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It has been critically argued by V. A. Leus (Sobolev Institute of Mathematics, Novosibirsk, Russia) that in my proof that Einstein’s Special Theory of Relativity is logically inconsistent and therefore false, I violated the basic tenets of Special Relativity and foisted an alternative theory upon Einstein’s. A careful study of the critical analysis reveals however a failure to address the key arguments I adduced to prove Special Relativity logically inconsistent, and a concomitant invocation of Einstein’s theory to try to argue that my analysis is incorrect because it does not concur with Einstein. There is therefore no proof advanced of any alleged error in my analysis. In my paper I did not introduce an alternative theory. The aforementioned critical paper affords opportunity in rebuttal to amplify the invalidity of A. Einstein’s tacit assumption, in constructing the Special Theory of Relativity, that systems of clock-synchronised stationary observers consistent with Lorentz Transformation can be mathematically constructed. Since such systems of observers have in fact no mathematical existence the Special Theory of Relativity is logically inconsistent. It is therefore invalid. The consequences for physics, astronomy, and cosmology, are significant.

The recent critical paper [

The two key arguments I adduced in [

1) Einstein defined time by means of his clocks. However clocks do not define time. Clocks no more define time than a pressure gauge defines pressure, than a speedometer defines speed, than a graded spring defines gravity. Measuring instruments are invented to measure something other than themselves. Einstein’s clocks measure only themselves. By defining time by means of his clocks, Einstein detached time from physical reality.

2) Einstein’s method of clock-synchronisation is certainly inconsistent with the Lorentz Transformation, recently proven by Engelhardt [

The paper [

In relation to this

“The Lorentz Transformation is the basis for Einstein’s time dilation and length contraction. It is regarded in general by physicists ( [

When compared with the actual passage in [

“A system of clock-synchronised stationary observers and the Lorentz Transformation are the bases for Einstein’s time dilation and length contraction. It is regarded in general by physicists ( [

Note that Einstein’s theory requires systems of clock-synchronised stationary observers and the Lorentz Transformation. This is the essence of his tacit assumption: that he can construct systems of clock-synchronised stationary observers consistent with the Lorentz Transformation. It has been proven in [

An objection to

“The depicted drawing is rather bewildering than helpful. Nothing similar can be going on if the order established in the special relativity is strictly kept. A correct illustration is delineated in the

I reproduce

The contraction of the clock faces into ellipses in the moving system is irrelevant to the issue, which is time, by Einstein’s false definition thereof. Moreover, my

The insinuation that I have drawn an inaccurate figure for relativist representations of Einstein’s clocks is not correct. Moreover, the point is that all these diagrams depicting stationary and moving clocks are meaningless.

In ( [

“Only at x = ξ = 0 do the clocks depicted read the same time in both systems, where t = τ = 0 .”

A vague and protracted objection is raised in relation this condition. However, this condition is clearly evident in Einstein’s

In ( [

“In the second section of his paper [

t σ i = t i = t 1 + ( σ i − 1 ) v x 1 c 2

where t i is at will.”

This is not correct. The expressions that actually appear in ( [

x σ = σ x 1 (1)

t σ = t 1 + ( σ − 1 ) v x 1 c 2 (2)

where in σ ∈ ℜ , x 1 ≠ 0 . Expressions (1) and (2) convey very different outcomes to those alleged in ( [

In relation to

“The time t = T is elapsed in the K system, so the origin of the k system ξ_{o} = 0 is located at the point x = vT. The event (x_{n}, t_{n}), where x_{n} = γvT/(γ + 1), t_{n} = T, is subject to the Lorentz Transformation (1):” ( [

The term x_{n} is the author’s “neutral point”. The term ξ_{o} and Equation (1) in ( [

τ = β ( t σ − v x σ / c 2 ) , x σ = σ x 1 , ξ σ = β ( x σ − v t σ ) = β [ ( σ β 2 + v 2 c 2 ) x 1 − v t 1 ] , t σ = t 1 + ( σ − 1 ) v x 1 c 2 , η = y , ς = z , β = 1 / 1 − v 2 / c 2 , σ ∈ ℜ . (3)

However, “The time t = T is elapsed in the K system” is inconsistent with Equation (3) above, because there is no common time t for the stationary observers of Equation (3)―they cannot be clock-synchronised, contrary to Einstein’s tacit assumption. The “time t = T is elapsed in the K system” is in fact just Einstein’s common time by his false tacit assumption. Furthermore, the “neutral point” x_{n} is moving:

“The neutral point is moving along the x-axis in positive direction with speed v n x = v γ / ( λ + 1 ) .” ( [

None of the observers x_{σ} in Equation (3) are moving―they are all stationary by mathematical construction (none are functions of time). The neutral point’ argument simply invokes Einstein’s Special Theory of Relativity, which is already proven false by Equation (3). There is nothing in the arguments in [

Paper ( [

“Either way, Einstein’s system of clock-synchronised stationary observers is inconsistent with the Lorentz Transformation.” ( [

The following remarks then appear:

“This conclusion is fatally wrong. The very notions “stationary-moving” are quite relative: from the point of view of any K-observer the K system is stationary and the k system is in motion, but from the point of view of any k-observer the k system is stationary and the K system is the moving one.” ( [

This passage clearly attests that neither my arguments nor those of Einstein have been understood by the author. Einstein’s systems K and k of clock-synchronised stationary observers are each clock-synchronised and stationary with respect to themselves. Einstein then sets his system k of clock-synchronised stationary observers into motion with respect to his system K of clock-synchronised stationary observers; his system K he calls “the stationary system”:

“Now, however, as we know how to judge whether two, or more, clocks show the same time simultaneously and run in the same way, we can very well imagine as many clocks as we like in a given CS. Each of them will help us to determine the time of events happening in its immediate vicinity. The clocks are all at rest relative to the CS. They are ‘good’ clocks and are synchronized, which means that they show the same time simultaneously.” ( [

“It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it ‘the time of the stationary system’.” ( [

“Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring rod, and the clocks.” ( [

It is plainly evident that Einstein’s systems of observers K and k are each clock-synchronised and stationary. That one system is then set into constant rectilinear parallel motion with respect to the other system does not alter this. Einstein’s moving system of clock-synchronised stationary observers is k and his stationary system of clock-synchronised stationary observers is K. There is a difference between systems of clock-synchronised stationary observers and the relative motion of such systems, which has not been recognised in [

I repeat, for emphasis, the objection in [

“This conclusion is fatally wrong.” ( [

However, my statement is correct. Engelhardt [

“We have so far defined only an ‘A time’ and a ‘B time’. We have not defined a common ‘time’ for A and B, for the latter cannot be defined at all unless we establish by definition that the ‘time’ required by light to travel from A to B equals the ‘time’ it requires to travel from B to A. Let a ray of light start at the direction of A, and arrive again at A at the ‘A time’ t'_{A}”._{ }

“In accordance with definition the two clocks synchronize if

t B − t A = t ′ A − t B .” ( [

“… let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in §1; similarly let the time τ of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in §1, of light signals between the points at which the latter clocks are located.

“To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values ξ, η, ζ, τ, determining that event relatively to the system k ...” ( [

Einstein began running his clocks from t = τ = 0, at x = ξ = 0:

“At the time t = τ = 0, when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K.” ( [

He then produced the Lorentz Transformation:

τ = β ( t − v x / c 2 ) , ξ = β ( x − v t ) , η = y , ς = z , β = 1 / 1 − v 2 / c 2 , (4)

where x, y, z, t, pertain to his “stationary system”. Elimination of x from Equation (4) gives:

τ = t β − ξ v c 2 (5)

Setting τ = 0 yields:

ξ = t c 2 β v (6)

Thus, for every t > 0 of the “stationary system K” there exists a point ξ ≠ 0 in the “moving system k” where τ = 0 . However, according to Einstein’s clock-synchronisation method this is impossible because all clocks in his moving system k are synchronised, so that when t > 0, τ > 0 too. Thus, Einstein’s clock-synchronisation method is inconsistent with the Lorentz Transformation [

Systems of stationary observers consistent with the Lorentz Transformation cannot be clock-synchronised. In §5 of [

In §3 of [

“Then in the Section 3 the author addresses the procedure of length measurement. There is a thin rigid rod fixed along the abscissa ξ in his own k system. Let (ξ_{1}, τ_{1}) and (ξ_{2}, τ_{2}) be the simultaneous event (τ_{1} = τ_{2} = τ) of measurement the location of its two ends, so that the rod’s length is L_{ξ} = ξ_{2} − ξ_{1}. The inverse Lorentz transformation gives us these events viewed from the K system:

( x 1 , t 1 ) = [ λ ( ξ 1 + v τ ) , γ ( τ + v c 2 ξ 1 ) ] ; ( x 2 , t 2 ) = [ λ ( ξ 2 + v τ ) , λ ( τ + v c 2 ξ 2 ) ] .

Here the procedure of measurement lost its simultaneity. Thus, the value (x_{2} - x_{1}) ≠ L_{x}_{ }because the rod has shifted during the time interval (t_{2} - t_{1}) for the distance ΔL = v(t_{2} - t_{1}). In this case the real rod’s length would be

L x = ( x 2 − x 1 ) − Δ L = γ ( ξ 2 − ξ 1 ) − γ v 2 c 2 ( ξ 2 − ξ 1 ) = γ ( 1 − v 2 c 2 ) ( ξ 2 − ξ 1 ) = L ξ γ .

The rod is contracted by the factor γ despite the author’s assertion.” ( [

This is not correct. The objection is merely Einstein’s theory, as the common time “(τ_{1} = τ_{2} = τ)” immediately attests. However, relative to the system k the times τ_{1} and τ_{2} cannot be equal because, by Equation (3), a system of stationary observers consistent with the Lorentz Transformation cannot be clock-synchronised. The inverse Lorentz Transformation adduced in [

t = β ( τ σ + v ξ σ / c 2 ) , ξ σ = σ ξ 1 , x σ = β ( ξ σ + v τ σ ) = β [ ( σ β 2 + v 2 c 2 ) ξ 1 + v τ 1 ] , τ σ = τ 1 − ( σ − 1 ) v ξ 1 c 2 , y = η , z = ς , β = 1 / 1 − v 2 / c 2 , σ ∈ ℜ . (7)

Neither Equation (3) nor Equation (7) have been directly addressed in [_{0}, as shown in

Now, following Einstein, impart constant motion at speed v > 0 to the system k as in _{0} according to observers in K.

By Equation (3) above,

Δ x = β l 0 = l 0 1 − v 2 c 2 . (8)

Thus, the moving rod is longer than the stationary rod. Einstein however maintained that the moving rod is shorter than the stationary rod, owing to his false assumption.

In ( [_{σ} of the stationary system K observes the same time interval in K and the same time-dilated interval as Einstein in the moving system of stationary observers k, but they do so at the expense of length contraction and of clock-synchronised stationary observers, which is irreconcilable with Einstein’s theory. The entire objection to this is simply:

“In Section 4 the author manipulates with a time interval.” ( [

A derivation of Einstein’s time-dilation is then presented in [

Section 3 of [

“The further analysis of the article would be senseless because it just seems to criticize the special relativity theory. The author neglects basic tenets of the SRT, foists his own and confuses this makeshift ‘theory’ with Einstein’s creature.” ( [

That [

No proofs are adduced in [

Clocks do not define time. Clocks no more define time than a pressure gauge defines pressure or a speedometer defines speed, or a graded spring defines gravity [

Einstein’s Special Theory of Relativity is certainly inconsistent with Lorentz Transformation. The reason why is somewhat subtle: Einstein’s tacit assumption that he can construct systems of clock-synchronised stationary observers consistent with the Lorentz Transformation is false [

The Special Theory of Relativity is logically inconsistent. Therefore it is false. The Lorentz Transformation is meaningless.

The consequences for physics, astronomy, and cosmology, are profound. All aspects of theoretical physics where the Theory of Relativity has been employed must be re-examined because they cannot hold good. Certain consequences have already been explored [

Crothers, S.J. (2018) Reply to “Critical Comments on the Paper ‘On the Logical Inconsistency of the Special Theory of Relativity’”. Journal of Applied Mathematics and Physics, 6, 1230-1241. https://doi.org/10.4236/jamp.2018.66103