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sapere aude: page two

Reclaiming the common sense foundations of knowledge:
The catastrophic consequences of counter-intuitive trends in mathematics and its philosophy.

Last revised: 10 October 2017 (main text as 14 October 2016)
url: http://home.btconnect.com/sapere.aude/page2.html


1. SR orthodoxy and Einstein's 1905 Transformation of Coordinates: Mathematics in transit from visualization to blind trust in equations.
2. Einstein's "Simple Derivation".
3. Tower of Babel: On the nature of relativistic effects.


The purpose of this page is to aid critics of the mathematics of SR, in view of the habit of reading equations in disregard of the geometric scenario. I argue in response to difficulties encountered by critics, with substantial revisions over the decades. Although parts of the present version of Section 1 suffer from a deplorable lack of concision, I have reached a point where I do not wish to waste yet more time and effort (except for minor editing, and, of course, unforeseen circumstances) on so absurd a topic. The philosophical background is more difficult and will therefore require further updates.
The bulk of objections to special relativity (SR) has tended to be made on physical grounds. Where critics do discuss the mathematical argument, with few exceptions, they fail to see what is radically wrong with the orthodox solution, and introduce contradictions where none exist (see my comments in page3). The reason for this is the fatal loss of the capacity of visual thinking, brought about by arguments of anti-empirist philosophers.
Russell, in concert with others, had finally succeeded in persuading the mathematics profession that the visual study of geometric figures, as fallacious because subject to perceptual error, has to be abandoned in favour of symbolic logic in abstraction from any reference (set theory; mathematical logic). Despite pleas and complaints (Archive, Bibliography, e.g. Arnheim, Ferguson), including mathematicians (e.g. Gray, with Kline's history an excellent source for clues), the regression to symbolic logic, on a priori anti-empiricist grounds, has now long been orthodox in mathematics.

Anti-empiricist thinking has been a perennial philosophical trend (philosophy as "footnotes on Plato"), with Aristotle alone dismissing the notion of a transcendent realm of truths. There is here clearly a lacuna in metaphysics in that it entirely ignores the role of the strategies of living things, culminating in the inventions of homo sapiens, as in early collections of achievements (Euclid), followed by the spectacular advances in Archimedes and Descartes' important creation of coordinate geometry. (Popper's "Third World" merely adds a further mystagogical level of transcendence to the a priori of Kant.)

The mathematical regression to symbolic abstraction is fatal for physics ("things moving in space"). The ability to understand equations as the language of space, namely "seeing", in terms of geometric figures, what an equation is saying about its referent, needs patient nurture (effort, practice). Older physicists may still have benefited from "outdated" mathematics teaching. Today, the old books are no longer even in print (with exceptions, e.g. Lamb's Dynamics). Younger physicists, inevitably, lack the visual capacity altogether because it has been deliberately disabled, bred out of homo sapiens, by the mathematical orthodoxy of the restriction to purely symbolic thinking. We need not wonder that the ability to "see" what might be wrong with the "equations" of mathematical physics, even in a case as grotesquely simple as SR, has been lost. It is therefore with the geometry (coordinate geometry specifically created for the purpose) that we must here begin.

1. SR orthodoxy and Einstein's 1905 Transformation of Coordinates: Mathematics in transit from visualization to blind trust in equations.

Readers would be expected to have the text of Einstein's 1905 paper to hand;
I will be referring to the text of the English translation in the Dover edition (1952).

Einstein, in 1905, while leading teams of mathematicians (conferences around Minkowski and Poincaré) were looking for a "solution", hit upon a set of equations (incomplete, easily seen to be invalid as well as inapplicable even if corrected) that fulfilled the requirement to form a "Poincaré group" (equivalence of all frames in unaccelerated relative motion). In the words of the fiercely anti-relativist mathematician, Prof. Umberto Bartocci:

