sapere aude: page three
A brief discussion of critics' arguments: The mathematics of Special Relativity (SR)

Update of 8 June 2016

Back to sapere aude. See also page 2.

1. Purpose and background.
2. Objections to Einstein's "Simple Derivation".
3. Objections to the derivation of the general form of the Lorentz Transformation.
3.1. Objections that proceed from the geometry of the case.
3.2. Objections based on operationalist misinterpretations and misconceptions.
3.3. Attempts at refutation by interpretation of the symbolism.

1. Purpose and background.

Following complete re-writing of my page 2, the comments to authors' arguments are in need of some editing. Please check for updates.

I collect here papers critical of the mathematics of SR, and append comments intended to advance comprehension in a case where the trend to abstraction (blind symbol pushing without attention to the geometric scenario) has misled even the mathematicians. In my homepage and my page2 I explain briefly the philosophical fallacy at the root of such truly tragic developments, namely the dismissal, as fallacious because subject to perceptual illusion, of the visual analysis of geometric figures as expressed in the equations of coordinate geometry. Training in the difficult skill of "seeing" the geometric meaning of equations has therefore been neglected, while the recent advances in mathematics have, as a matter of purpose, been confined to purely symbolic and formal analysis. It is to be expected that such a loss of of vital knowledge (physics as "things moving in space") would be a particular hindrance for physicists.

SR is central to the crisis in theoretical physics, for two reasons. Firstly, Einstein's apparent "proof", in Par.3 of his 1905 paper, of the dynamic nature of space and time, for reasons of mathematical necessity, is the explicit foundation for his General Theory (GR). The invalidity of that proof puts the foundation of GR in doubt. Secondly, rather more seriously, Einstein's "proof" consists in the inverse transformation for a set of equations linear in four "variables"; the success of the operation showed that that set forms a "Poincaré group", namely just that formal, abstract, purely symbolic set major mathematical conferences (around Minkowski and Poincaré) had been looking for. Precisely because of the shift in the philosophy of mathematics the geometric invalidity of the set (in reference to Einstein's actual, purely geometric "problem") had become "invisible". Only by close attention to Einstein's own "problem" and his "solution" are we able to recognize how here, at an absurdly simple level of geometric logic, the new orthodoxy in mathematics, blindly pursuing the path to symbolic abstraction, can be seen to have gone off the rails.

However, any naive "study" of Einstein's texts should be resisted. His logic is notoriously faulty. To quote Jacques Barzun (The House of Intellect, London: 1959):

"[When] Einstein's misguided friends publish his non-scientific essays, thereby exposing his intellectual inadequacy, naive astonishment is soon succeeded by excessive contempt." (p.21, and p.108:) "government contracts in aid of science are written by legislators and bureaucrats to whom Eddington and Einstein are ... wonderworkers".
It is therefore unsurprsing that critical students dismiss Einstein's transformations with contempt and, all too hastily, conclude that the "inadequacies" in his procedure demonstrate, once for all, the invalidity of the mathematics of SR. Briefly, the purpose of the 1905 transformation had been to find a change of time measurement such that the position vector OiP (points P on surface of a "light sphere" in reference to origin Oi) for all systemsi in unaccelerated relative motion has the form cti (ti = time elapsed, in units of the respective systemi). This is a purely geometric procedure, not influenced in any way by the "behaviour" of light signals; considerations from physics therefore merely distract from the weird logic of "relativistic" time measurement (see my note) and generate just that grotesque mystagoguery that befogs the debate.

For a detailed discussion of the geometric transformation in Einstein's 1905 paper see my page 2.

There is the assumption, in the "community" of critics, that we must all stand together, and that therefore open discussion of the weakness of our arguments constitutes a hostile act. This is counterproductive. The SR problem reveals a weakness in mathematical understanding, a weakness at the root of counter-intuitive developments in theoretical physics. Scrutiny is a valuable tool in the growth of understanding; scrutiny of the critical objections of critics is therefore as important as the scrutiny of SR mathematics itself. In sum, I show here briefly in what way individual papers fail to attend to Einstein's actual procedure and thus to recognize what exactly is wrong with the 1905 solution (today, transposed into set-theoretical jargon, accepted as mathematically orthodox), as also with the later "simple" derivation.
I presuppose acquaintance with source texts, e.g. Einstein's 1905 paper and his books Relativity, 1917 (for the "Simple transformation), and The Meaning of Relativity, 1921 (for his 4D interpretation). I will be referring to the 1952 Dover edition. Or see the Bibliography in my Archive.

