# Dichotomy Between Null Results from all Interferometer Experiments versus Special Relativity Expectations

## Steven D. Deines

## ____________________________________________

ABSTRACT

The fundamental principle of physics is that all observers of an event must have identical results, especially when the recorded data are transformed to a common inertial frame. The Michelson-Morley experiment demonstrates contradictions with length contraction when three or more independent inertial observers concurrently record the output. Length contraction has never been observed directly despite current technological precision. Special relativity does not explain the constant output of the Michelson-Morley and Kennedy-Thorndike interferometers, since Earth’s rotation causes unequal changing velocities for both arms. Displacements, instead of time intervals, for the split beams are analyzed when the interferometer is stationary and moving at a constant velocity. This paper reveals that any interferometer with equal or unequal arms can neither prove nor disprove the existence of a hypothetical medium for light transmission. All interferometer experiments indicate light velocity obeys vector velocity addition involving moving light sources. This property of light explains how measurements of light speed in any moving laboratory are precisely identical when sources and detectors are fixed relative to each other. The universal speed of light is an excellent approximation due to high speed, but not an exact constant when light sources and observers move independently. Many physical concepts will require reexamination.

Keywords: length contraction, time dilation, light speed, michelson-morley experiment, kennedy-

thorndike experiment.

Author: Donatech Corporation, Inc., Fairfield, Iowa USA.

- BACKGROUND

Hippolyte Fizeau [1, 2] measured the speed of light through moving water and analyzed it theoretically using the concepts of vector addition with c + v and c - v, where c is the speed of light in a vacuum and v is the speed of the moving water. Fizeau used a special inter- ferometer to compare the frequency shift by combining two light beams that went through moving water in different directions. Fizeau’s theory grossly overestimated the measured shift. Albert Michelson and Edward Morley replicated Fizeau’s water experiment with a more accurate setup and obtained Fizeau’s same residual nonzero result. They concluded, “The result of this work is therefore that the result announced by Fizeau is essentially correct; and that the luminiferous ether is entirely unaffected by the motion of the matter which it permeates.” (original italics)[3] In 1881, Michelson designed and demonstrated an interferometer to split a light beam into two perpendicular beams that traversed equal lengths and that recombined into one output light beam [4]. The analysis was based on Fizeau’s theory and predicted that the motion of the interferometer due to Earth’s orbital motion alone should cause the light beams to arrive at the recombination point at different times, which would produce destructive interference. It was hoped that rotating the interferometer would align one axis with Earth’s orbital velocity and produce a maximum destructive interference to determine the orbital velocity of the laboratory through the hypothetical medium for transmitting an electromagnetic wave as expected in Maxwell’s equations. With the collaboration of Morley, the updated and much larger interferometer was tested in 1886, and null results were published later in 1889 while using Fizeau’s original theory for comparison [5, 6].

Kennedy and Thorndike [7] modified the Michelson-Morley interferometer by making one arm shorter than the other, so that destructive interference was produced. The Kennedy- Thorndike setup photographed interference rings, which were virtually unchanging. The outcome was a static interference, which did not change in any orientation of the apparatus over months of monitoring.

Retests of both interferometer experiments using more precise equipment, laser light sources, and atomic clocks produced the same null results. Jaseva et al. published in 1964 that the frequency shift for the Michelson-Morley test is less than 1/1000 of the effect predicted with the assumption that light has a fixed velocity with respect to the assumed medium [8]. Hils, Deiter and Hall retested the Kennedy-Thorndike experiment in 1990 with a 300-fold improvement and found no variation of destructive interference above 2x10-13 [9]. Another recent retest of the Michelson-Morley interferometer is the Laser Interferometer Gravitational-Wave Observatory (LIGO).

- ORIGINAL THEORY APPLIED IN THE EQUAL-ARM INTERFEROMETER

In the laboratory frame, the Michelson-Morley setup [6] has the basic form in Figure 1, where the half-silvered (i.e., partially coated) mirror at A splits the laser beam into two beams, (partial transmission and partial reflection), each reflected off Mirrors B and C that are equally distant from A.

Figure 1: Michelson-Morley apparatus in inertial lab frame.

The reflected beams recombine at A, and output is monitored at O. With equal lengths L=AB=AC, the combined beams produce constructive interference, which indicates the split beams arrive at A at the same time. Except for multiple wavelengths, if the lengths AB or AC changed (as done in the Kennedy-Thorndike version), making arrival times different, destructive interference would have been expected. This was the classic test for the theoretical ether as the medium to propagate light waves. Figure 2 depicts the apparatus moving relative to an external observer, which simulates the apparatus moving at a constant velocity parallel to one arm relative to an absolute, stationary frame.

Figure 2: Moving Michelson-Morley experiment relative to inertial observer.

Here, A′ is the point where the beam is split into two beams by Mirror A. One beam travels from A′ to Mirror B at B′, reflects back to Mirror A, and recombines at point A″. A second beam moves from A′ to Mirror C at C′, is reflected to Mirror A, and arrives at point A″. The entire apparatus travels at a uniform velocity v along the X-axis during the test, which is a valid assumption when c » v. For clarity in Figure 2, the output through O is not shown, and the arrows A′C′ and C′A″ are displaced laterally apart. Let Observer 1 be fixed in an inertial frame while the apparatus is moving uniformly to the right at velocity v, and the Laboratory Observer is fixed relative to the apparatus, so the Laboratory Observer has a velocity v compared to Observer 1. The Laboratory Observer would witness the apparatus to operate as in Figure 1, while Observer 1 perceives the apparatus as in Figure 2.

If the apparatus is at rest relative to Observer 1, then 2L = c 2Δt for c the speed of light and Δt for the time interval for light to traverse the distance L. If the apparatus is modified to count intervals of 2Δt, a proper time scale for Observer 1 is defined. For light to travel from A′ to B′ to A″, 2d = c 2Δτ, where Δτ > Δt, since d > L. Letting v Δτ be the distance the moving apparatus is displaced to the right until the beam from A′ touches B′, one gets the following geometric relation between Δt and Δτ.

(1)

If one defines a master time as counting the time span between round trips of light bouncing between Mirror A and Mirror B for either observer, 2Δt is the time unit for Observer 1’s timescale, and 2Δτ is the Observer 1’s perceived time unit for the Laboratory Observer’s proper timescale. Equation (1) is the time dilation equation [6, p. 1200-1203] from special relativity that Observer 1 perceives between Δt and Δτ.

The time interval for light to traverse from A′ to C′ to A″ is assumed longer than Δt in Observer 1’s frame. Let D be the distance the apparatus moves until light from A′ touches Mirror C at C′. Light along the X-axis will have traveled L+D to Mirror C. Then, the reflected light will travel L–D back to A″ for a total distance of 2L for the two traverses in the laboratory frame , and

. (2)

Replace D with v ΔT in the external frame.

(3) (4)

(5)

Equation (5) implies that the traversing beam parallel to the X-axis takes 1/ longer than the other beam arriving at A″ by (1). The relativistic compensation by length contraction replaces L in (5) with , so that the resulting ΔT now equals Δτ in (1) when 2L/c = ΔT. Length contraction is the standard explanation that the parallel split beam must travel a shorter distance so that the merged light beams are synchronous.

Virtually all velocities of the interferometer in the XY plane will not be parallel solely to either the X- or Y-axis, but it is the vector sum of Vx and Vy. Start with L = Lx = Ly initially in the stationary frame. Define 1/ = γ. The component velocity Vx will cause Lx = = L/γ < L via length contraction in the laboratory frame. Apply the component velocity Vy to the interferometer. Length contraction should shorten the Y-axis arm to cause LY = = L/γ < L, but the perpendicular arm is not at the required length L to cause constructive interference, but the shorter arm of LX. To preserve the null result in the ratio of ∆Tx/∆Ty, the Y-axis arm must be shorter by a multiplicative factor of . In the original explanation for the null result as given above, the VX component of velocity will cause length contraction only in the X-axis and VY only in the Y-axis. Generally, length contraction affects both arms when V is not relegated to being parallel to either the X- or Y-axis. Carry the argument further and examine the X-axis arm again. The shorter arm on the Y-axis as the updated perpendicular arm now forces the X-axis arm to be further shorten by the multiplicative factor of . Continuing this argument of length contraction shows that both arms approach zero lengths, which is the trivial case to produce null results. Effectively, length contraction is applicable in the general velocity case only if the interferometer has no arm lengths and is a point. Length contraction leads to contradictory results and should be eliminated to explain the null results for the general case.