"Of course, there is no place for questioning the logical validity of the theory, since it presents itself in the guise of a mathematical theory (naturally, a mathematical theory with physical significance, namely, endowed with a set of codification and decodification rules, which allows to transform a physical situation into a mathematical one, and conversely, but a mathematical theory anyhow), and as that one has to confront it. As a matter of fact, one can prove that any two simply connected and complete flat space-times are isometric, and then both isometric to the space R4 with its canonical Lorentz structure. One calls this unique (up to isometries) space-time the Minkowski space-time, and from now on we shall frame our relativistic considerations in such a space-time, let us call it M. Of course, M is endowed with privileged coordinate mappings, or systems, (called Lorentz coordinate systems), which are the (time-orientation preserving) isometries M ® R4. These are physically interpreted as the coordinate mappings introduced by inertial observers, two of which are completely equivalent, in the sense that they differ up just to an isometry of R4 into itself (the transformations of the so-called Poincaré group). This can be considered a formulation of the Principle of (Special) Relativity."
As we shall see, Einstein's form is "linear" (in x,y,z - on the "variable" t see below) because derived for y, z = 0 only, with a Lorentz Factor that vanishes as soon as a particularly obtuse "mistake" is corrected: namely the failure to correct the relative velocity for the changed unit of time measurement in the second system (to my knowledge, a mistake spotted only by Miles Mathis). This linear set, including the here erroneous Lorentz Factor, "works" in the matrix logic of "R4 spaces" where the inapplicability of Einstein's "solution" to his actual geometric "problem" must forever remain literally "invisible".

To tackle the logic of Einstein's 1905 transformation, the source of orthodox formalisms, a sure grasp of coordinate geometry is needed. It is quite clear that at every step Einstein is thinking in terms of a figure as found in every traditional textbook of analytic geometry. It is true that these comprehensive texts, because of the rejection of visual thought by the new philosophy of mathematics, have long gone out of print. But rudimentary treatments with adequate figures (including translation of origin) are today found in dictionaries, collections of formulae (e.g. Beyer's, CRC Press), and textbooks for engineers. The excuse that one has not been taught geometry simply won't do: for that part of physics defined as "things moving in space" coordinate geometry is a sine qua non.

Einstein presupposes the existence of a sphere as given; we must correlate elements of this sphere, for a given time t, with the symbolisms used in mathematical physics and by Einstein. For any given time t, this sphere represents the location of points with the distance |ct| from the origin (|OP| = |ct| or |-ct|; see below for our choice of convention in this regard). Because of the tendency of image files to get lost in downloads, I confine myself to the crude preformatted text style. As we all know what a circle or sphere looks like, I merely show points on a sphere (or circle) with their coordinate components, with points on the sphere to the right as well as the left of O (that they are conveniently symmetrical about y is coincidental).


            P2  .  .  .  .  .  .    .  .  .  .  .  .  P1
            .                    |                    .

            .                    |                    .

            .                    |                    .

            .                    |                    .                                
    (x) ---- --------------------O-------------------- ---- (x)

                               Fig. 1
(I don't show the z-axis. In Einstein 1905, the scenario is symmetric about the x-axis, so that an appropriate rotation should enable us to let the z-component of OP vanish. To spell out the general case, in my equations here I include a z-component.)
At this point it is convenient to tackle at once the significance of the t in the equations of kinematics.
If we have a displacement along the x-axis, say x = vt, there is the tendency to confuse two different coordinate configurations: the one-dimensional displacement as a stretch on the x-axis, as against the two-dimensional graph of the displacement as a function of time. Where we are dealing with a case of translation of origin, even in the case of a one-dimensional displacement, the graph of the function is already completely useless.
In Einstein's case, the scenario is that of a sphere, of radius ct in the "stationary" system. By orthodox definition, the radius of a sphere - whether r or ct - is constant, or fixed. While the r or ct may assume any value, that value is constant; not only the c but also the t is a constant.
This is of particular importance in the case of a translation of origin (orthodox solution a set of linear equations in four variables). The detail of this will be shown below; but I want it to be clear form the start that the notion of the t here being a variable is a typical mistake to be expected in orthodox mathematical logic.
I had previously included elaborate discussions of "operationalism", namely thinking in terms of moving points, light signals, the expanding sphere, and the difficulty thus generated especially in application to the case of something as elusive as a lightray. These notions merely prevent us from looking at the ratios evident in a figure for any chosen time, ratios that, in fact, never change and are thus completely independent of the value of t chosen by us. Once we recognize that the t is constant, the matter is closed.
For the equation of the sphere we have
|OP| = ct = (x2 + y2 + z2)1/2.

Here the x-, y-, z-components of ct are x = cxt, y = cyt, z = czt, so that we have
|OP| = ct = t(cx2 + cy2 + cz2)1/2.
In the orthodox mathematical procedure this simple logical fact - fourth "variable" a constant - cannot be seen. For there, instead of lightpaths (displacements in 3D coordinate space), the geometry gives us, absurdly, a monster akin to the function graphs of velocities (v, c) in two 4D spacetime coordinate systems rotated about a common origin.
When engaging with Einstein we need to be aware that he is notorious for Before tackling his equations in detail, we need therefore a sure grip of the geometric meaning of the relevant symbolic expressions used in the literature (mathematical physics and Einstein).

