Challenge to the Mathematical Validity of the Voigt Transformation Equations and Hence the Theory of Relativity. swaterman@watermanpolyhedron.com [ introduction ] copyright december 2008 by Steve Waterman all rights reserved A well known problem re-worded...where is the missing ten bucks? Three inventors go to apply for their patents at the patent office.
The clerk charged them a hundred dollars each. Later, realizing that he overcharged them; since they had filed simultaneously, the actual cost should have been only two hundred and fifty dollars. He decided to take an early tip and pocketed a quick twenty, giving each inventor only ten dollars back. Therefore, each has paid ninety and three times ninety is two seventy. The clerk has pocketed twenty. Two seventy and twenty is two ninety - so where is the missing ten dollars? Regardless, of what truth the responder may speak, they hear back... "True, you are right" and the clerk just keeps repeating... "Therefore, each has paid ninety and three times ninety is two seventy. The clerk has pocketed twenty. Two seventy and twenty is two ninety - so where is the missing ten dollars?", as if you should hand him the ten bucks. The solution is to find the flaw in his statements...not in the stating of truths. The deception is in the phrase "plus the twenty which the clerk tipped himself". This twenty bucks should not be added to the two seventy; it should be subtracted from the two seventy. The twenty the clerk got is part of the two seventy the three inventors spent collectively. If you subtract the twenty from the two seventy you get the two fifty that the patent office received.. This exact same trick is employed in the Galilean equation x' = x - vt... when the correct answer is x' = x + vt. Actually...the Galilean transformation does the opposite by subtracting when he should be adding. some working definitions As with more things, it is always good to define things that upon occasion may seem evem simple or too obvious. This applies here and since the challenge is to the Galilean Coordinate Transformation Equations. It is logical to start with what an abscissa and a coordinate is and what a coordinate transformation is.
abscissal values 3 2 0 1.5 absciccsa You can look at a 2D Coordinate system as a system in which a point can be defined using two values. We do this by using two perpendicular and directed lines called the abscissa(x axis) and the ordinate(y axis). The point of intersection of these two lines is called the origin denoted O(0,0). Any point can be determined as P(x,y), where x is the value in the x-axis and y is the value in the y-axis. from Wikipedia..."coordinate transformation" In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis, and the y-axis), are specified, as well as the unit length, which is marked off on the two axes (see figure above). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions. A coordinate transformation is a conversion from one system to another, to describe the same space. * such that the new coordinates of the image of each point are the same as the old coordinates of the original point. For example, in 1D, if the mapping is a translation of 7 to the right, the coordinate of each point becomes 7 less. * such that the old coordinates of the image of each point are the same as the new coordinates of the original point. For example, in 1D, if the mapping is a translation of 7 to the left, the coordinate of each point becomes 7 more. Way back in 1887, before the THEORY of Relativty was submitted in 1905, a set of transformation equations were also submitted...called the Voigt transformations equations. In modern notation Voigt's transformation was
Now to attempt to expose that missing ten dollars... A very simple example...two frames are coincidental and have a point at 11,0. One frame moves to the right a distance of 7, while the other does not move. It is the comparison of these two frames AFTER the Voigt transformation that is questioned. Mathematical argument against the validity of the Voigt transformations of 1887. [ part 1 of 2 ] http://watermanpolyhedron.com/completedargument.html Short historical examination of the use of versions this equation...x' = X - XT This will involve looking at 4 cases that will consider a comparison of points in two frames. CASE 1 X' = X CASE 2 X' = X - VT
CASE 3 X' = X + VT CASE 4 X' = X - VT CASE 1 If a point is in a frame, then it must also exist in a coincident frame. CASE 2 point from frame into frame transformed mathematically, correctly, to the left by
Since the frame was moved to the left, the point at 11,0 as well, transforms to the left... passing through 10, then 9, 8, 7, 6, 5 and stopping at 4,0. Notice that the transformed TO point in the original frame is at in the new frame. CASE 3 point from frame into frame transformed mathematically, correctly, to the right by Since the frame was moved to the right, the point at 11,0 as well, transforms to the right... passing through 12, then 13, 14, 15, 16, 17 and stopping at 18,0. Notice that the transformed TO point in the original frame is at 11,0 in the new frame.
