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PHYSICS ESSAYS 37, 1 (2024)
Special relativity and the Lorentz equations. Errors in Einstein’s 1905 paper
Alasdair N. Beala)
10 King George Avenue, Chapel Allerton, Leeds LS7 4LH, United Kingdom
(Received 3 July 2023; accepted 18 December 2023; published online 9 January 2024)
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.” VC 2024 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-37.1.46]
Resume: L’explication de la theorie de la relativite restreinte d’Einstein dans son article original de 1905 est examinee. Son analyse pr^ete a confusion, car les termes x, y, z, t, etc. ont des signiﬁcations differentes a differents moments et il presente des equations basees sur des hypothe`ses differentes et incoherentes. L’ajout d’indices clariﬁe ces proble`mes mais induit des erreurs dans son raisonnement. Pour calculer ses equations de transformation, il selectionne une combinaison d’equations qui donne des resultats correspondant a la transformation de Lorentz mais il ignore les autres solutions valides possibles. Ses calculs contiennent egalement de graves erreurs. Il ne parvient donc pas a prouver que sa theorie conduit aux equations de Lorentz comme solution unique. L’analyse d’Einstein inclut des horloges ‘en onseque’ qui afﬁchent un temps ‘stationnaire’ t, donc l’idee selon laquelle une horloge en mouvement devrait fonctionner plus lentement qu’une horloge stationnaire est incompatible avec sa theorie. De plus, son calcul de la dilatation du temps contient de graves erreurs. En consequence, il ne parvient pas a fournir une justiﬁcation theorique a son fameux ‘paradoxe des horloges’.
Key words: Einstein; Special Relativity; Lorentz Equations; Time Dilation; Transformation Equations; Clocks Paradox; Mathematical Errors.

I. 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 theory2 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
a)a.beal@btinternet.com

positioning system (GPS), e.g., Refs. 4–16. The present paper examines the logic and validity of Einstein’s analysis and calculations in Part I (Kinematical Part) of his 1905 paper.
As terms such as x, x0, 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-reference, Einstein’s symbols and general notation are adopted rather than modern notation.
II. 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-ordinates,” “a stationary rigid rod,” “the time of the stationary system,” “stationary space,” “stationary clocks,” and “the time of the

ISSN 0836-1398 (Print); 2371-2236 (Online)/2024/37(1)/46/9/$25.00

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VC 2024 Physics Essays Publication

Physics Essays 37, 1 (2024)

47

stationary system;” he also compares the timekeeping of “travelled clocks” which have “moved” with clocks which “remained at rest.” To understand this it is necessary to refer to his deﬁnition of “stationary system” in Sec. I:
“Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the ‘stationary system.’”
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 Secs. II and III, as these are all in uniform translatory motion. This should be borne in mind whenever he describes an item as stationary.
In Sec. II, he deﬁnes the principle of relativity and the principle of the constancy of the velocity of light:
“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-ordinates in uniform translatory motion” (the principle of relativity).
“2. Any ray of light moves in the ‘stationary’ system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body” (the principle of the constancy of the velocity of light).
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).

III. SYNCHRONISING CLOCKS
At the heart of Einstein’s analysis is a procedure for synchronising clocks by light ﬂashes. In Sec. I, 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’ t0A. In accordance with definition the two clocks synchronise if

tB À tA ¼ tA0 À tB

(1)

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 Sec. II, 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.’”

IV. THOUGHT EXPERIMENT A

In Sec. III, 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 n, g, and f and clocks show time s.
In the ﬁrst thought experiment (referred to here as “experiment A”) a light ﬂash emitted at time s0 by an observer at the origin of system k is reﬂected at s1 from a clock on the X-axis remote from the origin and it returns to the origin at s2. As the remote clock is synchronised with the observer’s clock,

1=2ðs0 þ s2Þ ¼ s1:

(2)

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

x0 ¼ x À vt;

(3)

and states

“… it is clear that a point at rest in the system k must have a system of values x0, y, z, independent
of time.”