The geometry alone enables us to disentangle the illusive and highly deceptive ratios between distances (pathlengths, for instance); but we need to be clear as to the convention to be followed. This differs in the literature:
Jordan & Smith (p.2) have distances, such as OP, "always non-negative numbers" (i.e. regardless whether to the right or left of the origin), with the coordinates as "signed distances".
Sommerville, in contrast, has (p.1) "distances measured in one direction are positive, distances measured in the opposite direction being negative."
Finally, Lamb (p. 1, for a point on the x-axis): "The position of a point P at any given instant t ... is specified by its distance from some fixed point O on the line, this distance being reckoned positive or negative according to the side of O on which P lies."
There would be another option, not available in html-code: to indicate the direction by an arrow above OP (or PO), meaning "in direction from first to second point".
Unfortunately, the orthodox mathematical treatment has resulted in considerable confusion and uncertainty. I shall therefore use absolute values, where, e.g., |OP| = |PO|, |ct| = |-ct|.

We are now ready to consider Einstein's transformation. (As Einstein introduces an x' that differs from x, instead of our x', ..., t', in the following I shall use Einstein's x ... t.) As argued in his par. 2, the purpose of the procedure is to arrive at a t/t ratio that "synchronizes" clocks such that, instead of, for instance, (c-v)t (the distance measured from O' attained by points representing light signals along the x-axis to the right) as well as of (c+v)t (distance etc. to the left), we have now ct for both cases (and generally ct for yet different distances O'P for points P elsewhere on the "lightsphere"). Although this geometric operation can be done mathematically, it never occurs to Einstein that his "synchronized" clocks, as observed by numerous critics (e.g. Shi and Wankow), are an unrealizable phantasy. It is, then, the purpose of the transformation (par. 3) to find the general equation expressing this ratio ct/ct. By Einstein's definition of k, we have OO' = vt.

Right at the start, let's get rid of the confused notion that we have lengths (OP, x, etc.) as well as a time t. By definition, OP = ct; hence t = OP/c (and nothing else). Furthermore, vt = (v/c)ct = (v/c)OP.

Similarly, Einstein's t = O'P/c (and nothing else).
I had considered emphasizing this important fact by placing ct, vt, ct etc. in brackets, as well as inserting at every occurrence of t, ct, vt (similarly for t) in an equation the expression for the respective geometric quantity (OP, O'P, OO' or OP/c etc.). But this is not practical. It may suffice for us to remember at all times that, in the kinematic transformations of Einstein, expressions containing t or t represent a distance (with the t a constant).
The transformation will require Einstein to obtain equations for points generally on the sphere. It is crucial to adhere to the visual scenario. Our earlier figure must be expanded, for instance as follows.

                                (y)    (y') 

                                 |      |
            P2  .  .  .  .  .  .    .  .  .  .  .  .  P1
            .                    |      |             .

            .                    |      |             .

            .                    |      |             .

            .                    |      |             .                                
(x, x') ---- --------------------O------O'------------ ---- (x, x')

                               Fig. 2
Let's consider the equations for the position vectors here. We had already (Fig.1), for all points on the surface of the original sphere about O,
|OP| = ct = t(cx2 + cy2 + czt)1/2

For the position vectors (for all points on the surface of the sphere about O), we have

O'P = ct = (x2 + h2 + z2)1/2
where x = x - vt = t(cx - v), so that
O'P = ct = t[(cx - v)2 + cy2 + cz2)]1/2,
where the sign of the components of c depends upon direction: to the left of O we have |(cx - v)| = |(cx + v)|.
So let's turn to the start of Einstein's derivation: he has system K(x,y,z) ("stationary") and k(x, h, z) (moving), with the time t being that of the "stationary" system and t of k. On p.44 he introduces, in representation of the rod AB of his par.2, x' = x - vt with a point x' "independent of the time". A ray is to be emitted along the X-axis (to be reflected etc.).
Here our figure simplifies to

(x, x') ----P2 --------------------O------O'------------P1 ---- (x, x')
Fig. 3

There is no problem concerning the equation for the position vectors generally, namely O'P = x - vt. The problem, unrecognized in Einstein's defective logic, lies in the distances, because of the ambiguity of the x (ct, -ct), which gives us two different x', namely (c-v)t and -(c+v)t.
|O'P1| = |ct| = |OP1| - |OO'| = |ct| - |vt|,
|O'P2| = |-ct| = |OP2| + |OO'| = |ct| + |vt|.

