Physics Essays volume 7, number 4, 1994

An Absolute Theory for the Electrodynamics of Moving Bodies

D.J. Larson

Abstract

An axiomatic set employing Larmor time dilation alone is shown to be sufficient to derive the Lorentz kinematic transformations; FitzGerald length contraction is not required. Since the kinematic transformation leads solely and directly to the electrodynamic transformation, this paper presents the third independent axiomatic set (the others are due to Lorentz and Einstein) consistent with presently accepted electrodynamic theory. The theory proposed herein assumes classical concepts for space and time; Galilean relativity and an ether are assumed. The present experimental evidence relative to space-time theory is reviewed, and its relevance to the three axiomatic sets is discussed. Two new tests are proposed that could experimentally prove that the axiomatic set proposed herein is the set that best represents nature.

**Key words:** absolute theory, special relativity, kinematics, electrodynamics, Michelson–Morley, laser lunar ranging, Doppler shifts, Bell’s theorem

1. INTRODUCTION

This paper presents an absolute (as opposed to relative) theory for the electrodynamics of moving bodies. The paper will be restricted to a "special" theory; gravitation and acceleration are not included. The theory presented herein is similar to the absolute theories of Larmor^{(1)} and Lorentz^{(2)} in that a preferred frame and ether are assumed. The difference between the theory proposed here and that of Refs. 1 and 2 is that the theory proposed here is based on time dilation alone; no length contraction is required.

1.1 Purposes of this Paper

Two well-known axiomatic basis sets are consistent with presently accepted electrodynamic theory; they are the axioms of Lorentz^{(2)} and those of Einstein.^{(3)} The purposes of this paper are (1) to propose a new (third) axiomatic basis set for space and time; (2) to show that the new axiomatic set is consistent with all known experimental facts; and (3) to propose new experimental tests of space-time theory possibly capable of indicating that the new axiomatic set is the best representative of nature.

1.2 Nomenclature and Notation

This paper will assume that an ether exists. (The ether is the medium upon which light waves travel.) The terms "ether rest frame" and "preferred frame" are to have the same meaning and are used interchangeably.

There is an important distinction in absolute theories, hopefully made clear below, between the "equations of transformation of space and time" and "the coordinate systems inferred by moving observers with their velocity-altered measuring instruments." For that reason, this paper adopts the convention that uppercase letters, such as *X*, denote a variable that is measured by preferred frame observers, and lowercase letters, such as *y*, will denote a variable inferred by observers who move with respect to the preferred frame.

2.AXIOMS

The axiomatic basis of any theory consists of those basic assumptions upon which the rest of the theory is built. This section will present the axiomatic sets of the various competing space-time theories. Classifications of axioms will be divided into two types. Postulates are defined as axioms not classically assumed. Assumptions are defined as classically assumed axioms.

2.1 The Axioms of Einstein

The theory of the electrodynamics presently employed by the vast majority of physicists is based upon the work of Einstein.^{(3)} The special theory of relativity is well known and is advertised to be based upon the following axioms:

**Postulate AE-I:** The relativity principle (Ref. 3, p. 37): "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."

**Postulate AE-II:** The speed of light is *c* = 299^{ }792_{ }458 m/s in all inertial reference frames (Ref. 3, p. 38): "light is always propagated in empty space with a definite velocity *c* which is independent of the state of motion of the emitting body."

But in addition to the advertised postulates, Einstein’s theory also relies on the following assumptions:

**Assumption AE-I:** Space (in all inertial frames) is homogeneous. (This is an underlying assumption that leads to the linearity of the transformation equations for space and time. See Ref. 3, p. 44.)

**Assumption AE-II:** Time (in all inertial frames) is homogeneous. (This is an underlying assumption that leads to the linearity of the transformation equations for space and time. See Ref. 3, p. 44.)

**Assumption AE-III:** "Let the Maxwell–Hertz equations for empty space hold good for the stationary system K." (Ref. 3, p. 51; Einstein goes on to show that once this assumption is made for one inertial frame, the same equations hold good in all inertial frames.)

**Assumption AE-IV:** The electrodynamics of moving bodies in one inertial frame are governed by the Lorentz force equation. (See Ref. 3, p. 60; Einstein shows that once the basis for Lorentzian electrodynamics is assumed in one inertial frame, the equations have the same form in any inertial frame.)

**Assumption AE-V:** Charge and mass are invariant quantities, independent of reference frame. (Einstein, Ref. 3, p. 54, claims that charge invariance is a result of Postulate AE-I, but this is not correct. Assuming that an arbitrary scalar is invariant is in actuality an assumption, as here stated. Also, Einstein, Ref. 3, p. 63, defines what he calls "transverse mass" and "longitudinal mass" as relations involving the rest mass and kinematical quantities. Okun^{(4)} has shown that only one concept of mass, the so-called "rest mass," is relevant to special relativity, and it is this mass that is assumed invariant by Einstein.)

As a further addition to the advertised basis for special relativity, two additional postulates must be accepted in order to admit special relativity as a viable candidate for a space-time theory:

**Postulate AE-III:** Wave phenomena can be propagated without requiring a medium of propagation. (No ether is needed. While this statement results from Einstein’s work, it must be assumed that it is possible that no medium is required. Note that this assumption is a radical alteration of a classically assumed axiom.)

**Postulate AE-IV:** Classical concepts of absolute space and time may be incorrect.

Postulates AE-III and AE-IV embody the bold departure of Einstein’s work from the classical space-time theory. These unstated postulates form the basis for special relativity’s fame and its conceptual incomprehensibility. Modern physics assumes that Postulates AE-III and AE-IV are correct, but it must be admitted that they are, in the end, axiomatic assumptions.

2.2 The Axioms of Lorentz

Prior to the advent of special relativity, the classical concepts of space and time were quite different. The classical theory was developed by numerous contributors, including FitzGerald and Poincaré. While Larmor^{(1)} was the first to publish the equations that later became known as the "Lorentz kinematic transformations," Larmor gives credit for the approach to Lorentz, who later published a work on electrodynamics using the transformations.^{(2)} Note that Voigt^{(5)} was the earliest to formulate expressions similar to the Lorentz kinematic transformations, but as discussed by Kittel,^{(6)} Voigt’s formulations are not identical to what are known as the "Lorentz transformations." The classical axiomatic set consists of the following.

**Postulate HAL-I:** Larmor^{(1)} (also Lorentz^{(2)}) time dilation: When clocks travel in absolute space they run slow, recording a time advance of *t* = *T*/g, where *T* is the true time advance (recorded by clocks in the preferred frame), g = (1 - b^{2})^{-1/2}, b = *V*/*C*, where *C* is the speed of light, and *V* is the magnitude of the moving clock’s velocity as observed from the preferred frame.

**Postulate HAL-II:** FitzGerald (also Lorentz^{(2)}) length contraction (Ref. 2, p. 21): Bodies "have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming g*l* times and those in perpendicular directions *l* times smaller." (For *l* = 1 we have the usual FitzGerald–Lorentz physical contraction hypothesis.)

**Assumption HAL-I:** Absolute space is isotropic and homogeneous, described by a Euclidean metric, and has a unique rest frame (the preferred frame). (This assumption goes back to Euclid.)

**Assumption HAL-II:** Absolute time is simply the independent parameter that governs events. (Absolute time can be measured by clocks at rest in the preferred frame. This assumption probably predates Euclid.)

**Assumption HAL-III:** Free space electric and magnetic fields are governed by the Maxwell equations in the preferred frame.

**Assumption HAL-IV:** The electrodynamics of moving bodies in the preferred frame are governed by the Lorentz force equation.

**Assumption HAL-V:** Charge and mass are invariant quantities, independent of reference frame.

**Assumption HAL-VI:** A medium, the ether, exists upon which light waves are transmitted. The rest frame of the ether is called the preferred frame.

2.3 The Axioms Proposed Herein

The absolute theory proposed herein will be based on an axiomatic set very similar to the Lorentzian axiomatic set.

**Postulate DJL-I** is identical to Postulate HAL-I.

**Postulate DJL-II:** The Michelson–Morley apparatus mirrors alter the phase nature of light within the apparatus in such a way that nodes are enforced (zeros of the electric field amplitude function) that move along with the apparatus.