He then expresses s as a function of (x0, y, z, t):

 1=2 sð0; 0; 0; tÞ þ s 0; 0; 0; t þ

x0

þ

x0 !

cÀv cþv


¼ s x0; 0; 0; t þ x0 :

(4)

cÀv

In Einstein’s equations (1) and (2) different values of variables are identiﬁed by subscripts, without which the equations would be meaningless. However, in Eq. (4), there

48

Physics Essays 37, 1 (2024)

are no subscripts, despite some of its terms appearing to

have more than one meaning

First set: ðs ¼ s0Þ : x0 ¼ 0; y ¼ 0; z ¼ 0; t ¼ t;

second set: third set: ðs

ðs ¼ ¼ s1Þ

s2Þ : x0

: x0 ¼ ¼ x0;

0; y y¼

¼ 0;

0; z z¼

¼ 0;

0; t t¼

¼t tþ

cþÀx0 vc.Àx0v

þ

cþx0 v;

Before proceeding further, the meanings of these terms

need to be determined. In the ﬁrst set, it is fairly obvious that

the meaningless “t ¼ t” should actually be “t ¼ t0,” i.e., the

system K time when the ﬂash is emitted, corresponding to s0 in Eq. (2). Similarly, in the second set, having x0 ¼ 0 and also t ¼ t þ ðx0=ðc À vÞÞ þ ðx0=ðc þ vÞÞ makes no sense; logically, the ﬁrst x0 must be the coordinate of the system k ori-
gin but the second x0 (in the expression for t) must represent

some other value, e.g., the coordinate of the remote clock. Likewise, in the third set x0 and t must represent particular

values of these variables. Values of (x0, y, z, t) which would make sense of Eq. (4)

are listed below. In the analysis which follows, these are

assumed to have been Einstein’s intended meanings.

ﬁrst set: ðs ¼ s0Þ : x0 ¼ 0; y ¼ 0; z ¼ 0; t ¼ t0,

second set:

þ

x01 cÀv

þ

cxþ01v,

third set: ðs

ðs ¼ ¼ s1Þ :

s2Þ : x0 ¼

x0 ¼ x01; y

0; y ¼ ¼ 0; z

0; ¼

0;

z t

¼ ¼

0; t ¼ t2 ¼ t0 t1 ¼ t0 þ cxÀ01v,

where x01 is the K coordinate of the remote clock, t0 is the

system K time when the light ﬂash is emitted (corresponding

to s0), t1 is the system K time when the light ﬂash reaches the remote clock (corresponding to s1), and t2 is the system K time when the reﬂected ﬂash reaches the origin of k (corre-

sponding to s2) (Fig. 1). 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

1=2

sð0; 

0;

0;

tA0

Þ

þ

 s 0;

0; 0; 

tA0

þ

x0A1 cÀv

þ

x0A1 ! cþv

¼s

x0A1

;

0;

0;

tA0

þ

x0A1 cÀv

:

(4a)

FIG. 1. Experiment A (a) light ﬂash on X-axis as observed in system K. (b) Light ﬂash on X-axis as observed in system k.

For the system k observer, the corresponding equation with coordinates (n, g, f, s) is


 !

1=2 sð0; 0; 0; sA0Þ þ s

0;

0;

0;

sA0

þ

2nA1 c


¼s

nA1;

0;

0;

sA0

þ

nA1 c

;

(5)

where nA1 is the k coordinate of the remote clock. Note: In this experiment, the light ﬂash velocity is c rela-

tive to the system K observer and it is also c relative to the

system k observer, although these observers are moving rela-

tive to one another. To make the analysis easier to under-

stand, 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 x ¼ x0 þ vt) 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 posi-

tioned 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 Eq. (4), Einstein states

“Hence, if x0 be chosen inﬁnitesimally small:


1

1 @s @s 1 @s

1=2 c À v þ c þ v @t ¼ @x0 þ c À v @t

(6)

or

@s @x0

þ

c2

v À

v2

@s @t

¼

0:

(7)

…

Since s is a linear function, it follows from these equa-

tions that


s¼a

t

À

c2

v À

v2

x0

;