That is to say: the pathlengths O'P1 and O'P2 differ in magnitude.
Evidently, ct, that is to say O'P, cannot be the radius of a sphere.

Several authors have examined the logic of the argument on p.44 in great detail (e.g. Crivelli, Hannon, Rebigsol, Dr. Smid); Bob Hannon had once sent me an unpublished learned article on Einstein's approach by partial differentiation. A little circumspection does not come amiss.

What is Einstein's aim? He wants to sell in a competitive market. It had already been agreed that the speed of the two-way signal seems too high by a factor b2, involving an equation t' = (1/2)[L/(c-v) + L/(c+v)]. Lorentz had proposed as main elements in his solution L' = bL and t' = t/b. So the result of the labours on p.44, namely the b2 for both L and time, is not what Einstein wants (see below).

Einstein tries to use his x' (blind to the difference for x = ct as against x = -ct) in lieu of the fixed length L. For the outward ray, there should be no problem in choosing a time t such that ct = AB + vt; an x' "independent of the time" does not present a difficulty. The respective pathlengths in k and K, for the return journey, even if "started" at the time t when B and P coincide, by the definition of the x', correspond exactly to the case of such a ray starting at O: we have x' = |(c+v)|t. (There is no point in speculating about the meaning of the passage "ray ... reflected to the origin of the coordinates". Einstein's illogicality is notorious; there exists a serious cognitive deficiency in that he is unable to "see" what he is talking about.)

Incidentally, if x' is meant to be the L, in the argument where, on the right side, he uses x' he should instead of x'/(c-v) have x'/c. Unless this third x' is yet another "fixed" length L. For x' = (c-v)x/c and therefore x/c = x'/(c-v), the "fixed" length L might here be the x of K: he is likely to prefer x'/(c-v) to x/c because his symbols representing the L should at least look identical. With the x' for L, whether of two or three different magnitudes, his setup is not so much invalid as hilarious. He resorts to partical differentiation for x' "infinitesimally small".
He had already postulated that the solution "must be linear"; the first linear result appears on p.45.

As already pointed out, Einstein never uses this result. The ab2 on p.45 becomes jb (p.46), b (p.48). Mathematicians never paid any attention to the detail of Einstein's crude argument, and merely used his "proof" of reciprocity on p.47 because it "works" in matrix algebra. They never noticed that, on p.49, the time-equation changes yet again: Einstein finally succeeds in shifting the b from the unwanted enumerator into the desired denominator, namely to Lorentz's t = t/b.
Proceeding from p.44, more fun is awaiting us on p.45.
Some authors (e.g. Steven Bryant), arguing from modern algebra and ignoring the geometric reference, question the various symbolic permutations. Here x=ct or x=-ct, hence t=x/c etc. and the initial expressions immediately simplify to ab2[1 - (v/c)]x/c, with [1 + (v/c)] when x = -ct.
What is clear here is that Einstein has no notion of what is meant by a transformation. We want the general equation for the position vector for a point. Instead, he gets the position vectors for three different points: for x = ct, h = ct and z = ct.
Since, generally, the t/t ratio changes with the x,x-component, and as both h and z lie in the same plane orthogonal to the x-axis at x=vt, x=0, the t/t ratio is the same for Einstein's h and z.
Inevitably this new value of the t/t differs from the one previously derived. He ignores this here inconvenient circumstance on p.46 where the first draft of the complete set gives us only the t-equation derived for x=ct (x=-ct); it is assumed that this is the "general" solution for all points on the "lightsphere". The ab2 has become jb.

The only other point worth attention on p.46 is the proof that "The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system." The proof rests on the fact that substitution of x, y, z, t in the equations for x, h, z, t gives us

x2 + h2 + z2 = c2t2.
That is to say, the ct is assumed to be the radius of a sphere. One is tempted to laugh, were it not for the fact that even the majority of critics of the mathematical treatment cannot see anything wrong here, except for some queer contradictions. But the position vector for any point, on any curve, obeys this Pythagorean relation; this vector is the radius of a sphere only when it is constant; even in Einstein's set (t derived for y, z = 0) it is clear that the magnitude of the ct varies: for points "in general" on the lightsphere the ct is a non-linear function of x, y, z.

Text from here in a largely unrevised, overlong old version. Except for minor editing, I may not wish to waste yet more time on the topic.