**Assumptions DJL-I, DJL-II, DJL-III, DJL-IV, DJL-V, and DJL-VI** are identical to Assumptions HAL-I, HAL-II, HAL-III, HAL-IV, HAL-V, and HAL-VI, respectively.

**Assumption DJL-VII:** The dimensions of a physical rigid body, as determined from the preferred frame, are unaltered by motion through the preferred frame. (There is no physical length contraction. This assumption also goes back to early history, and required Lorentz to form Postulate HAL-II stating the negation of this assumption.)

2.4 Three Competing Axiomatic Sets - Which is Superior?

The advancement of natural philosophy as it relates to our understanding of the physical world requires that we choose from among various axiomatic sets when we set out to describe that physical world. Our choice involves both subjective and objective reasoning. On the subjective side, we desire the axioms to be "simple" and "reasonable," while on the objective side, it is a requirement that the axioms lead to a theory that correctly predicts the results of experimental tests.

The present choice to use Einstein’s axioms over those of Lorentz is almost entirely based upon subjective criteria. There is a common misrepresentation stating that Einstein’s theory is simpler than Lorentz’s theory because it employs fewer assumptions, but, as seen above, each of the competing space-time theories starts out with a similar number of axiomatic assumptions. On the "reasonableness" nature of the axioms, proponents of absolute theories would say that the postulate of the constancy of the speed of light is an absurdity, and indeed it was a long time before relativity won the day over the absolute theory of Lorentz. Arguments can rage over what is more truly "simple." Are conceptual simplicities simpler than formulational simplicities? Despite the love affair between modern philosophers and the late William of Occam, it is also possible that nature may not be simple or reasonable.

Fortunately, the objective reasoning of experimental testing can be used to decide between the competing axiomatic sets. The purpose of science is to build from the axioms a theory that makes experimental predictions unique to each axiomatic set, and then proceed with the crucial experiments in order to decide which axiomatic set best represents nature. It will be shown below that each of the above three axiomatic sets is sufficiently different from its competitors that a unique and objective determination of the best axiomatic set can be made.

2.5 Other Axiomatic Sets

It will be shown below that the above three axiomatic sets are consistent with present experimental data, although the Einstein and Lorentz theories will be shown to have some difficulties. Other axiomatic sets have been proposed over time, and these sets still have proponents.

All the nonclassical axioms began to appear due to the null result of the Michelson–Morley test,^{(7)} since the null result was incompatible with the classical axioms. One early theory was that the ether (the ether is the medium capable of sustaining light waves) was dragged along with the Earth. The ether-drag hypothesis was quickly discredited, since it could not explain the results of stellar abberation.^{(8)} Nonetheless, the ether-drag hypothesis still has modern-day proponents.^{(9)} A second attempt at reconciling the null result of the Michelson–Morley test with the classical axioms was made by Ritz^{(10)} who proposed that the speed of light in the ether is equal to *c* plus the velocity of the source. The Ritz theory was defended fairly recently by Fox,^{(11,12)} but an experiment by Alvager *et al*.^{(13)} showed that light emitted from high-velocity pions travels at speed *c* with respect to the laboratory, not the source, disproving the Ritz theory. Since the Alvager *et al*. result concerns the group velocity of light only, it is conceivable that the phase velocity of light is source-dependent, but that the group velocity is *c* with respect to the ether. I have shown elsewhere^{(14)} that such a proposal (and therefore also the Ritz proposal) cannot properly predict the Doppler shifts observed for light coming from rapidly moving sources.

In addition to the ether-drag and source-dependent velocity axioms, there are numerous other axiomatic sets that appear regularly. All such sets contain one or more of four common problems: (1) the "new" axioms are in essence identical to the Einstein axioms; (2) the "new" axioms are in essence identical to the Lorentz axioms; (3) the new axioms lead to a theory that is incompatible with the results of one or more experimental tests; (4) the development of the theory from the new axioms contains a logical or arithmetic error.

Of the many alternative space-time theories proposed, notable recent works include those of Beckmann,^{(9)} Wesley,^{(15)} Phipps,^{(16)} Cornille,^{(17)} and Huang.^{(18)} Since none of these works have made it into the mainstream scientific discourse, the present work is compared only with the Einstein and Lorentz theories, which are far better known.

3. THE STANDARD EXPERIMENTAL TESTS RELEVANT TO SPACE-TIME THEORIES

There is an enormous amount of experimental data supporting the special theory of relativity. This section will present a discussion on how each of the three axiomatic sets deals with the standard experimental evidence.

3.1 Michelson–Morley

The Michelson–Morley experiment^{(7)} and variations of that experiment^{(19,20)} fail to detect any motion of the Earth through a preferred frame. The theory of Einstein accounts for these results by stipulating that light travels at speed *c* independent of the motion of the apparatus, and when viewed from the frame of the apparatus, this very simply explains the null result. The theory of Lorentz accounts for the null result by postulating that the apparatus dimensions are shrunk in the direction of motion by just the amount needed to comply with the null result (Postulate HAL-II). The theory proposed herein accounts for the null result by postulating that the apparatus mirrors enforce nodes in the electric field that move along with the apparatus (Postulate DJL-II). The node-enforcement hypothesis has been discussed in great detail^{(21)} and will be summarized below.

3.2 Time Dilation

Time dilation is experimentally proven through the lengthening of decay time for rapidly moving unstable particles^{(22)} and also through an experiment involving atomic clocks transported around the globe.^{(23)} These experiments are accounted for in the Einstein theory by deriving a relativistic time dilation from Postulates AE-I and AE-II. The Lorentz theory and the theory proposed herein account for these results by postulating an absolute time dilation of clocks moving with respect to the preferred frame (Postulate HAL-I).

3.3 Doppler Shifts

Extremely accurate measurements indicate the correctness of the presently accepted formula for the Doppler shift of light emanating from moving sources and absorbed by moving receivers.^{(24–30)} Doppler shifts are the measured shifts in the frequency of light emitted by a moving source, and the frequency measurement is simply the result of counting wave crests that pass a given point over a unit time interval. Since no length is involved, Doppler shifts only rely on time dilation, and since all three theories (Einstein, Lorentz, and herein) agree that there is a time dilation, all three theories account for Doppler shifts in the same way once time dilation is established.

3.4 Stellar Aberration

Observations of stellar aberration preclude the possibility that the ether is dragged along with the Earth. The Einstein theory, the Lorentz theory, and the theory proposed herein all account for stellar aberration experiments in the same way. Viewed from an inertial frame, light will traverse a straight path from the star to a telescope. But the Earth’s motion through that inertial frame will move the telescope a small amount during the time it takes the light to pass through the telescope, causing an apparent angular shift in the star's position. Since this effect is due to the Earth’s motion through the inertial frame, the effect will have a yearly periodicity.

3.5 The Sagnac Experiment

Sagnac^{(31)} and Michelson–Gale^{(32)} have measured fringe shifts by interfering two light beams, one of which traverses a clockwise path, the other of which traverses the same path in a counterclockwise manner. The measurements indicate that the effect is caused by the rotation of the Earth. Again, when viewed in the inertial frame, this effect has the same explanation in the Einstein theory, the Lorentz theory, and the theory proposed herein. The rotation of the Earth during the transit time of the light will cause one path to be longer than the other and the fringe shift is expected.

3.6 The Alvager Experiment

Measurements by Alvager *et al*. show that the speed of light emitted from decaying pions precludes the possibility that the speed of light is equal to *c* plus the velocity of the source.^{(13)} This is accounted for in the Einstein theory by stipulating that the speed of light is *c* with respect to the observer. In the Lorentz theory, and in the theory proposed herein, this is accounted for by stipulating that the speed of light is *c* with respect to the ether.

3.7 Experiments Involving the Electrodynamics of Moving Bodies

The electrodynamics of moving bodies as described by the equations of relativistic electrodynamics correctly predicts the outcome of experimental tests. There is a vast array of such experiments, and these experiments provide a crucial experimental test for any space-time theory. The Einstein theory accounts for electrodynamics by assuming the Maxwell equations are valid in all inertial frames. The Lorentz theory accounts for electrodynamics by postulating that moving observers’ measuring instruments (clocks and metersticks) deform due to their motion. Using these deformed measuring instruments, moving observers arrive at the same electrodynamic equations. The theory proposed herein will show below that once moving observers accept the postulate of the constancy of the speed of light, time dilation alone is sufficient to reach the accepted electrodynamic equations. The theory proposed herein will thus demonstrate that only the moving observers' clocks need to deform in order for them to infer present electrodynamic theory; their metersticks are assumed to retain their length in all inertial frames.