(8)

where a is a function u(v) at present unknown, and where
for brevity it is assumed that at the origin of k, s ¼ 0 when t ¼ 0.”
Note: At the origin of k, n ¼ x0 ¼ x À vt ¼ 0, so if t ¼ 0
the origins of k and K coincide.
Again, Einstein fails to identify the meanings of terms clearly: In Eqs. (6)–(8) do x0, t, and s represent general varia-
bles, or particular values of these? If Eq. (4a) interprets Einstein’s intended meanings correctly, then in Eqs. (6)–(8) x0 should be xA0 1, the system K coordinate of the remote clock. However t is probably not the t which appears three times in
Eq. (4), which is the experiment start time t0: it is probably t1 but is it t1 ¼ t0 þ ðx01=c À vÞ, or is it t1 ¼ 1=2ðt0 þ t2Þ¼ 1=2ðt0 þ ðx01=ðc À vÞÞ þ ðx01=ðc þ vÞÞÞ? This is unclear, as these are both combined in Einstein’s subsequent calcula-
tions and (as discussed below) there are incompatibilities

Physics Essays 37, 1 (2024)

49

between the “events” t1 and t2 are based on. To reﬂect this

uncertainty, the subscript 1? is applied here to t and s:


sA1? ¼ a

tA1?

À

vx0A1 c2 À v2

:

(8a)

Equation (4a) includes three pairs of events:

(i) outward ﬂashes emitted by the system K and k observers at s ¼ t ¼ 0;
(i) reﬂection of these ﬂashes from the remote clock at sA1 and tA1; and
(iii) the reﬂected ﬂashes in both systems reaching the system k origin at sA2 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

¼

1

x0A1 À v=c

;

(9)

tA1

¼

xA1 c

¼

x0A1 cÀv

:

(10)

In system k, the ﬂash has path length nA1 and arrival time

sA1 ¼ nA1=c:

(11)

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

ðxA1

À

xA2Þ

¼

1

x0A1 þ v=c

;

(12)


tA2 ¼

tA1

þ

x0A1 cþv

¼

x0A1 þ x0A1 : cÀv cþv

(13)

In k the ﬂash path length is nA1 and its arrival time is

sA2 ¼ ðsA1 þ nA1=cÞ ¼ 2nA1=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 s2 ¼ 2s1 and if it leaves and returns to the system K origin t2 ¼ 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 ¼ ðx0A1=ðc À vÞÞ and ðtA2 À tA1Þ ¼ ðx0A1=c þ vÞ. Therefore, sA2 ¼ 2sA1 but tA2 6¼ 2tA1; also 1=2ðs0 þ s2Þ ¼ s1 but 1=2ðt0 þ t2Þ 6¼ t1.
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)]—and vice versa. Therefore, events (ii) and (iii) are incompatible.
Einstein’s time calculations in Eqs. (4)–(8) are based on a two-way light ﬂash, including both its outward and return paths. However, he now considers a light ﬂash which travels in only one direction:

“For a ray of light emitted at time t ¼ 0 in the direction of increasing n:

n ¼ cs

(15)

or


n ¼ ac

t

À

c2

v À

v2

x0

:

(16)

But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c–v, so that

x0 ¼ t:

(17)

cÀv

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

n

¼

a

c2

c2 À

v2

x0:

(18)

[With subscripts added, these equations become

nA1 ¼ csA1;


nA1 ¼ ac

tA1?

À

c2

v À

v2

x0A1

;

(15a) (16a)

x0A1 cÀv

¼

tA1;

nA1

¼

a

c2

c2 À

v2

x0A1?:

(17a) (18a)

Note: X0 ¼ x0A1? in (18a) because t ¼ tA1? in (16a).] Einstein does not explain his reasons for the change
from a 2-way ﬂash to a 1-way outward-only ﬂash, or how the
obvious differences between these scenarios are to be recon-
ciled in his calculations. The change to a 1-way ﬂash does
not signiﬁcantly affect equation (15), as the corresponding equation for a 2-way ﬂash is: sA2 ¼ 2sA1 ¼ 2nA1. However, Eq. (17) is affected by both the change to a 1-way ﬂash and
the choice of ﬂash direction: [see Eq. (4)] the time for a 1-way return ﬂash is t ¼ ðx0=c þ vÞ and the mean of the 2way outward and return ﬂash times is t ¼ 1=2ððx0=c À vÞ þðx0=c þ vÞÞ. Therefore, Eq. (8), which is based on a combination of 2-way ﬂash times, is not compatible with (17) and
Eq. (18), which Einstein derives by combining Eq. (8) with
(17), is invalid.