- CONTRADICTION OF LENGTH CONTRACTION

Einstein [10, §2] derived the apparent round-trip transport over the length L using (3) and (4), but he did not replace his c – v or c + v with his addition of relativistic velocities that he derived in §3 with c = c – v = c + v. Instead, Einstein followed Fizeau’s derivation, since both aberration of light and Fizeau’s water experiment were most meaningful to him [11]. Einstein never published any analysis of the Michelson-Morley experiment, even in his book, which he deferred to Lorentz, who “showed that the result obtained at least does not contradict the theory of an aether at rest”. [12].

Einstein considered two inertial reference frames K and K′ with a relative velocity v between them. His second postulate would imply that observers fixed in either frame would see a spherical shell of light expanding from a flash point with a constant speed of c. The Lorentz transformation and its inverse will convert time and space coordinates between the frames with the X coordinate parallel to v by using length contraction and time dilation, which is a straightforward proof [13, p.516; 10, §3]. The perpendicular components to the velocity are unchanged while the parallel coordinate and time are transformed. Although the Lorentz trans- formation includes length contraction, it has never been observed directly since it was proposed over a century ago, despite the high precision now achievable with measurement equipment.

To demonstrate the contradiction with length contraction, the Michelson-Morley apparatus is modified so that 5 inertial observers can witness the results concurrently, as shown in Figure 3 below. To simplify the numerical gamma factors, let Observer 1 and 3 be traveling at 0.6 c and Observers 2 and 4 be moving at 0.8 c. The velocity of Observers 1 and 2 is parallel to the X body axis of the apparatus, and Observers 3 and 4 are moving parallel to the Y body axis. The laser outputs a continuous, monochromatic light.

Figure 3: Five inertial observers concurrently viewing output.

Partially silvered mirrors are located at the exit output for the Laboratory Observer and the other Observers 1 through 4. This allows some light to go through those mirrors and reflect other light to be shared with all observers. The merged light beam from A passes through the first partially silvered mirror to the Laboratory Observer, who sees a coherent beam of maximum intensity due to the constructive interference. The core postulate of physical observation requires that all observers witness the maximum energy of the coherent beam, since the same output is shared. Granted, the moving observers will see a shift in frequency, but the received beam is at maximum intensity for all observers. For calibration of the light sensors, a simple change in AB over one wavelength of the continuous monochromatic laser beam would verify the intensity changed from maximum (full constructive interference) to nothing (full destructive interference) and back to full intensity for all observers when AB is reset.

However, the moving observers have contradictory explanations. Comparing ΔT/2 from Equation (5) to (1) requires that length contraction occurred inside the apparatus for the moving observers to witness the coherent output. Any length contraction applied to the merged beam past the Laboratory Observer would only change the coherent wavelength, but not the intensity. For example, the apparatus must be contracted along the x axis to 0.8L for Observer 1 and concurrently to 0.6L for Observer 2 to observe maximum intensity. Length contraction also requires that the apparatus must be contracted along the y axis to 0.8L for Observer 3 and concurrently to 0.6L for Observer 4. “There is no length contraction perpendicular to the direction of relative motion.” [14, p. 872]. The calculations made by Observers 1 and 2 would also prove that no length contraction could be even observed in the perpendicular y axis, which Observers 3 and 4 are moving parallel to the y-axis, while Observers 3 and 4 would make the same claim against Observers 1 and 2 moving parallel to the x-axis. All of this is contradictory, because there is complete disagreement in magnitude and in direction where length contraction could be applied so that all observers recorded a maximum energy of the output laser beam from the experiment.

Length contraction is unnecessary. When light is measured perpendicular to the velocity between inertial frames, the universal speed of light in a nongravitated, vacuum environment is equivalent mathematically to time dilation using this perpendicular velocity. Assume a moving observer transmits a laser pulse perpendicular to the velocity between the moving frame and a stationary frame over the length L marked as AB in Figure 4 with L = c Δt. In the stationary frame, the laser pulse traverses the hypotenuse A′B where D = c′ Δτ. The perpendicular length AB moves constantly to the right in Figure 4 to create the triangle AA′B. The laser pulse originates at point A′ and the perpendicular length moves sideways to the right at a constant velocity V. Light reaches the end at B. Light would travel the length AB in the time interval of Δt, but light travels the longer distance of the hypotenuse in c′ Δτ. Light reaches B when the base A′ is directly below B at point A after the time interval Δτ.

Figure 4: Constancy of light and time dilation.

The Pythagorean theorem [15] leads to the equation:

(6)

Immediately, c′ = c ⇔ Δτ = Δt/ =γ Δt. Under special relativity conditions (i.e., no gravity) with inertial frames, the speed of light in a vacuum is a universal constant in all directions if and only if the time dilation equation using this perpendicular velocity is correct. Length contraction is not involved when light is measured perpendicular to the velocity between reference frames.

A real problem surfaces when considering two reference frames with mutually parallel x, y, and z axes, but a general constant velocity V that moves one frame’s origin relative to the other. For the general case, velocity V is at angles (θ, φ, ψ) from the respective (x, y, z) axes so that Vcos(θ) = vx, Vcos(φ) = vy and Vcos(ψ) = vz, but insisting each component velocity differs from the others. Light is measured in the first frame along each axis of length L where cx = cy = cz and the corresponding proper time units, Δτ = Δτx = Δτy = Δτx. That numerical value is transformed with the proper time unit Δτ from the first frame into the corresponding time units of the second frame. The above proof states Δτ = γxΔt = γyΔt = γzΔt. This states that time dilation γxΔt is applied to the coordinate time units along the y and z axes in the second frame, γxΔt is applied to the coordinate time units along the x and z axes, and γzΔt is applied to the coordinate time units along the x and y axes of the second frame. Time is a scalar unit, but this example shows that time dilation is now directionally oriented based on the different velocity components. The contradiction is that two different coordinate time units for each of the x, y and z axes are designated by the Lorentz time dilation equation. Special relativity is inconsistent, because light speed is transformed from the first frame into different values in the second frame with conflicting time dilation for coordinate time based on different velocity components.

It was proven by Terrell [16] and Penrose [17] that a sphere moving at high velocity would remain visually a sphere [18] that would rotate by an angle θ = sin-1 (v/c) to the observer [19]. Even a cube centered on the origin that presented only one face when stationary to an observer on the +X-axis would appear to rotate by the same angle θ if the cube moved at a velocity v along the Z-axis [19]. This contradicts Einstein’s calculation that a moving sphere would appear to be an ellipsoid of revolution [10, §4], because he only analyzed length contraction along the X axis and ignored the Y and Z axes. All diameters of a moving sphere remain the same lengths in three dimensions to the inertial observer. To emphasize this, consider an observer on the positive X axis and a rod that is parallel to the X axis while its center can move on the Y axis. The stationary rod appears to be a point at the origin. When the rod moves at a uniform velocity along the Y axis and crosses the origin, the observer sees the rod of nonzero length along the Y axis. The rod rotates by an angle θ = sin-1 (v/c) as predicted [16-19]. Such a moving rod exhibits length preservation, not length contraction. The issue is a rigid rod never changes length, regardless how fast it is moving relative to the observer. The moving rod rotates relative to the observer due to the light’s finite speed emanated from the end points, but the three-dimensional length is preserved according to the Pythagorean theorem [15] when recording instantly the projection in three dimensions.

Can light speed be a universal constant as assumed by special relativity? Let Observer 1, who is fixed in inertial Frame 1, measure the light speed c. The fixed length span and fixed time interval are divided up into length units of a meter (ΔL1) and one time unit of a second (Δτ1) to obtain the real number c. Observer 2 has the identical length unit and time unit when at rest in Frame 1. Accelerate Observer 2 until a fixed relative velocity V is achieved by moving parallel to the length AB that Observer 1 uses to measure c. Concurrently when Observer 1 conducts the light speed test, Observer 2 measures the moving length AB shortened by length contraction and the time duration that light traveled along AB in Frame 2. Since AB is parallel to the velocity, length contraction should occur. Observer 2 measures the speed of light concurrently with Observer 1 during the test and compares it numerically to Observer 1’s value. Defining γ = 1/, time dilation requires between time units of Δτ1 and Δτ2, respectively, in frames 1 and 2. Length contraction requires . The transformations show numerically c1 ≠ c2, because the units changed sizes between observers.

