The history of the development of the Lorentz transformation (LT) is
reviewed, starting with the original suggestion of Voigt in 1887 for a
modification of the longstanding classical (Galilean) relationship between
space and time coordinates. His conjecture has led to the currently accepted
view by theoretical physicists that space and time are inextricably mixed
and are thus merely two components of a single entity “space time.” The
LT itself, which retains the space-time mixing characteristic, was first
introduced by Larmor. He recognized that Voigt’s transformation needed
to be amended in order to conform to the requirements of the Relativity
Principle (RP). It is pointed out that Newton’s First Law indicates that
clock rates must remain fixed in the absence of unbalanced external
forces, which therefore implies that the ratio of two such rates in different
inertial rest frames should be time-independent as well (Δt’=Δt/Q). It
is critical in this discussion to note that the non-simultaneity of events
demanded by space-time mixing is not consistent with the proportionality
of the time-dilation prediction of the LT; it is impossible for Δt and Δt’
to be proportional to one another without both of them vanishing at
the same time. This contradiction removes the LT from contention as a
physically valid transformation. Lorentz showed at the end of the 19th
century that there was a degree of freedom in the definition of the LT
that could be explored to eliminate this inconsistency. By choosing a
particular value for a normalization constant, it is possible to obtain a
different transformation (GPS-LT) which eliminates space-time mixing
while still satisfying both of Einstein’s two postulates of relativity and
remaining consistent with Newton’s First Law. The *asymmetric* time
dilation observed in many experiments and assumed in the operation of
the Global Positioning System indicates that clock-rate proportionality
should be an essential component of relativity theory, in agreement with
the GPS-LT assumption of a strict proportionality between the rates of
clocks in different inertial systems.

**Keywords:** Time Dilation; Remote Non-Simultaneity; Lorentz
Transformation (LT); Universal Time-Dilation Law (UTDL); Alternative
Global Positioning System-Lorentz Transformation (GPS-LT)

#### Introduction

Confusion ran high among physicists in the latter half of the 19th century
because of their inability to explain the results of a number of experiments
that had been recently carried out with light waves [1]. It had started with
the Fresnel light-drag experiment, which not only showed that light is
slowed as it moves through a transparent medium but, by extrapolation
of the value of the medium’s refractive index n to a unit value, that the
observed light speed in the laboratory should be completely independent
of the speed v of the medium in the limit of free space [c(v) = c]. Maxwell’s
theory of electricity and magnetism published in 1864 also indicated that
the speed of light had the same constant value c in each rest frame in
which it is observed. This result was clearly at odds with the traditional
application of the classical space-time (Galilean) transformation which
indicates that speeds should be additive and therefore that c + v ≠ c. This
led to a frantic search for”ether” which serves as a rest frame for the light
waves analogous to that known for sound waves. Michelson and Morley
[2] used their newly developed interferometer to test this theory, but it
merely verified the conclusion that the speed of light is independent of
the rest frame through which it moves, in particular that it is directionally
independent at all times of the year.

Voigt [3] then stepped into the fray with what in retrospect must be seen
as both a daring and ingenious proposition. He speculated in 1887 that
the problem lay with the Galilean transformation itself. He attempted to
resolve the issue by using nothing more than a free parameter and a little
algebra. The resulting transformation was ultimately rejected on other
physical grounds, namely it violates Galileo’s Relativity Principle (RP),
but it is nonetheless deserving of more than just a footnote in history. This
is because it introduced for the first time the concept of space-time mixing,
which remains to the present day to be a dogmatic principle of theoretical
physics. It contradicts one of Newton’s most cherished beliefs, which held
sway with the physics community for several centuries, namely that space
and time are completely separate entities, one measured with a yardstick
and the other with a clock. The consequences of this aspect of Voigt’s
conjecture will be discussed in the following.

#### Derivation of the Voigt Transformation

The starting point of Voigt’s derivation is the classical or Galilean
transformation (GT). It relates the measured values of space (x,y,z) and
time (t) for a given object obtained by two observers in relative motion
to one another. It is assumed that the two observers are separating with
constant speed v along the common x,x’ axis of the their respective
coordinate systems. The relationship between their measured values is
given below in terms of their respective coordinates, x,y,z,t and x’,y’,z’,t’,
whereby it is assumed that the two systems are coincident at t=t’=0:

$$\text{B}=\frac{\text{A}}{\gamma}\text{}\left(1a\right)$$
$$\text{x'\u2032}=\text{x}-\text{vt}\left(1b\right)$$
$$\text{y=\u2032}=\text{y(1c)}$$

$$\text{z}=\frac{\prime}{=}$$

By construction, the velocity of the object in each coordinate system is
obtained by division of the space and time coordinates at any instant.
Using equations. (1-a-b), one therefore obtains the key relationship
between the measured speeds of the object when it moves along the x,x’
axis:

$$\frac{\text{x \u2032}}{\text{t \u2032}}{\text{= u \u2032}}_{\text{x}}\text{}=\frac{\text{x}}{\text{t}}\text{}-{\text{v=u}}_{\text{x}}-\text{v(2)}$$