V. THOUGHT EXPERIMENT B
After Eq. (18), Einstein states:
“In an analogous manner we find, by considering rays moving along the other two axes ….”
In this scenario, the origins of k and K coincide at t ¼ s ¼ 0 and the system k origin moves along the X-axis at velocity v but instead of the light ﬂash travelling to a clock

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Physics Essays 37, 1 (2024)

nB1 ¼ 0:

(23)

FIG. 2 Experiment B Light ﬂash along Y-axis of system k.

at coordinate n on the X-axis, it travels to a clock at coordinate g on the system k Y-axis. Einstein compares how these events are seen by the system k and K observers but, unlike in experiment A Eq. (4), he considers only a 1-way outward ﬂash. As this scenario is signiﬁcantly different from experiment A (see discussion below), it is considered here separately as “experiment B” (Fig. 2).
Einstein’s equations are

g ¼ cs;

(19)

pﬃﬃﬃﬃyﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ t;

(20)

c2 À v2

x0 ¼ 0:

(21)

Unfortunately, once more he 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 ¼ ct1 but in B (see below) x1 ¼ vt1 (22);
• in experiment A, x01 ¼ t1ðc À vÞ (17) but in B, x01 ¼ 0 (21); • in experiment A: y1 ¼ 0pﬃ(ﬃ4ﬃﬃﬃaﬃﬃ)ﬃ,ﬃﬃﬃsﬃﬃ1ﬃ ¼ n1=c (11) and g1 ¼ 0
(4a); but in B: y1 ¼ t1 c2 À v2 (20a), n1 ¼ 0 (23), and g1 ¼ cs1 (19a).
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

gB1 ¼ csB1;

pﬃﬃyﬃﬃﬃBﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃ c2 À v2

¼

tB1;

x0B1 ¼ 0;

(19a) (20a) (21a)

where sB1 is the system k time when the light ﬂash reaches the remote clock, gB1 is its system k coordinate, tB1 is the corresponding system K time and yB1 is the system K coordinate.
Also

xB1 ¼ vtB1;

(22)

VI. 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-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x0, y, z …”

He provided no evidence or analysis to support this

assertion but he relies on it in his subsequent analysis, where

he assumes that equations from experiment A will be valid

in experiment B and vice versa.

He combines experiment B equation (19) g ¼ cs with

experiment A equation (8) to obtain


g ¼ ac

t

À

c2

v À

v2

x0

:

(24)

He combines Eq. (24) with (21) to obtain

g ¼ act:

(25)

He combines Eq. (25) with (20) to obtain

g ¼ a pﬃﬃﬃﬃcﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y:

(26)

c2 À v2

He then substitutes

a ¼ uðvÞ=b;

(27)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b ¼ 1= 1 À v2=c2;

(28)

to obtain

g ¼ uðvÞy:

(29)

To obtain his transformation equations Einstein com-

bines (27) and (28) with experiment A equations (8) and (18)


s¼a

t

À

c2

v À

v2

x0

(8)

À

Á

¼> s ¼ uðvÞb t À vx=c2 ;

(30)

n

¼

a

c2

c2 À

v2

x0

(18)

¼> n ¼ uðvÞbðx À vtÞ;

(31)

g ¼ uðvÞy;

(32)

f ¼ uðvÞz:

(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 s, an additive constant is to be placed on the right side of each of these equations” (see Refs. 17 and 18).

Physics Essays 37, 1 (2024)

51

With subscripts added, Eqs. (30) and (31) are

À

Á

sA1 ¼ uðvÞb t1? À vx1=c2 ;

n1 ¼ uðvÞbðx1 À vt1?Þ:

(30a) (31a)

Einstein then introduces “a third system of co-ordinates K0” which, according to the Perrett and Jeffery translation,

“… relatively to the system k is in a state of parallel transla-

tory motion parallel to the axis of X, such that the origin of coordinates of system k moves with velocity Àv ….”19 How-

ever this is incorrect: The phrase “of system k” is not in the

original German text, so it should read: “… relatively to the

system k is in a state of parallel translatory motion parallel to

the axis of X, such that the origin of coordinates moves with velocity Àv ….” Thus K0 is moving at Àv relative to k and it

is simply system K viewed from system k.