(c ) (7)

Numerically, c1 ≠ c2 as γ > 1. Einstein proved that a spherical light wave propagated from a point is still a sphere either in a stationary frame fixed at the origin of the expanding sphere or in a moving inertial frame. He also proved the Lorentz transformation for length and time preserves the sphere [10,§3]. Special relativity uses the variable c, which incorporates both a numerical number and units of length and time. If length and time units change in a separate reference frame compared to a standard frame due to a mutual velocity, then the number for light speed must change. For example, light speed is 299792458 m/s that can be converted to yards/minute where a yard < meter (illustrating length contraction) and a minute > second (demonstrating time dilation). Our technology is so precise now that such numerical differences should have been observed, especially with two LIGO observatories (i.e., ultraprecise Michelson-Morley interferometers) at different latitudes. So, how does a laboratory with a fixed light source and fixed light detector measure the same precise value for light speed? How do the retests of the interferometers always produce the same null results regardless how the interferometer is oriented or rotated? A careful reexamination is warranted.

- EVALUATION OF INTERFEROMETER TESTS

Runners know that a footrace covers more (or less) ground if the finish line is moved away (or toward) the runners during the race. The same is expected in Zeno’s paradox of Achilles and the tortoise. Zeno’s paradox relates how the tortoise challenged Achilles to a race with a head start of L, claiming the tortoise would win. Achilles conceded the race without running due to the tortoise’s logic that Achilles was always behind the tortoise for an infinite number of time intervals, even though those intervals were approaching zero. Embellish this paradox with two tortoises starting together. The original tortoise is retreating from Achilles at speed v, and the second tortoise approaches Achilles at the same speed v, but Achilles will overtake both with his speed of V > v. These two cases will be solved analytically and applied later to the parallel beam in the equal-arm interferometer.

The tortoises begin at a distance L ahead of Achilles, who runs toward the tortoises at his uniform velocity V. Consider the retreating tortoise first, which Achilles will overtake after running a distance L→ > L. When Achilles runs a distance L over the time L/V, the first tortoise is out of reach by being ξ(1) distance further. When Achilles runs the extra distance ξ(1), the first tortoise has moved a further distance ξ (2) over the same time interval of ξ(1)/V. Over n repetitions of this, the tortoise has moved a distance of L + ξ(1) + ξ(2) + … + ξ(n) where ξ(1) = v × (L/V) and ξ(i+1) = v × ξ(i)/V. Substitute the individual terms with v < V, and the series is:

(8)

Chasing the approaching tortoise, Achilles runs a shorter distance L← < L as the second tortoise moves toward Achilles. Note that the velocity of this tortoise is now in the opposite direction of Achilles’s velocity, so the ratio of velocities is now negative as -v/V for each term in this series.

(9)

Consider a different race. Place the second tortoise beside Achilles at his starting line and place the first tortoise ahead of both at the distance L. A rod of length L could be tied to both tortoises to ensure both crawl at the same slower velocity v and maintain their mutual distance L. Achilles runs after the first tortoise, tags it, turns back, and finally tags the second tortoise. The total distance that Achilles runs in this second race is:

(10)

Algebra can directly verify (8) and (9) by solving for D, the distance the receding tortoise moved before Achilles tagged it. In the equation, solve for D in D/v = (L+D)/V with replacement of D in . Also, solve for d, the distance the approaching tortoise moved before Achilles touched it in d/v = (L-d)/V and replace d in = L - d.

Transition from the paradox to the Michelson- Morley interferometer where local gravity is perpendicular to the plane containing the two perpendicular arms in the x axis and y axis. At any instant of time, there will be changing velocities of vx and vy for each arm moving relative to outer space due to the rotation and orbit of the Earth plus the Milky Way’s motions. Michelson and Morley knew the Earth orbited the Sun at approximately 30,000 m/s, but were oblivious that the solar system was in the Milky Way galaxy with additional rotation and movement within the cosmos. They hoped to rotate their platform on a pool of mercury to find the maximum interference when one axis was virtually zero in the projection of Earth’s orbital velocity relative to the Sun compared to the maximum projected orbital velocity of the other axis from splitting and merging light beams. Their null result was that maximum intensity was always output from constructive interference regardless of the orientation of the x axis or y axis of the apparatus. The high speed of light would ensure the laboratory frame was effectively inertial, because the apparatus would have no practical change during the few ns to output the merged light.

To illustrate, consider placing the laboratory on Ascension Island or anyplace near the equator. Twice a year, the solar ecliptic will pass through the local plumb line, which is affected by all gravitational sources from the Earth, Moon, Sun, etc. Let the vector S to the Sun be directly overhead so the plumb line and body axis z are parallel to S, and set up the y axis arm of the interferometer to be parallel with SxV, where V is the orbital velocity of Earth in the solar barycenter frame. The x axis arm is parallel to X = YxZ. Vector V will be nearly perpendicular to S, so vx will be nearly 30000 m/s and vy roughly zero. Six hours later, Earth rotation will cause vx to be nearly zero m/s. The largest that factor γ can be is 1 + 5.0069E-9.

Equation (10) gives the distance that light with velocity V=c traverses a round trip on the x axis arm experiencing a velocity of vx. That velocity vx displaces the y axis arm sideways as shown by the isosceles triangle in Figure 2. Divide the isosceles triangle into 2 equal right triangles (e.g., see Figure 4 for one triangle).

The author believes the horizontal axis of Figure 4 is vx ∆t, where ∆t is the time needed for light to traverse the length Ly only, as vector displacements require. In that case, the hypotenuse is Ly′=Ly(1+vx2/c2)0.5, and the total distance traveled by the y axis beam in the stationary frame is 2Ly(1+vx2/c2)0.5 as the vx velocity moves the interferometer along the x axis. The y axis beam is forced to traverse a longer distance of Ly′. The alternate explanation for the sideway displacement is vx ∆τ, where ∆τ is the time needed for light to traverse the hypotenuse only, resulting in the hypotenuse to be Ly′=γxLy for light to traverse the y axis with a given vx in the external inertial frame.

In the general case, vy also exists, so the round trip displacement by the y axis beam along the y axis is (9) with vy inserted. This also means the x axis beam is displaced sideways due to vy, forcing the x axis beam to travel along twice the distance of the hypotenuse. That longer distance is either 2γyLx or 2Lx(1+vy2/c2)0.5, which both interpretations will be applied to the two interferometers.

For the Kennedy-Thorndike interferometer, the time intervals between splitting and recombining the light beams will be simply the total displacement divided by the light speed c, which is the author’s interpretation.

(11)

Earth’s rotation would continually change the speed of the interferometer’s body axes relative to the stationary frame (e.g., solar barycentric frame with Earth orbiting the Sun). In the example of placing the laboratory on the equator where Earth rotates at a tangential velocity of 465 m/s, the apparatus’s x axis would vary between 30,000 m/s and zero in speed and the y axis would vary between zero and 30,000 sin(23.5°) m/s every six hours. The factor γx would vary between 1 and (1+ 5.0069E-9), and the γy would vary in the opposite change of γx with a smaller maximum. The experimental retests done in 1990 showed no change of destructive interference above 2E-13 [9]. Neither length contraction nor time dilation can explain the output in (11). In terms of the Michelson-Morley interferometer, Lx and Ly would cancel in (11). This ratio must be exactly one as observed, which special relativity still fails to explain.

Using the alternate interpretation for the sideway displacement, the formula of the time ratio with γ factors for the Kennedy-Thorndike inter- ferometer would be:

(12)

Any time dilation inside the moving interferometer should be common to both axes. However, γx ≠ γy as both factors are continually changing during extended periods of testing as the Earth rotates the interferometer relative to Earth’s orbital velocity. The experimental tests showed no change of destructive interference above 2E-13 [9]. In the Michelson-Morley interferometer, the arm lengths cancel, and the ratio in (12) must equal one as observed, forcing γx = γy, but vx ≠ vy. This contradiction results from assuming light speed is a universal constant. Can light speed vary and still produce the identical precise constant measured in laboratories on Earth?