3.8 The Trouton and Noble Experiment

Trouton and Noble^{(33)} have measured no torque on a delicately suspended condenser, although early arguments indicated that motion through a preferred frame would cause a torque. This experiment is historically important, and therefore worthy of mention, but for the present work it falls under the class of Sec. 3.7, and so it is equally well accounted for by all three of the competing theories.

3.9 The Standard Experiments - A Discussion

This section has now shown that the theory of Einstein, the theory of Lorentz, and the theory proposed herein can equally well describe the standard experiments relevant to space-time theory provided the theory proposed herein can show how moving observers "infer" present electrodynamic theory. This proviso will be addressed in Sec. 7 below. But first it is important to deal with the difference between "inferring" the electrodynamic relations and with specifying what those electrodynamic relations "really are." This difference is also a key difference between the Einstein theory and the Lorentz theory, and it is this key difference that allows an experimental test to determine whether the Einstein theory or the Lorentz theory is most representative of nature.

4. THE NATURE OF SPACE AND TIME IN ABSOLUTE THEORIES - GALILEAN RELATIVITY

In spite of the fact that Lorentz first derived the "relativistic" transformations of kinematics and electrodynamics, both the Lorentz theory and the theory proposed herein (the absolute theories) assume that in actuality the kinematics of moving bodies is described by Galilean transformations, and that the electrodynamics of moving bodies is Maxwellian.

4.1 Galilean Kinematics

The kinematics of space and time within absolute theories is described by Galilean kinematics:

* X¢
* = *X* + *U _{X}T*, (1)

*Y¢
* = *Y* + *U _{Y}T*, (2)

* Z¢
* = *Z* + *U _{Z}T*, (3)

* T¢
* = *T*. (4)

In Eqs. (1) to (4), *X*, *Y*, and *Z* represent the three coordinates of a point in the three-dimensional space which is at rest with respect to the ether. The values for *X*, *Y*, and *Z* can be determined in the frame of the ether by using metersticks to measure the projection of the distance between an arbitrary origin and the point (*X*,*Y*,*Z*) along three arbitrarily assigned, mutually orthogonal axes. *T* represents the time advance in the ether rest frame. Time can be measured using clocks at rest in the ether. *X¢
*, *Y¢
*, and *Z¢
* represent the three coordinates of a coordinate system set up by preferred frame observers that moves with a velocity *U* with respect to the ether rest frame. *T¢
* is the time associated by preferred frame observers with the moving coordinate system. The numerical values of *X¢
*, *Y¢
*, *Z¢
*, and *T¢
* can be determined for the moving coordinate system by using Eqs. (1) to (4).

Equations (1) to (4), the Galilean transformations, are the only transformations of space and time that retain the classical meaning of space and time. Thus it is assumed in both the Lorentz theory and the theory proposed herein that coordinate representations of time and space transform via Eqs. (1) to (4), but modifications of measuring instruments caused by motion through the ether cause moving observers to infer incorrect coordinate systems for their moving frames.

Note that this is the first crucial conceptual difference between relativity and the absolute theories. Relativity assumes that time and space are transformed by relative motion, and that all observers are on an equal footing with regard to determinations of space and time. Absolute theories assume a fixed space and one universal time with one well-defined simultaneity. Since absolute theories assume a single well-defined space and time, no transformations of these concepts are allowed. It is possible, however, to transform to a moving coordinate system and use such a system to describe the single space and time. In absolute theories one can set up a coordinate system in a moving frame via a Galilean transformation, but this is not a transformation of space; rather, it is a transformation of the coordinate system used to define the one existing space.

4.2 Maxwellian Electrodynamics (Electrodynamics Based on a Galilean Kinematics)

Absolute theories also assume that the original Maxwellian electrodynamics best represents nature. Since electrodynamics based upon a Galilean kinematics has virtually disappeared from physics education, this section will quickly rederive the formulas. The inverse equations of the Galilean kinematic transformations are

* X * = *X¢
* - *U _{X}T¢
*, (5)

* Y *= *Y¢
* - *U _{Y}T¢
*, (6)

* Z *= *Z¢
* - *U _{Z}T¢
*, (7)

* T *= *T¢
*. (8)

The absolute theories assume that Maxwell’s equations are valid in the ether frame (Assumptions HAL-III and DJL-III). Here the ether frame is defined by the unprimed variables, leaving Maxwell’s equations in free space as

** Ñ
**_{ ×
}**E** = 4pr , (9)

** Ñ
**x **E** + (1/c)¶
* ***B**/¶
* *T = 0, (10)

** Ñ
**_{ ×
}**B** = 0 , (11)

** Ñ
** x **B** + (1/c)¶
* ***E**/¶
* *T = 4p**j**/c, (12)

In Eqs. (9) to (12) **E** is the electric field vector, **B** the magnetic field vector, r the charge density, *j* the current density, and *c* the speed of light in the ether. All quantities are evaluated in the ether rest frame. The expressions all employ Gaussian units.

In order to transform Maxwell’s equations to the moving (primed) frame, the chain rule of calculus will be used:

*¶
*E_{x}/¶
* *X’ = (¶
* *E_{x}/¶
* *X)(¶
* *X/¶
* *X’) + (¶
* *E_{x}/¶
* *Y)(¶
* *Y/¶
* *X’) +

(¶
* *E_{x}/¶
* *Z)(¶
* *Z/¶
* *X’) + (¶
* *E_{x}/¶
* *T)(¶
* *T/¶
* *X’) = ¶
* *E_{x}/¶
* *X . (13)

And, since the above chain rule can be applied to all three Cartesian components of the electric and magnetic field vectors,

Ñ ’ = Ñ . (14)

Now, dealing with the partial derivative with respect to time, and applying the chain rule,

*¶
*E_{x}/¶
* *T’ = (¶
* *E_{x}/¶
* *X)(¶
* *X/¶
* *T’) + (¶
* *E_{x}/¶
* *Y)(¶
* *Y/¶
* *T’) + (¶
* *E_{x}/¶
* *Z)(¶
* *Z/¶
* *T’) + (¶
* *E_{x}/¶
* *T)(¶
* *T/¶
* *T’)

= ¶
* *E_{x}/¶
* *T - U_{x}(¶
* *E_{x}/¶
* *X) - U_{y}(¶
* *E_{x}/¶
* *Y) - U_{z}(¶
* *E_{x}/¶
* *Z) . (15)

By again noting that the above chain rule can be applied to all three Cartesian components of the electric and magnetic field vectors,

* ¶
*/¶
* *T’ = (¶
* */¶
* *T) - **U**_{ ×
} Ñ
or ¶
* */¶
* *T = (¶
* */¶
* *T’) + **U**_{ ×
} Ñ
’ (16)

Applying Eqs. (14) and (16) to the Maxwell equations of the preferred frame [Eqs. (9) to (12)] leaves the Maxwellian transformation of the electric and magnetic fields as assumed by the absolute theories:

** Ñ
**’_{ ×
}**E** = 4pr’ , (17)

** Ñ
**x **E** + (1/c)¶
* ***B**/¶
* *T + (1/c)(**U**_{ ×
Ñ
}’)**B** = 0 , (18)

** Ñ
**’_{ ×
}**B** = 0 , (19)

** Ñ
**’x **B** + (1/c)¶
* ***E**/¶
* *T’-(1/c)(**U×
Ñ
**’)**E** = 4p**j**’/c, (20)

where

r’ = r and **j**’ = r_{P}(*V*_{P} - *U*) - r_{N}(*V*_{N} - *U*). (21)

Expression (21) is the classical transformation of the differential charge and current densities. Since no length contraction of space itself is assumed, a finite volume element retains its numerical value of volume under a Galilean transformation, and since charge is assumed as an invariant quantity as well, charge density is invariant under a Galilean transformation. Currents result from charge densities in motion. Currents can arise even if the charge density is zero, since a charge density of one sign can move within a stationary charge density of equal magnitude and opposite sign. Thus it is necessary to include two charge densities (one positive, subscript P, the other negative, subscript N) in the calculation of the current density with each charge density having its own velocity (*V*) as measured from the rest frame of the ether.

5. KINEMATIC AND ELECTRODYNAMIC TRANSFORMATIONS INFERRED BY MOVING OBSERVERS WITHIN THE LORENTZ THEORY

It is well known that the time dilation and length contraction postulates of the Lorentz theory lead moving observers to infer the same kinematic and electrodynamic transformation equations as those that result from the special theory of relativity, and no lengthy discussion of this will be done here. The interested reader is referred to the development of the Lorentz transformations as given in Ref. 2. Note that throughout Ref. 2 Lorentz takes great pains to emphasize that the quantities being transformed do not represent a real transformation of time and space, but rather a transformation inferred by moving observers due to their malfunctioning measuring instruments. Lorentz assumes that there is only one absolute time and space, and that coordinate systems would properly transform in a Galilean manner as discussed above. But Lorentz postulates that moving observers infer local coordinate systems that lead them to the incorrect transformation equations that now bear his name. This is a large conceptual difference with the Einstein approach, since Einstein assumes that all observers' coordinate systems are on equal footing. It is this difference that can be tested in the laboratory.

6. SHERWIN’S EXPERIMENT: A CRUCIAL TESTABLE DIFFERENCE BETWEEN THE EINSTEIN AND LORENTZ THEORIES

An enormous conceptual gulf separates the Einstein and Lorentz theories. The Lorentz theory postulates that a rigid object, originally at rest in the ether (frame E), when accelerated into a moving frame M, will shrink by a factor of g in the direction parallel to its motion. The Einstein theory is equivocal on this point. While observers situated in frame E will observe a shrinkage of the rigid object, observers who accelerate with the object will observe no such shrinkage, and observers forever at rest in frame M will observe the object to expand. In the Einstein theory the shrinkage or expansion of the object is relative, and depends on one’s point of view. There is no force associated with the change in length in the Einstein theory, because the object's length is relative to one’s motion with respect to it. In the Lorentz theory, however, the shrinkage of the object is well defined and absolute. Lorentz’s proposal is that the electromagnetic fields of the moving object alter during the acceleration in such a way that forces are induced that cause the object to shrink.

Recently, Sherwin proposed that if the Lorentz theory is correct, and there is an absolute physical length contraction, the Earth’s velocity through the ether could be observed by rotating a spring at a velocity high enough such that the forces that cause the Lorentz contraction are resonant with the oscillations of the spring.^{(34)} Sherwin made a crucial assumption that a rotating spring would not change its length in a time period much less than the vibrational period of the spring. If the rotation rate was in resonance with the vibrational period, this leads to a prediction that an accelerometer will detect the Earth's motion through the ether.^{(34)} (In the Einstein theory, viewed from an inertial frame moving with the center of the spring, there is no shrinkage, and hence no resonant force.) As a best guess for where the rest frame of the ether would be, Sherwin assumed that frame in which the 3 K background radiation is isotropic.^{(35)} With such an assumption for the preferred frame, Sherwin saw no effect. The Sherwin experiment thus shows that nature is better represented by Einstein’s theory than that of Lorentz if one assumes an ether at rest with respect to the 3 K background, and if one assumes that the contraction of a rotating spring occurs on a time scale near the vibrational period.

The theory proposed herein assumes there is no physical length contraction of objects as they move with respect to the preferred frame, and therefore would naively predict the same outcome of Sherwin’s experiment that the Einstein theory predicts (no effect should ever be observed).

Yet even with continued null results of Sherwin's experiment, a case can be made that the Lorentz theory is still the best representative of nature, because it may be that the spring shrinkage occurs over time periods associated with the spring length divided by the speed of light. Also, a full analysis must include the effects of mass change during the rotation, and involves questions about quantum mechanical bonds.

Therefore, the most that should be claimed from Sherwin’s experiment is that under one set of assumptions the experiment indicates the absence of a physical length contraction in the Earth's frame of reference.

7. THE KINEMATIC TRANSFORMATION INFERRED BY MOVING OBSERVERS FOR THE THEORY PROPOSED HEREIN

As discussed above in Sec. 4, the kinematics of space-time in this absolute theory is assumed to be Galilean (and moving observers will agree once they correct for the effect that motion through absolute space has on their measuring instruments). That is, space and time are assumed conceptually to be classical. However, by Postulate DJL-I, clocks that move with respect to the ether run slow as compared to clocks at rest in the preferred frame. Therefore, while space and time may indeed be Galilean, should moving observers incorrectly believe that their moving clocks are capable of making accurate measurements of time, they will arrive at an incorrect kinematics. Present relativistic dogma dictates further that the speed of light is a frame-independent constant. The purpose of this section is to calculate the kinematics that moving observers will set up if (1) space and time are governed by the axioms proposed herein (moving clocks run slow), and (2) moving observers incorrectly assume that the speed of light is *c* in their frame.

7.1 Operationalism: Setting up the Inferred Kinematics Using Light Pulses

Consider two sets of observers, one set at rest in the preferred, stationary frame K, and the other set, in the frame k, moving with a uniform velocity *V* with respect to the preferred frame. Here, uppercase letters denote quantities evaluated in the preferred frame, and lowercase letters denote quantities evaluated in a moving frame. Each observer in each set is equipped with an identical clock and a means to receive, send, and record the arrival and departure times of light pulses. Clocks are defined operationally either by counting the number of wave crests associated with electromagnetic transitions between two atomic states (atomic clocks) or by observing the exponential time constant associated with the decay of an ensemble of unstable particles, such as muons.

The Einstein approach will be used to set up coordinate systems. The origin of both the moving and preferred frames will be arbitrarily chosen within the constraint that the two origins must coincide at some point in time. Clocks at the origins will be set to zero at the moment the two origins coincide. Three orthogonal axes will be chosen as *X*, *Y*, and *Z* in the preferred frame and *x*, *y*, and *z* in the moving frame. Without loss of generality it can be assumed that the direction of relative motion is *X _{ }*(-

Special relativity assumes that the speed of light is invariant. The following sections will analyze what moving observers infer about their coordinate system once they make the *c* = *C* assumption in a space-time actually governed by the axioms proposed herein. The inferences of moving observers will be compared to observations made by preferred frame observers of the same process, and since preferred frame observers arbitrarily close to moving observers can relate their observations to each other with arbitrarily small errors, this will lead to an understanding of how observers relate *X*, *Y*, *Z*, and *T* to *x*, *y*, *z*, and *t*. Using curly brackets to denote functional dependence, this process will find the kinematic transformations *X*{*x*,*y*,*z*,*t*}, etc., between the preferred frame and coordinate systems inferred by arbitrarily moving observers. Since present natural philosophy assumes *c* = *C*, we may be in the role of the moving observers now to be discussed.

7.2 Thought Experiment for Light Pulses Traveling Perpendicular to *x*

Let a moving observer anywhere in the *x*–*z* plane, (*x*_{0},0,*z*_{0}), send out a light pulse at any time *t* = *t*_{0} in such a way that the *x* and *z* coordinates of the pulse do not change during its flight. The calculation for the round-trip is the same as the traditional calculation for the light pulse in the arm of the Michelson–Morley apparatus which is oriented perpendicular to the direction of motion. The moving observer will infer that the pulse is traveling perpendicular to the *x*–*z* plane, since no change in *x* or *z* is recorded; the direction of pulse motion is *y*. A second moving observer located at (*x*_{0},*y*,*z*_{0}) will receive and return the pulse to (*x*_{0},0,*z*_{0}). As a result of the assumption that the speed of light is *c* in the moving frame, the moving observers will infer that the distance from (*x*_{0},0,*z*_{0}) to (*x*_{0},*y*,*z*_{0}) is *y* = *c*(*t*_{R} - *t*_{0})/2, where *t*_{R} is the time the pulse returns to (*x*_{0},0,*z*_{0}), as observed in the moving frame. A preferred frame observer will view the pulse as traveling with a horizontal velocity equal to that of the moving observer, so that the pulse always maintains a value of *x* = *x*_{0} in the moving frame. Thus a preferred frame observer sees the pulse travel a distance in the *X* direction of *V*(*T _{y}* -

