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```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.

Both sets of transforms are 'translations' - lateral
movements of an axis, increasing over time in these
cases - but with the SR transform also involving 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.  In general, rescaling does not effect scale
quality in the size-of-units sense we have here.

SR likes to consider its transforms just rotations,
however - in spite of the fact Einstein correctly said
they were 'translations' (movements) - and in the case
of 'good' rotations, ratio scale data quality is indeed
preserved, but SR violates the conditions of good ro-
tations; they are not rigid rotations and they don't
appropriately rescale all the axes that must be rescaled
to preserve compatibility.

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 temperatures
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 (difference)
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 1.00 mark there instead of the zero,
and you have interval scale measures.

Let A,B,C,   be 10, 20, 40.
Let a,b,c    be x' at v=.5, t=10.

x'=x-vt.

A   B   C         a      b      c
----------------  --------------------
10  20  40         5     15     35
----------------  --------------------
B/A = 2           b/a = 3
C/A = 4           c/a = 7
C/B = 2           c/b = 2.333

Obviously, the transformed
values are no longer ratio
scale. The effect is less on
the greater values.

C-A = 10          b-a = 10
C-A = 30          c-a = 30
C-B = 20          c-b = 20

Obviously, the transformed
values are now interval scale.
This will hold true for any
value of time or velocity.

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

Obviously, the ratios of the
differences are ratio scale,
being identical to the ratios
of the corresponding original
- ratio scale - differences.

The main difference between these results and the SR
results is that the differences do not correspond so
neatly to the original, ratio scale, differences.

This is due only to the rescaling by 1/sqrt(1-(v/c)^2).
The ratios of the differences on the transformed values
do correspond neatly and exactly to the ratio scale
results.

Using the generalized coordinate form, such as (x-x0),
the transform produces an interval scale x' and an
interval scale x0'. That gives us a ratio scale (x'-x0'),
just like - and equal to - (x-x0).

```

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Top Document: Invariant Galilean Transformations On All Laws
Previous Document: 5. Transformations on Generalized Coordinate Laws
Next Document: 7. The Crackpots' Version of the Transforms.

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