By combining transformation equations for these sys-

tems, he obtains

À

Á

t0 ¼ uðÀvÞbðÀvÞ s þ vn=c2 ¼ uðvÞuðÀvÞt;

(34)

x0 ¼ uðÀvÞbðÀvÞðn þ vsÞ ¼ uðvÞuðÀvÞx

(35)

(NB: Here x0 is the system K0 coordinate of the system K origin, not x0 ¼ x À vt as elsewhere.)
Einstein concludes that K0 and K must be identical, so
uðvÞuðÀvÞ ¼ 1 and, because the length of a rod perpendicular to the X-axis must be the same for þv and Àv,
uðvÞ ¼ uðÀvÞ, so:

uðvÞ ¼ 1:

(36)

Thus his transformation equations become

À

Á

s ¼ b t À vx=c2 ;

(37)

n ¼ bx0 ¼ bðx À vtÞ;

(38)

g ¼ y;

(39)

f ¼ z;

(40)

where

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b ¼ 1= 1 À v2=c2

(28)

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 ¼ 0.
In these equations x and n are x1 and n1, the system K and k coordinates of the remote clock when the light ﬂash is reﬂected back from it; t and s also relate to this event but, as noted earlier, Einstein does not deﬁne their meanings clearly in Eqs. (6)–(8).

VII. 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) s ¼ aðt À vx0=ðc2 À v2ÞÞ with Eq. (15) n ¼ cs to obtain Eq. (16) n ¼ acðt À vx0=ðc2 À v2ÞÞ; and

(ii) he also combines (8) with experiment B equation (19) g ¼ cs to obtain Eq. (24) g ¼ acðt À vx0=ðc2 À v2ÞÞ.
Therefore if Eqs. (16) and (24) are both true:
g ¼ cs ¼ n.
However,
(iii) in experiment A, n1 ¼ cs1 (11) and g1 ¼ 0 (5), so g 6¼ n; and
(iv) in experiment B, n1 ¼ 0 (23) and gB1 ¼ csB1 (19a), so again g 6¼ n.

Therefore, Einstein’s assumption g ¼ cs ¼ n is false and his calculations based on this are invalid.
Also, as noted earlier, some of his equations are based on a 2-way light ﬂash in experiment A, some are based on a 1-way ﬂash in experiment A and some are based on a 1-way ﬂash in experiment B. As these are not compatible with one another, when combined they will produce false results.
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 Eq. (16) n ¼ acðtÀvx0=ðc2 Àv2ÞÞ
and he also combines (8) with experiment B equation (19) g ¼ cs to obtain (24) g ¼ acðtÀvx0= ðc2 Àv2ÞÞ;
(ii) he combines (24) with experiment B equation (21) x0 ¼ 0 topoﬃbﬃﬃtﬃaﬃﬃﬃiﬃnﬃﬃﬃﬃ(ﬃﬃ2ﬃﬃ5ﬃﬃ) g ¼ act and combines this with (p20ﬃﬃ)ﬃﬃﬃﬃﬃﬃyﬃﬃ=ﬃﬃﬃﬃﬃﬃﬃðﬃ c2 À v2Þ ¼ t to obtain (26) g ¼ acy= ðc2 À v2Þ;pbyﬃﬃﬃﬃcﬃﬃoﬃﬃmﬃﬃﬃﬃbﬃﬃﬃiﬃnﬃﬃﬃing this with (27) a ¼ uðvÞ=b, (28) b ¼ 1= 1 À v2=c2 and (36) uðvÞ ¼ 1 he obtains
his transformation equation (39) g ¼ y;
(iii) by combining experiment A equation (8) with (27),
(28), and (36), he obtains his transformation equation (37) s ¼ bðt À vx=c2Þ.

