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. See reader questions & answers on this topic!  Help others by sharing your knowledge 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 sizeofunits 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 xaxis, 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 xvalues 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/meterstick: 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'=xvt. 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. CA = 10 ba = 10 CA = 30 ca = 30 CB = 20 cb = 20 Obviously, the transformed values are now interval scale. This will hold true for any value of time or velocity. (CA)/(BA) = 3 (ca)/(ba) = 3 (CB)/(BA) = 2 (cb)/(ba) = 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 (xx0), 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  (xx0). User Contributions: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. Single Page [ Usenet FAQs  Web FAQs  Documents  RFC Index ] Send corrections/additions to the FAQ Maintainer: Thnktank@concentric.net (Eleaticus)
Last Update March 27 2014 @ 02:12 PM

Comment about this article, ask questions, or add new information about this topic: