sapere aude: page three

A brief discussion of critics' arguments: The mathematics of Special Relativity (SR)

Entries here are a small selection from hundreds of objections I have been able to find online. Because of my workload, my comments are uneven, often much too impatient or brief, and in need of editing and revision.

1. Objections to Einstein's "Simple Derivation"

I commence with the "Simple Derivation" because, as Dr. Turzyniecki writes (see below), the essential faults of [Einstein's] reasoning come here to light more clearly.

(PDF) Asquith, P.R.: Einstein’s Logical Errors.

Smid, Dr. Thomas: Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation.

Turzyniecki, Kazimierz: On the problems concerning the Lorentz Transformation.

I base my discussion on a version of Dr. T.'s paper in pdf format which had been sent to me but which I have been unable to find online.
Dr. T. examines the Simple Derivation only briefly as part of a learned paper on the history of the Lorentz Transformation. As mentioned above, Dr. T. writes that there "the essential faults of [Einstein's] reasoning come to light more clearly", a judgment which makes the acceptance of his paper by the ultra-orthodox PIRT astonishing. Unfortunately, Dr. T. confines himself to a listing of the numerous anomalous equations that are adduced in this derivation; as in the paper by Asquith, listed above, he objects to the notion of snapshots being taken. Einstein commences with the definitions x = ct = -ct, x' = ct' = -ct', with the further uncritical assumption that, although the systems, by definition, are in relative motion, we have |ct| = |-ct| as well as |ct'| = |-ct'|. It is thus immediately evident that he uses identical symbols for different quantities. The derivation derails virtually at the first steps, and none of the anomalous equations actually follow.

(Comment: V. imputes to Einstein lack of math skills: the opposite is the case - we see Einstein here having achieved mastery as a fixer of maths problems, by the deliberate abuse of symbolic procedure (e.g. using identical symbols to represent different quantities).

2. Objections to Einstein's co-ordinate transformations (1905 and later).

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

On the fundamental distinction between the expressions for the co-ordinates of kinematics, on the one hand, and algebraic equations, on the other, see the Introduction to my page2.

(In an earlier edition, I had placed here by mistake the comment to the entry for Prof. Denisov, see below: My apology.)
Like Prof. Thim (see below), Prof. Arteha misreads the equation for the position vector ct' in the LT as the equation for a sphere. By Einstein's own definition, the x', y', z' are the coordinates of a point on the sphere with radius ct; that they necessarily obey the Pythagoras-equation does not turn the figure about the origin O' of the second system into a sphere.

L. pays no attention whatever to Einstein's explicit expositions (1905: points on a sphere about O; the unspecified later text: "signals" moving along the x-axis only). He sets up an elaborate scenario with an additional, redundant, moving "object", finds the t'/t ratio for the case when x = vt (i.e. intersection of the y'-z'-plane with the sphere about O), namely t'= t/g, and concludes that this time-equation is "always true" and independent of the moving "object". He judges that Einstein's LT is either the result of a logical error or a hoax.

Neil Munch deserves respect for his awesome effort to invalidate the mathematical foundation of SR. Unfortunately, he lacks the expertise to correlate Einstein's mathematical treatment with the imagined physical scenario. This shows up in his assumption that contraction/dilation (by the Lorentz factor) refers to the difference between the displacements in the two systems in relative motion (for a signal on the x-axis, the difference between x and x'). But contraction/dilation by the Lorentz factor refers to the peculiar (reciprocal!) difference between elements that ought to be identical, namely x' and x - vt. (This paradoxical effect arises in consequence of Einstein's failure to correct the relative velocity for the changed time scale in S': the inverse transformation does not work because vt' is too short; the facile mathematical remedy is to multiply by a stretch factor.)
M. is impeded by his "operationalist" approach: imagining a light signal and its "marks" to indicate position: we can make progress only if we look critically at a sphere with radius ct, and observe what happens to our geometry if we impose on the position vector r' in the second system the "postulate" r'=ct'.
M. draws attention to an apparent contradiction in Einstein's verbal exposition: light path versus fixed length. This is a desperate objection: in an experimental situation we cannot evaluate indefinite light paths but must confine ourselves to some definite extension; even a moving target, at a definite time t, is at some definite point in space. (From the shoddy logic of Einstein's mathematics one would not expect high precision in his verbal explanations; but "fixed length" is adequate to describe what we do in an experiment.)

Dr. Smid dismisses the LT as invalid because the solution of the quadratic equation for spherical propagation is indeterminate. However, although the co-ordinates of all points on the sphere obey the quadratic equation, the solution for a given point is not obtained by finding the root of the equation but by attention to the geometric scenario. Only grossly blind symbol pushing would lead anyone to pair the co-ordinates of points in different octants (e.g. +ct with -ct').

Prof. Thim pays no attention to Einstein negligent use and reading of symbols: if Einstein reads the quadratic equation for ct' as the equation of a sphere we need not follow him. After all, by definition, the x',y',z' were to be the co-ordinates of points on the sphere with radius ct. As such, these co-ordinates necessarily obey the quadratic equation for the position vector ct'. The equation for t' in the LT is clearly direction-dependent; for points on the x-axis only (even with the fallacious Lorentz factor) we have, on the right, t' = gt(1 - v/c), and on the left, t' = gt(1 + v/c). Close attention to the LT, in its own geometric scenario, therefore shows that we have not two spheres, but one only, for which the symbols x,y,z and x',y',z' respectively, merely give the coordinates in respect of the origin of the two systems of co-ordinates. (The further argument in Prof. Thim's paper is rendered redundant by the invalidity of the LT itself.)

2.2.Operationalist misinterpretations and misconceptions

de Hilster, David: Carezani Frame Reduction

Dr. Carezani thinks in terms of physical frames where coincidence is clearly impossible. He finds that the transformation does not require two frames. (The transformation is one of systems of co-ordinates; it necessarily determines the co-ordinates of a given point in reference to two systems.) Dr. C. dismisses kinematics as non-existent; perhaps the topic, as elementary for physics as basic arithmetic, had never featured in his physics education.

Prof. D. has fallen victim to extreme operationalism. An item of information such as ct would then have been obtained by means of a reflected signal; this means that the input "ct" refers to an event at t/2 since when, as the second system is moving, the actual scenario we believe to be investigating would have changed. There would, then, be two spheres: at the times of t/2 as well as t. (Einstein's own negligent talk of light signals is misleading; see above. At the start of the transformation he tries to conjure up the then already orthodox g by superimposing on the one-way signal the average speed of the two-way signal: the displacement, however, remains that of a point indicating the position of a hypothetical one-way signal.)

The main objection in Dr. Kalanov's paper is that "there are no coordinates and no transformation of coordinates in general, and there exist the coordinates and transformation of the coordinates of the object only". To conflate the mathematical point with the object has been a perennial temptation. In any quantitative investigation, a point indicates a hypothetical position in some reference space: conventionally conceptualized as a coordinate system. SR does exactly what Dr. K demands: it tries (unsuccessfully) to transform the co-ordinates of one specific point (corresponding to the hypothetical position of a light signal at a given time t). Dr. K., more specifically, tries to invalidate SR by the contention that the transformation involves two different physical objects in Michelson's experiment; this objection ignores the explicit symbolic treatment (the position of one point, not two).

2.3. Various objections by appeal to mathematical considerations

Ekkehard Friebe's apparently learned argument from the constants of integration may well impress amateurs ignorant of the concepts and methods of mathematics; but his argument reveals ignorance of theory and application. There are here two fundamental misconceptions.
First, F. confuses two different types of linear equation: the 3D linear equations of geometry and the 4D linear equations of analysis. The first kind represent the displacements of kinematics (in SR as an application, e.g. the x-, y, z-components of the "light path" ct): t is here not a fourth "dimension"; quantities like vt, ct lie in 3D space. The second kind of equation is that of displacements as a function of t; x = ct or x = vt would here be lines in 4D mathematical space ("space-time"), at an angle in the x-t plane. F.'s graphs show that this important distinction is not understood. Trying to upgrade 3D geometry into 4D functional analysis would render any kind of transformation unintelligible; it certainly is no help in trying to discover what is wrong with the SR transformation.
Second, the discussion of the constants of integration is nonsensical. Since the derivative of a constant is zero, the integral of a derivative must take account of an unknown constant in the original function: the so-called constant of integration. Even if the SR equations were expressions of x, y, z as functions of t, we would here not be dealing with derivatives from which we would have to reconstrue the required equations; the equations including any constants are already known because given by definition; in SR, by definition, there are no constants. If we were to mistake the 3D equations explicitly given in SR for 4D expressions of x, y, z as functions of t, their graphical representation would show lines such as x=ct, x=vt lying through the 4D origin. The attempt to demonstrate any uncertainty in the SR equations from the constants of integration is grotesque and does not enlighten.

I might here mention the numerous objections to the supposed failure to argue in terms of differentials. In kinematics, as in SR, all velocities are constant; in the analysis of the equations, whether we are referring to displacements (3D; t an auxiliary variable, a kind of counter on the space-axes), or to displacements as functions of the time (4D; t conventional variable visualized as a 4th dimension in "space-time"), differentials are redundant. More seriously, they obscure the scenario, namely of systems of reference in relative motion. Objections of this kind constitute a serious error of judgment.

Dr. Laski's objection - a common one - is misconceived. (Elsewhere - see his profile on the NPA database - Dr. L. shows that he has fallen into the trap of misreading the LT as a formalism in 4D space.) Einstein uses the conventional symbolism of analytical geometry, the method of classical kinematics. The quantities in the transformation are one-dimensional displacements (the x-, y-, z-components of ct, vt); their direction is indicated by the plus- or minus-sign. (There is here no fourth t-co-ordinate; t merely measures quantities on the space-axes.) Quantities representing velocities (c, v, w) are here scalars, corresponding to the norms of vectors in vector algebra. The grotesque problems of SR merely arise because Einstein's argument is invalid at a purely quantitative level; but the treatment of quantities like c or v as scalars is in accordance with correct mathematical procedure.

2.4. Attempts at refutation by interpretation of the symbolism

Objections listed here accept the LT as a given set of equations, without attention to their derivation in explicit reference to a specific geometric scenario.

B. adduces the equations for x' and t' with b in the denominator, which might be an oversight. He finds that "the denominators [b] may be eliminated as they are common to both frames of reference"; for x = ct and x' = ct' the time equations then reduce to t' = t(1 - v/c) and t = t'(1 + v/c). He does not appear to notice that the inverse transformation from his reduction introduces the b all over again because then t = t/b².

Comment: B. approaches SR from the standpoint of abstract mathematics where rules are independent of the meaning of terms. This is useless for the bulk of mathematics where, to the contrary, rules are determined by the subject matter, not to mention the fundamental difference between the "equations" of kinematics and algebraic equations.
B. finds fault with the transformation presented by Einstein (1905 and elsewhere). He focuses on Einstein's x = ct, in disregard of its meaning in the present context, namely as the x-component of the position vector, in the second system of co-ordinates, of the point of the sphere on the positive x-axis, in which case, necessarily, for the first system we have x=ct. According to B., the "function" t(x,t) must be generally valid for all values of x, t; he finds Einstein's t-equation incorrect because it is valid only for x = ct. (The argument is presented at great length. Because of his blind adherence to rules in disregard of context and meaning, B. fails to recognize the actual fatal error here in Einstein's argument, namely the assumption, in his typical logical negligence, that the time equation derived for that single point may be used for all other points on the sphere.) In the 2007 paper B. goes so far as to adduce a counter-example (x=50, t=10, v=5) which he believes to invalidate Einstein's time equation; since he has not understood that in the present geometrical case necessarily x = ct, he fails to observe that his example has v=c, and that therefore his proof involves division by zero.

Dr. Lange ignores the geometric meaning of the symbolism of SR, and tries to exploit the inevitable contradictions that have arisen in consequence of actual errors in the derivation to which he pays no attention. He assumes that x = ct, which is not the case in 3D. The paper shows the futility of a purely symbolic examination.