- REANALYSIS OF MICHELSON-MORLEY TEST

Without loss of generality of the interferometer experiment, assume the external observer is absolutely stationary. (Another constant velocity could be added to both v and V for the general case and not change the results.) Since there is no net dynamical acceleration or different gravity between frames, special relativity mandates the time units do not change sizes during the test. The laboratory timescale has a time unit that converts precisely to the external observer’s time unit by a multiplicative factor. Arbitrary times could be chosen for a common epoch, such as the onset of splitting the light beam at the half-silvered mirror A. The laboratory timescale can be replaced with another timescale defined by a new epoch (t0 = 0 when the beam is split at A) and original time unit divided by the γ scale factor so that the new timescale is identical to the external observer’s timescale. So, even if time dilation makes the original time unit different, a conversion is available to create a new laboratory timescale with identical time units of the external observer so that one timescale is needed for both reference frames. It is assumed that both observers have synchronized clocks at the needed locations within the domains of the two reference frames to record coordinate time using this common timescale.

The basic tenant of physics is that observers must witness identical results of the same event when results are transformed to one common inertial frame. The author defines simultaneous to apply only when multiple phenomena or events separate or combine at a point at the same time. When different events occur at separate locations at the same coordinate time, those events are synchronous without being simultaneous. For example, a formation of swimmers performing a synchronous routine are independent of each other as any swimmer can stop moving independently of the other swimmers without interfering. Two novae are synchronous when they explode at the same coordinate time, but if the transmission times to convey the light to an observer were different, the reception at the observer would be nonsimultaneous. Usually, two novae are not synchronous by exploding at different coordinate times, but simultaneous novae might be recorded serendipity if the two light beams from the stars arrived at the same instant at the observation point.

When light is split at the half-silvered Mirror A, both observers record the same time instant and the respective X coordinates in each frame when a beam moving parallel to the X-axis was created. When the parallel beam touches Mirror C, both observers record the same time instant and the corresponding X coordinate in each frame. (The constant speed of light, V, is slightly slower due to the atmosphere by the index of refraction, but it is nearly c, the light speed in a vacuum.) In this example, the stationary observer records the parallel beam traveled longer than L to overtake the receding Mirror C. Still, the moving laboratory observer makes no compensation for the laboratory’s movement through outer space and surmises the parallel beam only traveled a fixed distance L to touch Mirror C in the laboratory frame. However, both observers must record the same identical time interval between the creation of the parallel light beam and the later instant of time when the parallel beam touched Mirror C, since one common time scale was created for both frames.

Referring to Figure 1, the distances between mirrors A and C and between mirrors A and B are both L. When the mirrors A and C move, a light pulse is flashed toward C from A and reflected to A″ as shown in Figure 2. The distances the parallel beam covers between the mirrors in the stationary frame are:

, and (13)

. (14)

The same length L is retained in the moving laboratory frame. In the stationary frame, light traverses a longer, , or shorter, , than L due to the movement of the apparatus during the test. In (13) and (14), the speed V from (8) and (9) is replaced by c for the speed of light in a vacuum.

A footrace will cover different ground if the finish line moves toward or from the runners during the race. Achilles runs further than L to overtake the receding tortoise by (8) or shorter than L toward the approaching tortoise by (9) with a total footrace > 2L. The external observer witnessing the moving interferometer should encounter the same effects in the parallel beam’s displacements.

The laboratory observer makes no compensation for the moving endpoints of either arm in the interferometer moving through outer space. In the stationary frame, the round trip parallel distance is longer than 2L by the addition of (13) and (14). The speed of light is assumed a constant c as based on past measurements of light speed. If the stationary observer measured the one-way transmission interval precisely, the predicted time intervals would be (15) and (16).

, and (15)

. (16)

The time span begins when light beams separate at Mirror A, which is simultaneous between both observers, because the split occurred at one point at one time instant. The two beams later combine at Mirror A simultaneously for both observers at one time instant at a point location. The time spans from splitting a light beam to later combining two light beams must be identical for both observers using the common timescale described earlier.

When light touches Mirror C or Mirror A at A″ in either frame, either event is simultaneous for both observers. Thus, the time span when light overtakes Mirror C moving one-way parallel to Vx is (15) in the laboratory if light’s speed is c. When light moves antiparallel to Vx, the one-way timespan is (16) in the laboratory. The laboratory observer calculates T = L/c. The variation in time spans to traverse the parallel arm must be absorbed in the apparent speed of light, which incorporates time since length has no time units. As the interferometer is stationary in the laboratory frame, the apparent one-way velocities in the laboratory frame are surprisingly:

, and (17)

. (18)

The beam overtaking the perpendicular arm in Figure 2 of the external stationary frame travels a round trip distance of 2d, along the hypotenuses AB′ and B′A″, which either is longer than L, a perpendicular leg of the triangles. If c is the emitted speed of light in the stationary frame in all directions, then the transmission intervals to B′ and A″ should be longer than L/c, which would predict destructive interference at the combination point, but both observers must record constructive interference by previous tests. Since light obeys vector addition of parallel velocity by (17) and (18), then it should also obey vector addition of velocity in general. In the stationary frame, the light would have to travel faster to B′ and then to A″ under vector velocity addition where . From Figure 2, the distance d along the hypotenuse is d = . In the stationary frame, the perpendicular beam traverses the longer distance 2d of AB′A″ with a faster light velocity c′ to complete the time interval:

(19)

Again, the laboratory observer makes no compensation for the moving arm perpendicular to the velocity Vx of the laboratory through outer space. Still, the time interval for light to traverse the perpendicular arm in the stationary frame is the same as 2L/c, which the laboratory observer deemed.

A faster speed of light would elongate the wavelength along the hypotenuse, but the frequency would remain unchanged so that c′ = fλ′. If one tuned continuous monochromatic light to create n integer waves in each arm in the laboratory frame, then the wavelengths in the stationary frame along the hypotenuse would also have n longer wavelengths of λ′ due to the faster light speed, but a projection onto the perpendicular leg would obtain the same wavelengths of λ recorded by the laboratory observer. Both observers record the same n wavelengths on the perpendicular arm, but this can only happen if the speed of light obeys vector addition of velocity induced by the interferometer’s velocity at emission. A standing wave of n wavelengths on the perpendicular leg must be recorded by both observers in this setup, which will invalidate the γ interpretation for the sideway displacement by (12). This is also true for the parallel arm. For the external observer, the parallel leg is longer by c/(c-v) by (13) and is shorter when antiparallel by c/(c+v) in (14). The external observer counts n wavelengths on the parallel leg only when light’s velocity obeys vector velocity addition as c ± v, respectively.

In the stationary frame, the parallel beam travels further than L to overtake the receding Mirror C by (13) and shorter than L by (14) to intercept the approaching Mirror A. The stationary observer allows the endpoints of the parallel arm to move. The laboratory observer considers the parallel light traveled only L in the parallel direction and L in the antiparallel direction, but the stationary observer notes that the respective distances are given by (13) and (14). The round trip distance by adding (13) and (14) > 2L, but the speed of light within the parallel arm is not c in the stationary frame, which would force the parallel beam to travel longer for the round trip than the perpendicular beam and force the recombined beams to exhibit destructive interference. Since the interferometer effectively elongated the parallel distance by the factor c/(c-v) and shortened the antiparallel distance by c/(c+v), the interferometer alters the speed of light by the same ratios due to velocity containing length units in its numerator. The laboratory observer failed to incorporate the ratio c/(c-v) for the parallel distance and c/(c+v) for the antiparallel distance. Therefore, the speed of light in the stationary frame is affected by the same ratios. The parallel beam’s velocity in the stationary frame is faster by c2/(c-v). After reflection, the parallel beam’s velocity is slower by the ratio of c2/(c+v). The actual time spans in each direction for the stationary observer are:

, and (20)

. (21)

Thus, the recorded parallel and antiparallel times in the stationary frame are precisely L/c in each direction, which makes the total round trip time equal to 2L/c as surmised by the laboratory observer in the laboratory frame.

If a hypothetical medium exists for light transmission with a velocity Vx and Vy relative to the stationary frame while the interferometer moves with vx and vy, then the results are unchanged using the total Ꝟx = Vx+vx and Ꝟy = Vy+vy. Both the stationary observer and laboratory observer will record constructive interference for the output of the Michelson-Morley interferometer, regardless of the speed of a hypothetical medium relative to either observer. This demonstrates that neither interferometer test can prove nor disprove the existence of any hypothetical medium.

This also explains why precise measurements of light speed appear to be a constant no matter what the velocity may be of the moving inertial frame. The movement of the test apparatus relative to outer space is not compensated during laboratory measurements of light speed, so that both the length traversed and the velocity of light are affected equally to maintain the apparent ratio of L/c as a constant time span in any external inertial frame. This means that the measured speed of light is precisely the standard value c if both the light source and light detector are mutually fixed with respect to each other. When the distance between the light source and detector is fixed at a distance L during a measurement, then (20), (21), or half of (19) will produce the standard light speed c = L/(L/c).

Some may argue that prior tests measured the same standard speed of light emitted by a moving particle. For example, γ rays from the decay of π0 mesons with more than 6 GeV were measured absolutely by timing over a known distance [20]. The test was intended to measure c + kv, and the result was k = (-3±13)x10-5 for mesons moving near light speed (γ>45). The team used two detectors spaced 31.450 m to measure the time interval the γ rays traveled. Photons are absorbed in a material and subsequently emitted by the atoms of that material. The first detector absorbed the light from high-speed γ rays and emitted light rays afterward at the usual speed of light. The measured speed recorded by the second detector after light passed through the first detector (i.e., absorbed and reemitted at c) was the standard light speed. This and many other tests must be reexamined carefully to ensure that the photon speeds were directly measured without interception to eliminate any mismeasurements causing misinterpretations of the results.

The constructive interference output of the Michelson-Morley interferometer demonstrates that photons obey velocity vector addition as analyzed above. No observer reported that one beam moved off the reflection mirror or the recombination point during the test. This is because the photons move in the required direction mandated by vector displacement. A simple test would demonstrate this. Set a laser to point horizontally at a partially silvered mirror that is angled at 45 degrees relative to the local plumb line. The reflected beam is aimed vertically to a hemisphere mirror that is a distance, d, of about 10 meters above the partially silvered mirror, so that the hemisphere is centered along the line of transmission. The vertical beam is reflected from the hemisphere of 2 cm radius, r, to the partially silvered mirror below, and some light is transmitted through it to the ground below. Observe if the impact point varies over time as Earth rotates. The hemisphere mirror has a sideway displacement due to the velocity v of Earth around the Sun, forcing the beam to miss the nadir of the hemisphere mirror by an angle of θ ≈ sin(θ) = v ∆t / r = vd/(cr) in radians. The reflected beam from the hemisphere mirror should touch the ground by a maximum of d θ = 50 cm from the plumb line assuming a maximum v ≈ 30000 m/s from Earth’s orbit. Even assuming a perpendicular projection of Earth’s velocity to the 10 m arm that is a slower speed of 15000 m/s, this would still cause a 25 cm displacement from the plumb line. If no observed displacement occurs over a day, then the beam is constantly touching the nadir of the hemisphere mirror. This would prove light obeys vector addition of velocity, not only in magnitude, but also in horizontal displacement (i.e., v⊥∆t) identically as the hemisphere mirror undergoes vector addition of displacement relative to an inertial frame in outer space (i.e., L′ = L(1+v⊥2/c2)0.5 and c′ = c(1+v⊥2/c2)0.5).

A more rigorous demonstration reveals light does obey vector velocity addition by the operation of a precise version of the Michelson-Morley interferometer over the last three years. The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment with twin observatories to detect cosmic gravitational waves [21]. The observatories are near Hanford, Washington, and Livingston, Louisiana, with 4 km long arms within ultrahigh vacuum chambers allowing laser beams to detect gravity waves. A continuous laser beam is amplified from 40 watts to 750 watts with power reflecting mirrors. The signals are also magnified with signal recycling mirrors. LIGO has enhanced vibration absorption mechanisms to remove ground vibrations, tremors, solid Earth tides, etc. to isolate the signals. To increase the arm lengths from 4 km, Fabry-Perot cavities are installed near the beam splitting mirror and near the hanging reflection mirror at the end of each arm so that 300 reflections inside the cavities increase the effective arm length to almost 1200 km. An ultrahigh vacuum is maintained so that any gaseous molecule is removed promptly to avoid interference with the light beams. Also, one of the split signals is inverted so that destructive interference is created when recombining the two beams. The original Michelson-Morley interferometer gives constructive interference, but this enhancement allows far easier detection of variations after merging signals. Any light is readily seen against an absolute black background, but hard to detect any light variation against a bright light, much like the inability to see sunspots when looking directly at the Sun. LIGO’s signal output is complete darkness (i.e., destructive interference) to indicate the recombined beams arrived at the same time. Any deviations from destructive interference (e.g., any light) are the subtle signals that gravity waves should produce according to general relativity. “At its most sensitive state, LIGO will be able to detect a change in distance between its mirrors 1/10,000th the width of a proton.” [21]

The standard argument that the Michelson-Morley interferometer produced null results is that special relativity causes the arm parallel to the velocity of the moving interferometer to be shorter by the γ factor than the perpendicular arm (i.e., ratio of (5) and (1)). LIGO reveals special relativity fails to explain its null results (excluding the rare anomalies which LIGO was built to detect). As already mentioned, LIGO signals are not compensated for any speed due to the translational velocities of the Local Group that includes the Milky Way relative to the cosmos (627 km/s [22]), the Milky Way’s own translational motion (estimated at 552 km/s [22]), or the solar system’s tangential speed from our galaxy’s rotation (220 km/s [23]), or Earth’s orbital velocity (29.3 to 30.3 km/s between perigee and apogee) that would cause a sideway displacement of either arm that light traverses. LIGO has the extreme sensitivity to detect the different velocity within each arm due to Earth’s rotation.

Pick any suitable geocentric inertial frame for Earth (e.g., J2000 after transforming for rotation, nutation, precession and pole wander since January 2000 to the present or the GPS Earth-Centered, Earth-Fixed (ECEF) frame set to an arbitrary time tag near when LIGO is operating). Let the ECEF frame rotate to map the rotation of LIGO in longitude and geodetic latitude coordinates or equivalent xyz coordinates relative to the fixed geocentric frame during the test. Assume the LIGO arms are oriented north-south (NS) and east-west (EW). (If not, project both arms into NS and EW for each leg within each arm and sum the projected legs in NS and EW distances for a similar analysis.) Each arm has lengths that will be called legs that the two split beams traverse. The first leg is between the beam splitter and the entry point of the nearbyt Fabry-Perot cavity. The beam is reflected 150 times to cover almost 4 km each way and exits from the opposite cavity to the hanging mirror near the end of the arm. On the return, the reflected beam retraces the leg, reenters the cavity to reflect 150 times and exits the cavity to the splitting mirror. In the round trip, the beam traverses the two short legs twice and reflects a total of 300 times inside the cavity. At the atomic level, photons are absorbed by molecules and emitted so that virtually all emitted photons are aimed in the direction given by Snell’s law to return or transmit light through reflective or transparent materials. The photons start each leg at the initial mean location within the reflective material and reach the other end of the leg by impacting molecules at the opposite end without interference from free molecules floating in the pathway due to the ultrahigh vacuum kept in each arm.

In the EW arm, the geodetic latitude is the same for each end of every leg, L(i), so that Earth’s rotation imparts the same tangential speed of m/s relative to the geocentric frame. The beam bounces through 150 legs heading east and 150 legs heading west inside the Fabry-Perot cavity, and the beam traverses each short leg twice. Using (10) for each round trip within each leg, the total EW is:

(22)

In the NS arm, the geodetic latitude on the south end is less than the latitude on the north end of each leg. Each emission of a photon traveling from south to north begins from the southernmost latitude, and vice versa. Earth’s rotation will cause a sideway displacement of each leg, so that the photons move at c along the hypotenuse as shown in Figure 4. Earth’s rotation causes two tangential speeds of m/s faster than m/s for photons emitted respectively on opposite ends of each leg. Use (1) multiplied by c to get each individual leg length, LS(i) and LN(i). The total round trip in the NS arm is summed over all legs:

(23)

Since total LEW = total LNS for each LIGO observatory on Earth’s surface, the output (excluding anomalies) is identical in arrival times for both beams at the recombination point. The ratio of (22) to (23) is not γEW from length contraction as γN < γS = γEW. A constant light speed in all directions predicts nonsynchronous output from LIGO for the geocentric observer, but LIGO observers record synchronous output.

