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Hannon, R. (2006). Einstein's 1905 derivation of his Transformation of Coordinates and Times. PHILICA.COM Article number 16.

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Einstein’s 1905 derivation of his Transformation of Coordinates and Times

Robert J. Hannonconfirmed user (Independent Researcher)

Published in physic.philica.com

An invalidating algebraic defect in Einstein’s 1905 derivation of his Transformation of Coordinates and Times is revealed.

Article body


I. Einstein derived the equations now known as "the Lorentz transformation" in the first three sections of his 1905 article "On the Electrodynamics of Moving Bodies" (OEMB).[1]  Einstein actually performed his derivation in §I-3 of OEMB, having provided necessary information in the preceding paragraphs.  Einstein's analysis is predicated on his two Postulates, plus his beliefs about simultaneity, the relativity of lengths, and the relativity of times.

The First Postulate: "…to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."  Einstein then says, "(the purport of which will hereafter be called the ‘Principle of Relativity')"

The Second Postulate: "…that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

Why did Einstein believe that a transformation is necessary? He didn't say. We may surmise that he believed that we cannot perceive nature as it really is in systems that are moving at great translatory speeds relative to us, necessitating transformation of our perceptions into reality.

The last phrase of Second Postulate attributes an unique property to light. Einstein said that the Second Postulate is "only apparently incompatible with" the First.  Eliminating the apparent incompatibility of the two postulates appears to be the purpose of all that follows.  The purpose of the transformation equations is to equalize the distances and travel-times, respectively, of the ray of light in Einstein's kinematic model.  Those unequal distances and times being due to our misperception of nature.

II. Einstein's kinematic model does not assume that constant relative translatory motion causes distances and intervals of time, respectively, to differ. Instead, it is intended to illustrate the conflict between his two postulates so that the transformations of coordinates and times required to eliminate that conflict are defined.  Einstein's model consists of two systems of Cartesian coordinates in "stationary" space: system K, (x,y,z, where time is t) and system  k (X,Y,Z, where time is T).   The x and  X  axes coincide, and at some initially-undefined time, the X-axis is set in translatory (lengthwise) motion at constant speed v along the x-axis, in the direction of increasing values of x.  Thereafter, system k is defined as the "moving" system, and system K as the "stationary" system.  Times t and T are measured by  "clocks" that are "synchronous" in accord with Einstein's definition of synchronism, placed along the respective axes. 

Then Einstein said, "To any system of values, x,y,z,t, which completely defines the place and time of an event in the stationary system, there belongs a system of values X,Y,Z,T, determining that event relatively to the [moving] system k, and our task is to find the system of equations connecting these quantities.

In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time. If we place x'=x-vt, it is clear that a point at rest in the [moving] system k must have a system of values x',y,z, independent of time.  We first define T as a function of x',y,z, and t.  To do this we have to express in equations that T is nothing else than the summary of the data of clocks at rest in {moving] system k, which have been synchronized [in accord with Einstein's definition of synchronism]."

What is x'?  Since it is independent of time, x' must be independent of both time t and time t, because those times are functionally related.  If we rewrite x'=x-vt as x=vt+x', it is clear that x' is both the constant distance between points x and vt, and a point located at the end of distance vt+x' that is farthest from x=0..  Point x' and distance x' exist on both of the coincident x and x  axes.  Distance and point x' are at rest on the x-axis, and because that axis is moving at speed v along the x-axis, distance x' and point x' are also moving at speed v along the x-axis.   The value of the symbol x in x=vt+x' cannot be a function of any time-related parameter other than v (such as c), otherwise distance x' could not be independent of time.  This value of x should have a special symbol, such as x1, to distinguish it from the generic x of the stationary system.   Then its algebraic definition is x1=vt+x', and x'= x1-vt.  When t=0, x1 = x'.

What is the purpose of x'?  Einstein didn't say.  We surmise that it is a necessary parameter whose value is always the same in both systems.  Being independent of times t and t, the value of x' cannot change if t and t differ or vary.

Einstein continued, "From the origin of [moving] system k let a ray be emitted at the time t0 along the X-axis {i.e., the coincident x and X axes] to x', and at the time t1 be reflected thence to the origin of the coordinates, arriving there at the time t2; we then must have

                         ½[T0 +  T2] =  T1 , "                                                 (1)

As  stated by Einstein, T0 ,  T1 , and  T2  are instantaneous readings of clocks in the moving system.  This contrasts with all of Einstein's previous equations involving time (in §I-1 and I-2 of OEMB), which are concerned with intervals  (differences between instantaneous readings of clocks).

Einstein continued,  "or, by inserting the arguments of the function T  and applying the principle of the constancy of the velocity of light in the stationary system:—

   ½{T[0,0,0,t]+T[0,0,0,t+x'/(c-v)+x'/(c+v)]} = T[x',0,0,t+x'/(c-v)]. "       (2)

III. What is (2)?  It is (1), written in values of the function T corresponding with values of its arguments, x',y,z,t, given in that order within the square brackets.  As written (2) is an inequality, and requires a transformation of coordinates and times to make it into an equality.  Values of the function T are values of time T, but in (2) they are not equal to T1  or T2 in (1) .  A bit later, Einstein said, "…for brevity it is assumed that at the origin of k, t=0, when t=0", which reduces the three symbols "t" to zeros. Then (2) may be written:.