The next point of attention is the inverse transformation on 47, that is to say the transformation back into what turns out to be K (at rest with K).
This is the proof of the equivalence of frames of reference and of the reciprocity of effects. The success of the operation (performed upon a farcically invalid set of equations, inapplicable even if corrected) means mathematically that the ludricous set forms a Poincaré group, subsequently re-written in orthodox forms such as
x'i = Saijxij (i, j = 1, 2 ... 4).
Readers should be aware of a mistranslation in the Dover edition (long known and probably corrected in other editions). The introductory text presents a third system K' relatively to which the second, k, moves to the left (-v on the x-axis); this is clearly at variance with the subsequent equations. What is meant is that K' moves with speed -v on the x-axis. (N.B.: Needless to say, Einstein has not recognized the interdependence of speed and time measurement, and therefore uncritically believes the relative speed to be "the same" regardless of the different units of time measurement used in the different systems.) The equations presented for the system taken to be moving with speed -v are false; the relative speed must be corrected; let's call it v'. This depends on the location of the point on the surface (spherical about O); we obtain the magnitude by way of ratios in our sketch. For instance, if x = ct, we have

-----Q--------------------O-----O'--------------P----- (x, x')

Fig. 2


OO' = vt = v't', OP = ct and O'P = (c - v)t = ct',
so that
OO' : O'P = vt : (c - v)t = v't : ct'
and therefore (for this point only!)
v' = vc/(c - v).
For Q we get
v' = vc/(c + v).

The relativistic assumption that coincident lengths in S and S' differ in extension does not alter the ratios. If we use the correct v' (for each point), we find that, if we use in our equations a reciprocal b, then the result of the inverse operation is
not at all Einstein's t" = t, x" = x, ect.,
but t" = b2t, x" = b2x,
so that, either, b = 1,
or if, e.g. we assume x = b(x - vt),
then x = (x + v't)/b.

That is to say, if the frames are "equivalent", then any relativistic "contraction" is mathematically non-existent. For equivalent frames SR actually rules out that relative movement as described can produce a dynamic effect. If, on the other hand, we assume, for lengths l and l' in S and S', l' = bl , then l = l'/b . And so the entire relativistic mystery vanishes into thin air.

With the inverse transformation we have come to a point where the nonsense of Einstein's procedure must be fully evident; Pars. 4 and 5 are also instructive. As to Par. 5, for instance, attention to the geometry exposes the hilarious nonsense of Einstein's equations (obediently copied even by the "critical" Dingle). By definition, we have

-----Q--------------------O-----O'----|---------P----- (x, x')

namely a point (here represented by the |) such that O'| = x = wt. As for the v' above, what with the t/t ratio depending upon direction, there is no difficulty whatsoever in deriving the wanted (and again completely useless because inapplicable) value for the w.

It is instructive to put the geometric referents for the expressions used here by Einstein into his equations, e.g. his t=ct/c derived for x=ct (respectively -ct). I remember having previously written this out in full, but don't have the time to do so again. There should be no difficulty whatsoever in recognizing clearly the grotesque nature of this kind of blind symbol pushing by our greatest ever mathematical genius.
Earlier versions of this webpage had included extensive presentations of the formalisms of the orthodox 4D SR transformation, as well of similar sets of equations elsewhere in algebra. This stuff has been omitted here as not serving any rational purpose.

2. Einstein's "Simple derivation"

Einstein's 'Simple Derivation of the Lorentz Transformation' forms Appendix I to Relativity. First published in German in 1917, the book was written for the amateur reader (English tranlation published in 1920 by Methuen). According to the bibliography in A.P. Schilpp (Albert Einstein: Philosopher - Scientist, Open Court, 1949 & 1969; 706), the book constitutes the only comprehensive survey by Einstein of his theory, and is his most widely known work. (One gathers from Schilpp that, even in 1949, the debate about simultaneity had disintegrated into irreconcilable philosophical factions: clearly a waste of time.)
Einstein's great mathematical fame rests on his work on the General Theory (one might add that the mathematics for that theory was provided by Marcel Grossmann, and that, as usual, one looks in vain for an acknowledgement). The simple derivation of 1917 was therefore written at the time of Einsteins greatest mathematical mastery. (The 'mastery' is immediately evident if one compares the 'simple' derivation with the inchoate transformation of 1905). The derivation uses only the most elementary "algebra", and should present no difficulty whatsoever even to the amateur reader. Yet it has been entirely ignored; if one draws the attention of expert philosophers to it, they refer such supposedly technical stuff to mathematical experts.