* Y*{*x*,*y*,*z*,*t*} = *y*. (22)

Since moving observers assume that light travels at speed *c* in their frame, they further assume that the time it takes the light pulse to travel from (*x*_{0},0,*z*_{0}) to (*x*_{0},*y*,*z*_{0}) is the same as the time it takes for the return trip. Thus the clock moving with the observer at (*x*_{0},*y*,*z*_{0}) must be set to a value of *t*_{0} plus half the round-trip flight time when the pulse arrives at (*x*_{0},*y*,*z*_{0}) for moving observers to agree that the clocks are correctly synchronized. Half the round-trip flight time, as observed in the moving frame, is *t*_{1/2} = (*t*_{R} - *t*_{0})/2 = *y*/*c*. Preferred frame observers see half the round-trip flight time as *T*_{1/2} = (*T _{y}* -

* Z*{*x*,*y*,*z*,*t*} = *z*. (23)

* T*{*x*,*y*,*z*,*t*} = *T*{*x*,*t*}. (24)

7.3 Thought Experiment for Light Pulses Traveling Parallel to *x*

Let a light pulse emanate from the origin of the moving system at *t* = *T* = 0 toward (*x*_{0},0,0). The calculation for the round-trip is the same as the traditional calculation for the light pulse in the arm of the Michelson–Morley apparatus which is oriented parallel to the direction of motion. Let (*x*_{0},0,0) and (*X*_{0},0,0) coincide at *T* = 0, *X*_{0} = *X*{*x* = *x*_{0},*T* = 0}. The light pulse travels from the origin to (*x*_{0},0,0), and (*x*_{0},0,0) moves from (*X*_{0},0,0) to (*X*_{0} + *VT*_{1},0,0) during the oneway pulse transit time, *T*_{1}. Preferred frame observers record the time the pulse arrives at (*x*_{0},0,0) to be *T*_{1} = (*X*_{0} + *VT*_{1})/*C* = *X*_{0}/(*C* - *V*). Let the pulse be reflected from (*x*_{0},0,0) back to the moving origin. Preferred frame observers record the time for the return trip to be *T*_{2} = (*X*_{0} - *VT*_{2})/*C* = *X*_{0}/(*C* + *V*). Clocks in the preferred frame will advance

* T*_{3} = *T*_{1} + *T*_{2} = [*X*_{0}/(*C* - *V*_{ })] +[*X*_{0}/(*C* + *V*_{ })] = 2*CX*_{0}/(*C*^{ 2} - *V*^{ 2}) = 2*X*_{0}g^{2}/*C*

during the round-trip. The clock at the moving origin advances *t*_{3} = *T*_{3}/g = 2*X*_{0}g/*C* during the round-trip, due to the postulated Larmor time dilation. Since observers in k assume that light travels at speed *c* in their frame, they assume that light takes an equal amount of time for both the forward and return trips, so they will synchronize their clocks by having the (*x*_{0},0,0) clock set to *t*_{1} = *t*_{3}/2 = *X*_{0}g/*C* when the pulse arrives at (*x*_{0},0,0). Also based on the assumption that light travels at speed *c* in the moving frame is the inference by moving observers that

* x*_{0} = *ct*_{1} = g*X*_{0}. (25)

Preferred frame observers record

* T*_{1} = *X*_{0}/(*C* - *V*_{ }) = *X*_{0}(*C* + *V _{ }*)/(

when the pulse arrives at (*x*_{0},0,0). Since the moving clocks run slow, preferred frame observers know that the moving clock will have advanced by *t*_{1} = *T*_{1}/g = *X*_{0}g(1 + *V*/*C*)/*C* during the time it takes for the pulse to travel to (*x*_{0},0,0). Thus, in addition to running slow, moving clocks are mis-synchronized by an amount -g*X*_{0}*V*/*C*^{ 2} = -*Vx*_{0}/*C*^{ 2}. At *T* = 0 moving observers thus mis-synchronize the clock at (*x*_{0},0,0) to *t*{*x* = *x*_{0},*T* = 0} = -*Vx*_{0}/*C*^{ 2}, and after *T* = 0 the moving clock will incrementally change by the postulated *T*/g as *T* increases; hence *t*{*x*_{0},*T*} = (-*Vx*_{0}/*C*^{ 2} + *T*/g), or *T* = g (*t* + *Vx*_{0}/*C*^{ 2}). With *v* = -*V*, and since *x*_{0} can be any *x*,

* T*{*x*,*t*} = g (*t* - *vx*/*C*^{ 2}) = g (*t* - b*x*/*C*). (26)

Equation (26) uses *v* = -*V*. This identity will now be proven. As observed from the preferred frame, the moving origin will coincide with the stationary origin at *T*_{4} = 0 and it will arrive at *X*_{0} at the time *T*_{5} = *X*_{0}/*V*. As observed in the moving system, the observer at *X*_{0} coincides with (*x*_{0},0,0) at *t*_{4} = -g*VX*_{0}/*C*^{ 2}, and arrives at the moving origin at the time *t*_{5} = *T*_{5}/g = *X*_{0}/g*V*. Moving observers thus infer that it takes

* t*_{6} = *t*_{5} - *t*_{4} = *X*_{0}[(1/g*V*_{ }) + g*V*/*C*^{ 2}] = (*X*_{0}/*V*g)[1 + g^{2}*V*^{ 2}/*C*^{ 2}] = g*X*_{0}/*V*

for *X*_{0} to travel from (*x*_{0},0,0) to the moving origin. The velocity observed for *X*_{0} (and hence the preferred frame) is thus the observed distance traveled, given by Eq. (25) as -*x*_{0} = -g*X*_{0}, divided by *t*_{6}, *v* = -g*X*_{0}/(g*X*_{0}/*V*) = -*V*.

* X*_{0} coincides with (*x*_{0},0,0) at *x* = g*X*_{0}, when *t*_{4} = -g*VX*_{0}/*C*^{ 2}, or *X*{*x* = g*X*_{0},*t* = -g*VX*_{0}/*C*^{ 2}} = *X*_{0}. Since *X*_{0} is arbitrary, we have *X*{*ax*,*at*} = *aX*{*x*,*t*}. *X*_{0} coincides with the moving origin at *x* = 0, when *t*_{5} = *X*_{0}/g*V*, or *X*{*x* = 0,*t* = *X*_{0}/g*V*} = *X*_{0}. Since *X*_{0} is arbitrary we have *X*{0,*at*} = *aX*{0,*t*}. Therefore, *X*{*x*,*t*} is linear, *X*{*x*,*t*} = *Ax* + *Bt*. Now, *X*_{0} = *X*{0,*X*_{0}/g*V*} = *BX*_{0}/g*V*, or *B* = g*V*. Also, *X*_{0} = *X*{g*X*_{0},- g*VX*_{0}/*C*^{ 2}} = *A*g*X*_{0} - *B*g*VX*_{0}/*C*^{ 2}, and with *B* = g*V*, *A* = g. Using *v* = -*V*,

* X*{*x*,*t*} = g(*x* - *vt*) = g(*x* - b*ct*). (27)

7.4 Transformation Between Arbitrarily Moving Observers

Equations (22), (23), (26), and (27) are the standard Lorentz kinematic transformation equations. The equations are well known, with one result being their internal self-consistency. The self-consistency of the Lorentz transformations allows us to transform from one arbitrarily moving frame to the preferred frame, and then transform from the preferred frame to a second arbitrarily moving frame with a result that is the same as if we had transformed directly from one arbitrarily moving frame to the second, provided we use the appropriate velocity in the transformation equations and rotate the axes appropriately. The intermediate observer in the preferred frame can clearly understand how the two comoving observers arrive at the appropriate angle and velocity, and thus once the basis for the Lorentz transformation is understood for a transformation between one arbitrarily moving frame and the preferred frame, the basis for the Lorentz transformation between two arbitrarily moving frames is understood as well. Therefore, the Lorentz kinematic transformation equations have now been derived from an axiomatic set based on time dilation alone; no length contraction is required.