   ½{T[0,0,0,0]+T[0,0,0,x'/(c-v)+x'/(c+v)]} = T[x',0,0,x'/(c-v)].        (2a)

a) T[0,0,0,0] is the value of the function T when the values of its arguments are x'=0, y=0, z=0, and t=0.  

b) T[0,0,0,t2] is the value of the function T when the values of its arguments are x'=0, y=0, z=0 and t = t2 = x'/(c-v)+x'/(c+v).  

c) T[x',0,0,t1] is the value of the function T when the values of its arguments are x'=x', y=0, z=0, and t=t1=x'/(c-v).  

1) What is x'/(c-v)?  In the stationary system, the ray departs x=0 at speed c toward point x', which at t0 is at distance x'.  While the ray is in transit, point x' moves away from the ray at speed v.  The ray arrives at point x' at time (t1-t0), so c(t1-t0) = x'+v(t1-t0), or (t1-t0) = x'/(c-v).  So x'/(c-v) is the interval of time the ray requires to travel distance x'+v(t1-t0) =x'+vx'/(c-v).  It is also the interval of time something other than light would require to travel distance x' at speed (c-v).

2) What is x'/(c+v)?  In the stationary system, the ray is reflected at point x' at instant (t1-t0), and travels at speed -c toward point X=0.  While the ray is in transit, point X=0 moves toward the ray at speed v.  The ray arrives at point X=0 at time (t2-t1), so -c(t2-t1) = -[x'-v(t2-t1)], or (t2-t1) = x'/(c+v). So x'/(c+v) is the interval of time the ray requires to travel distance x'-v(t2-t1) = x'-vx'/(c+v).  It is also the interval of time something other than light would require to travel distance x' at speed (c+v).

Why doesn't the ray return to its point of emission in the stationary system, x=0?    Einstein didn't say.

3) The total distance the ray travels in the stationary system is x'[1+v/(c-v)+1-v/(c+v)] = 2x'(beta)2.

x'[1+v/(c-v) ¹ x'[1-v/(c+v)] so the ray's round trip is asymmetrical in the stationary system, but symmetrical in the moving system. 

4) Einstein's values of the x'-arguments are not the actual distances traveled by the ray in the stationary system, nor are they the ray's coordinates on the x-axis at three different times..  The x'-arguments are the instantaneous locations (x'=0 or x'=x') of the ray on the moving distance x' in the stationary system. These odd values are not coordinates values in the stationary system; indeed, Einstein appears to have abandoned the x-axis. 

By definition x'=x1-vt.  At instant t=t0, x'=x1-vt0=x1, but the ray has not yet moved from x=0.  According to Einstein in (2), the ray is at x'=0, which is true only if x' has been redefined to mean:

a)  "the end of distance x1 nearest the origin of the x-axis is x'=0".

Then what is the definition of x'=x' at instant (t1-t0)=x'/(c-v)?  It may be either: 

b) "the point x' " or

c) "the end of distance x' farthest from the origin of the x-axis".

While definitions b and c are mutually compatible, neither agrees with definition a.

What is the definition of x' in the expressions x'/(c-v) and x'/(c+v)?  It is:

d)  "a constant distance on the x-axis extending from point x1=vt to point x1=vt+x' ",

Which disagrees with definitions a, b, and c.

At instant x'/(c-v)+x'/(c+v), Einstein again says the value of x' is 0, in accord with definition a, above.

In addition, Einstein previously told us:

e) x' is "independent of time",

which does not permit the symbol x' to have different values or meanings at different times t or t..   So, in (2), the symbol which represents the value of the x'-argument has at least three different and unequal definitions, violating the Transitive Law of Algebra, and invalidating equations (2). 

CONCLUSION:  Einstein's multiple unequal definitions of the symbol x' in his equation (2) invalidate that equation.  Since the remainder of his analysis is predicated on equation (2), his derivation of his Transformation of Coordinates and Times is meaningless.

REFERENCE:  1. Einstein, Albert: ON THE ELECTRODYNAMICS OF MOVING BODIES, Annalen der Physik, 17, 1905.  English translation in THE PRINCIPLE OF RELATIVITY, Dover Publications, Inc., New York,1952.

Information about this Article
Peer-review ratings (from 1 review, where a score of 100 represents the ‘average’ level):
Originality = 22.93, importance = 25.00, overall quality = 26.11
This Article was published on 4th September, 2006 at 21:18:58 and has been viewed 5321 times.

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The full citation for this Article is:
Hannon, R. (2006). Einstein’s 1905 derivation of his Transformation of Coordinates and Times. PHILICA.COM Article number 16.

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1 Peer review [reviewer #7116confirmed user] added 5th September, 2006 at 14:24:47

It appears to me that the author has misread Einstein’s mention of the older Galilei value for x’, which he mentions in passing, as part of the Lorentz transformation. In all events, the problem of Minkowski-space dynamics are derived numerous times throughout Einstein’s work and the wider literature. A paper setting out to topple special relativity needs to cast a rather wider net.

Papers of this nature—narrowly focused and esoteric, but suggesting sweeping changes in theory—present a real change in terms of reviewer patience. Philica is ideally equipped to look for the treasure in those hills, and, perhaps, especially nervous about getting lost there. I hope we can avoid venting that anxiety on the author. (Ethan Mitchell)

Originality: 2, Importance: 2, Overall quality: 2

2 Author comment added 6th September, 2006 at 20:56:38

My article pertains only to Einstein’s 1905 derivation, and points out an invalidating mathematical error therein. Valid science cannot be predicated on incorrect mathematics.

I have been a serious student of Einstein’s “Special Relativity” for more than 15 years

I refer the Reviewer to my article EINSTEIN’S “SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION” (Physics Essays, Vol 17, No.1), in which I point out the numerous mathematical flaws in that derivation.

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