However, if each photon has a light speed that obeys vector velocity addition based on the velocity of the molecules emitting the photons, then one obtains the same predicted synchronous output from LIGO for a laboratory or inertially moving observer. For the photons moving within the NS arm, each one-way transmission over each leg is expressed as L(i) after dividing 2 from all sides of (19). Photons travel over the hypotenuse of distance d, which is longer by L(i) (1+v2/c2)0.5, where vi = speed at the latitude of emitted light. Also, the modified light speed, c′, is faster as c (1+v2/c2)0.5 by vector addition. The result is d/c′ = L(i)/c, which means the observer in the nonrotating geocentric frame will record the same time intervals that photons traverse each perpendicular leg in the moving LIGO observatory as the LIGO observer recorded time spans in the Earth fixed frame. For photons moving within the EW arm, each one-way transmission over each leg L(i) with the modified light speed is given in (20) for parallel transmission and in (21) for antiparallel transmission. In all cases, the geocentric observer will record the same time intervals for each leg as the LIGO observer recording the time spans of L(i)/c.

- RAMIFICATIONS

It will take time for others to verify the results of this research, but if the velocity of photons does obey vector velocity addition, then several concepts of physics must be addressed. Readers will recall that most textbooks state that no particle travels faster than the speed of light in a vacuum. Countless electromagnetic experiments seem to demonstrate this. One of the best videos of a rigorous test using a linear accelerator demonstrated that electromagnetic fields alone do not accelerate free electrons faster than c [24]. As the electromagnetic energy was increased to accelerate the free electrons, the velocity of the electrons approached an asymptote of c. The film verified the timing cables were calibrated. It also showed that the colliding electrons did impart heat to the target that nearly equaled the total energy given to the electrons, but the speed of the electrons was virtually c, but less than c. The test is just as valid today as then. Unfortunately, the conclusion has been overgeneralized to state that nothing can go faster than the speed of light. Photons generated by electromagnetic fields are limited to the standard light speed as emitted from the molecules to impart increases in momentum to the charged electrons. The free electrons can be nudged forward by momentum transfer if slower than the moving photons. Once free electrons obtain the limit of the standard light speed, the photons can no longer overtake the electrons to transfer momentum or nudge the free electrons any faster, although more energy can be given to the electrons. Such tests prove that an electrodynamic force alone does not accelerate charged particles faster than the standard light speed in a vacuum relative to the equipment in the laboratory frame.

This paper demonstrated that the measured speed of light will be the standard speed when light sources and detection equipment are mutually fixed. However, when the source has an additional velocity relative to the detector, then Maxwell’s equations must be modified to allow c to vary outside of the laboratory frame. This means that Einstein’s assumptions for the axioms of special relativity theory are indeed excellent approximations, but not exactly accurate. For example, the last section of Einstein’s 1905 paper contained the dynamics of the slowly accelerated electron [10, §10]. He derived that longitudinal mass would differ from transverse mass. No test has confirmed this even exists—directly or indirectly. No one has shown Einstein’s derivation has any mathematical flaw concerning this subject. This unverified topic should have raised doubts about the accuracy of these two postulates of relativity. Most electrodynamic experiments do not approach the precision in significant figures to test Maxwell’s equations when the sources are moving independent of the detectors.

Some standards of physics will need to be restored. The meter is now defined in terms of the time span it takes light to traverse a meter. With a varying light speed, the physical meter standard should be reestablished as the international unit of length. Some other standards may need reexamination.

General relativity shows that gravity waves and light waves have the same universal speed [25, chapter 5]. “The existence of gravitational waves is an immediate consequence of special relativity and, to some extent, the experimental discovery of gravitational waves would merely confirm the obvious.” [25, p. 242] If photons can move at different speeds, then gravity could also have different speeds other than the universal standard speed. General relativity predicted the solar deflection of light and Mercury’s perihelion precession. Classical physics can predict the same 1.75″ of solar deflection [26] by allowing the Sun to move as well as the photon. Previous classical derivations only obtained half of the total observed effect using a stationary Sun. General relativity predicts the Mercury perihelion has an excess precession of 42.98″/century [27, 28], but Misner, Thorne and Wheeler published their 1973 evaluation that this residual was 41.4″±0.9″/century [28, p. 1112] after updating for the revised aberration, but they offered no explanation for this discrepancy that general relativity falls outside of the error bounds. “The deviation from the theoretical prediction is not considered significant.” [27, p.512] However, the author disagrees, as dynamical astronomers are not that imprecise. A post-Newtonian approximation of an accelerated, rotating frame in a gravitational field yields 5/6 of the general relativity prediction [29]. Both Icarus and Mercury fall within the error bounds of the observations utilizing this post-Newtonian approximation with the appropriate coordinate timescale. This reinforces the author’s judgment that relativity is an excellent approximation, but it is not exact.

A variable light speed could modify two particles in the standard model. The muon and tauon are identical in all characteristics to the electron except their masses are different. If a high-energy particle after a collision can exceed the standard light speed, then E = mc2 may be misinterpreted for the same electron with its usual rest mass as a different particle, because c is assumed a constant. A faster speed of light than the standard light speed c would explain all the entanglement experiments where two or more objects at significantly long distances, d, interact with each other within a measured short time interval, ∆t (i.e., c′Δt = d > c ∆t).

A varying light speed would imply that the Lorentz time dilation is incorrect for many precise timing applications. Hafele and Keating claimed they demonstrated time dilation precisely using four atomic clocks flown in commercial aircraft in westward and eastward circumnavigation of the world [30]. Essen, one of two horologists at the National Physical Laboratory (NPL) who operated the original cesium clock in the calibration effort that defined the atomic second, had reviewed the Hafele-Keating report and concluded the alterations in drift rates of the atomic clocks made the results useless [31].

The author examined the clock rates in Tables 1 and 2 [32] of the Hafele-Keating report released in 1972. The fact that the drift rates of 3 of the clocks varied significantly before and after each circumnavigation in Table 1 completely casts doubt that any average of the ensemble demonstrated relativistic time changes, as 3 of 4 clocks did not individually drift according to relativistic predictions. Table 2 proves the stability of those 3 clocks was not maintained throughout the test. Hafele had released his results at a 1971 conference, in which Hafele even doubted his own test. He admitted, “Most people (myself included) would be reluctant to agree that the time gained by any one of these clocks is indicative of anything….” [33, p.273]. By averaging the time gain with 4 clocks, Hafele did get the eastward circumnavigation with error bounds to agree with the predicted theoretical result, but there was no such fit between theory and the westward time gain [33, p. 282]. Keating later worked with Hafele to write the second report with claims that they verified time dilation [31], but careful examination uncovers many discrepancies.

The author’s numerical findings on the Hafele-Keating drift rates were duplicated earlier by A. G. Kelly [34]. Kelly obtained the original raw data from the US Naval Observatory (USNO) to review against the published 1972 report and found some data in the tables differed very significantly from raw data with no explanation. Kelly determined Clock #120 performed very irregularly. He wrote, “Discounting this one totally unreliable clock, the results would have been within 5ns and 28ns of zero on the Eastward and Westward tests respectively. This is a result that could not be interpreted as proving any difference whatever between the two directions of flight”. Kelly was not condemning relativity, but he was critical that the flight tests were not rigorous, and the claims were unverifiable based on the actual raw data. If light speed is not a universal constant, then clocks may exhibit different time effects than the Lorentz time dilation, which would warrant a more rigorous retesting.

If light obeys vector velocity addition, then a different interpretation of the Sagnac effect is needed. Light is inserted into a ring interferometer and splits in opposite directions. The beams exit the ring at the start/end point and undergo interference. The destructive interference determines how much the ring interferometer rotated after the beams were split. The diagram on the left of Figure 5 [35] shows a nonrotating ring of radius R would output constructive interference as each beam would travel the same length of 2πR and exit at the same time. If the ring interferometer rotated as shown on the right side of Figure 5, one beam would travel further than the other, so both beams would exit with destructive interference.