2. Objections to Einstein's "Simple Derivation"

I commence with the "Simple Derivation" because, as Dr. Turzyniecki writes (a paper read at the orthodox international 2009 Budapest conference, but no longer online), "the essential faults of [Einstein's] reasoning [here] come to light more clearly". For my own discussion, see page 2. None of the papers listed below pays attention to the crux of the matter, namely Einstein's addition/subtraction as quantitatively equal of |ct'| and |-ct'|. Instead they discuss at length the absurd symbolic permutations which, after Einstein, have become a hallmark of the procedures of theoretical physics. Some purely algebraic points mentioned (e.g. problems of roots) never arise in regard of the equations of kinematics.

(PDF) Anderton, Roger J.: Einstein’s simple derivation of Lorentz transformation: a critique.

(PDF) Asquith, P.R.: Einstein’s Logical Errors.
This paper is also available here.


Dissler, Walter: Führt der Glaube an Einsteins Relativitätstheorie zu einer gewissen Art geistiger Invalidität

D. is guilty of one mistake himself, one which would get him on the blacklist of any professional mathematician, namely that for OQ and O'Q we have -x = -ct, -x' = -ct'. I should know, for I once got thus blacklisted. The symbol x is used to denote all values, negative as well as positive (points on the x-axis, numbers).
Pagels, Kurt (go to link "Denk- und Rechenfehler eines Jahrhundertgenies"): „Mathematische Kritik der Speziellen Relativitätstheorie“, in an excerpt quoted by Ekkehard Friebe.
P. does not recognize that the equations of kinematics refer to geometric scenarios; they are not "algebraic" and most particularly do not represent "functions". As a matter of fact, this misreading of the one-dimensional equations of kinematics as "functions" is at the root of the "space-time" fallacy. Such a reading, typical of professional mathematicians who dismiss the conventions of kinematics as "primitive", renders the queer logic of the kinematic/geometric case wholly unintelligible. P.'s critique, sadly, does not help to advance understanding.
Smid, Dr. Thomas: Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation.

(PDF) Vukelja, Aleksandar: "Mathematical Invalidity of the Lorentz Transformation in Relativity Theory".

3. Objections to the derivation of the general form of the Lorentz Transformation.

3.1. Objections that proceed from the geometry of the case (the transformation as a procedure of classical kinematics)

The geometric scenario is set by the sphere with radius ct. What is wanted is the coordinates of points on this spherical surface in reference to a second system of coordinates shifted through vt along the x-axis, namely (points Q2r, P2r on a surface spherical about O)

                          (y)  (y')

                           |    |
           Q2r                             P2r
                           |    |

                           |    |

                           |    |

                           |    |                              
-----Q1x--------------------O-----O'--------------P1x----- (x, x')
where, e.g., for points on the x-axis (representing the location of a light signal at the time t), OP1x = ct, O'P1x = ct', OO' = vt, OQ1x = -ct, O'Q1x = -ct'. See my page 2.

The main difficulty of critics arises from the failure to attend to the asymmetry of the figure, and thus the direction-dependence of the position vector ct' (and of the "clock rate" t' = ct'/c). Some even assert explicitly that ct' is the radius of a sphere.

Arteha, Dr. Sergey N.: The Lorentz Transformations.
(Ch. 1.4 of his book Criticism of the Foundations of the Relativity Theory, see also full version).

