A Revealing Test of the Compatibility of Special Relativity Postulates

by David Bryan Wallace
Cape Coral, Florida, USA
Copyright © 2015-06-11

Introduction

Rarely is a scientific theory so enthusiastically acclaimed as has been the relativity theory of Albert Einstein. Although special relativity is often associated in our thinking with the Michelson-Morley experiment, it was the elaboration of a remarkable intuition rather than an explanation of an empirical result. Einstein presented special relativity as a deduction based on two postulates, and he declared the motivation was his firm conviction that there was no reality corresponding to the notion of absolute rest. This conviction is open to challenge.

The Postulates of Special Relativity
I. The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
II. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

Together these two postulates imply that the speed of light is isotropic in every inertial frame of reference. However, there is imperfect empirical confirmation of the constancy of light speed for every inertial frame. Only for round-trip average speed of light has the second postulate been established, never for one-way speed. That the round-trip average speed of light is the constant c with respect to any inertial frame of reference is now established by definition, by defining length measure in any inertial frame by the time of a round trip of light, length = c × time‑round‑trip / 2. One-way speed of light has not been measured. When one sets time‑forward = time‑back to synchronize clocks with light signals, as special relativity theory requires, it vitiates timing the one-way speed of light between the synchronized clocks. Today's atomic clocks make one-way light speed measurable without such vitiating assumptions.

When this aspect of relativity is finally tested, will it fail? There is reason believe so. Astronomical observations support the expectation that the frame of reference in which one-way speed of light is isotropic is unique[1][2]. At rest in such a unique frame of reference is worthy of being considered absolutely at rest.

This paper presents both a way to compare one‑way speed of light in opposite directions and also the beginnings of a modified relativity theory that accommodates a unique stationary frame of reference.

Part 1: A Realizable Test of the Isotropy of Light Speed

Assume for the present that the notion of absolute rest does have a basis in reality. The velocity of an experimental inertial frame relative to the absolutely stationary frame can be expressed as a fraction β (beta) of the speed of light. The apparent one-way speed of light in that frame of reference will be the same for any evacuated light path the same angle φ (phi) from β provided the clocks at opposite ends of the paths are truly synchronized. Thus, perpendicular to β the ratio time-difference / time-round-trip = ρ (rho) must be zero, while parallel to β the maximum ρ is expected, consistent with ρ = β⋅cos(φ) and independent of path length. In contrast, the compatibility of the special relativity postulates requires ρ = 0 always. We test the second postulate by using stable accurate atomic clocks that move slowly with respect to each other.

β = v/c = velocity of the inertial frame / speed of light (1.1 beta)
σ = time‑round‑trip = time‑forward + time‑back (1.2 sigma)
time‑difference = time‑forward − time‑back (1.3)
ρ = time‑difference / time‑round-trip = β⋅cos(φ) (1.4 rho)
α = right ascension (celestial coordinate angle east from the vernal equinox) of the forward direction or of β (1.5 alpha)
δ = declination (celestial coordinate angle north from the equatorial plane) of the forward direction or of β (1.7 delta)
φ = measure of the angle from β to the forward direction of the light path (1.8 phi)

The three dimensional celestial coordinate graph of ρ = β⋅cos(φ) for every possible direction, with clocks absolutely synchronized, is a spherical surface with radius β/2 and center at β/2. In the absence of synchronization error, β can be found simply by finding the center as the point equidistant from all data points (ρ, α, δ) with a minimum of four independent data points. A constant synchronization error would add a fixed ρerror to the magnitude of each data point, presenting an expanded surface for ρraw of one sign and a contracted surface for ρraw of the other sign; there would remain a point of tangency where ρraw = 0. The two unknowns, β and synchronization error, ½⋅σ⋅ρerror, can both be determined with as few as five data points.

As Earth turns, an earth-bound light path of fixed length not parallel to the earth's rotational axis will present a cone of orientations of like declination that includes (1) a path of maximal ρraw at φmin, (2) a path of minimal ρraw at φmax, and (3) equal φ pairs with equal ρraw. This is a pattern of bilateral symmetry about a plane parallel to both β and the earth axis. A lack of synchronization does not alter the orientation of the symmetry plane provided the synchronization error is constant. Figures 1.13 and 1.14 show a dark trace for such a conical sample with 45° declination and ρerror = 0.0003 on a surface representing all possible orientations; in figure 1.13 β of magnitude 0.001 is shown with declination 32.7°, and in figure 1.14 with declination −24.6°.

ρraw = ρtrue + ρerror (1.9)
error of second clock time relative to first clock = ½⋅σ⋅ρerror (1.10)
Plot_3d (figure 1.11)
Plot_3d (figure 1.12)

Due to Earth's motions an earth-bound experiment site has inconstant β, so the determination of β must be referred to a specific time and place. In the final analysis, data must be adjusted for deviations from this β. For clarity, the preliminary exposition above neglected the variations of velocity due to Earth motion.

The ranges of significant known oscillatory motions that are superimposed on the unknown β include up to 60,000 meter per second (0.0002 light speed) due to the earth-solar orbit, up to 920⋅cos(latitude) meter per second due to earth rotation, and up to 25 meter per second due to the earth-lunar orbit. The effect of these motions can be removed from the raw data by subtracting the parallel component of vdeviation/c. For example, if the raw datum (ρ=0.000010, α=75°, δ=45°) is taken when the motion differs from the sought unknown β by vdeviation/c = (0.000003, 15°, 50°) = (light speed fraction, α, δ), one finds adjusted datum magnitude (direction unchanged) ρadjusted = 0.000010 − 0.000003⋅cosine(angle difference) = 0.000010 − 0.000003⋅((cos(75°−15°)+cos(90°−45°)⋅cos(90°−50°))/(sin(90°−45°)⋅sin(90°+50°)) = 0.0000031, where the spherical law of cosines for sides has been used to find the cosine of the angle difference.

How accurate must clocks be to make a decisive test? Let us suppose a test site situated at the Bonneville Salt Flats (Lat. 40.736556, Long. -113.411537) with path ninety kilometer long and forward direction about 19° east of true north to give a constant 45° declination. The relative velocity of the endpoints of the path would be less than 5 meter per second. The value of σ is 0.6 millisecond. Further, suppose β = (β, α, δ) = (0.001, 10°, −34°). The expected 0.00117 range of ρ, from 0.001⋅cos(169°) = -0.00098 to 0.001⋅cos(79°) = 0.00019, implies a one-way timing range of 0.35μs. Nanosecond timing accuracy would resolve this range into 350 parts, enough for a decisive test.

Ultimately, considering the much longer light paths achievable with a space-based test, an earth bound test may be deemed uninteresting. A space based test would involve variable path lengths, but ρ is independent of path length after all.

Part 2: Revising Relativity Theory to Accommodate an Absolutely Stationary Frame of Reference

If it is found that isotropic light speed is not an attribute of every inertial frame of reference, then relativity theory needs revision, and a caveat is needed with the principle of relativity: "In any inertial frame of reference the laws of physics are the same for all phenomena except those that depend on isotropy of the one-way speed of electromagnetic propagation."

The Michelson Morley experiment involved no clocks and so reveals no such thing as time dilation. The apparent null of the Michelson Morley experiment is explained by length contraction of moving solids being precisely such as to produce constancy of locally determined interferometric length, (1−β2) parallel to β and sqrt(1−β2) perpendicular thereto. This should not be a great surprise as it is implicit in the use of interferometric standards for length measure. A curious feature of Einstein's special relativity paper is that these were the contraction ratios he derived before an unexplained switch to the Lorentz transformations. The Lorentz transformations are incorrect because they fail to recognize contraction transverse to motion. Equations 2.1 through 2.4 show correct transformations between rest frame coordinates (x0, y0, z0) and moving frame coordinates (xβ, yβ, zβ), where x-axes are parallel to β, origins coincide at t0 = tβ = 0 and y- and z-axes respectively are parallel. Figure 2.5 expresses these transformations in matrix form.