There is thus a linear relation connecting the two values of the speed
of the object. More generally, the GT predicts that the corresponding
velocities u and u’ are related by vector addition when the object travels
in a direction which is not parallel to the separation velocity of the two
observers. According to equation. (2), if a light wave has the speed u_{x} = c,
it follows that the corresponding value for the other observer is u′x = c − v.
It is exactly this expected relationship that is not supported by the above
experiments. Voigt’s solution [3] to this problem is simply to add an extra
term to equation. (1a) which depends on x and which also contains a free
parameter a to be determined by requiring that u_{x} = u′_{x} = c for a light wave
moving along the x axis:

$$\text{t \u2032}=\text{t}+\text{ax(3)}$$

Combining this relation with equation. (1b) of the GT, one obtains the
following equation for the two values of the speed of light:

$$\frac{\text{x \u2032}}{\text{t \u2032}}=\text{c}=\frac{\text{x}-\text{vt}}{\text{t}+\text{ax}}\text{}=\frac{\frac{\text{x}}{\text{t}}-\text{v}}{\text{1}+\frac{\text{ax}}{\text{t}}}=\frac{\text{c}-\text{v}}{\text{1}+\text{ac}}\text{(4)}$$

from which one concludes that a = −vc^{−2} in equation. (3). The Voigt
replacement for equation. (1a) of the GT is thus determined to be:

$$\text{t \u2032}=\text{t}-{\text{vc}}^{-\text{2}}\text{x(5)}$$

The above derivation needs to be extended to apply to motion of the light
waves in an arbitrary direction. This can be done most simply by first
forming the quantity ( x′^{2} − c^{2}t′^{2} ) using both equation. (1b) and (5), which
is seen to have the following result

Where γ = (1−v^{2} c^{-2})0.5. Equation. (6) is seen to not only verify the above
light-speed relation for motion along the x axis, but also to give a clear
indication of how to obtain the desired generalization for the case when
the light waves travel in a different direction than along the x axis. By
assuming instead of equations. (1c-d) that y′ = γ^{-1}y and z′ = γ^{-1}z, one
arrives at the following relation between the two sets of coordinates for
light waves moving in an arbitrary direction:

$$\left({\text{x \u2032}}^{\text{2}}+{\text{y \u2032}}^{\text{2}}+{\text{z \u2032}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t \u2032}}^{\text{2}}\right)={\gamma}^{-\text{2}}\left({\text{x}}^{\text{2}}+{\text{y}}^{\text{2}}+{\text{z}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t}}^{\text{2}}\right)\text{(6)}$$

Both sides of this equation vanish for a light wave regardless of its
direction in space, which was the goal of Voigt’s derivation [3]. The
corresponding transformation is thus:

$$\text{t \u2032}=\text{t}-{\text{vc}}^{-\text{2}}\text{x(}8a,\text{}5\text{)}$$

$$\text{x \u2032}=\text{x}-\text{vt(}8b,\text{}1b\text{)}$$

$$\text{y \u2032}={\gamma}^{-\text{1}}\text{y(}8c\text{)}$$

$$\text{z \u2032}={\gamma}^{-\text{1}}\text{z(}8d\text{)}$$

It can be seen that this set of equations reduces to the GT of equations.
(1a-d) in the limit of null relative velocity of the two observers, i.e. if
we ignore the fact that the equations are useless in this case (with v =
0). More significant is the fact that the same equations reduce to the GT
when c is assumed to have an infinite value. One can say then without
qualification that the classical transformation (GT) contains the implicit
assumption that the speed of light is infinite.

#### Taking the Relativity Principle into Account

The space-time transformation that Voigt [3] presented is successful
in satisfying the light-speed constancy condition, but it fails on other
grounds. This can be seen by evaluating the inverse transformation, obtained by Gauss elimination from equations. (8a-d):

$$\text{t}=\text{}{\gamma}^{\text{2}}\left(\text{t \u2032}+{\text{vc}}^{-\text{2}}\text{x \u2032}\right)\text{(9}a\text{)}$$

$$\text{x}={\gamma}^{\text{2}}\left(\text{x \u2032}+\text{v t \u2032}\right)\text{(9}b\text{)}$$

$$\text{y}=\gamma \text{y \u2032 (9}c\text{)}$$

$$\text{z}=\gamma \text{z \u2032 (9}d\text{)}$$

According to Galileo’s Relativity Principle (RP), the inverse
transformation should be obtained by simply exchanging the primed and
unprimed subscripts in the forward set of equations and substituting −v
for v. This is a mathematical procedure that mimics the situation when
the observers change positions; it will be referred to as Galilean inversion
in the following. It is clear that equations. (9a-d) do not satisfy this
relationship relative to equations. (8a-d), hence showing that the Voigt
transformation is not consistent with the RP and thus must be rejected
as a physically valid set of equations. It is nonetheless a simple matter to
modify the transformation in a way which satisfies both the RP and the
light-speed constancy condition. Before doing this, it is helpful to make
a change in variables to intervals for two different events: Δx = x_{2} − x_{1},
Δx′ = x′_{2} − x_{1}′ etc. This change allows each observer to choose his own
coordinate system without the necessity of having it coincide at some
point with the other coordinate system. Intervals are of course required
in order to compute speeds, which remains the center of attention in this
discussion. This being done, one merely needs to multiply each of the
corresponding equations. (8a-d) by a factor of γ on the right, with the
result:

$$\Delta \text{t \u2032}=\gamma \left(\Delta \text{t}-{\text{vc}}^{-\text{2}}\Delta \text{x}\right)=\gamma {\eta}^{-\text{1}}\Delta \text{t(10}a\text{)}$$

$$\Delta \text{x \u2032}=\gamma \left(\Delta \text{x}-\text{v}\Delta \text{t}\right)\text{(10}b\text{)}$$

$$\Delta \text{y \u2032}=\Delta \text{y(10}c\text{)}$$

$$\Delta \text{z \u2032}=\Delta \text{z(10}d\text{)}$$

with $\eta ={\left(\text{1}-{\text{vc}}^{-\text{2}}\frac{\Delta \text{x}}{\Delta \text{t}}\right)}^{-\text{1}}$
in eq. (10a).

This space-time transformation was first presented by Larmor [4]. It is
what we know today as the Lorentz transformation (LT). The inverse set
of equations is obtained by the Galilean inversion procedure as well as by
Gauss elimination:

$$\Delta \text{t}=\gamma \left(\Delta \text{t \u2032}+{\text{vc}}^{-\text{2}}\Delta \text{x \u2032}\right)=\gamma {{\eta}^{\prime}}^{-\text{1}}\Delta \text{t \u2032 (11a)}$$

$$\Delta \text{x}=\gamma \left(\Delta \text{x \u2032}+\text{v}\Delta \text{t \u2032}\right)\text{(11b)}$$

$$\Delta \text{y}=\Delta \text{y \u2032 (11c)}$$

$$\Delta \text{z}=\Delta \text{z \u2032 (11d)}$$

${\eta}^{\prime}={\left(\text{1}+{\text{vc}}^{-\text{2}}\frac{\Delta \text{x \u2032}}{\Delta \text{t \u2032}}\right)}^{-\text{1}}$
in eq. (11a); note that η’ and η are related
by Galilean inversion. The same procedure as for eq. (7) for the Voigt
transformation when applied to the LT gives:

$$\left({\text{x \u2032}}^{\text{2}}+{\text{y \u2032}}^{\text{2}}+{\text{z \u2032}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t \u2032}}^{\text{2}}\right)=\left({\text{x}}^{\text{2}}+{\text{y}}^{\text{2}}+{\text{z}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t}}^{\text{2}}\right)\text{(12)}$$

This relationship is referred to as Lorentz invariance. On this basis it is
obvious that the LT satisfies both the light-speed constancy requirement
and the RP.

The relativistic velocity transformation (RVT) is easily obtained
from the LT by dividing the three equations for Δx’, Δy’, and
Δz’ in equation. (10b - d) by Δt’ in equation. (10a):

$${\text{u \u2032}}_{\text{x}}={\left(\text{1}\u2013{\text{vu}}_{\text{x}}{\text{c}}^{-\text{2}}\right)}^{-\text{1}}\left({\text{u}}_{\text{x}}-\text{v}\right)=\eta \left({\text{u}}_{\text{x}}-\text{v}\right)\text{(13a)}$$

$${\text{u \u2032}}_{\text{y}}={\gamma}^{-\text{1}}{\left(\text{1}\u2013{\text{vu}}_{\text{x}}{\text{c}}^{-\text{2}}\right)}^{-\text{1}}{\text{u}}_{\text{y}}=\eta {\gamma}^{-\text{1}}{\text{u}}_{\text{y}}\text{(13b)}$$

$${\text{u \u2032}}_{\text{z}}={\gamma}^{-\text{1}}{\left(\text{1}\u2013{\text{vu}}_{\text{x}}{\text{c}}^{-\text{2}}\right)}^{-\text{1}}{\text{u}}_{\text{z}}=\eta {\gamma}^{-\text{1}}{\text{u}}_{\text{z}}\text{(13c)}$$
$${\text{u \u2032}}_{\text{z}}={\gamma}^{-\text{1}}{\left(\text{1}\u2013{\text{vu}}_{\text{x}}{\text{c}}^{-\text{2}}\right)}^{-\text{1}}{\text{u}}_{\text{z}}=\eta {\gamma}^{-\text{1}}{\text{u}}_{\text{z}}\text{(13c)}$$