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

a. If experiment B equations (19) g ¼ cs and (25) g ¼ act

are combined with Eqs. (27), (28), and (36) the result is

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s ¼ t 1 À v2=c2:

(41)

b. If experiment A equations (8) s ¼ aðt À vx0=ðc2 Àv2ÞÞ and (17) x0=ðc À vÞ ¼ t are combined with Eqs. (27),

(28) apndﬃﬃ(ﬃﬃ3ﬃﬃ6ﬃﬃﬃ)ﬃﬃﬃtﬃhﬃﬃe result is

s ¼ t p1ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃvﬃﬃ=ﬃﬃcﬃﬃ :

(42)

1 þ v=c

c. If experiment A equations (8) and (17) are combined with experiment B equation (21) x0 ¼ 0, the result is

s ¼ t ¼ 0:

(43)

These alternative transformation equations are all calculated from combinations of Einstein’s equations, yet they differ from his Eq. (37). [Note: In (37) s appears to depend on x, t and v/c, whereas in Eqs. (41) and (42) it depends on only

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Physics Essays 37, 1 (2024)

t and v/c and in Eq. (43) it depends on neither. However this difference is illusory: x and t can be interchanged via his Eqs. (3) x0 ¼ x À vt and (17) x0=ðc À vÞ ¼ t.] Einstein does not explain why he selected only the particular combination of equations which produces transformation equation (37) and ignored other possible combinations. Alternative transformation equations (41)–(43) are all equally valid solutions to his analysis.
Thus Einstein has failed to prove that calculations based on his theory produce the Lorentz equations as a unique solution.

VIII. LENGTH SHORTENING AND TIME DILATION

In Sec. IV, “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-dimension will appear shortened in the ratio 1:ͱ(1 À v2/c2) … It is clear that the same results hold good of bodies at rest in the ‘stationary’ sys-
tem, viewed from a system in uniform motion.”

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-ordinates of k and so adjusted that it marks the time s. What is the rate of this clock, when viewed from the stationary system?

Between the quantities x, t, and s, which refer to the position of the clock, we have, evidently, x ¼ vt and

s

¼

pﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

À t

À

Á vx=c2 :

(37)

1 À v2=c2

Therefore,

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s ¼ t 1 À v2=c2 ¼ t À 1 À 1 À v2=c2 t;

(44)

whence it follows that the time marked by the clock p(vﬃiﬃeﬃﬃwﬃﬃﬃﬃeﬃﬃdﬃﬃﬃﬃiﬃnﬃﬃﬃﬃthe stationary system) is slow by
1 À 1 À v2=c2 seconds per second, or— neglecting magnitudes of fourth and higher order—by 1=2v2=c2.”
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 Sec. II.

(ii) His equation x ¼ 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— and in experiment A the relevant equation for this is (10) x1 ¼ ct1.
(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 n ¼ g ¼ f ¼ 0 and its K coordinates are x0 ¼ y ¼ z ¼ 0. In experiment A, if nA1 ¼ x0A1 ¼ 0 then according to Einstein’s equations (4), (15), and (17): sA1 ¼ tA1 ¼ 0. In experiment B, if gB1 ¼ yB1 ¼ fB1 ¼ zB1 ¼ 0 then according to his Eqs. (19) and (20) sB1 ¼ tB1 ¼ 0. Therefore if the clock’s timekeeping is analysed correctly on the basis of experiments A and B only one conclusion is possible: s ¼ t ¼ 0 (cf. Ref. 20).
Therefore, Einstein’s calculation of time dilation is ﬂawed and does not support his claipm ﬃtﬃhﬃﬃﬃaﬃﬃtﬃﬃﬃaﬃﬃﬃcﬃﬃlﬃoﬃﬃﬃck at the system k origin will run slow by 1 À 1 À v2=c2 s/s relative to a clock at rest in system K.
IX. 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 1=2tv2=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 1=2tv2=c2when it returns to its starting position; and
d. a clock on the Earth’s Equator will run slow by1=2tv2= 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., Refs. 4 and 8–13). Kelly21 lists 54 different explanations of the “twins paradox” proposed by various authors (including Einstein). The present analysis adds further objections.
(i) In his main Sec. III 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 Sec. II. 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

Physics Essays 37, 1 (2024)

53

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 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 slow compared with B.
requires a clock at the system k origin to run slower than system K clocks by1=2tv2=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. 3(a)] 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. 3(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 ¼ 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.
X. 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 synchronize a moving clock to show stationary time t and this is implicitly accepted by Einstein in Sec. II. 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, t0A, t0B, s0, s1, and s2 by subscripts in Eqs. (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 experi-

ments. In the ﬁrst (referred to here as experiment A) all

observers, clocks and light ﬂashes are on the X-axis but

his equations are based on a variety of different assump-

tions. In the second (experiment B) the remote clock

and light ﬂash are on the system k Y-axis.