Transformation of Coordinates

t0 = tβ (2.1)
<i>x</i><sub>β</sub> = (<i>x</i><sub>0</sub> − β⋅<i>c⋅t</i>) / (1 − β<sup>2</sup>) (2.2a)
<i>x</i><sub>0</sub> = <i>x</i><sub>β</sub>⋅(1 − β<sup>2</sup>) + β⋅<i>c⋅t</i> (2.2b)
<i>y</i><sub>β</sub> = <i>y</i><sub>0</sub> / sqrt(1 − β<sup>2</sup>) (2.3a)
<i>y</i><sub>0</sub> = <i>y</i><sub>β</sub>⋅ sqrt(1 − β<sup>2</sup>) (2.3b)
<i>z</i><sub>β</sub> = <i>z</i><sub>0</sub> / sqrt(1 − β<sup>2</sup>) (2.4a)
<i>z</i><sub>0</sub> = <i>z</i><sub>β</sub>⋅ sqrt(1 − β<sup>2</sup>) (2.4b)
[<i>x</i><sub>β</sub> <i>y</i><sub>β</sub> <i>z</i><sub>β</sub> <i>t</i>] = [<i>x</i><sub>0</sub> <i>y</i><sub>0</sub> <i>z</i><sub>0</sub> <i>t</i>]dot Matrix(4, 4, {(1, 1) = 1/(1-beta^2), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 1/sqrt(1-beta^2), (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1/sqrt(1-beta^2), (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1})-Vector[row](4, {(1) = beta*c*t/(1-beta^2), (2) = 0, (3) = 0, (4) = 0}) (2.5a)
[<i>x</i><sub>0</sub> <i>y</i><sub>0</sub> <i>z</i><sub>0</sub> <i>t</i>] = [<i>x</i><sub>β</sub> <i>y</i><sub>β</sub> <i>z</i><sub>β</sub> <i>t</i>] dot Matrix(4, 4, {(1, 1) = 1-beta^2, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = sqrt(1-beta^2), (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(1-beta^2), (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1})+Vector[row](4, {(1) = beta*c*t, (2) = 0, (3) = 0, (4) = 0}) (2.5b)

The variant of relativity being offered here is essentially classical or Galilean relativity modified by the inclusion of contraction of solids induced by absolute velocity. Every moving coordinate system is to be set up using the contracted length measure proper to the system, corresponding to the defined interferometric standard applied locally. Time is well defined and shared by all frames of reference. A distinct variant of time called "local time" may be defined for a specific inertial frame of reference. If clock rate is affected by motion or gravitation, that shall be regarded as a peculiarity of the clock, not a peculiarity of time.

A solid rod which at absolute rest would have length l contracts when moving. If moving at fraction of light speed β with the angle φ0β in the moving frame) between β and the axis of the rod, the absolute span s0 of the rod is given by s = `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(sqrt(`+`(1, `-`(`*.... The proper length of the rod, measured in the inertial frame of the rod itself, remains l.

In the moving frame we find the angle φ transformed, tan(phi[beta]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(tan(phi[0]))); if phi[0] = `+`(`*`(`/`(1, 2), `*`(Pi))) then phi[beta] = `+`(`*`(`/`(1, 2), `*`(Pi))).

In the moving system, a point at absolute rest (at rest in the rest frame) will have velocity v = `+`(`-`(`/`(`*`(c, `*`(beta)), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))))). On the other hand, a point at rest in the moving system will in the rest frame have velocity βc. Time in all frames of reference is the same as in the rest frame and might be called universal time or absolute time. However, for some applications it will be preferable to ignore β and synchronize clocks with light signals assumed to take the same time each way; such synchronization is outside the practices of our modified relativity and shall be called "local synchronization" giving rise to "local time." If a moving system has its x-axis parallel to β and local time at the origin is equal to universal time, then local time at other points is related to universal time by tuniversal = tlocal + β x/c. Importantly, this relationship allows synchronization to absolute time by light signals once β is known.

tuniversal = tlocal + β x/c

(2.6)

Transformation of Measure

tan(φ)

tan(phi[0]) = `/`(`*`(tan(phi[beta])), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))
tan(phi[beta]) = `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(tan(phi[0])))

(2.7a)


(2.7b)

cos(φ)

cos(phi[0]) = `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`^`(tan(phi[beta]), 2)), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))))))), `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`^`(tan(phi[beta]), 2)), `*`(`+`(1, `-`(`*`(`^`(bet...
(cos(phi[beta]) = `/`(1, `*`(sqrt(`+`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2))))))), (`/`(1, `*`(sqrt(`+`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2...