The same definitions for η and γ are used as in the LT; ${\text{u \u2032}}_{\text{x}}=\frac{\Delta \text{x \u2032}}{\Delta \text{t \u2032}},{\text{u}}_{\text{x}}=\frac{\Delta \text{x}}{\Delta \text{t}}$
etc. Note that the *same set of equations* results from the Voigt transformation when the analogous divisions are made using equation. (8a-d).The corresponding inverse of the RVT can be obtained using Galilean inversion, demonstrating that these equations also satisfy the RP. The following identity is useful in proving the inverse relationship [5]: ηη′ = γ2. The proof given below relies on equation. (13a):

$$\eta \eta \text{'}={\left[\left(\text{1}-{\text{u}}_{\text{x}}{\text{vc}}^{-\text{2}}\right)\left(\text{1}+{\text{u \u2032}}_{\text{x}}{\text{vc}}^{-\text{2}}\right)\right]}^{-\text{1}}={\left[\left(\text{1}-{\text{u}}_{\text{x}}{\text{vc}}^{-\text{2}}\right)\left(\text{1}+\eta {\text{vc}}^{-\text{2}}\left({\text{u}}_{\text{x}}-\text{v}\right)\right)\right]}^{-\text{1}}$$

$$={\left[\left(\text{1}-{\text{u}}_{\text{x}}{\text{vc}}^{-\text{2}}\right)\frac{\text{1}-{\text{u}}_{\text{x}}{\text{vc}}^{-\text{2}}+{\text{vc}}^{-\text{2}}{\text{u}}_{\text{x}}-{\text{v}}^{\text{2}}{\text{c}}^{-\text{2}}}{\text{1}-{\text{u}}_{\text{x}}{\text{vc}}^{-\text{2}}}\right]}^{-\text{1}}\text{(14)}$$

$$={\left(\text{1}-{\text{v}}^{\text{2}}{\text{c}}^{-\text{2}}\right)}^{-\text{1}}={\gamma}^{\text{2}}$$

This identity will also prove useful in Sect. IV.

Lorentz [6] took note of the experience of Voigt and Larmor with
relativistic space-time transformations, especially that both results are
consistent with the light-speed constancy requirement. He pointed out that
there is a degree of freedom [7] in the definition of such transformations
that can be expressed in terms of a type of normalization constant which
he referred to as ε. The resulting general transformation (GLT) is given
below:

$$\Delta {t}^{\prime}=\text{}\gamma \epsilon \left(\Delta t\u2013v\Delta x{c}^{-2}\right)=\gamma \epsilon {\eta}^{-1}\Delta t\text{(15a)}$$

$$\Delta {x}^{\prime}=\text{}\gamma \epsilon \left(\Delta x\u2013v\Delta t\right)\text{(15b)}$$

$$\Delta {y}^{\prime}=\epsilon \Delta y\text{(15c)}$$

$$\Delta {z}^{\prime}=\text{}\epsilon \Delta z\text{(15d)}$$

The original transformation given by Voigt [3] is obtained from the GLT
by setting ε = γ^{−1}, whereas Larmor’s LT results for ε = 1. The fact that
this degree of freedom exists is easily understandable because light-speed
constancy only puts a restriction on the ratio of space and time variables.
Any proportionality constant therefore suffices to fulfill this condition. As
a consequence, the RVT can also be obtained from the GLT by appropriate
division of its space and time variables.

The inverse of the GLT is obtained by Galilean inversion:

$$\Delta \text{t}=\gamma {\epsilon}^{\prime}\left(\Delta \text{t \u2032}+{\text{vc}}^{-\text{2}}\Delta \text{x}\right)=\gamma {\epsilon}^{\prime}{{\eta}^{\prime}}^{-\text{1}}\Delta \text{t \u2032 (16a)}$$

$$\Delta \text{x}=\gamma {\epsilon}^{\prime}\left(\Delta \text{x \u2032}+\text{v}\Delta \text{t \u2032}\right)\text{(16b)}$$

$$\Delta \text{y}={\epsilon}^{\prime}\Delta \text{y \u2032 (16c)}$$

$$\Delta \text{z}={\epsilon}^{\prime}\Delta \text{z \u2032 (16d)}$$

which also defines the relationship between ε and ε’ in the two sets of
equations. It is easy to show that this relationship places a condition on the
value of ε required for a particular1transformation to also satisfy the RP,
namely $\epsilon {=}_{\epsilon}{}_{\prime}.$
This condition is not satisfied by the Voigt1transformation
of equation. (8a-d), in which case both ε and ε’ are equal to 1
γ . On the other hand, it is obvious that the LT satisfies the RP since ε = ε′ = 1 in that case
The degree of freedom in the GLT raises the key question of whether the
value of ε = 1 is a unique solution for determining the desired relativistic
space-time transformation. Poincaré [8] at least made an attempt to
justify the latter choice. He pointed out that the set of LT 4x4 matrices for
different relative speeds v forms a mathematical group. Multiplication of
any two of them leads to a third which has the same form and a different
value of the relative speed which agrees with the RVT. This argument is
still a popular justification for the LT value of ε = 1, but there is a problem
with it nonetheless. The group character only results when the two relative
velocities are in the same direction, a fact which Thomas [9] used 20
years later to predict the precession of electronic spins. Another argument
for the ε = 1 choice is based on the assertion that distances measured
transverse to the direction of relative motion of the two observers should
be the same for both, as in equation. (10c, d). Yet, the analogous argument
for the corresponding RVT equation. (13b-c) does not hold; the ratio of
two such components is equal to ηγ-1 in the general case.