The "Simple Derivation" is a typical illustration of Einstein's capacity to turn a farcically elementary problem into mathematical esoterics and thus to render it completely unintelligible - a capacity which has earned him the appellation of genius and the admiration of mystagogues. The problem can be solved without any difficulty whatsoever; see below. Here is Einstein's text:

For the relative orientation of the coordinate systems indicated in [an earlier figure for 3 space axes], the x-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the x-axis. Any such event is represented with respect to the coordinate system K by the abscissa x and the time t, and with respect to the system K' by the abscissa x' and the time t'.

We require to find x' and t' when x and t are given.

A light signal, which is proceeding along the positive axis of x, is transmitted according to the equation

x = ct or x - ct = 0 [1].

Since the same light signal has to be transmitted relative to K' with the velocity c, the propagation relative to K' will be represented by the analogous formula

x' - ct' = 0 [2].

Those ... events ... which satisfy [1] must also satisfy [2]. Obviously, this will be the case when the relation

(x' - ct') = l(x - ct) [3]

is fulfilled in general; where l indicates a constant; for, according to [3], the disappearance of x - ct involves the disappearance of x' - ct'.

If we apply quite similar considerations to light rays along the negative x-axis, we obtain the condition

(x' + ct') = m(x + ct) [4].

By adding (or subtracting) equations [3] and [4], and introducing for convenience the constants a and b in place of l and m, where

a = (l + m)/2, b = (l - m)/2,

we obtain the equations

x' = ax - bct, ct' = act - bx [5].

We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.

Comment: Let's pause at this point and look at an appropriate figurative representation. Care is here needed because any figure would necessarily reflect whether we assume propagation to be isotropic in K or in K'; it cannot be isotropic in both. To be on the safe side I use two versions:

Fig. 3.a: Isotropy in K

_Q___________________________O_____O'____________________P_ (x, x')

Fig. 3.a: QO = OP

Fig. 3.b: Isotropy in K'

_Q_____________________O_____O'__________________________P_ (x, x')

Fig. 3.b: QO' = O'P

Now let's look at Einstein's text. Notice how we are slowly getting into trouble. [1] and [2] are perfectly in order, and compatible with either version of our figure. For regardless whether QO = OP or QO' = O'P, we may say that OP = ct and O'P = ct'.

So far, so good. Although not 'false', the indeterminate zero equation [3] warns of trouble ahead. The derivation derails fully with equation [4]. As here explicitly defined, the symbols x and x' [4] denote quantities which differ from those previously used in [1] and [2]. We have, in [4], x = -ct, x' = -ct', whereas, in [1] and [2], x = ct, x' = ct'.

But that is not all. There is, first, the problem that symbols like x and x' may appear ambiguous, in that it is not immediately evident whether they represent positive or negative values, that is to say, in the case of geometry, the displacements of points moving to the right or left. Second, the question of isotropy must be now be faced. Although equations [1] and [2] are compatible with either version of the figure, careless symbol use here leads us to assume that QO = OP as well as QO' = O'P; addition and subtraction can only result in mathematical nonsense. Note that the presence of l and m serves to assure the negligent reader that the difference between the ratios QO/OP and QO'/O'P is properly being taken into account. For the quantities l and m, and presumably therefore the ratios QO/OP and QO'/O'P, are assumed to differ, for otherwise b = 0. But Einstein's actual treatment of the symbols x, x', ct and ct' is at variance with the assumption that these ratios differ. The vague assumption that the ratios QO/OP and QO'/O'P are equal as well as different is a typical instance of Einstein's logic.

Let's first sort out the ambiguity of symbols like x or x'. In the case of a 3D displacement OP(x,y,z), the variables x, y, z denote the components of ct. For a point on the x-axis such that x = ct, the expressions x and ct (x' and ct' respectively) are alternative names for one and the same displacement. Of these alternatives ct (ct' respectively) is preferable, for the direction of movement is clearly indicated by the sign. In contrast, shoddy thinkers like Einstein easily forget a definition like x = -ct (movement to the left). To avoid this kind of confusion, let's eliminate x and x' in favour of their safer alternatives.

Einstein's equations [3] and [4] should then read:

(ct' - ct') = l(ct - ct)
(-ct' + ct') = m(-ct + ct).

We could stop here, for all operations can already be seen to cancel. But let's continue.