(2.8a)


(2.8b)

sin(φ)

`and`(sin(phi[0]) = `/`(1, `*`(sqrt(`+`(1, `/`(`*`(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(`^`(tan(phi[beta]), 2))))))), `and`(`/`(1, `*`(sqrt(`+`(1, `/`(`*`(`+`(1, `-`(`*`(`^`(beta, 2))))), `*`(`^`(tan(...
`and`(sin(phi[beta]) = `/`(1, `*`(sqrt(`+`(1, `/`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(phi[0]), 2)))))))), `/`(1, `*`(sqrt(`+`(1, `/`(1, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))), `*`(`^`(tan(p...

(2.9a)


(2.9b)

absolute span of moving length

l[beta] = `/`(`*`(s[0], `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `*`(`+`(1, `-`(`*`(`^`(beta, 2))))))
s[0] = `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2), `*`(`^`(sin(phi[0]), 2)))))))), `/`(`*`(l, `*`(`+`(1, `-`(`*`(`^`(beta, 2))))))

Derivation (with x-axis parallel to β)

(2.10a)


(2.10b)

ψ dihedral angle between two angles φ1 and φ2

ψ0 = ψβ

(2.11)

θ angle between the vectors β1 and β2 of two inertial frames

`and`(cos(theta[0]) = `/`(`*`(Typesetting:-delayDotProduct(beta[1], beta[2])), `*`(abs(beta[1]), `*`(abs(beta[2])))), `/`(`*`(Typesetting:-delayDotProduct(beta[1], beta[2])), `*`(abs(beta[1]), `*`(abs...

(2.12)

Part 3: Stellar Aberration

An observer's motion causes stellar aberration, an angular shift α of the apparent angular position of a star toward β from the rest frame angular position of the star. Analysis of this phenomenon for an observer of known β for a star observed at an angle φβobserved begins by transforming the observed stellar position to the angle measure of the rest frame. Then, as shown in figure 3.2, sin(α0), the sine of the rest frame measure of the aberration, equals β⋅sin(φ0observed), and φ0true = φ0observed + α0. Finally, by comparing sin(φ0true) with φβobserved, we find αβ = φ0true − φβobserved.

sin(φ<sub>0<i>observed</i></sub>) = sin(φ<sub>β<i>observed</i></sub>)/sqrt(1−β<sup>2</sup>⋅cos<sup>2</sup>(φ<sub>β<i>observed</i></sub>)) (3.1)[applying 2.9a]
aberration (figure 3.2)
sin(α<sub>0</sub>) = β⋅sin(φ<sub>0<i>observed</i></sub>) (3.3)
φ<sub>0<i>true</i></sub> = φ<sub>0<i>observed</i></sub> + α<sub>0</sub> (3.4)
sin(α + β) = sin(α)cos(β) + cos(α)sin(β) (trigonometric identity)
sin(φ<sub>0 <i>true</i></sub>) = sin(φ<sub>0  <i>observed</i></sub>) cos(α<sub>0</sub>) + cos(φ<sub>0 <i>observed</i></sub>) sin(α<sub>0</sub>) (3.5)
α<sub>β</sub> = φ<sub>0 <i>true</i></sub> − φ<sub>β <i>observed</i></sub> (3.6)
messy equation (3.7)

[1] Wallace, David Bryan, "Improving the Definition of Simultaneous Events" Academia.edu (2013) https://www.academia.edu/7273747/Essay_Improving_the_Definition_of_Simultaneous_Events

[2] de Sitter, Willem, “Ein astronomischer Beweis für die Konstanz der Lichgeshwindigkeit," Physik. Zeitschr, 14, 429 (1913). “A proof of the constancy of the velocity of light,” Proceedings of the Royal Netherlands Academy of Arts and Sciences 15 (2): 1297-1298, online in English http://www.datasync.com/~rsf1/desit-1e.htm..

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