Einstein [10] derived the same set of GLT equations as Lorentz [6, 7] but
referred to the normalization constant as φ instead of ε. He tackled the
problem of justifying the LT choice of ϕ = 1 as follows. He stated that
“φ is a temporary undefined function of v.” He then used a symmetry
argument to prove that *under these circumstances* the only allowed value
of φ is unity, thereby arriving at the definition of the LT in equation. (10ad).
It has gone largely unnoticed, however, that Einstein’s assertion that
φ can only depend on v is an assumption that also requires justification.
In short, the derivation of the LT, which is the basis for the strongly held
belief that space and time are inextricably mixed, rests on an unproven
assumption about the value of a normalization constant in the GLT of
equation. (15a-d)

#### Contradiction Between Remote Non-Simultaneity and Einsteinean Time Dilation

Is space-time mixing the inevitable consequence of the empirical fact of
light-speed constancy, as originally proposed by Voigt [3] in 1887? Is
the LT the unique solution to the goal of merging the two concepts in the
relativistic theory of kinematics? Both questions posed a clear challenge
to Newtonian mechanics, and so there is merit in considering what light
can be shed upon each of them by the classical theory.

A good place to start with is Newton’s First Law of Kinematics (Law of
Inertia). Both the observers and the object of their measurement to be
described in terms of the space-time transformation are assumed to move
with constant speed and direction. This assumption is consistent with the
Law of Inertia, and is a result of the complete absence of unbalanced forces
acting on them. Under the circumstances, what should one expect for the
properties of the object of the measurement? In accordance with the Law
of Causality, it can be assumed that all these properties *should remain
unchanged* for an indefinite amount of time, *including the rates* of its
stationary clocks. The same conclusion holds for all other inertial systems,
no matter what their speed and direction might be. The respective rates of
the two sets of clocks can be different, however, but their ratio must then
be a *constant* as well. On this basis, one can deduce a clear relationship
between the elapsed times Δt and Δt’ measured by inertial clocks for any
given event, namely where Q is a proportionality constant which
does not depend in any way on the object of the measurement. This result
is clearly at odds with equation, (10a) of the LT.

Does the above proportionality relationship prove that Newton’s First
Law is in violation of the LT? Or is the opposite the case? To consider this
question, it is well to recall two of the predictions about time relationships
that result from the LT: remote non-simultaneity of events and time
dilation. Poincaré [11] recognized that equation, (10a). indicates that
pairs of events that are simultaneous for one observer might not be so for
another moving relative to him, and on this basis he began to question
the traditional belief that everything occurs at the same time throughout
the universe. For example, suppose that observer O finds that two events
occur at the same time A on his stationary proper clock ( Δt = 0 ), but that
observer O’ finds instead that they occur at different times B and C on his
stationary proper clock, i.e. B ≠ C and Δ t′ ≠ 0 . This situation is allowed
according to equation. (10a).

The prediction of time dilation is derived [10, 12] by considering the
relationship between elapsed times Δt and Δt’ for the same event that are
measured by two observers in relative motion with speed v along the x
axis. Attention is centered on a clock that is stationary in the rest frame of
one of the observers (Δx′ = 0). The other observer (O) finds that this clock
moves during the measurement from his vantage point, specifically that
the change in its position is Δx = vΔt. Substitution of the latter relationship
into equation. (10a) of the LT leads to the following equation:

$$\Delta \text{t \u2032}=\gamma \left(\text{v}\right)\left(\Delta \text{t}-{\text{v}}^{\text{2}}{\text{c}}^{-\text{2}}\Delta \text{t}\right)={\gamma}^{-\text{1}}\Delta \text{t(!7)}$$

According to equation. (17), the elapsed times in the above example
are always strictly proportional to one another. Thus for the first event,
since observer O found it to occur at time A on his stationary clock, the
corresponding value obtained by the other observer (O’) must be $\text{B}=\frac{\text{A}}{\gamma}$
Similarly for the second event, eq. (17) leads one to conclude that $\text{C}=\frac{\text{A}}{\gamma}$
. In other words, B = C according to this argument. This equality stands
in contradiction to the original inequality based on the non-simultaneity
assumption, namely B ≠ C. The same result can be obtained more directly
by considering time differences for the two events. Since they occur at
the same time for O (Δt = 0), it is impossible according to equation, (17)
that the corresponding time difference for O’ (Δt’) not be equal to zero.
Both the predictions of remote non-simultaneity and proportional time
dilation are obtained from equation. (10a) of the LT. The inescapable
conclusion is therefore that the LT is not a physically valid space-time
transformation. It is simply unacceptable that the same theory gives
diametrically opposite answers for the same question. The conflict
between Newton’s First Law and the LT mentioned at the beginning of
this section is thus resolved in favor of the former. The question is thus
open as to whether the First Law is itself compatible with light-speed
constancy and the RP. It will be shown in the following that there is
nothing standing in the way of this possibility