Dr. Lange's objections fails: he ignores that both x and x', by definition, are displacements (here ct, ct') and that, therefore, we cannot evaluate Einstein's equations by assuming x' to be constant.

The majority of critics of SR assert that Einstein's mathematical treatment is above reproach; C. Rebigsol deserves respect for trying to invalidate the mathematics of SR by drawing attention to inconsistencies and contradictions. In view of the invalidity of Einstein's mathematical treatment (the solution, if it did follow, would be self-contradictory), there is no dearth of inconsistencies and contradictions; pointing them out does not solve our real and very urgent problem, namely to discover what are the actual errors in the treatment, and why are they not being seen.
Unfortunately, R. keeps writing huge papers where he belabours non-existent inconsistencies and contradictions.
In the paper listed first (5196.pdf), R. assembles in great detail part of the symbolism of Einstein's 1905 transformation, and tries to show that elementary applications fail (his equations (4) to (13)). But he is not paying attention to the meaning of Einstein's equations. He has ignored the essential definitions of x and x, namely that they denote the components of the position vectors of a point moving with velocity c; for points moving on the x-axis, we have x = ct and x = ct (for movement to the left, we have respectively x = -ct and x = -ct. The vague diagram is misleading; to see what we are really doing we need, for each of the "events" a clear representation of the change of position of both the origin of k as well as of the point representing the hypothetical light signal. R. misreads his equation (4) which reduces to x'₁ = bt₁(c - v) (for movement to the left, x'₁ = -bt₁(c + v)). R.'s equations (7) should read x₂ - x₁ = c(t₂ - t₁); all the following equations are equally false.
In the paper listed second (5457.pdf) R. selects one expression from the verbal description before the actual co-ordinate transformation, namely r_AB, but ignores the context. For a length in the rest system, in conventional time, we had there previously AB. Both the AB, and the r_AB ("in the time of the stationary system", i.e. before the - invalid - relativistic "correction") refer to the displacements (not to be confused with L), respectively in K and k. The transformation appears to give us two expressions for L: the L in K, namely x - vt, to which corresponds in k x = b(x - vt).
C. ignores this difference clearly spelled out in Einstein's paper. C.'s entire paper rests on this misconception.

Dr.S. suggests that a proper treatment must start with x(x',t') rather than x'(x,t), i.e. perform the inverse BEFORE the original transformation. But we are able to express x etc. as a function of the derived quantities only AFTER these have been obtained.

In the chapter on the LT, Erich Wanek adduces equations for x' and x that have the signs reversed (+v when we should expect -v); this bodes ill for his understanding of the meaning of the transformation. Since c as the limiting velocity in either system of co-ordinates is the very purpose of the transformation, the elaborate proof that the LT yields c' = c is somewhat misconceived. Interpretation of the equations in terms of observers and clocks suggests that W. has paid no attention to the geometric scenario from which Einstein had (unsuccessfully!) attempted to derive the equations.

(Comment: Included here because of the perverse claim of a contradiction between quantities before and after the transformation, namely c-v or c+v, before, and c, after. The change from the one sytem of measurement to the other is, of course, the sole purpose of the exercise, namely to achieve the "PIVL"!)

3. The two time equations and the twin paradox

The full time equation is inconvenient because time units in the second system, the prescription for clock rates, would shorten for "signals" in one direction, but lengthen for "signals" moving in the other direction: an impossibility for physical clocks. (Incidentally, the g either reduces to 1 if the systems are equivalent, or is no longer reciprocal and the systems cannot be equivalent. The paradoxically reciprocal g appears to be proven by the inverse transformation - Dover ed. p. 47 - only because Einstein forgets to correct the relative velocity for the new time unit - see my page 2).

While clocks that obey the LT would be completely useless, if applicable to the twins, it would merely mean that, according to SR, the rate of their body clocks depends on direction: any effect shown on the journey outward would be reversed on the return journey. (In any case, since the orthodox Einsteinian Lorentz factor is believed to have been mathematically proven to be reciprocal, either twin would always be getting younger than the other. The twin debate does not commend itself to the critical mind.)