8. ELECTRODYNAMICS

To determine the electric and magnetic fields in any frame experimentally, it is necessary to note the change in velocity of a moving small test charge and define the fields through the Lorentz relation (proposed by Lorentz prior to relativity and a relation Lorentz based on observations alone):

* d***p**/*dt* = *d*(g*m***v**)/*dt* = *q*(**E** + **v**/*c* ´
** B**). (28)

[Note that for Eq. (28) to be strictly correct it must be modified to include radiation reaction effects. In the following discussion it is assumed that the radiation reaction effects are accounted for in the presently accepted manner.] Here, *m* is the mass and *q* the charge of the displaced object, **E** is the electric field vector and **v ´
B** is the vector (cross) product of **v** and **B**, where **v** is the velocity of the test charge as measured in the reference frame being used to determine the fields, and **B** is the magnetic field vector. Many test particles, with different velocities, can be used to obtain **E** and **B**, and such measurements will uniquely determine the electromagnetic fields.

Present (relativistic) electrodynamic theory assumes that Maxwell’s equations are obeyed in all frames. Calling one frame a preferred frame, and using the relativistic approach to transform the fields, the observed accelerations of test charges can be understood from either a preferred frame or an arbitrarily moving frame once the kinematic variables are transformed via Eqs. (22), (23), (26), and (27).

It is assumed here that Maxwell’s equations hold in the preferred frame, so accelerations caused by fields as evaluated in the preferred frame will be the same as under present theory. Since the kinematic transformation is also the same, accelerations observed in moving frames are also the same as in the present theory. (Accelerations are purely kinematic.) Since measurements of the fields rely solely on measurements of accelerations, the inferred transformation of the fields by moving observers is therefore the same as in the presently accepted theory.

The purpose of this section has been to show that the experimental basis for the Lorentz electrodynamic transformations is based on kinematics alone. However, if the reader desires a derivation of the Lorentz electrodynamic transformations from the Lorentz kinematic relations, it can be found in Ref. 2, where Lorentz assumes that Maxwell’s equations are valid in the ether and further assumes that moving observers infer incorrect kinematic relations, identical to what is assumed used herein.

It has now been shown that moving observers will arrive at the presently accepted kinematic and electrodynamic transformations in a space and time governed by the axioms proposed herein if they set up their coordinate systems using light pulses under the incorrect assumption that *c* = *C* in all frames. All the experiments done in our laboratories with pointlike particles agree with present relativistic dynamics, but present theory also assumes *c* = *C* in all frames, so the theory proposed herein also agrees with these experiments.

9. THE MICHELSON–MORLEY TEST

The theory proposed herein is based on time dilation alone; no physical length contraction occurs for objects moving with respect to the ether. If there is no physical length contraction, a non-null result of the Michelson–Morley (MM) test^{(7)} is expected, via the traditional arguments.

Lorentz’s hypothesis of a length contraction was the first proposed solution to this dilemma. If, as measured from the preferred frame, the length of a rigid body in the direction of motion, D*X*, is contracted by a factor of g while the two perpendicular lengths are unaltered, the null result of MM is explained. It is commonly assumed that this physical length contraction is a trivial consequence of Eqs. (22), (23), (26), and (27), but the assumption is incorrect. As shown above in Sec. 7, Eqs. (22), (23), (26), and (27) arise from time dilation and an assumption of *c* = *C* alone. Observations of length contractions based on these equations are merely illusions caused by time dilation; these equations say nothing about what happens to the length of a rigid physical body upon acceleration or rotation. If a physical rigid body does not contract when moving (as proposed here), preferred frame observers will observe a body to have D*X* = *L _{X}* whether it moves or not, but an observer comoving with that same body will say D

It is possible that the MM apparatus may affect the intended measurement, and that the null result can be explained within an absolute theory without recourse to a postulated physical length contraction. While not known at the time Larmor, Lorentz, and Einstein proposed their theories, it is now well understood that a measuring apparatus can often affect the result of the intended measurement. Postulate DJL-II proposed herein proposes that the MM mirrors force nodes of the electric field to move along with the apparatus, leading to the null result of the famous experiment. The postulate is based on a simple analogy. If a person holds a guitar string with thumb and forefinger, and the string is plucked, a standing wave will ensue that has enforced nodes at her fingers. If she now moves her fingers slowly along the string, the oscillation will continue with the nodes moving along with her fingers. Since the electric field associated with reflected light must be zero at the mirrors, the mirrors act to enforce nodes in the electric field in a way analogous to fingers enforcing nodes on a vibrating string. For an apparatus at rest in the preferred frame, and for a single well-defined frequency, the enforced boundary condition that the electric field have nodes at each of two mirrors separated by a distance *L* is satisfied if the electric field between the mirrors is a standing wave,

* E* = *E*_{0 }cos^{ }(*kX* - w*T*) - *E*_{0 }cos^{ }(*kX* + w*T*) = 2*E*_{0 }sin^{ }(*kX*)^{ }sin^{ }(w*T*), (29)

where *L* = *n*l/2 = *n*p/*k*. If the apparatus is moved, the boundary conditions will still be satisfied if

* E *= 2*E*_{0 }sin^{ }[*k*(*X* - *VT*)]^{ }sin^{ }(w_{d}*T*) = *E*_{0 }cos^{ }[*k*(*X* - *VT*) - w_{d}*T*] - *E*_{0 }cos^{ }[*k*(*X* - *VT*) + w_{d}*T*]. (30)

Noting that w_{d}/*k* = *C*, where w_{d} is the drive frequency (w_{d} is the number of wave fronts hitting the semitransparent mirror per unit time), Eq. (30) reveals a forward wave traveling at *C* + *V*, and the return wave traveling at *C* - *V*, as though the phase velocity of light were the vector sum of *C* and *V*, which clearly leads to the null result of the MM test.

Note that the simple analogy between light within the MM apparatus and a standing wave on a string cannot be strictly true. Standing waves on a string are oscillations in three-dimensional space of a one-dimensional medium, while light waves in an ether are assumed to be an oscillating displacement of the three-dimensional ether. Also, while the thumb and forefinger clearly hold the displacement of the string to zero, the partially transparent mirror does not require the electric field to be zero. The physical pictures are not completely equivalent.

For the interference to occur, and since the interference pattern can be built up one interaction at a time, it is necessary that each individual wave packet of light interfere with itself. The physical picture being proposed here is that the single wave packet encounters the semitransparent mirror and excites an electric field oscillation in each half of the MM apparatus. In each half of the apparatus the field induced by the wave packet interacts with two mirrors, once with the fully reflecting mirror at the terminus of the apparatus arm, once with the semitransparent mirror. At the fully reflecting mirror the electric field is null, since the sum of the incident field and the reflected field is zero. At the semitransparent mirror, that portion of the field that is reflected (the portion in the one apparatus arm) is exactly equal to the negative of that portion of the incident field that enters (or leaves) the arm in question. Thus, analyzing each portion of the field independently within each apparatus arm there are two enforced nulls for each arm, and the enforced node analogy is valid.

This section has attempted to explain the physical picture behind Postulate DJL-II so that the postulate appears reasonable. However, from a scientific point of view, the postulate is one of the axioms, and arguments over "reasonableness" of starting assumptions are not crucial. What is crucial is that the set of axioms proposed herein lead to predictions of experimental reality that are different from competing theories so that tests can be done to determine which set of axioms best represents nature. While I hope readers now consider the postulate reasonable, the issue of testability is more important and will now be dealt with.

10. NONSTANDARD TESTS OF SPACE-TIME THEORY

In addition to the standard experimental basis for space-time theory discussed in Sec. 3 above, there are a few other tests of nature that have been done that have a bearing on which axioms of space and time best represent nature.