#### An Alternative Version of the Lorentz Transformation

The GLT of equation. (15a-d) offers a surprisingly easy means of
incorporating the clock-rate proportionality implied by Newton’s Law
of Inertia into relativistic theory. One simply needs to evaluate the
normalization constant under the condition of Δt’=Δt/Q derived in Sect.
IV. Starting with equation. (15a) one arrives at the following equation
[13-16]:

$$\Delta \text{t \u2032}=\gamma \left(\text{v}\right)\epsilon \left(\Delta \text{t}-{\text{vc}}^{-\text{2}}\Delta \text{x}\right)=\gamma \epsilon {\eta}^{-\text{1}}\Delta \text{t}=\frac{\Delta \text{t}}{\text{Q}}\text{(18)}$$

Upon solving for ε the result is:

$$\epsilon =\frac{\eta}{\gamma \text{Q}}\text{(19)}$$

Inserting this value in the GLT equations then gives the desired spacetime
transformation, originally referred to as the Alternative Lorentz
Transformation (ALT [13-16]):

$$\Delta \text{t \u2032}=\eta \frac{\Delta \text{t}-{\text{vc}}^{-\text{2}}\Delta \text{x}}{\text{Q}}=\frac{\Delta \text{t}}{\text{Q}}\text{(20a)}$$

$$\Delta \text{x \u2032}=\eta \frac{\Delta \text{x}-\text{v}\Delta \text{t}}{\text{Q}}\text{(20b)}$$

$$\Delta \text{y \u2032}=\left(\frac{\eta}{\gamma \text{Q}}\right)\Delta \text{y(20c)}$$

$$\Delta \text{z \u2032}=\left(\frac{\eta}{\gamma \text{Q}}\right)\Delta \text{z(20d)}$$

Does the ALT satisfy the RP, however? The condition for that is εε ′ = 1
[see the discussion after equation. (16a-d)]; hence, from equation. (19):

$$\frac{\eta {\eta}^{\prime}}{{\gamma}^{\text{2}}\text{Q Q \u2032}}=\text{1(21)}$$

where Q’ is the constant obtained from Q by Galilean inversion. After
applying equation. (14), this reduces to simply QQ′ = 1. Since there is
no other restriction on the choice of Q, this condition is easily met. One
can think of the constant Q as *a conversion factor *between the clock rates
in the two inertial systems, so taking Q’ to be its reciprocal is exactly
what one expects for the corresponding conversion factor in the reverse
direction. The inverse set of equations for the ALT is obtained by Galilean
inversion because the RP is satisfied:

$$\Delta \text{t}={\eta}^{\prime}\frac{\Delta \text{t \u2032}+{\text{vc}}^{-\text{2}}\Delta \text{x \u2032}}{\text{Q \u2032}}=\frac{\Delta \text{t \u2032}}{\text{Q \u2032}}\text{(22a)}$$

$$\Delta \text{x}={\eta}^{\prime}\frac{\Delta \text{x \u2032}+\text{v}\Delta \text{t \u2032}}{\text{Q \u2032}}\text{(22b)}$$

$$\Delta \text{y}=\frac{\eta \text{'}}{\gamma \text{Q}\text{'}}\Delta \text{y}\text{'}\text{(22c)}$$

$$\Delta \text{z}=\frac{\eta}{\gamma \text{Q \u2032}}\Delta \text{z \u2032 (22d)}$$

One can also derive the ALT by combining the time relation in equation
(20a) with the RVT of Equation. (13a-c). By definition, ${\text{u \u2032}}_{\text{x}}=\frac{\Delta \text{x \u2032}}{\Delta \text{t \u2032}}$
for
example, so multiplication of the left-hand side of equation. (13a) by Δt’
gives equation. (20b) after multiplying the right-hand side with $\frac{\Delta \text{t}}{\text{Q}}$
In
some ways the RVT is the most important transformation in relativity
theory because the corresponding space-time transformation in equation.
(20a-d) can so easily be derived from it. It has the additional advantage
of being completely independent of the clock-rate ratio connecting the
two inertial systems. It also should be noted that the RVT “automatically”
satisfies the RP since it can be derived from either the LT, which does
satisfy the RP, or the Voigt transformation, which does not. It is obvious
that any experiment which is consistent with the RVT is also described
correctly by the ALT. This is an important observation since many
of the greatest successes of relativity theory can be obtained from the
RVT without making use of the LT. The derivation of the aberration of
starlight at infinity [1, 17] can be obtained directly by applying the RVT,
for example, whereby the corresponding derivation based on the LT is
notably more complicated [18]. The RVT is also sufficient for obtaining
the Fresnel light-drag expression [19], so the ALT is also compatible with
this result. The same is true for Thomas spin precession [9], since that
relationship is also independent of the ε normalization constant in the
GLT.