Figure 5: Sagnac effect in circular loops

It is easy to see as an external observer that one beam takes a longer trip than the oppositely traveling beam when the interferometer rotates. The rotating interferometer is a frame that is sufficiently inertial when comparing the constant angular velocity ω with the speed of light. The ring is a conduit that bends both beams to traverse in a circular path. A perpendicular acceleration is required to force the light beams to traverse a circle in the rotating interferometer. Such an acceleration is no different than the perpendicular gravity that exists for the Michelson-Morley interferometer, as any acceleration to the linear paths will be equal to both beams, so the acceleration effects cancel out when combining the two beams. As the beams are bent in circles, the distances covered by both beams in the external frame are Equations (13) and (14) for L→ and L← when setting L = 2πR and v = ωR. The time difference between the beams is:

(24)

However, the usual application of the Sagnac effect is found in inertial navigation systems where the interference measurement device is fixed at the exit or end of the rotating ring interferometer. If an observer is fixed with the rotating ring, special relativity requires that each

beam originates at the same constant light speed c, and each beam travels the circumference of 2πR in the inertial rotating frame. This set of assumptions of special relativity for the rotation of the ring undergoing constant angular velocity, ω, would predict both beams exit simultaneously (i.e., constructive interference in the output), but reality contradicts that theoretical prediction. Similarly for the inertial laboratory undergoing a constant linear velocity, v, equations (17) and (18) show that the inertial ring interferometer undergoing constant angular velocity will experience different one-way velocities of the light beams to be c-ωR and c+ωR. This explanation will obtain (24) in the rotating ring frame as output. The identical observed interference of output is witnessed in both inertial rotation frames (i.e., the time difference of opposite circular light beams in an interferometer ring is due to different distances traversed in the stationary rotation frame or due to different velocities of the beams with respect to the inertial rotating frame that is fixed with the rotating ring interferometer). Thus, light obeys vector velocity addition in both linear and rotating reference frames.

This is an incomplete list of possible ramifi- cations if light speed is not a universal constant. In any case, it will take time for the scientific community to review these results.

- CONCLUSION

It appeared special relativity was consistent, but some peculiar properties of special relativity reveal unresolved issues. For example, the Lorentz transformation for length and time does not appear to preserve the numerical speed of light. Let light be measured between two points A and B in one inertial frame by an observer fixed in that frame. A second observer calibrates duplicate instruments to the same units. That second observer is accelerated to a desired constant speed, turns around, and travels parallel to AB. The first observer records the speed as the standard numerical value c in m/s, and the second observer measures the light speed concurrently with the first observer to obtain the same quantity. Length contraction and time dilation of the second observer’s units will cause the second observer’s numerical result to be γ2c by (7), where γ = 1/ with different sizes of units. For example, 299792458 m/s can be converted numerically in yards/minute units, which illustrate the length contraction and time dilation of the modified units. In physics, the same standard number for c is used regardless how other inertial frames move. This violates the real meaning of a universal light speed by assuming only one numerical value for all cases. Einstein [10, §3] assumed an algebraic symbol as a constant quantity of light speed, which is a numerical value in units of length and time, but the numerical value must change when the sizes of the units change to maintain the same quantity. Physicists must decide either: (A) the numerical constant for light speed is permanently fixed with no changes in the length and time units of moving inertial frames, (B) the speed of light must change its numerical value whenever the units of measurement change sizes in moving inertial frames, or (C) an alternate model is required.

This paper shows that measuring light perpendicular to the relative motion between inertial frames results in the mathematical identity that the universal speed of light is correct if and only if the Lorentz time dilation equation is correct when using that perpendicular velocity, making length contraction superfluous. This is expected as “There is no length contraction perpendicular to the direction of relative motion.”[14] This leads to multiple time dilations used in the transformation between inertial frames. Consider two 3-dimensional inertial frames with (x, y, z) axes mutually parallel, but the origins are moving apart at a relative velocity V with components vx ≠ vy ≠ vz. The observer in Frame 1 measures light speed over three separate lengths of L set parallel to each axis and transforms that light speed of cx = cy = cz to Frame 2. Because light speed was measured in three different directions, the time dilation of the proper time unit ∆τ in Frame 1 is transformed to Frame 2 as γx∆t, γy∆t, and γz∆t. Note that γx ≠ γy ≠ γz where γx is computed using vx, etc. The derivation is based on Figure 4, which shows a time dilation of γx∆t is appropriate for adjusting a timed quantity in Frame 1 of any phenomena perpendicular to the x axis in Frame 2. The same applies to the time dilation γy∆t for actions in Frame 1 perpendicular to the y axis, and γz∆t for events perpendicular to the z axis. Observer 2 has the dilemma of having three different time dilation terms to perform a Lorentz transformation for any (x, y, z) event in Frame 1 to Frame 2, which is untenable. The three identical values for light speed in Frame 1 are transformed as three different light speeds perpendicular with each of the axes in Frame 2, which then contradicts the universality of light speed.

Special relativity produces contradictions when 3 or more inertial observers are concurrently observing the Michelson-Morley interferometer. As shown in the paper, four inertial observers in spacecraft concurrently witness the shared output of maximum intensity from constructive interference with the laboratory observer. Special relativity would predict no length contraction for the stationary laboratory observer, but it predicts two different length contractions along the x-axis and two separate length contractions parallel to the y-axis for the observers in their spacecraft. Contradictions exist when the five predictions from length contraction have no agreement in direction or size for concurrent observations.

The standard description using special relativity to explain the simultaneous null output (i.e., constructive interference) of the Michelson- Morley test assumes one axis, such as the x axis, is always parallel to the constant velocity of the apparatus. For the external observer, the round trip along the x-axis is 2γ2L′, and the y-axis round trip is 2γL. Length contraction obtains L′ = L/γ, which forces the ratio of the round trip intervals to equal one for the null output. However, the orientation of the y-axis is very rarely perpendicular to the velocity vector, resulting in vx ≠ vy ≠ 0. The null output must have the ratio ∆tx/∆ty = 1 = γx/γy by (12), but γx ≠ γy in virtually all orientations, except the trivial case that both arms have zero lengths.

This paper derives the total displacement of each beam in each direction for both the Michelson- Morley interferometer with equal arms and the Kennedy-Thorndike interferometer with unequal arms. The linear movement on the x axis by a velocity vx causes a sideway displacement of the y axis as shown in Figure 2 for the external observer. A similar effect of vy causes a sideways movement of the x axis. The total distances are calculated using an infinite series. Based on vector addition of displacements divided by c, the ratio of time intervals for the round trip excursion over either arm by (11) or (12) is determined for both types of interferometers. Special relativity fails to predict the static output except for the trivial case when the interferometer is a point (i.e. no arms) or in the very rare case that one leg is constantly perpendicular with the constant velocity of the interferometer through space. The latter case is routinely disproved as Earth’s rotation and orbital velocity changes vx and vy of the interferometer arms relative to the supposed constant velocity fixed in outer space, but the output remains unchanged.

This paper illustrates the ratio of time intervals by dividing light’s speed into each beam’s round trip excursion. Equation (12) predicts the output of ∆tx/∆ty = γxLx/γyLy. As vx ≠ vy and varies continually, then γx ≠ γy. The output should change continually for the Kennedy-Thorndike interferometer, contrary to test results of unchanging interference. Any time dilation is common to both unequal arms. Likewise, for the Michelson-Morley interferometer, the predicted output for the external observer is a ratio ∆tx/∆ty ≠ 1, but 1 is required to obtain the observed constructive interference. Special relativity does not explain the output in general from either test.

The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale version of the Michelson-Morley interferometer with twin observatories to detect cosmic gravitational waves [21]. Both observatories have operated for months outputting null results, except for the rare anomalies that they were designed to detect. Earth’s freely falling frame is considered an inertial frame, so any geocentric, nonrotating frame of Earth should suffice for comparison to the LIGO on a rotating Earth using special relativity to predict null results. LIGO is positioned generally with one arm oriented east- west (EW) and the perpendicular arm oriented north-south (NS). Earth’s rotation creates a tangential EW velocity in the geocentric frame. The total EW time of transmission is 2γEW2L/c. The sideways motion of the perpendicular arm causes the NS interval to be (γSL + γNL)/c. In most laboratories, γN ≈ γS as the arms are short. However, LIGO has 4 km long arms, and the tangential velocity from rotation on the north end of an arm is slightly slower than the south end that also has the same velocity as the EW arm endpoints. For LIGO, γN < γS = γEW. LIGO is designed so that nothing interferes with the emitted beam from the originating end of an arm. LIGO has the extreme sensitivity to detect any subtle difference, such as γN < γS, but LIGO outputs null results (except for the rare anomalies). Thus, special relativity fails to adequately explain the null output from this ultraprecise Michelson-Morley interferometer.