10.1 The Experiment of Aspect *et al*.: A Crucial Testable Difference Between the Einstein Theory and the Absolute Theories

In 1935 Einstein, Podolsky, and Rosen (EPR)^{(36)} wrote a paper asking the question, "Can quantum mechanics be considered complete?" Basically, EPR show that under certain experimental circumstances quantum mechanics predicts effects that imply a faster-than-light signaling between distant events. If faster-than-light signaling exists, by the postulates of special relativity a cause preceding an effect in one frame would be an effect preceding the cause in another. Since causality is generally assumed, such a violation of causality is deemed problematic. More recently, Bell^{(37)} extended the work of EPR to show that quantum mechanics predicts the violation of inequalities not allowed classically unless faster-than-light signaling is possible. In the Bell thought experiments a source sends out two photons to distant detectors. The two photons leave the source in a correlated state. Measurement of the linear polarization of one photon implies a correlation of the measurement of linear polarization of the other. In order to preclude the possibility that the detectors influence the source prior to the emitting of the correlated photons (such an influence could be done with signals traveling at the speed of light or less), Bell emphasized that the detector settings of EPR experiments should be rapidly changed during the flight of the photons to the detector as proposed earlier by Bohm and Aharonov.^{(38)} Aspect, Dalibard, and Roger (ADR)^{(39)} have performed tests of Bell’s inequalities using quickly switched detectors, and the results of those tests have verified the quantum mechanical predictions.

The correct interpretation of the meaning of the ADR results is a topic of intense present debate. But the simplest possible explanation of the ADR results is to conclude that faster-than-light signaling is an experimental reality, and that special relativity is not the theory that best represents the nature of space and time. In both the absolute theory of Lorentz and the absolute theory proposed herein, faster-than-light signaling can be achieved without any problems regarding causality. The absolute theories have a well-defined concept of simultaneity valid in all reference frames, and instantaneous actions-at-a-distance are in no way "spooky" within the confines of absolute theories.

In any discussion of the theoretical ramifications of a single experiment, it must be readily admitted that many alternative explanations are allowable, and that no single experiment is by itself justification to set aside a tremendously successful theory. But it must be emphasized that the simplest explanation of the ADR results leads to a conclusion that is damaging to the special theory of relativity. Indeed, it was the author of special relativity himself who, together with Podolsky and Rosen, first proposed a similar test, and the EPR reasoning was as simple and straightforward as the discussion of the results presented herein.

10.2 The Two-Slit Experiment

Another experiment with relevance to space-time theory is Young’s two-slit experiment. The inability of modern-day physics to easily understand this problem continues to be celebrated.^{(40)} Yet if special relativity is set aside, a simple explanation for observations is possible. If all entities (photons, electrons, etc.) are wavelike diffuse entities, with the usual wave function equal to the square root of the density, the two-slit experiment can be understood as a simple interference of these waves, if it is further assumed that an instantaneous collapse of the wave function occurs when the wave function meets the wall. It is only because of special relativity that such a simple explanation cannot be accepted, since special relativity does not allow a well-defined simultaneity. The absolute theories do not have this difficulty, since simultaneity is well defined within their axioms.

Note that the two-slit experiment and the EPR tests are closely related. The simplest interpretation of both experiments involves action-at-a-distance, a prospect that is not allowed by the axioms of special relativity.

10.3 The Experiment of Krisher *et al*.

A recent experiment by Krisher *et al*. has tested for the anisotropy of the oneway speed of light by using two hydrogen maser standards separated by 21 km.^{(41)} The light from each maser is split, with one-half sent to a local detector and the other half used to modulate a laser carrier signal that is sent to a detector at the distant location. The light from the local maser and the distant maser are combined, and their relative frequency difference monitored. Since all light propagation is oneway in this experiment, the node enforcement hypothesis, Postulate DJL-II, is no longer easily motivated by an analogy with a pinned string, and it is possible that the Krisher *et al*. experiment could yield a non-null result. (There are no longer mirrors enforcing boundary conditions at both ends of a light path, so nodes may no longer be forced to move along with the apparatus in this case.) An analysis of the Krisher *et al*. result using the theory presented herein shows that experimental noise is too large at present to be able to detect the Earth’s motion through an ether at rest with respect to the 3 K microwave background radiation. However, further refinements in the experiment may detect such motion.

10.4 Laser Lunar Ranging Experiments

Retroreflectors placed on the Moon by Apollo astronauts and (Russian) Lunakhod spacecraft have allowed a capability for sending short laser pulses on round-trips between the Earth and Moon. By recording the round-trip traversal time of such pulses, the distance to the Moon can be inferred using an assumption about the speed of light during the transit; study of this topic is termed laser lunar ranging (LLR). LLR tests have become very accurate in recent years, with present round-trip traversal times being resolvable to about 0.2 ns. Such accuracy has allowed LLR to become a versatile experimental tool, with applicability toward studies ranging from observations of solid Earth tides to investigations of general relativity.^{(42–44)} It is natural to question whether the extremely accurate LLR tests are capable of providing experimental evidence in support of one of the competing space-time theories.

One aspect of the LLR data is a daily variation in the light pulse round-trip traversal time due to the Earth’s rotation. When the Moon is on the horizon, the Earth–Moon distance is approximately *L* + *R*, where *R* is the radius of the Earth, and *L* is the Earth–Moon distance when the Moon is overhead. Thus the round-trip light pulse traversal time will vary by

D*T* = 2*R*/*V* (31)

during the day, where *V* is the average group velocity of light during the round-trip. It is being proposed here that there is no length contraction and that the free space group velocity of light is *C* with respect to an ether. Therefore (unless the Earth is at rest with respect to the ether), it is expected that the average group velocity of light will be dependent on the orientation of the Earth–Moon axis with respect to the Earth’s velocity through the preferred frame, and D*T* should therefore vary with a two-week period. No such variations have been seen in published analysis of the LLR data. This situation has led to the misunderstanding that the LLR experiments have proven the isotropy of the group velocity of light.

The problem with the above analysis is its two-dimensional nature. In actuality, the Moon does not always pass from the horizon to a position directly overhead, due to the inclination of the Earth’s polar axis with respect to the lunar orbit plane. As a simple example, consider the case where, during Sunday of week 1, the Earth–Moon system is aligned with the direction of the Earth’s motion through the ether. Further assume that the Moon passes directly overhead of a detector located on the Earth’s equator on Sunday of week 1. In this case the outgoing light pulse will have a velocity *C* - *V* with respect to the Earth, and the return light pulse will have a velocity *C* + *V* with respect to the Earth, for a daily variation in round-trip light traversal time of

D*T*_{1} = *R*/(*C* - *V*) + *R*/(*C* + *V*) = 2g^{2}*R*/*C*. (32)

About one week later, during week 2, the Earth–Moon system will be aligned perpendicular to the Earth’s motion through the preferred frame, and the velocity of a light pulse will be *C*/g with respect to the Earth for both the outgoing and return trips. During Sunday of week 2, the Moon will no longer pass directly overhead, but rather will make an angle q with respect to vertical when it is at its maximum ascent, where q is the angle between the Earth’s polar axis and the normal to the lunar orbit plane. Thus during week 2 the daily variation in round-trip traversal time is

D*T*_{2} = 2*R*g^{ }cos^{ }q/*C*. (33)

Therefore, there is a variation in D*T* that has a two-week period (just what would be expected if there is an angular dependence of the group velocity of light). Unfortunately, the main contributor to the signal is the variation in cos^{ }q, which varies between 1.0 and about 0.9. Assuming an ether at rest with the 3 K background radiation, the average velocity of light with respect to the Earth would only change by about one part in 10^{6} from week 1 to week 2. Thus in order to determine such an effect in the LLR data, it would be necessary to know cos^{ }q to such an accuracy. At present, q is one of several parameters fit to the LLR data under the assumption that the group velocity of light is isotropic and equal to *c* as stipulated by special relativity. Even if it was desired to measure q independently to check the isotropy of the speed of light, such a measurement would be extremely difficult, since the angular spread due to the surface roughness of the Moon is somewhat larger than the required angular accuracy. Therefore, the LLR data have no bearing on which of the competing space-time theories best represents nature.

10.5 The Silvertooth–Whitney Experiment

Silvertooth and Whitney^{(45)} have published the results of an experiment that appear to indicate a nonuniformity in the speed of light as observed from Earth. Should this result hold up under scrutiny, such a result would be incompatible with the Einstein and Lorentz theories. Unfortunately, an analysis of the Silvertooth–Whitney result using the theory proposed herein would indicate that the Earth moves through the ether at a velocity of *v* = 0.0355*c*, and while by itself such a result would be possible, the results of the Krisher *et al*. experiment do have a signal-to-noise ratio sufficient to preclude such a large velocity.