The Lorentz invariance condition in equation. (12) is also invalid
because it suffers from the same contradictory relationship as the LT. The
corresponding invariance relation obtained from equation. (20a-d) is:

$$\left({\text{x \u2032}}^{\text{2}}+{\text{y \u2032}}^{\text{2}}+{\text{z \u2032}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t \u2032}}^{\text{2}}\right)={\left(\frac{\eta}{\gamma \text{Q}}\right)}^{\text{2}}\left({\text{x}}^{\text{2}}+{\text{y}}^{\text{2}}+{\text{z}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t}}^{\text{2}}\right)\text{(23)}$$

Multiplication by $\frac{{\eta}^{\prime}}{\gamma \text{Q \u2032}}$
leads to the more symmetric invariance relation given below:

$$\left(\frac{{\eta}^{\prime}}{\text{Q \u2032}}\right)\left({\text{x \u2032}}^{\text{2}}+{\text{y \u2032}}^{\text{2}}+{\text{z \u2032}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t \u2032}}^{\text{2}}\right)=\left(\frac{\eta}{\text{Q}}\right)\left({\text{x}}^{\text{2}}+{\text{y}}^{\text{2}}+{\text{z}}^{\text{2}}-{\text{c}}^{\text{2}}{\text{t}}^{\text{2}}\right)\text{(24)}$$

The distinction between the ALT and the LT becomes critical when times
are measured. A prime example is the study of atomic clocks carried out
by Hafele and Keating [20, 21] in 1971 with circumnavigating airplanes.
They found that the elapsed times on the various clocks employed in the
experiment decreased with their speed relative to the “non-rotating” polar
axis, or simply the earth’s center of mass (ECM). The corresponding
empirical formula expressed in terms of the present notation is:

$$\Delta \text{t \u2032}\gamma \left(\text{u \u2032}\right)=\Delta \text{t}\gamma \left(\text{u}\right)\text{(25)}$$

where u and u’ are the speeds of two such clocks and Δt and Δt’ are the
respective measured elapsed times over the same distance. Because of
the earth’s rotation, the clocks on the eastward-flying airplane returned to
the airport of origin with less elapsed time than those left behind there,
which in turn had more elapsed time than the clocks on the other airplane
flying in the westerly direction. The same relationship was found earlier
in experiments [22-24] with an x-ray source and absorber mounted on
a high-speed rotor. In this case frequencies were measured and a blue
shift was found when the absorber is mounted farther from the rotor
axis than the x-ray source. The relationship in equation. (25) can thus be
looked upon as the Universal Time-dilation Law (UTDL [25]). In order
to evaluate it, it is first necessary to determine a rest frame (objective rest
system or ORS [26]) relative to which the speeds of the various timing
devices are measured. The UTDL can be used directly to compute the
value of Q in the ALT of equation. (20a-d), namely as $\text{Q}=\frac{\gamma \left(\text{u}\text{'}\right)}{\gamma \left(\text{u}\right)}$

It is easy to confuse the clock-rate proportionality relationship in
equation. (20a) with the LT time-dilation formula in equation, (17). The
latter states expressly that the ratio of the two clock rates depends on
the speed v by which the observers are moving relative to one another.
The proportionality constant is fixed by the LT to be γ(v) in all cases.
Nonetheless, it is completely unclear which of the two clocks runs at the
slower rate. The derivation of equation. (17) can be changed by reversing
the roles of the two observers, in which case:

$$\Delta \text{t}={\gamma}^{-\text{1}}\Delta \text{t \u2032}\left[17\text{'}\right]$$

Equation. (17, 17’) are not related by algebra. The indication from the
LT instead is that it is always the “moving clock” that runs slower than
that in the rest frame of the observer [23]. Time dilation is supposedly
symmetric, and therefore *subjective*, in this respect
On the other hand, the time dilation indicated by equation. (20a) is perfectly
*asymmetric*, in accord with Newton’s Law of Inertia. The constant Q is
fixed for all time, at least until some unbalanced external force is applied
to the clocks. It can take on any value, greater or less than unity, consistent
with the UTDL of equation. (25). Accordingly, Q is a ratio of γ factors in
the usual case and is thus a characteristic of the relationship between the
two rest frames, and therefore is not necessarily equal to γ(v). Timing
experiments of different kinds have always been in agreement with the
UTDL and therefore with equation. (20a). It was recognized by Sherwin
[27] that the LT prediction of symmetric time dilation was violated by the
results of the x-ray frequency measurements [22-24]. His explanation,
which can be recognized as ex *post facto* in nature, was that the existence
of a unique inertial rest frame in the experimental arrangement, namely
the rotor axis, changed the way in which the LT was to be used to arrive at
its prediction. The same argument was used later by Hafele and Keating
[20]. The implication is that in the absence of unbalanced external forces
the situation would be different, that it would become “ambiguous,” to
use Sherwin’s term, which of the two clocks runs slower. *This type of
symmetric time dilation has never been observed.
The engineers who developed the Global Positioning System (GPS)
recognized the importance of asymmetric time dilation of the atomic
clocks carried onboard satellites. A “pre-correction factor” is applied
[28-29] to the standard frequency of the clocks prior to launch to insure
that their rates are nearly the same in orbit as for their counterparts on
the earth’s surface. This procedure allows for the accurate measurement
of the time required for a light signal to travel between the two
positions, and this is essential for obtaining the high accuracy for
distance measurements required. The correction for time dilation can
be determined using the UTDL (another correction is required for the
effect of gravity on clock rates). For this reason a more descriptive
name for the relativistic space-time transformation in equation. (20a-d)
is the Global Positioning System - Lorentz transformation (GPS-LT).
It is the only transformation that satisfies both of Einstein’s postulates
of relativity [10] and is also consistent with Newton’s First Law.*