A detailed reexamination of both interferometers is made. In the moving interferometer, light travels further than the arm L to overtake the receding mirror and shorter than L to intercept the approaching mirror on the other end of the arm, but the total round trip is greater than 2L. The laboratory observer makes no compensation for the moving endpoints of either arm as the interferometer moves through outer space. The laboratory observer assumes light speed is c based on previous measurements, so the expected time interval in the laboratory frame is this expected distance through outer space divided by c, which the external observer measures as the precise one-way transmission interval. If the laboratory observer assumed the arm distance is L, then L/(expected time interval) = c ± vx by (17) or (18) in either direction. More details are derived for the light beam’s traverse along the perpendicular arm. In all cases, the (actual distance traversed)/ (modified speed) = L/c, which is the one-way time interval surmised by the laboratory observer. If the laboratory observer tuned the light so that exactly n wavelengths are standing on the equal arms, then vector velocity addition is the required result in the external frame so that both external and laboratory observers record the same nodes of light on both arms.

Consequently, light not only obeys vector velocity addition in magnitude, but also in displacement. No report has been made that the light beams inside the interferometers exhibited a diurnal drift away from the recombination point after months of continuous operation, which LIGO has conducted. As described in the paper, a simple test using a laser beam, half-silvered pane, and a small hemisphere mirror could demonstrate that the laser beam stays on the nadir of the hemisphere mirror, even though the mirror exhibits perpendicular displacement to the laser beam due to Earth’s motion through the domain of an inertial frame in outer space. This would demonstrate that not only does the speed of light vary in magnitude due to vector velocity addition, but the direction of the light beam also obeys vector addition of displacement.

The concept that light’s velocity obeys vector velocity addition is simply an improvement over Einstein’s excellent approximation that light speed was a universal constant in all directions. At its inception, special relativity had a slight shortcoming that cast doubt on its accuracy. Special relativity predicted that longitudinal mass is γ3μ and transverse mass is γ2μ for a particle’s velocity in an inertial frame where μ is the rest mass of the particle [10, §10]. No test has identified any directional dependence when mass is measured, either directly observed or indirectly inferred from other phenomena. As demons- trated in the paper, the measured speed of light is the standard value when the light source and light detector are mutually fixed.

Ironically, neither the Michelson-Morley nor the Kennedy-Thorndike experiments can prove or disprove the existence of a hypothetical ether in the transmission of electromagnetic light. Regardless of the velocity of the hypothetical ether relative to the interferometer, identical results are produced. It is also shown that the ring interferometer that utilizes the Sagnac effect has the same property. Thus, light obeys vector velocity addition in both inertial reference frames and inertial rotating rings.

If others verify that light obeys vector velocity addition, then several physics concepts will need to be revised. Previous test results should be reviewed to determine if updates are necessary, but the precision in significant figures will generally not change most results. Several ramifications have been listed in the paper, such as old standards would need to be reestablished, like the physical meter, if light speed is not a fixed, exact numerical constant in all reference frames. It would not be surprising that other axioms of physics, concepts, or physical models may need revision in the future.

REFERENCES

- Fizeau, H., (1851) "Sur les hypothèses relatives à l’éther lumineux". Comptes Rendus 33: 349–355.
- Fizeau, H., (1859)"Sur les hypothèses relatives à l’éther lumineux". Ann. de Chim. et de Phys 57: 385–404.
- Michelson, A. A. and Morley, E. W., (1886) "Influence of Motion of the Medium on the Velocity of Light". Am. J. Science 31: 377–386.
- Michelson, A. A., (1881) "The Relative Motion of the Earth and the Luminiferous Ether". Am. J. Science 22: 120–129.
- Michelson, A. A. and Morley, E. W.,(1889) "On the feasibility of establishing a light-wave as the ultimate standard of length". Am. J. Science 38: 181–186.
- Serway, R. A. and Jewett Jr., J. W., (2014) Physics for Scientists and Engineers, Cengage Learning, 9th ed., p. 1196-1198.
- Kennedy, R. J.; Thorndike, E. M., (1932) "Experimental Establishment of the Relativity of Time". Physical Review, 42 (3): 400–418.
- Jaseva, T. S., Murray J., and Townes, C. H., (1964) “Test of Special Relativity or of the Isotropy of Space by Use of Infrared Masers” Physical Review, 133 (5A):1221.
- Hils, D., and Hall, J. L., (1990) “Improved Kennedy-Thorndike experiment to test relativity”, Phys. Rev. Lett. 64 (15): 1967.
- Einstein, A., (1905) “On the Electrodynamics of Moving Bodies” Ann. Phys. 17: 549-560, (translated from 1923 edition of W. Perrett and G. B. Jeffery in The Principle of Relativity:…, Methuen, London.)
- Shankland, R. S., (1963) "Conversations with Albert Einstein". American Journal of Physics 31 (1): 47–57.
- Einstein, A., (1961) Relativity: The Special and the General Theory, Three Rivers Press, New York, 15th ed., ISBN 0-517-88441-0.
- Jackson, J. D., (1975) Classical Electrodynamics, 2nd ed., John Wiley & Sons, New York.
- Young, H. D., Adams, P. W., and Chastain, R. J., (2016) Sears and Zemansky’s College Physics, Pearson, 10th ed., p. 869-872.
- Euclid, Elements, Book 1, Proposition 47.
- Terrell, J., (1959) “Invisibility of the Lorentz Contraction”, Phys. Rev. 111, (4): 1041-1045.
- Penrose, R., (Jul 1958) “The Apparent Shape of the Relativistically Moving Sphere”, Proc. Camb. Phil. Soc. 55.
- Marion, J. B., (1970) Classical Dynamics of Particles and Systems, Academic Press, NY, 2nd ed., 326-327.
- Weisskopf, V. F., (1960) “The Visual Appea- rance of Rapidly Moving Objects”, Phys. Today 13 (9): 24.
- Alväger, T. A., Farley, F. J. M., Kjellman, J., and Waller, I, (1964) “Test of the Second Postulate of Special Relativity in the GeV Region”, Physics Letters, 12 (3): 260-262.
- LIGO.caltech.edu website (operated by Caltech and MIT), Facts subpage lists the extreme engineering to construct LIGO and its operation.
- Kogut, Alan; et al., (December 10, 1993). "Dipole anisotropy in the COBE Differential Microwave Radiometers First-Year Sky Maps". The Astrophysical J., 419: 1-6.
- Fix, J. D., (2004) Astronomy: Journey to the Cosmic Frontier, 3rd ed., McGraw-Hill, p. 532,
- Bertozzi, W., “The ultimate speed: and explo- ration with high energy electrons”, Educational Development Center, Newton, Mass.
- Ohanian, H. C. and Ruffini, R. (1994) Gravitation and Spacetime, 2nd ed, W. W. Norton and Co., New York,
- Deines, S. D., (2016) “Classical Derivation of the Total Solar Deflection of Light”, Int. J. App. Math. and T. Phys., 2(4): 52-56, DOI 10.11648/ j.ijamtp.20160204.15.
- Goldstein, H., (1981) Classical Mechanics, 2nd ed., Addison-Wesley Publishing Co., Reading, MA, p. 511-512.
- Misner, C. W., Thorne, K. S., and Wheeler, J. A., (1973) Gravitation, W. H. Freeman and Co., San Francisco, p. 1112-1113.
- Deines, S. D., (2017a) “Comparing Relativistic Theories to Observed Perihelion Shifts of Mercury and Icarus”, Int. J. App. Math. and T. Phys., 3(3): 61-73, DOI 0.11648/j.ijamt. 20170303.14.
- Hafele, J. C. and Keating, R. E., (1972) “Around-the-World Atomic Clocks: Predicted Relativistic Time Gains”, Science, 177: 166-177. Essen, L., Creation Res. Society Quarterly, (1977) 14: 46.
- Essen, L., (1977) Creation Res. Society Quarterly, 14, 46.
- Deines, S. D., (2017b) ‘Vector Addition of Light’s Velocity Versus the Hafele-Keating Time Dilation Test’, Int. J. App. Math. and T. Phys., 3(3):50-55, DOI 10.11648/j.ijamtp. 20170303.12.
- Hafele, J. C., (1971) “Performance and Results of Portable Clocks in Aircraft”, Third Annual Department of Defense (DOD) Precise Time and Time Interval (PTTI) Strategic Planning Meeting, 16-18 Nov 1971, Washington, DC, p. 261-288
- Kelly, A. G., (2000) “Hafele and Keating Tests: Did They Prove Anything?”, Physics Essays, 13(4): 616, DOI: 10.4006/1.3025451.
- https://www.mathpages.com/rr/s2-07/2-07. Htm3