Additional refinements and confirmations of both the Krisher *et al*. result and the Silvertooth–Whitney result are needed before one can say which axiomatic space-time set best explains their results. But even if well-controlled experiments of this type yield consistent null results, it may still be that the apparatus enforces the result of these phase measurements. A test relying on amplitude measurements is therefore very desirable.

11. PROPOSAL FOR NEW CRUCIAL TESTS OF SPACE-TIME THEORY

11.1 A Presently Feasible but Equivocal Test

Section 9 above has proposed Eq. (30) as the form of the electric field within the arms of the MM apparatus,

* E* = 2*E*_{0 }sin^{ }[*k*(*X*_{ }-_{ }*VT*)]^{ }sin^{ }(w_{d}*T*) = *E*_{0 }cos^{ }[*k*(*X*_{ }-_{ }*VT*)_{ }-_{ }w_{d}*T*]_{ }-_{ }*E*_{0 }cos^{ }[*k*(*X*_{ }-_{ }*VT*)_{ }+_{ }w_{d}*T*]. (30)

If the electric field within the apparatus does have the form given in Eq. (30), the null result of the MM test is expected, as discussed in Sec. 9.

But Eq. (30) involves only a single light frequency. A short pulse of radiation is composed of many frequencies, multiplied by an amplitude function. Assuming Eq. (30) for each individual frequency, the electric field of a monodimensional wave packet can be written as

* E* = ò
*g*{(*X* - *CT*), w_{d}}^{ }sin^{ }(w_{d}*T*)_{ }sin^{ }[(w_{d}/*c*)(*X* - *VT*)]*d*w_{d}. (34)

Therefore, while the mirrors may enforce nodes on the allowed electric fields to force a null result of the MM test, it remains possible that the velocity of propagation of the amplitude function (which is the velocity of energy propagation) within the apparatus travels at a speed of *C* with respect to a preferred frame only. A non-null result of a group velocity equivalent of the MM test is thus possible, and such a test is therefore a critical new test for space-time theory.

A group velocity equivalent of the MM test is possible with the recently developed technology of ultrashort laser pulses.^{(46)} One hundred and fifty femtosecond laser pulses of fundamental angular frequency w can be split in two by a beam splitter, sent through two mutually perpendicular paths and recombined. The recombined pulse can be sent through a non-linear doubling crystal to produce second harmonic (2w) photons, and then through a filter to eliminate the fundamental. (Here, the entire fringe pattern passes through the doubling crystal.) The 2w photon intensity can be monitored over time. Since the number of 2w photons produced in the doubling crystal is nonlinear, many more photons will be produced if the two pulses arrive at the crystal simultaneously than will be produced if the two pulses arrive separately. Thus by measuring the number of 2w photons produced, it is possible to measure the relative arrival time of the two pulses. If the velocity of energy propagation is *C* with respect to the ether, as proposed here, rotation of the device will cause a shift in the relative arrival time (following the traditional MM arguments), and this will cause a change in the number of 2w photons produced by the crystal. If the Einstein or Lorentz theory is more representative of nature, the relative arrival time of the pulses will not change during rotation, and there will be no change in the number of 2w photons produced by the crystal.

Since it is known that if the path lengths of the two apparatus arms are held fixed, no fringe shift is observed, this information can be used to lock the path lengths to a constant relative value by sending a continuous wave HeNe laser beam into the apparatus through a dichroic beam splitter, detecting the interference pattern of the HeNe beam, and using a piezoelectric spacer to lock the interference pattern onto a single fringe. Thus with such a locked apparatus, the Earth’s rotation can rotate the device.

The phase of reflected light is dependent upon frequency, both in metallic and dielectric mirrors, but this effect can be overcome for the proposed test. Efforts in the last ten years on ultrafast laser pulse generation have relied on the development of mirrors that reflect ultrafast pulses without broadening them to any significant extent. The relevant point is that the mirrors have flat amplitude and phase response over the bandwidth of the pulses. Such mirrors have already been developed for the pulse lengths required for a group velocity equivalent of the MM test.

Note that while the axioms presented herein do allow for a non-null result of a group velocity equivalent of the MM test, the axioms proposed herein do not require a non-null result. It is, of course, possible that the MM apparatus mirrors affect the group velocity of light as well. While it is best for theories to make firm predictions upon which they can be judged, in this case the theory proposed herein merely allows a non-null result of a group velocity equivalent of the MM test; no firm prediction can be made; the test is equivocal. But the Einstein and Lorentz theories do not allow a non-null result, so should a non-null result be experimentally obtained, it would be strong evidence that the best theory of space and time is that proposed herein.

11.2 An Experimentally Demanding but Unequivocal Test

The difference between the theory proposed herein and that of Lorentz is that the theory presented herein proposes that no physical length contraction exists. This can, in principle, be tested directly. If one can arrange to send laser pulses at photodetectors and place an object between the lasers and the photodetectors, a value for the object’s length can be obtained by looking at the length of the shadow. (The shadow’s length can be determined from those detectors that receive no light.) If the same object could be accelerated to high speed, and its shadow measured while it is moving, one could compare the two lengths to see whether or not length contraction actually occurs. Unfortunately, acceleration of non-point-like bodies to relativistic speeds is very difficult, and one would need to have extremely short laser pulses fire across the moving object at the right time in order to measure the shadow. But as technologies advance, this unequivocal test of the theory proposed herein should eventually be possible.

12. CONCLUSION

At the time of this writing three theories of space-time are consistent with the standard experimental evidence: the theory of Lorentz, the theory of Einstein, and the theory proposed herein. Under one set of assumptions the Lorentz theory has difficulty explaining the results of Sherwin’s^{(34)} experiment. The Einstein theory makes it difficult to understand the experiments of Aspect *et al*.^{(39)} and the two-slit experiment. Neither the Einstein nor the Lorentz theory is compatible with the results of the Silvertooth and Whitney^{(45)} experiment. The theory proposed herein only has difficulty in explaining the Silvertooth and Whitney result. The theory proposed herein allows for a non-null result of a group velocity equivalent of the Michelson–Morley experiment, while the theories of Einstein and Lorentz do not. A group velocity equivalent of the Michelson–Morley experiment is therefore a crucial new test for space-time theories. The theory proposed herein predicts that no length contraction of physical bodies will occur. A direct measurement of length contraction is therefore a second crucial new test for space-time theories. It must be admitted that none of the three possible space-time theories are ruled out at the moment, since the experimental picture is not completely clear in those areas where the Lorentz and Einstein theories appear to have difficulties. Nonetheless, it is clear that there are experiments that can be done in the future that can clarify which of the three theories best represents nature.

Acknowledgment

An anonymous reviewer has mentioned that the enforced node hypothesis discussed herein bears resemblance to work done by Jennison. See Ref. 47 and citations therein for details.

Numerous individuals have provided constructive criticism, including G. Bauer and J. Bell, then of CERN, A. Gara, Columbia, F. Mills, Argonne, L. Elias, University of Central Florida, the editors and anonymous reviewers of *Physics Essays*, *Physical Review Letters*, and *Physical Review* *A*, J. Griffin, Fermilab, retired, E. Marx of the National Institute of Standards and Technology, and T. Garavaglia, A. Chao, S. Dutt, D.E. Johnson, J. Palkovic, X.Q. Wang, and N.K. Mahale of the SSC. I thank all of them for strengthening the scope and the presentation of my work.

Received 22 December 1993.

**Résumé**

*Un ensemble axiomatique employant seulement la dilatation du temps de Larmor suffit à dériver les transformations cinématiques de Lorentz; la contraction de longueur de Fitzgerald n'est pas nécessaire. Puisque la transformation cinématique conduit seulement et directement à la transformation électrodynamique, on présente le troisième ensemble axiomatique (les deux autres sont celles de Lorentz et d'Einstein) compatible avec la théorie reconnue de l'électrodynamique. La théorie suppose les concepts classiques d'espace-temps; la relativité de Galilée et le concept d'éther sont présumés être valides. On examine l'évidence expérimentale par rapport aux concepts d'espace-temps. Une vitesse de groupe pourrait prouver que l'ensemble axiomatique est celui qui représente le mieux la nature.*

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D.J. Larson

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