#### Conclusion

Voigt’s answer to the problem of finding a consistent explanation for the
surprising results of experiments carried out in the 19th century has had
lasting results. He suggested that the difficulties could be traced directly to
the Galilean transformation. What he then proposed is a classic example
of how to amend a theory to describe new results without affecting its
previous successful predictions. To do this, however, he broke with the
traditional Newtonian view that space and time are distinct entities.
This space-time mixing still remained after Larmor further modified the
coordinate transformation so that it (the LT) also satisfies the Relativity
Principle (RP) as well as the light-speed constancy condition Voigt had
introduced. Since that time it has been the established view among the
physics community that light-speed constancy can only be incorporated
into relativity theory by insisting that space and time are no longer distinct
when high speeds are involved. It is ironic that Lorentz’s observation
that there was a degree of freedom in the definition of the relativistic
space-time transformation only resulted in a series of suggestions to
justify the existing version, with its assertion that the mixing of space
and time is the inevitable consequence of light-speed constancy. From a
purely mathematical point of view, however, it is clear that the coefficient
of mixing can be eliminated by simply making the proper choice of the
normalization constant ε in equation. (15a).

An interesting aspect of the present discussion is the way Newton’s
thoughts on the subject were handled. The process began with Poincaré’s
observation that the mixing of space and time implies that events that
are simultaneous for one observer may not be so for another. He pointed
out that there had never been any definitive test that would eliminate
this possibility. Yet, a few years later Einstein showed that the mixing of
space and time in the LT also implies that clock rates in different inertial
rest frames must be strictly proportional to one another (symmetric time
dilation). It is impossible for two observers to disagree whether events are
simultaneous (Δt = 0 and Δt′ ≠ 0) and at the same time have proportional
rates, *so there is an inherent contradiction in these two LT predictions*.

On the other hand, Newton’s First Law asserts that a clock moving under
the absence of external unbalanced forces will do so with constant speed
and direction. It defies both common sense and experiment to believe
that the rate of the same clock will not remain constant as well. The
rates of two such clocks in different inertial rest frames might not be the
same under these circumstances, *but the ratio of their rates has to be a
constant,* thereby insuring that space and time are not mixed for them.
*Clock-rate proportionality is the antithesis of remote non-simultaneity.
*There is strong evidence for this conclusion from experience with the
Global Positioning System. It would make no sense to adjust the rates
of satellite and earth-bound clocks to be equal if events did not occur
simultaneously for them.

One of Einstein’s greatest advances was the prediction of time dilation,
the slowing down of clocks because of their motion/acceleration. The
version derived from the LT is flawed, however, because it claims that
two clocks in motion can both be running slower than one another at the
same time. Experiment has consistently shown instead that the elapsed
times of clocks satisfy the Universal Time-dilation Law of equation.
(25) and therefore that time dilation is an *exclusively asymmetric*
phenomenon. As with Newton’s Laws of Kinematics and the three Laws
of Thermodynamics, the UTDL cannot be derived from some theory.
Rather, it summarizes the results of all relevant observations and serves
as a stimulus for new experiments to further test its reliability.

A simple way of describing the results of this phenomenon is in terms
of a unit of time; a rest frame with systematically slower clock rates has
a larger unit of time than its counterpart. Such an organizing principlecannot be used to describe the symmetric time dilation of the LT because
it is meaningless to speak of units when there is no agreement about which
clock is slower. This situation is in complete agreement with Galileo’s
original statement of the RP. He foresaw that it would be impossible
based solely on in situ observations alone for passengers locked in the
hold of a ship moving on a perfectly calm sea to distinguish their state
of motion from that at the dock from which it departed. He did not rule
out the possibility that clocks on the ship run at a different rate than those
on shore, however, only that it is not possible to measure this difference
without looking outside. *There is merit in making an addendum [13, 30]
to the RP on this basis: The laws of physics are the same in all inertial rest
frames, but the units in which they are expressed can and do differ from
one frame to another.*

The GPS - LT of equation. (20a-d) subscribes to this version of the RP
through the introduction of the constant Q in its definition. The latter
serves as a conversion factor between the two units of time for a given
pair of rest frames. Conversion factors for all other physical properties
can be determined as integral powers of Q [31]. An analogous set of
relationships can be formulated [32] for conversion factors describing the
effect of gravitational forces on physical properties.

The present revised theory has many potential applications. Primary
among these are tests of the UTDL of eq. (25). For example, it would be
interesting to measure the ratio of the rates of clocks subject to different
ORSs, such as is the case when one is in the gravitational field of the moon
while the other is earth-bound. The main purpose of the present work is
not technological in nature, however. It contains an iron-clad proof that
the Lorentz transformation does not conform to physical reality. Since
this set of equations is the cornerstone of Einstein’s version of relativity,
this means that every standard text book dealing with this subject needs
to be revised in an essential way. The answer to the question in the title
of ref. 28 is clearly “No.” A significant part of Einstein’s legacy is at
stake. After a century of hearing unflinching support from experts in this
field, it is high time to challenge them to either refute the present claim or
acknowledge that it is correct once and for all time.

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