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Physics Essays 37:1, March 2024 pp. 46-

© Physics Essays Publication 2024

Special Relativity and the Lorentz Equations:

Errors in Einstein’s 1905 Paper

Alasdair N Beal BSc CEng FICE FIStructE

Abstract

The explanation of Einstein’s Special Theory of Relativity in his original 1905 paper is examined. His analysis is confusing, as terms x, y, z, t etc. have different meanings at various points and he presents equations based on different and inconsistent assumptions. Adding subscripts clarifies these issues but exposes errors in his reasoning. To calculate his transformation equations he selects a combination of equations which gives results matching the Lorentz Transformation but he ignores other possible valid solutions. Also his calculations contain serious errors. Therefore he fails to prove that his theory leads to the Lorentz equations as a unique solution.

Einstein’s analysis includes ‘moving’ clocks which show ‘stationary’ time t, so the idea that a moving clock should run slower than a stationary clock is incompatible with his theory. Also, his calculation of time dilation contains serious errors. As a result, he fails to provide a theoretical justification for his famous ‘clocks paradox’.

Keywords

Einstein, Special Relativity, Lorentz Equations, time dilation, Transformation Equations, Clocks Paradox, mathematical errors.

Introduction

Einstein’s Special Theory of Relativity was first published in 1905 [1]. In this paper he presented calculations to demonstrate that his theory leads to transformation equations for time and distance matching those previously derived by Lorentz from ether theory [2] and thus the idea of an ether is ‘superfluous’. Most scientists now accept Einstein’s theory and his paper is regarded as one of the most important in modern physics. In 2007 Hawking stated:

‘The details of Einstein’s reasoning, and the simple algebra behind it, are explained nowhere better than as found here, in Einstein’s own words’ [3].

However there are still some who have raised theoretical objections to Einstein’s theory and also questioned its reconciliation with results from experiments and the satellite global positioning system (GPS), e.g. [4-

As terms such as x, x', y and t have more than one possible meaning in his calculations, subscripts are added in the present paper to identify these and clarify the analysis. For ease of cross-

Definitions

In his introduction, Einstein states:

‘... the view to be developed here will not require an ‘absolutely stationary space’ provided with special properties’.

However in subsequent passages he refers to ‘the stationary system’, ‘the stationary system of co-

‘Let us take a system of co-

Thus Einstein’s ‘stationary system’ is not ‘stationary’ as normally understood: to comply with his definition it need only be in uniform translatory motion. Also it is not unique: it could be any of the coordinate systems he introduces in §2 and §3, as these are all in uniform translatory motion. This should be borne in mind whenever he describes an item as ‘stationary’.

In §2 he defines two principles:

‘1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to one or the other of two systems of co-

and:

‘2. Any ray of light moves in the ‘stationary’ system of co-

Based on Einstein’s definition of ‘the stationary system’, principle ‘2’ could be more clearly stated as:

‘In a system of coordinates in uniform translatory motion, any ray of light moves with the determined velocity c, whether the ray be emitted by a body which is stationary or moving relative to that system’ (‘the principle of the constancy of the velocity of light’).

Synchronising Clocks

At the heart of Einstein’s analysis is a procedure for synchronising clocks by light flashes. In §1 he states that if observers and clocks at A and B are at rest in the ‘stationary’ coordinate system:

‘Let a ray of light start at the ‘A time’ tA from A towards B, let it at the ‘B time’ tB be reflected at B in the direction of A, and arrive again at A at the ‘A time’ t'A. In accordance with definition the two clocks synchronise if

tB -

Einstein then states:

‘It is essential to have time defined by means of stationary clocks in the stationary system’.

For this synchronisation method to work, the velocity of light must be constant relative to the observers carrying out the measurements, so these observers and their clocks must be at rest in the relevant coordinate system. However if clock B has a reflective mirror face, observer A can read time tB directly in its reflected image and it is unnecessary to have an observer at B. Time tB depends only on the location of clock B at the moment when the light flash reaches it, not its state of motion, so, although Einstein does not discuss this, his light flash synchronisation method will work equally well if clock B is moving.

Therefore it is possible for a ‘moving’ clock to be synchronised to show ‘stationary system’ time. Einstein implicitly accepts this in §2, where he describes a ‘moving’ rod on which:

‘ ... clocks are placed which synchronise with the clocks of the stationary system, that is to say that their indications correspond at any instant to the ‘time of the stationary system’ at the places where they happen to be. These clocks are therefore ‘synchronous in the stationary system’’.

Thought Experiment A

In §3 Einstein describes thought experiments which compare time and space measurements in ‘stationary’ system K with those in ‘moving’ system k travelling at velocity v relative to K. Each system has an observer and clock at rest at its origin. In system K, space coordinates are x, y and z and clocks show time t; in system k, space coordinates are ξ, η and ζ and clocks show time τ.

In the first thought experiment (referred to here as ‘experiment A’) a light flash emitted at time τ0 by an observer at the origin of system k is reflected at τ1 from a clock on the X-

½(τ0 + τ2) = τ1 (2)

Einstein then analyses this scenario as seen by the system K observer. He defines:

x΄ = x -

and states:

‘... it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time.’

He then expresses τ as a function of (x',y,z,t):

½[τ(0,0,0,t) + τ(0,0,0,t + x'/(c-

In Einstein’s equations (1) and (2) different values of variables are identified by subscripts, without which the equations would be meaningless. However in equation (4) there are no subscripts, despite some of its terms appearing to have more than one meaning:

first set (τ = τ0): x' = 0, y = 0, z = 0, t = t;

second set (τ = τ2): x' = 0, y = 0, z = 0, t = t+x'/(c-

third set (τ = τ1): x' = x', y = 0, z = 0, t = t+x'/(c-

Before proceeding further, the meanings of these terms need to be determined. In the first set it is fairly obvious that the meaningless ‘t = t’ should actually be ‘t = t0’, i.e. the system K time when the flash is emitted, corresponding to Τ0 in equation (2). Similarly, in the second set, having x' = 0 and also t = t+x'/(c-

Values of (x',y,z,t) which would make sense of equation (4) are listed below. In the analysis which follows, these are assumed to have been Einstein’s intended meanings.

first set (τ = τ0): x' = 0, y = 0, z = 0, t = t0;

second set (τ = τ2): x' = 0, y = 0, z = 0, t = t2 = t0+x'1/(c-

third set (τ = τ1): x' = x', y = 0, z = 0, t = t1 = t0+x'1/(c-

where x'1 is the K coordinate of the remote clock, t0 is the system K time when the light flash is emitted (corresponding to τ0), t1 is the system K time when the light flash reaches the remote clock (corresponding to τ1) and t2 is the system K time when the reflected flash reaches the origin of k (corresponding to τ2).

With these subscripts added to identify the meanings of individual terms and ‘A’ added throughout to identify this as an ‘Experiment A’ equation, (4) becomes:

½[τ(0,0,0,tA0) + τ(0,0,0,tA0+x'A1/(c-

For the system k observer, the corresponding equation with coordinates (ξ,η,ζ,τ) is:

½[τ(0,0,0,τA0) + τ(0,0,0,τA0+2ξA1/c)] = τ(ξA1,0,0,τA0+ξA1/c) (5)

where ξA1 is the k coordinate of the remote clock.

FIG. 1

(a) Experiment A on X-

(b) Light flash on X-

Note: in this experiment the light flash velocity is c relative to the system K observer and it is also c relative to the system k observer, although these observers are moving relative to one another. To make the analysis easier to understand, in the following discussion the flash is described as if it is a pair of flashes, one travelling at velocity c relative to system k and the other at c relative to system K. In system k the flash travels between points which are all at rest in the system. However in system K the flash is reflected at t1 from a point which is moving at velocity v (it is ‘at rest in the system k’ and has coordinate ) and it returns at t2 to the origin of system k, which is also moving at velocity v. Times t1 and t2 could be determined by stationary system K observers positioned beside the locations where these occur or, alternatively, moving system K clocks could be positioned alongside the system k clocks and the times on their reflected images could then be read directly by the observer at the system K origin.

After equation (4) Einstein states:

‘Hence, if x' be chosen infinitesimally small:

½(1/(c-

or ∂τ/∂x' + (v/(c²-

...

Since τ is a linear function, it follows from these equations that

τ = a(t -

where a is a function φ(v) at present unknown, and where for brevity it is assumed that at the origin of k, τ = 0 and t = 0.’

Note: at the origin of k, ξ = x' = x-

Again, Einstein fails to identify the meanings of terms clearly: in (6), (7) and (8) do x', t and τ represent general variables, or particular values of these? If equation (4a) interprets Einstein’s intended meanings correctly, then in equations (6), (7) and (8) x' should be x'A1, the system K coordinate of the remote clock. However t is probably not the ‘t’ which appears three times in equation (4), which is the experiment start time t0: it is probably t1 but is it t1 = t0+x1/(c-

τA1? = a(tA1? -

Equation (4a) includes 3 pairs of ‘events’:

(i) outward flashes emitted by the system K and k observers at τ = t = 0;

(ii) reflection of these flashes from the remote clock at τA1 and tA1; and

(iii) the reflected flashes in both systems reaching the system k origin at τA2 and tA2.

For event (i), the origins of k and K coincide, so both flashes are emitted simultaneously from the same location.

For event (ii), in system K the flash path length and arrival time are:

xA1 = x'A1/(1-

tA1 = xA1 = x'/(c-

In system k the flash has path length ξA1 and arrival time:

τA1 = ξA1/c (11)

For event (iii), in K the flash path length and arrival time are:

(xA1 -

tA2 = tA1 + x'A1/(c+v) = x'A1/(c-

In k the flash path length is ξA1 and its arrival time is:

τA2 = τA1 + ξA1/c = 2ξA1/c (14)

As noted earlier, when synchronising clocks the flash should have equal outward and return path lengths and travel times. Thus if it leaves and returns to the system k origin τ2 = 2τ1 and if it leaves and returns to the system K origin t2 = 2t1. However in Einstein’s equation (4) both flashes return to the origin of system k. Therefore the outward and return travel times of the flash are equal in system k but in system K they are unequal: tA1 = x'A1/(c-

Therefore if the outward flashes are assumed to reach the remote clock simultaneously in both systems (event (ii)), the reflected flashes will not arrive simultaneously at the origin of k (event (iii)) -

Einstein’s time calculations in equations (4) -

‘For a ray of light emitted at time t = 0 in the direction of increasing ξ:

ξ = cτ (15)

or ξ = ac(t -

But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-

x'/(c-

If we insert this value of t in the equation for ξ, we obtain:

ξ = ax'c2/(c2-

(With subscripts added, these equations become:

ξA1 = cτA1 (15a)

ξA1 = ac(tA1? -

x'A1/(c-

ξA1 = axA1?'c2/(c2-

Note: x' = x'A1? in (18a) because t = tA1? in (16a).)

Einstein does not explain his reasons for the change from a 2-

Thought Experiment B

After equation (18) Einstein states:

‘In an analogous manner we find, by considering rays moving along the other two axes …’.

FIG. 2

Experiment B Light flash along Y-

In this scenario, the origins of k and K coincide at t = τ = 0 and the system k origin moves along the X-

Einstein’s equations are:

η = cτ (19)

y/√(c²-

x'B1 = 0 (21)

Unfortunately, he again fails to identify the meanings of individual terms with subscripts or to identify these as experiment B equations. This is important because:

* in experiment A: x1 = ct1 but in B (see below) x1 = vt1 (22);

* in experiment A: x'1 = t1(c-

* in experiment A: y1 = 0 (4a), τ1 = ξ1/c (11) and η1 = 0 (4a); but in B: y1 = t1√(c²-

In each case, it is clear that the experiment A and experiment B equations are incompatible, so any results obtained by combining them will not be meaningful.

If suffix ‘B’ is added generally, together with numerical suffixes as in experiment A, Einstein’s experiment B equations become:

ηB1 = cτB1 (19a)

yB1/√(c²-

x'B1 = 0 (21a)

where τB1 is the system k time when the light flash reaches the remote clock, ηB1 is its system k coordinate, tB1 is the corresponding system K time and yB1 is the system K coordinate.

Also:

xB1 = vtB1 (22)

ξB1 = 0 (23)

Combining equations from Experiments A and B

After experiment A equation (7), Einstein stated:

‘It is to be noted that instead of the origin of the co-

He provided no evidence or analysis to support this assertion but he relies on it in his subsequent analysis, assuming that equations from experiment A will be valid in experiment B and vice versa.

He combines experiment B equation (19) η = cτ with experiment A equation (8) to obtain:

η = ac(t -

He combines (24) with (21) to obtain:

η = act (25)

He combines (25) with (20) to obtain:

η = acy/√(c²-

He then substitutes:

a = φ(v)/β (27)

β = 1/√(1-

to obtain:

η = φ(v)y (29)

To obtain his transformation equations Einstein combines (27) and (28) with experiment A equations (8) and (18):

τ = a(t -

=> τ = φ(v)β(t -

ξ = ac²x'/(c²-

=> ξ = φ(v)β(x -

η = φ(v)y (32)

ζ = φ(v)z (33)

He states:

‘If no assumption whatever be made as to the initial position of the moving system and as to the zero point of τ, an additive constant is to be placed on the right side of each of these equations’ [see 18, 19].

With subscripts added, equations (30) and (31) are:

τA1 = φ(v)β(t1? -

ξ1 = φ(v)β(x1 -

Einstein then introduces ‘a third system of co-

By combining transformation equations for these systems he obtains:

t' = φ(-

x' = φ(-

(NB: here x' is the system K' coordinate of the system K origin, not x' = x -

Einstein concludes that K' and K must be identical, so φ(v)φ(-

φ(v) = 1 (36)

Thus his transformation equations become:

τ = β(t -

ξ = βx' = β(x -

η = y (39)

ζ = z (40)

Where β = 1/√(1-

Although he does not mention it, Einstein’s previous statement again applies: a constant must be added if the origins of k and K do not coincide at t = 0.

In these equations x and ξ are x1 and ξ1, the system K and k coordinates of the remote clock when the light flash is reflected back from it; t and τ also relate to this event but, as noted earlier, Einstein does not define their meanings clearly in equations (6), (7) and (8).

Errors in Einstein’s analysis

There is a major error at the heart of Einstein’s calculation of transformation equations:

(i) he combines experiment A equation (8) τ = a(t -

(ii) he also combines (8) with experiment B equation (19) η = cτ to obtain

(24) η = ac(t -

Therefore if equations (16) and (24) are both true: η = cτ = ξ.

However:

(iii) in experiment A, ξ1 = cτ1 (11) and η1 = 0 (5), so η ≠ ξ; and

(iv) in experiment B, ξ1 = 0 (23) and ηB1 = cτB1 (19a), so again η ≠ ξ.

Therefore Einstein’s assumption η = cτ = ξ is false and his calculations based on this are invalid.

Also, as noted earlier, some of his equations are based on a 2-

Furthermore, Einstein calculates his transformation equations by selecting one particular combination of experiment A and experiment B equations and he ignores other possible combinations. His analysis is summarised below, with relevant equations included for ease of reference.

(i) As noted above, he combines experiment A equations (8) and (15) to obtain (16) ξ = ac(t -

(ii) he combines (24) with experiment B equation (21) x' = 0 to obtain (25) η = act and combines this with (20) y/√(c²-

(iii) by combining experiment A equation (8) with (27), (28) and (36), he obtains his transformation equation (37) τ = β(t -

However Einstein has ignored other possible combinations of his equations, which produce different results. Examples are listed below.

a. If experiment B equations (19) η = cτ and (25) η = act are combined with equations (27), (28) and (36) the result is:

τ = t√(1-

b. If experiment A equations (8) τ = a(t -

τ = t√((1-

c. If experiment A equations (8) and (17) are combined with experiment B equation (21) x' = 0, the result is:

τ = t = 0 (43)

These alternative transformation equations are all calculated from combinations of Einstein’s equations, yet they differ from his equation (37). (Note: in (37) τ appears to depend on x, t and v/c, whereas in (41) and (42) it depends on only t and v/c and in (43) it depends on neither. However this difference is illusory: x and t can be interchanged via his equations (3) x΄ = x-

Thus Einstein has failed to prove that calculations based on his theory produce the Lorentz equations as a unique solution.

Length Shortening and Time Dilation

In §4 ‘Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks’ Einstein states that if a rigid body moving at velocity v is viewed from system K:

‘the X-

Thus its length only appears shortened and reciprocity applies: a rigid body at rest in the ‘stationary system’ will also appear shortened when viewed from the ‘moving system’. This is logical as, by Einstein’s definition, the ‘stationary system’ could be either system K or system k.

Einstein now considers a clock:

‘... located at the origin of the co-

Between the quantities x, t, and τ, which refer to the position of the clock, we have, evidently, x=vt and

τ = (t -

Therefore,

τ = t√(1-

whence it follows that the time marked by the clock (viewed in the stationary system) is slow by (1-

If reciprocity applies to relativistic shortening, logically it should also apply to time dilation but Einstein does not mention this. Also there are errors in his calculations.

(i) As noted previously, it is possible for a ‘moving’ clock to be synchronised to show ‘stationary time’ t and this was implicitly accepted by Einstein in §2.

(ii) His equation x=vt applies to a clock at the origin of system k but in his transformation equation (37) t is the time on a clock which is remote from the origin -

(iii) As shown previously, Einstein’s calculation of his transformation equation contains serious errors.

(iv) Furthermore his time dilation calculation is fundamentally flawed. If a clock is at the origin of k, its k coordinates are ξ = η = ζ = 0 and its K coordinates are x' = y = z = 0. In experiment A, if ξA1 = x'A1 = 0 then according to Einstein’s equations (4), (15) and (17): τA1 = tA1 = 0. In experiment B, if ηB1 = yB1 = ζB1 = zB1 = 0 then according to his equations (19) and (20) τB1 = tB1 = 0. Therefore if the clock’s timekeeping is analysed correctly on the basis of experiments A and B only one conclusion is possible: τ = t = 0 (cf. Kassir [20]).

Therefore Einstein’s calculation of time dilation is flawed and does not support his claim that a clock at the system k origin will run slow by (1-

Einstein’s ‘clock paradox’ thought experiments

After his time dilation calculation, Einstein states:

‘From this there ensues the following peculiar consequence’:

a. a clock moving at velocity v between two synchronised stationary clocks will be found to run slow by ½tv²/c² where t is its journey time;

b. the same is true if the clock moves on a polygonal line, including if its start and finish points coincide;

c. if a clock moves at speed v around a closed curve, it will be slow by ½tv²/c² when it returns to its starting position; and

d. a clock on the Earth’s Equator will run slow by ½tv²/c² relative to a clock on a Pole.

These are his famous ‘clocks/twins paradox’ thought experiments, which have led to much argument over the years [e.g. 4, 8-

(i) In his main §3 analysis Einstein defines v as the relative velocity of coordinate systems, not the relative velocity of clocks, or the velocity of a clock relative to a coordinate system.

(ii) It is possible to synchronise a ‘moving’ clock by the light flash method to show ‘stationary’ time t and this is implicitly accepted by Einstein in §2. However if a ‘moving’ clock shows time t, his predicted ‘peculiar consequence’ will not occur.

(iii) Because of the errors in Einstein’s calculation of time dilation, he fails to justify his claim that his theory requires a clock at the system k origin to run slower than system K clocks by ½tv²/c².

(iv) Einstein does not say whether reciprocity applies to time dilation. If it does, this would invalidate his ‘clock paradoxes’.

(v) The classic ‘clocks paradox’ scenario (fig. 3a) is asymmetrical, as accelerations are applied to the ‘moving’ clock, but Einstein does not discuss the possible effects of these. In a revision proposed by Ives to eliminate this asymmetry [22], clock B travels away eastwards and returns at speed v/2 and clock A travels away westwards and returns at v/2 (fig. 3b).

FIG. 3 (a) Clock paradox: clock A stationary, clock B travels away and returns at velocity v. (b) Clock paradox as amended by Ives: clock A travels away and returns at velocity v/2; clock B travels away and returns at velocity v/2.

Relative to observer A, clock B travels away and returns at speed v, as in the classic scenario, so, by Einstein’s reasoning, at the finish it should be ½tv²/c² slow compared with A. However, relative to observer B, clock A travels away and returns at speed v, so at the finish it should be ½tv²/c² slow compared with B.

(vi) In Einstein’s comparison of clocks on the Equator and Pole what is velocity v? As both clocks are at rest in a system rotating about the Earth’s axis and the distance between them is constant, it could be argued that v=0. Alternatively, if the equatorial clock is thought of as moving in a series of short straight lines at v relative to the polar clock, then the latter is also moving at v in short lines relative to the former. Einstein’s assumption that the equatorial clock has velocity v and the polar clock has velocity 0 is only true if there is a preferred ‘stationary system’ which travels with the Earth but does not rotate.

Conclusions

1. By Einstein’s definition, ‘the stationary system’ can be any system in uniform translatory motion. In his analysis v is the relative velocity of the observers’ coordinate systems, not the velocity of a clock relative to a coordinate system. Using light flashes, it is possible to synchronise a ‘moving’ clock to show ‘stationary’ time t and this is implicitly accepted by Einstein in §2. Therefore it cannot be true that his theory requires a ‘moving’ clock to run slow relative to a ‘stationary’ clock.

2. Einstein identifies times tA, tB, t'A, t'B, τ0, τ1 and τ2 by subscripts in equations (1) and (2) but although terms such as t and x also have multiple possible meanings in his later equations, these are not identified by subscripts. Similarly, he does not identify equations derived from different assumptions to distinguish them from one another. This leads to confusion and errors in his analysis.

3. Einstein’s main analysis considers two thought experiments. In the first (referred to here as ‘experiment A’) all observers, clocks and light flashes are all on the X-

Einstein then combines equations from experiments A and B, despite the fact that these are based on incompatible assumptions. He assumes η = cτ = ξ but this is false: in both experiment A and in experiment B, η ≠ ξ. Furthermore, to calculate his transformation equations he selects a particular combination of equations which produces a result matching the Lorentz equations and ignores other possible combinations of equations which produce different results.

Therefore Einstein fails to justify his claim that calculations based on his theory lead to the Lorentz Transformation equations as a unique solution.

4. Einstein’s calculation of time dilation τ = t√(1-

5. As Einstein has failed to prove that his theory requires a ‘moving’ clock to run slow by ½tv²/c² relative to a ‘stationary’ clock, any detailed discussion of his ‘clock paradox’ thought experiments is largely pointless. However these also contain further illogicalities and anomalies of their own.

Acknowledgements

Thanks to David Watson for German translation. Thanks to Gertrud Walton, the late Prof. John Field, Dr John McMillan, Steve Bannister, Des McLernon, David Sellers, Nicholas Percival and Dr Philip Webber for advice and comments.

References

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22. H. E. Ives, Journal Optical Soc. America, vol. 27 (1937), p. 305. https://DOI.org/10.1364/JOSA.27.000263

Online lecture (available 2 March 2024)

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