Inattentive to the asymmetry of the ct', A. argues that there must be two spheres.
(PDF) Crivelli, Franco: Mathematical Refutation of the Formulas of Special Relativity.
C. is to be applauded for his close attention to the detail of Einstein's 1905 argument (Par. 3, as well as parts of Pars 4 and 5). He stumbles over E.'s weird entity x' "independent of time". In fact, though E.'s discussion of the x' illustrates his shoddy logic, the transformation concerns a sphere for a specific time; in such a sphere (static!) all ratios are in fact independent of t (they depend only on the x-component; see my page 2). If in our thinking we are not to be derailed by Einstein, clear recognition of what exactly we are "looking at" is crucial.
(PDF) Termathe, Gerd (Dipl.-Ing.): A Simple Refutation of Special Relativity.
T. is among the few to notice that the t'/t ratio differs for signals moving to the right or left. He argues that "Time cannot run at different rates" and concludes "It is shown that the Theory of Special Relativity, applied to the simple case of two light fronts moving in opposite directions, leads to a contradiction and is thus refuted." But what is shown is merely that the "theory" is inapplicable (there are no such clocks). The chaos created by SR is much more serious, namely that any such (if inapplicable) change of the unit of time measurement shows that coincident lenghts reciprocally "contract", a circumstance, taken by Einstein as a fact as the basis for the General Theory. T.'s dismissal much underrates the implications, of the acceptance of SR as mathematically valid, for the entirety of mathematical physics.
Li, Prof. Zifeng:
(PDF) Special Relativity Arising from a Misunderstanding of Experimental Results on the Constant Speed of Light, pp. 8 - 20, ed. Smarandache et al., "Unsolved Problems in Special and General Relativity", and
(PDF) Special Relativity Being from Misunderstanding of Principle of Constant Speed of Light.
Both these papers quote some orthodox equations from the SR transformation ("The derivations and the mistakes involved in the Lorentz transformation and Einstein's original paper are analyzed.") But L. uncritically copies equations, inattentive to the invalidity of their derivation on purely mathematical grounds.
(PDF) Munch, Neil: Simple Assumption Errors Invalidated Relativity.
M.'s critique is based on a misreading of the meaning of the relativistic "contraction", namely the difference, by a reciprocal factor b, between coincident lengths, not as M. believes, a difference between the pathlengths (e.g. x, x'). M. had been among the first to suggest problems with the mathematics; his indefatigable labours, alone in a hostile climate, are to be honoured.
Pagels, Kurt (go to link "Denk- und Rechenfehler eines Jahrhundertgenies"): „Mathematische Kritik der Speziellen Relativitätstheorie“, in an excerpt quoted by Ekkehard Friebe (see link Denk- und Rechenfehler eines Jahrhundertgenies).

Smid, Dr. Thomas: Regarding the 'Light Sphere' Derivation of the Lorentz Transformation.

Sprecic, Mustafa: Aritmeticka, geometrijska i harmonijska sredina velicina u STR. (Caution: do not click the link to the title which loads the papere without the equations and figures.)

S.'s papers have not been tranlated into English, and I am unable to guess what might be the geometric reference of the many equations he adduces. A figure in "Einstein, Lorentz i Euclid" shows two spheres; but without access to the text it is impossible to see what they are meant to represent. I include the paper here, for the attention of critics conversant with S.'s language.
(PDF)Thim, Dr. Hartwig: Einstein’s Light Speed Postulate is Illogical.

(PDF) Thim, Prof. Hartwig W.: Three Major Inconsistencies of the Lorentz Transformations.

(PDF) Zeng, Prof. Qingping: Einstein’s Lorentz Transformation is a Mathematical Game.
This paper is also available here.

Z. presents a rendition of the SR transformation as given in a secondary text, where the "algebra" is already taken to be 4D. Although Z. has a sketch of two coordinate systems, the equations he cites can no longer be put in reference to their geometric meaning. Einstein's own derivation is of interest precisely because there, though ignored by himself, the geometric reference is evident so that his logical (strictly mathematical) errors are easy to see. Secondary expositions can be "read" only if the original transformation (and what is wrong with it) has been understood.