Einstein then combines equations from experiments A

and B, despite the fact that these are based on incompat-

ible assumptions. He assumes g ¼ cs ¼ n but this is

false: in both experiment A and in experiment B, g 6¼ n.

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 calcula-

tions based on his theory lead to the Lorentz transforma-

tion equations as a unique solution.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4. Einstein’s calculation of time dilation s ¼ t 1 À v2=c2

for a clock at the origin of system k contains further fun-

damental errors in addition to those in “3”.

5. As Einstein has failed to prove that his theory requires a moving clock to run slow by 1=2tv2=c2 relative to a sta-

tionary clock, any detailed discussion of his clock para-

dox thought experiments is largely pointless. However

these also contain further illogicalities and anomalies of

their own.

ACKNOWLEDGMENTS
Thanks to David Watson for German translation. Thanks to Gertrud Walton, the late Professor John Field, Dr. John McMillan, Steve Bannister, Des McLernon, David Sellers, Nicholas Percival, and Dr. Philip Webber for advice and comments.
1A. Einstein, Ann. Phys. 17, 891 (1905). H. A. Lorentz, A. Einstein, H. Minkowski, and W. Weyl, The Principle of Relativity, translated by W. Perrett and G. B. Jeffery (Methuen, London, 1923/Dover reprint, Mineola, NY, 1952). 2H. A. Lorentz, Proc. Acad. Sci. Amsterdam 6, 809 (1904). H. A. Lorentz, A. Einstein, H. Minkowski, and W. Weyl, The Principle of Relativity, translated by W. Perrett and G. B. Jeffery (Methuen, London, 1923/Dover reprint, Mineola, NY, 1952). 3The Essential Einstein: His Greatest Works, edited by Stephen Hawking (Penguin, London, 2008, p. x); (Published in USA as A Stubbornly Persistent Illusion, Running Press, 2007.) 4H. Dingle, Science at the Crossroads (Martin Brian & O’Keefe, London, 1972). 5J. H. Field, arXiv:physics/0612041 (2011). 6G. Walton, see https://www.gsjournal.net/Science-Journals/Research%20Papers/View/9346 for “The reciprocal Lorentz Factor.” 7W. Engelhardt, Int. J. Sci. Res. Methodol. 9, 159 (2018). 8L. Essen, The Special Theory of Relativity—A Critical Analysis (Oxford University Press, Oxford, UK, 1971). 9H. Ricker, see https://www.naturalphilosophy.org/site/harryricker/2015/ 07/09/special-relativity-is-irksome/ for “Special Relativity Is Irksome.” 10A. Kelly, Challenging Modern Physics: Questioning Einstein’s Relativity Theories (Brown Walker Press, Boca Raton, FL, 2005), pp. 160–165. 11I. McCausland, Phys. Essays 3, 157 (1990); reprinted in I. McCausland, A Scientiﬁc Adventure: Reﬂections on the Riddle of Relativity (C. Roy Keys Inc., Montreal, QC, Canada, 2011).

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12R. M. Kassir, Mathematical Conﬂicts in the Special Theory of Relativity,
3rd ed. (CreateSpace, North Charleston, SC, 2018). 13N. Percival, J. Phys.: Conf. Ser. 2197, 012022 (2022). 14G. N. Pellegrini, and A. R. Swift, Am. J. Phys. 63, 694 (1995). 15R. R. Hatch, Escape from Einstein (Kneat Kompany, 1992). 16R. P. Hatch, GPS Solutions 8, 67 (2004).

17Einstein, Ref. 1, p. 46. 18J. H. Field, arXiv:1210.2270v1 (2012). 19Einstein, Ref. 1, p. 47. 20R. M. Kassir, Phys. Essays 27, 198 (2014). 21Kelly, Ref. 10, pp. 279–289. 22H. E. Ives, J. Opt. Soc. Am. 27, 305 (1937).