0)'  x0'
----------------    --------------------------------
10  20  40    0     14.14    28.28    56.57   -10
----------------    --------------------------------
B/A = 2             b'/a' =  2
C/A = 4             c'/a' =  4
C/B = 2             c'/b' =  2

C-A = 10            b'-a' = 14.14
C-A = 30            c'-a' = 42.42
C-B = 20            c'-b' = 28.28

(C-A)/(B-A) = 3     (c'-a')/(b'-a') = 3
(C-B)/(B-A) = 2     (c'-b')/(b'-a') = 2.

The above data table shows us that focusing on (x-x0),
instead of just plain x, will give us ratio scale data
in any equation the transforms are applied to.

Use of the generalized coordinate form verifies the interval
nature of the transforms. Just as one x' subtracted from
another on the same scale is a ratio scale result, just so
does subtracting x0' from every x' create a ratio scale result.

There is absolutely nothing about the SR transform
derivation that says to not use the generalized
coordinate form, absolutely nothing to gain by insisting
- so to speak - on using interval scale data in your
equations. To do so is absolutely absurd.

Doing so is a sufficient cause of the obvious simultaneity
problem of Special Relativity, which is itself the cause
of the absurd Twins Paradox mess.

```

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Top Document: Einstein (1905) Absurdities
Previous Document: 7. Simultaneity and Measurement Prologue.
Next Document: 9. The absolute simultaneity SR transforms.

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