3.2. Objections based on operationalist misinterpretations and misconceptions

Operationalism was a trend in Einstein's time to interpret geometric arguments in terms of physical entities, such as light signals "seen" by "observers". It is, in fact, this trend which introduces errors into purely geometric thinking. Critics listed here object to, or reject completely, the "frames" or co-ordinate systems in the SR transformation. The procedure by signalling, also, is taken to be in conflict with the LT.
(PDF) de Hilster, David: Carezani Frame Reduction (Appendix: Frames in Relative Movement).
I had previously included a long comment here because the Cesarani Frame Reduction has been claimed to be THE final, authoritative, refutation of SR, with the implication that any further attention to the mathematics of SR would be a waste of time and effort. Attention to Einstein's own argument suggests that the notion of a "transmission" of physical effects across frames rests on a misreading of Einstein's attempt to quantify mere geometric distances.
Kalanov, Dr. Temur Z.: On Logical Errors Underlying the Special Theory of Relativity.
and Kalanov, Dr. Temur Z.: Letter to all Physicists, including "On Logical Errors Underlying the Special Theory of Relativity".

3.3. Attempts at refutation by interpretation of the symbolism

Comments in this section have been added in stages. Having had to re-read all these papers, aged 86, I am exhausted (and running out of patience). This part is in need of some editing.

Babin, Walter: Relativistic Transformation Equations - Additional Support for the Existence of Dual States.

B. quotes the LT, with corrections, in support of his physical theory. I am here concerned only with his discussion of the mathematical form. He uses the b in its inverse form and believes that the "reciprocal" effect - l' > l as well as l > l' - cancels. But he retains the false expression vt' and seems unaware that the b must re-emerge with a vengenance because, without it, the inverse transformation fails. He objects to the asymmetry of the time equation (signals on the x-axis only) as an ambiguity; he pays no attention to the general invalidity of Einstein's transformation for all points on the sphere.
Steven Bryant:
(PDF) Reexamining Special Relativity: Revealing and correcting SR’s mathematical inconsistency (2003),
(PDF) Communicating Special Relativity Theory’s Mathematical Inconsistencies (2005),
(PDF) Understanding and Correcting Einstein’s 1905 Time Transformation (2005), and
(PDF) A Brute-Force Mathematical Challenge to Special Relativity (2007-2008).