In order to distinguish between the symbols used in the different equations let's re-write [3] and [4] using subscripts:

(x'3 - ct'3) = l(x3 - ct3), [3*]

(x'4 + ct'4) = m(x4 + ct4). [4*]

If we now eliminate the ambiguous x and x' in favour of the safer alternatives ct, ct', -ct and -ct', these equations become

(ct'3 - ct'3) = l(ct3 - ct3), [3*]

(-ct'4 + ct'4) = m(-ct4 + ct4). [4*]

Clearly, addition and subtraction cannot lead to Einstein's equation [5] because all operations cancel. This is the case regardless whether movement is to be isotropic in K or K'. Even though, with the invalidity of [5], the 'Simple Derivation' has lost its foundation, we may look in passing at some of the subsequent equally brilliant considerations adduced to conjure up the LT. The main lines of the argument are these:

The coordinates of O' are x' = 0 and x = vt. From [5], we find avt = bct. Further progress can be made by evaluating [5] for t = 0 and t' = 0, when we find x' = ax and x' = a(1 - v2/c2)x. From the Principle of Relativity we have x'/x = x/x', therefore a = (1 - v2/c2)-1/2. Q.E.D. Some Q.E.D.

To conclude: After the revealing start of the derivation, namely from [1] to [5], it should be clear that nothing of value is to be expected of Einstein's mathematical brilliance. Need one wonder if admirers like Reichenbach believed Einstein (of EPR) to have proven the insufficiency of classical logic? Yet academic physics would persuade us to purchase from this "genius" the claim that, by recourse to non-Euclidean geometry and tensor calculus, he has obtained results that transcend the powers of the Newtonian metric.

A simple solution of the problem of Einstein's "Simple Derivation"

The given case restricts points to the x-axes, namely

where OP=x=ct, O'P=x'=ct' (t=x/c, t'=x'/c) and OO'=vt.

Despite the absence of dynamic effects, this appears to lead directly to the Lorentz Transformation, as follows. The entire literature assumes that, despite change of the time unit, the relative velocity is "the same" in both systems, so that OO'=vt=vt'. It is this uncritital assumption that leads to the mysterious effect, Minkowski's "gift from above". (Correctly we should put OO'=vt=v't', where the magnitude of v' can easily be obtained from the figure. If we use v't' instead of vt' the "gift from above" vanishes, for then k=1; but let's proceed as the relativists do.)

Assuming reciprocity of any effect, "we" put x'=k(x-vt), t'=k(t-vx/c2) [Ex.1a,1b],
x=k(x'+vt'), t=k(t'+vx'/c2) [Ex.2a,2b].
Entering x' and t' [Ex.1a,1b] in equations [Ex.2a,2b] we have x=k2x(1-v2/c2) etc. and thus k=(1-v2/c2)-1.

3. Tower of Babel

This is the edited version of an item distributed among critics in 1997. Once we begin to understand what has gone wrong, we may marvel at the ingenuity of generations of thinkers in rationalizing the outcome of an utterly simple but faulty geometric argument.

On the nature of relativistic effects

Note: Checking on a source revealed a discrepancy which, apart from deleting the offending item, I have solved by the wholesale demotion of all quotations to the status of summaries.
The reciprocal effect of length contraction and time dilation, which appears by logical necessity to emerge from the kinematic part of the special theory of relativity, has been variously explained as

1. true but not really true (guess who)
2. real
3. not real
4. apparent
5. the result of the relativity of simultaneity
6. determined by measurement
7. a perspective effect
8. mathematical.

Here is a small selection from the literature; for references see the Archive. Unless placed in quotation marks, authors' assessments are summarized.

1. Effects are true but not really true:

Pride of place goes to Eddington [1928, 33-34]:

"The shortening of the moving rod is true , but it is not really true."
(Thanks to Prof. I. McCausland, Toronto, for contributing this gem.)

2. Effects are real:

Arzelies [1966, 120-121]:

The Lorentz Contraction is a Real Phenomenon. ...
Several authors have stated that the Lorentz contraction only seems to occur, and is not real. This idea is false. So far as relativistic theory is concerned, this contraction is just as real as any other phenomenon. Admittedly ... it is not absolute, but depends upon the system employed for the measurement; it seems that we might call it an apparent contraction which varies with the system. This is merely playing with the words, however. We must not confuse the reality of a phenomenon with the independence of this phenomenon of a change of system. ... The difficulty arises because we have become accustomed to the geometrical concept of a rigid body with a definite shape, whatever the measuring system. This idea must be abandoned. ... We must use the term "real" for every phenomenon which can be measured ... The Lorentz Contraction is an Objective Phenomenon. ...
We often encounter the following remark: The length of a ruler depends upon its motion with respect to the observer. ... From this, it is concluded once again that the contraction is only apparent, a subjective phenomenon. ... such remarks ought to be forbidden.