If a point at X,0 in a frame is transformed to a new frame, then the new point must also be at X,0 in its' new frame, in order to be a mathematically valid coordinate point transformation. incorrect Voigt transformation< [ part 2 of 2 ]
The actual problem has specifically to do with both frames being stationary after the one frame has completed moving. This is where the problem is. Adding motion in, is post aut propter, to this completely mathematical challenge. That is, if we substituted D for distance, to replace VT, then the mathematically inequality would still be generated in X' = X - D CASE 4 Voigt coordinate point transformation...moving a frame to the right by Where four art thou... in, art thou the length going from backwards to?
The mechanics of Voigt's transformation are shown by the smaller green arrow. 12, 13, 14, 15, 16, 17, in a coordinate length of must commence from is not at instead is at in neither nor determine coordinate lengths in to to or, in, art thou the length going from to?
Voigt's "proof" that the transformation was done correctly. in does not commence from instead it commences from in Voigt never assigns a point name for the transformed point in called X' - to match the points' coordinate value. frame. It must to to
coordinate X' = coordinate X - coordinate VT Since the two shorter line segments have the same total length as the longest one. We are led to believe that the transformation was done correctly, It is not at all obvious then why his "proof" is wrong...as certainly the lengths of those line segments do add up properly. Voigt should not be transforming the line segment length of the corresponding coordinates for a point at to the right by 7.. The task was to determine Granted, this point in has an abscissa in its' moved to frame of. However, the line segment length of completely FAILS to have an abscissa in the corresponding unmoved frame. ALL X coordinate lengths start counting from 0,0,0 in their own frame. Voigt however, starts his length in at VT,0,0. No can do. is not an origin in. Nor is he allowed to select as an origin in to establish his length by counting backwards towards either VT,0,0 or to 0,0,0 in the unmoved frame. moved to the right correctly coordinate X' = coordinate X + coordinate VT
moved to the left correctly coordinate X' = coordinate X - coordinate VT Below, Voigt has one constant which has 2 different values being possible...no can do. AND Given a point at 11,0 then the correct coordinate transformation to the right by 7 is 18,0 not 4,0 not ( 11,0-7,0 ) Given a point at 11,0 then the correct coordinate transformation to the left by 7 is 4,0 This is the missing ten bucks...
Voigt's vt length must be added and not subtracted from the x length Additionally, it is questioned why only an ( X,0,0 ) is allowed and not ( X,Y,Z ) What if we wanted to transform to a new origin at 1,2,-5...all these points originally at (2,3,15) and (6,-2.5,3.4) and (0,-5,-2.34). Quite simply, due to it being a strictly mathematical process, then the is merely Conclusion - the Voigt transformations were malum in se in 1887 and are still wrong today and cannot be used as a proper basis for the accepted Theory of Relativity. Correspondingly, the validity of both time dilation and length contraction are reductio ad adsurdum. Without the Voigt equation, Relativistic conclusions are unsubstantiable. Historical use of the Voigt transformation equations. Woldemar Voigt's 1887 equation is challenged x' = x - vt
In two papers of 1888 and 1889, Oliver Heaviside calculated the deformations of electric and magnetic fields surrounding a moving charge, as well as the effects of it entering a denser medium. This included a prediction of what is now known as Cherenkov radiation, and inspired Fitzgerald to suggest what now is known as the Lorentz-Fitzgerald contraction. derived from Voigt's equation the square root of 1 - ( v 2 - c 2 ) The contraction of a moving body in the direction of its motion. In 1892 George F. FitzGerald and Hendrik Antoon Lorentz proposed independently that the failure of the Michelson-Morley experiment to detect an absolute motion of the Earth in space arose from a physical contraction of the interferometer in the direction of the Earth's motion. According to this hypothesis, as formulated more exactly by Albert Einstein in the special theory of relativity, a body in motion with speed v is contracted by a factor (as seen on left) in the direction of motion, where c is the speed of light.. (where x* = x - vt) Early approximations of the transformation were published by Voigt 1887 and Hendrik Antoon Lorentz (1895). (where x* = x - vt) His final transformations were completed by Joseph Larmor (1897, 1900)
x* must be replaced by x - vt and Lorentz (1899, 1904) and were brought into their modern form by Jules Henri Poincaré (1905) Albert Einstein 1905 Contrary to Lorentz, who considered "local time" only as a mathematical stipulation, Einstein showed that the "effective" coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. This was in some respect also done by Poincaré who, however, continued to distinguish between "true" and "apparent" time. Einstein's version was identical to Poincaré's ( Einstein didn't set the speed of light to unity ): The contents of this article are Copyright Steve Waterman or a third party contributer where indicated. You may print or save an electronic copy of parts of this article for your own personal use. Permission must be sought for any other use.