According to his CNPS profile, B. works with models involving coordinate systems. It is therefore strange that he seems unacquainted with coordinate geometry (analytical geometry including parts of kinematics) as that branch of mathematics specifically developed for problems of physics as the science of "things moving in space". He pays no attention to Einstein's elaborate setting of the geometric scene for his derivation, in par. 3 of his 1905 paper, of the t/t ratio needed for clock synchronicity as defined in par. 2. Instead, he bases his analysis on interpretations of some 1905 equations popular among physicists. (The pathlength x becomes a wave-front.) He is apparently ignorant of the fundamental difference between the equations of coordinate geometry, on the one hand, and equations in their many purely symbolic uses in mathematics elsewhere (e.g. mathematical logic, algebra), on the other; in the former, validity rests on visual logic, while in the latter ("in mathematics we don't know what we are talking about") the logical procedure must be defined by rules. B. therefore, in disregard of Einstein's text, confronts his equations, out of context, from the perspective of algebra. The purpose of the exercise (t/t ratio, dependent on the ratio of displacements within the "light sphere") wholly escapes him. The significance of the x, x', and the x within that sphere must remain a mystery. Instead of grotesque errors that expose Einstein's spectacular illogic, B. finds mistakes where none exist. (His "Brute Force Challenge" sets x = 50, t = 10, v = 5; but here, for the pathlength OP along the x=axis, by definition, x = ct!)
Nicholas Chavarga of mathematical errors Special Theory of Relativity.
I include the paper here because of the claim of "errors". Unfortunately, the paper loads without the equations discussed in the text. The main dispute seems to be with the assumption that, for small values of v, the LT reduces to the Galilean Transformation. (The dispute concerning the formal validity of the LT, never mind the application of its anisotropic unit of time measurement, is not attended to by C.)
(PDF) Crivelli, Franco: (Letter to universities, re.: inconsistencies that infringe the elementary algebra rules in Einstein's 1905 paper).
Even for a discussion among friends, this "letter" taxes one's patience. From his elaborate calculations following some of the objections it is clear that C. has not thought it advisable to familiarize himself with the mathematical background (coordinate transformations; geometry of the sphere) and conventional practice (e.g. substitutions like the j) for operations as in Einstein's 1905 paper. For a letter to leading international universities his approach is alarming. Relentlessly and in minute detail, he discusses items in Einstein's text, regardless of relevance, more often than not revealing only his own incomprehension of a not particularly difficult subject matter (e.g. the objection to the "different sizes" of the x'-component of different position vectors). If it were the case that Einstein's crucial proof of the effective curvature of space is valid, any further discussion of any inadequacy later on in Einstein's paper (e.g. significance for "solid bodies") is redundant. If, on the other hand, Einstein's logic is faulty and the proof invalid, why would one want to confront busy people with the discussions of the rest of Einstein's paper.
Hannon, Robert J.: Einstein’s 1905 derivation of his Transformation of Coordinates and Times.
H. pays no attention to Einstein's preceding text, to conventions in kinematics, and the historical background. He asks "Why did Einstein believe that a transformation is necessary? He didn't say." "What is x'?" "What is the purpose of x'? Einstein didn't say." If we are to judge the validity of Einstein's symbolic arguments their accepted usage must be taken into account. A stance of perfect ingenuousness defeats its purpose.
But the stance enables H. to spot one probable error important for the meaning of Einstein's definition of a ray "emitted at the time t0 along the x-axis to x', and at the time t1 ... reflected thence to the origin of the coordinates, arriving there at the time t2". H. asks "Why doesn't the ray return to its point of emission in the stationary system, x=0? Einstein didn't say." According to this reading of the passage, Einstein, despite the definition of x' in terms of x (light path in the stationary system S), in fact, at least for the return journey, completely ignores the ray in S. (An error so weird that I had never thought of imputing it even to the illogical Einstein; H. reminded me that I had to include it in my discussion in page2).
H. further asks "What is x'/(c-v)?" and "What is x'/(c+v)?" But both are standard forms (with length L instead of Einstein's x') in the Lorentz-FitzGerald-Larmor debate. Critics would be expected to be familiar with elementary terms; the validity of Einstein's quantitative treatment depends on legitimate and consistent usage. H. is thus mislead into laborious quantative analyses based on false premisses; he is therefore enable to "see" what is really wrong with Einstein's argument here.
Kassir, Radwan M.:
(PDF) On Special Relativity: Root Cause of the Problems with Lorentz Transformation (Sep. 2013),
(PDF) The Real Consequence of the Speed of Light Postulate: Failure of the Special Relativity (Oct. 2013), and
(PDF) Special Relativity Simply Debunked - in Five Steps! (2014)
K.'s main reference is Einstein's 1905 paper, yet his analysis of uncritically accepted equations indiscriminately adduces forms from the Minkowki and linear algebra traditions. By algebraic substitutions incompatible with Einstein's kinematic scenario, K. believes to be identifying "fatal contradictions" so that SR predictions "turn out to be overwhelming refuted" (5090), and "fatal mathematical contradictions leading to ... [the] refutation" of the LT. (5337)
Lange, Erik J.: Proof of the falsity of the Special Theory of Relativity.
L. subjects SR to a two-part analysis: from a mathematical standpoint, and with a "focus more on non-mathematical, experimental and philosophical arguments". Although Analysis 2 raises points commonly found in the "debate", I confine myself here to L.'s Analysis 1.
This is based on the transformation in a version of the general algebraic form for a system of linear equations where the aij represent the coefficients that appear in a particular set of equations. The general form is preferable for matrix algebra. L. claims that the central error in these equations is made also by Einstein in his 1905 paper. This central mistake, according to L., is the assumption "that x'=0 has to correspond with x=vt". L. is subject to two misconceptions.
(PDF) Lange, Dr. Wolfgang: Einstein's Error in the special theory of relativity.
L. assumes the x' in Einstein 1905 to be constant. Subjecting Einstein's equations on p.44 to integration he finds that t = t (+ integration constant). He applies this to the "Galilean transformation" as it would be expressed in the formalism of four-dimensional "space-time" (matrices). (His inadequate command of the language makes it difficult to decipher his meaning.) "That is all, folks! ... I think, all the problems with length contraction, time delation, twin paradox and others are to forgotten."
(PDF) Milnes, Dr. Harold Willis: The Death of the Lorentz Transformation.
The paper merits attention, for the late Dr. M. was a professor of mathematics. M. examines the LT in four different ways: I discuss here only his examination from a mathematical standpoint.
M. reads the transformation as the representation He agonizes at length, with figure, over the problem of relative movement as perceived by observers: what is the direction; is the quantity v or v'? "A piece of information is carried between them: that is, the velocity between their relative apparent motions is the same for both observers, O and O'. This still carries time with it". This is odd. Since times immemorial, homo sapiens had understood that we are able to measure the distance traversed by a moving object and, by use of some reliable time piece, the time elapsed; one could thus calculate a derived quantity v=ds/dt. SR explicitly changes the unit of time measurement; it seems extraordinary that M. accepts the suspect SR equations in confirmation that, despite of this change, we have for OO' vt as well as vt'.
As to the direction of v', the problem is solved by using expressions "contrary to convention": for x(x',t') and t(t',x') he uses the forms x=b(x'-vt') and t=b(t'-vx'/c2). (M.'s setup thus corresponds to a Galilean transformation for the premiss x'=x-vt, x=x'-vt: either of two frames moving, relatively to one another, with the speed v in the positive direction of the x-axis.) After use of much heavy mathematical machinery, M. finds that the only solution for this conundrum is v=0. (There follow more lenghty mathematical arguments; but it is already clear that nothing of use can be expected of this kind of mathematical expertise.)
Rebigsol, Cameron (aka C. Wong):
(PDF) Wong, C.: Mathematical Invalidity of Relativity. (Paper presented at 2000 St. Petersburg Conference)
(PDF) Rebigsol, C.: Mathematical Inconsistency in Relativity’s Original Paper of 1905. (2010)
(PDF) Rebigsol, C.: Relativity’s Length Measurement Inconsistency. (2010)
R. (W.) has been publishing on the topic, with detailed mathematical quotations and examples, at least since 1996; throughout, he reads the LT in disregard of context and conventions in 1905, and introduces contradictions where none exist. Believing to do science a favour by his supposed demonstrations of invalidity, over the years he has extended his arguments so that his papers, with elaborate numerical examples, get ever longer and more complicated. If I disscuss some of R.'s assertions here in greater detail than merited, it is to show that the assumed contradictions are of his own making, while he ignores altogether the "real" problem of SR mathematics.
Briefly, the "real" problem is twofold. Firstly, even when corrected of "mistakes", the time-equation inevitably renders the unit of time measurement anisotropic and hence inapplicable, for we should need separate clocks for rays in different directions. Secondly, the supposed proof of equivalence (p.47), with its paradoxical reciprocity of effects, appears to succeed, but only because of an error, namely the uncritical assumption, in the entire literature, that the relative speed must be the same for both frames, despite the change of time measurement in the second frame. (If this error is corrected, we find, as common sense logic would expect, that b=1: we have x'=x-vt, and no b in the time equation - the latter, as in Einstein, linear but only for y, z = 0, and generally not linear for nonzero y, z: t'[x(t), y(t), z(t)].)