Krane [1983, 23-25]:

It must be pointed out that time dilation is a real effect that applies not only to clocks based on light beams but to time itself. All clocks will run more slowly as observed from the moving frame of reference. ...
The length measured by the moving observer is shorter. It must be emphasized that this is a real effect.

Matveyev [1966, 305]:

The dimensions of bodies suffer contraction in the direction of motion ... A body is, therefore, "flattened" in the direction of motion. This effect is a real effect ...

Møller [1972, 44]:

Contraction is a real effect observable in principle by experiment. It expresses, however, not so much a quality of the moving stick itself as rather a reciprocal relation between measuring-sticks in motion relative to each other. ... According to relativistic conception, the notion of the length of a stick has an unambiguous meaning only in relation to a given inertial frame. ... This means that the concept of length has lost its absolute meaning.

Pauli [1981, 12-13]:

We have seen that this contraction is connected with the relativity of simultaneity, and for this reason the argument has been put forward that it is only an "apparent" contraction, in other words, that it is only simulated by our space-time measurements. If a state is called real only if it can be determined in the same way in all Galilean reference systems, then the Lorentz contraction is indeed only apparent, since an observer at rest in K' will see the rod without contraction. But we do not consider such a point of view as appropriate, and in any case the Lorentz contraction is in principle observable. ... It therefore follows that the Lorentz contraction is not a property of a single rod taken by itself, but a reciprocal relation between two such rods moving relatively to each other, and this relation is in principle observable.

Schwinger [1986, 52]:

Each will observe the other clock to be running more slowly. This is an objective fact. It is not a property of clocks but of time itself.

Tolman [1987, 23-24]:

Entirely real but symmetrical.

3. Relativistic effects are not physically real:

Taylor & Wheeler [1992, 76]:

Does something about a clock really change when it moves, resulting in the observed change in the tick rate? Absolutely not! Here is why: Whether a clock is at rest or in motion ... is controlled by the observer. You want the clock to be at rest? Move along with it. ... How can your change of motion affect the inner mechanism of a distant clock? It cannot and it does not.

4. Relativistic effects are apparent:

Aharoni [1985, 21]:

The moving rod appears shorter. The moving clock appears to go slow.

Cullwick [1959, 65, 68]:

[A] rod which is at rest in S' ... appears to the observer O to be contracted ... Similarly, a rod at rest in S will appear in S' to be contracted....

Jackson [1975, 520]:

The time as seen in the rest system is dilated.

Joos [1958, 243-244]:

The interval appears to the moving observer to be lengthened. A body which appears to be spherical to an observer at rest will appear to a moving observer to be an oblate spheroid.

McCrea [1954, 15-16]:

The apparent length is reduced. Time intervals appear to be lengthened; clocks appear to go slow.

Nunn [1923, 43-44]:

A moving rod would appear to be shortened. An interval is always less than measured by the other observer.

Whitrow [1980, 255]:

Instead of assuming that there are real, i.e. structural, changes in length and duration owing to motion, Einstein's theory involves only apparent changes, and these are independent of the microscopic constitution and hidden mechanisms controlling the structure of matter. [Unlike]... real changes, these apparent phenomena are reciprocal.

5. Relativistic effects are the result of the relativity of simultaneity:

Bohm [1965, 59]:

When measuring lengths and intervals, observers are not referring to the same events.

French [1968, 97],
Rosser [1967, 37],
Stephenson & Kilmister [1987, 38-39]:

Measurements of lengths involve simultaneity and yield different numerical values.

6. Relativistic effects are determined by measurements:

Schwartz [1972, 113]:

Each observer determines distances to be foreshortened.

7. Relativistic effects are comparable to perspective effects: Rindler [1991, 25-29]:

Moving lengths are reduced, a kind of perspective effect. But of course nothing has happened to the rod itself. Nevertheless, contraction is no illusion, it is real. Moving clocks go slow, a 'velocity-perspective' effect. Nothing at all happens to the clock itself. Like contraction, this effect is real.

8. Relativistic effects are mathematical:

Eddington [1924, 16-18]:

The connection between lengths and intervals are problems of pure mathematics. A travelling clock gives a low reading.

Minkowski [1908, 81]:

[The] contraction is not to be looked upon as a consequence of resistances in the ether, or anything of that kind, but simply as a gift from above, - as an accompanying circumstance of the circumstance of motion.

Rogers [1960, 496]:

Thus we have devised a new geometry, with our clocks and scales conspiring, by their changes, to present us with a universally constant speed of light.

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