The paper listed first had been presented at the 2000 St. Peterburg Conference (Volume 1 of the Congress-2000 Proceedings); it refers to a more detailed argument in R.'s earlier "Award" page (no longer online). Ignoring the "real" contradiction of "reciprocity", W.'s (R.'s) demonstrations here merely, by erroneous algebraic substitutions, introduce contradictions into SR equations where none exist. For example, he shows that, since x-vt = ct-vt = (c-v)t, therefore x' = (x-vt) = (c-v)x/(c+v) is "is no longer a function of time", and therefore the full equation for x' reduces to bx. He forgets that the t is equally in x as well as vt; omitting the vt only inevitably leads to a different result for his two equations for x'. (He elaborates with numerical examples.) His second argument concerns the meaning of x': a point from what axis, how "seen" by the "observers"? Like other critics, he is impeded by "operationalist" misconceptions: that geometry becomes easier if we speculate about what "observers" "see". The geometric scenario is plainly that of a sphere about the origin of Einstein's K with radius ct; what we want is the ratio ct'/ct, with OO'=x=vt, which, like all ratios in the sphere, is a function of x(t), y(t), z(t) only, the same for any value of t. He asks whether, if OO'=x=vt, we may assume (O'O=)x'=-vt'. Using the equation for x' that correlates the pathlengths of a ray x = ct (x'=ct'), he puts there -v=x'/t', and unsurprisingly finds that v=c. (There follow numerical examples for this supposed contradiction in SR.).

The two other, much more detailed papers had been presented at the 2010 NPA Confererence (Proceedings of the NPA, Volume 7, pp. 416-420 and 421-427); I list them here in that order.
The first 2010 paper examines in detail the argument at the start of Par.3 of Einstein's 1905 paper (Dover edition, p.44). In the passage in question, Einstein clumsily reformulates the problem as defined in the debate ever since 1887. The lightspeed had appeared to high by a factor of b2; instead of L/(c-v) + L/(c+v) we appear to have 2L/c. (Einstein, as observed by R., inadmissibly uses x' for L, with an additional mistake on the right side where Einstein should have x'/c. But the solution for L would be applicable to any length.)

In any case, Einstein, in his final equations, never uses the result, so we should not get too hysterical about any "invalidity" here. Lorentz had proposed to use L'= bL and t'=t/b; Einstein, meanwhile, is stuck with his b2 for his t (p.45); one b is immediately eliminated (p.46); in the t-equation, hilariously, the b is finally shifted from the enumerator into the denominator in Par.4 (p.49).
R. misreads the statement of the problem (his equation 5), and throughout keeps referring to it, as an "essential function". He obtains the apparent contradiction by presupposing the "contraction" factor, here b2, that only follows from the mathematical argument. With equation 7, invented by R., the solution, instead of t = b2L/c, would be c2 = c2 - v2, hence his later finding that, if c=nv, 1=0.
In his discussion of the validity of using x' for the moving rod, he is impeded by the assumption, elaborated in two figures, that points are moving, hence a difficulty as to the location of A and B at t=0 and at the chosen value for the radius ct of our sphere. There is no difficulty in choosing a value such that AB=(c-v)t. A, at the origin of k, would at the chosen time t be at vt. In an argument about a supposed change of position he introduces a length x2 - vDt, vDt - in transit anywhere between t=0 and the value chosen for the analysis of ratios in our sphere - radius ct - about the origin of K. As as in the 2000 paper, he introduces contradictions (elaborated in numerical examples) by using his vDt to represent length (here L) in the equation where x (L) is the pathlength ct. As in the 2000 paper, he elaborates upon the contradiction imputed to "relativity" (1=0).

For the second 2010 paper I confine myself to R.'s discussion of SR. Here, as in the earlier papers, at extraordinarly length and with figures extending over three A4 pages, he considers the supposed problems presented by "moving" points or lengths. Once again, he inserts equations of the form Dx=vDt into the equation where x is the pathlength ct, with the inevitable finding, as before, that 1=0.
He further investigates this non-existing problem of "moving" points by analysis of the general form for a system of linear equations. He states that here the equation for x' "boils down" to the nonsensical expression also derived by E. Lange in his 1999 paper (see comment above).

Smid, Dr. Thomas: Mathematical Flaws in Einstein's'On the Electrodynamics of Moving Bodies'.
Dr. S. solves Einstein's setup on p.44 by applying the chain rule, and finds that the implication is v=0. He asserts that "there can not be any transformation of the coordinates of a light signal between different reference frames"; he thus misses the point. While inapplicable (anisotropy of time), a correct mathematical solution is not only possible but perfectly easy; the accepted relativistic solution (linear time equation, reciprocal b) has been achieved by blind symbol pushing at the highest mathematical level in disregard of the kinematic scenario.
(PDF) Wanek, Erich: Paradoxe Relativität.
W.'s book is included because, although he does not further examine the premisses for and the validity of the LT in his Ch.2 "Die Lorentz-Transformation", he misquotes there both the Galilean and the relativistic transformation equations. This suggests a degree of negligence. W. is however among the few critics to have noticed that, in order to obtain the direction-dependent relativistic clock readings even for signals along the x-axis only, for each direction a separate clock is required. Although this absurdity is immediately evident from the form of the "time"-equation, W. discusses the implications at length in Ch.3 of the book.

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