To Measure Light Speed Anisotropy.

One-way speed of light can be measured.

A method for measuring one-way light speed anisotropy, thereby determining velocity relative to the so-called luminiferous ether, is the topic.

The constancy of round-trip-average speed of light does not demonstrate or test constancy of one-way speed of light. When timing the round trip of a light signal over a path of a certain length, the average speed of light will always be found to be the defined constant c, because length measure is defined in terms of round-trip light time. Clocks synchronized using an assumption of light speed isotropy must, because of that assumption, yield the assumption only and not a measure of one-way speed of light.

The equivalence of solid and interferometric standards of length, now recognized by standards organizations,^[1] fully accounts for the failure of the Michelson Morley experiment^[2] to detect motion relative to the luminiferous ether.

An experimental plan, simplified:

Determination of one-way speed is possible between clocks at the two ends of a one-way light path if the clocks can be depended on to measure time at the same rate. Synchronization of clocks is unnecessary, because the correction for synchronization error can be discovered from the data.

Let conditions known or suspected to affect clock rate be the same for the two clocks. Let the clocks be subject to slow relative motions, as by the change of path direction resulting from earth rotation while the clocks remain in fixed locations on the earth. If the change of light-path direction is due to earth rotation, the clock rate match can be verified at intervals of one sidereal day. A direction dependant variation in the one-way timing as a fraction of the round-trip timing is a demonstration of light speed anisotropy. If t_ratio, equal to time difference (for the opposing directions) over time sum (direct round-trip time,) is plotted against celestial coordinate orientation for sampled light-path orientations, the plot would reveal the right ascension of a plane of symmetry parallel to velocity. The clock synchronization error could be found from time difference when the clocks are aligned perpendicular to the plane of symmetry. Declination and speed would be revealed by best fit of data to theory.

However, if velocity parallels the earth axis, relying on earth rotation to shift orientation will limit the sample to a single angle between light path and velocity. In that case, it would not be possible to distinguish the effects of synchronization error and velocity.

Assumptions.

We shall provisionally adopt the length contraction conjecture of FitzGerald^[3]:

I have read with much interest Messrs. Michelson and Morley's wonderfully delicate experiment.... I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocity to that of light.... (1889 letter to the editor of Science)

The fraction of light speed of an inertial frame's velocity relative to the conjectured rest frame is designated beta, β = v / c, (bold to indicate a vector.) Using time and length measure of the rest frame, the expected contraction of material bodies in FitzGerald's theory would be by a factor (1 − β²) parallel to the motion and by (1 − β²)^½ perpendicular to the motion. For convenience, we define gamma, γ = 1/(1 − β²)^½. These FitzGerald factors are derived below, in the subsection titled "Derivations", and, as mentioned above, were also derived by Michelson. If moving bodies do contract, spaces between unconnected co-moving bodies would appear greater relative to their proper frame than they do relative to the rest frame.

A time dilation effect would act as a scaling factor independent of direction and would have no effect on the ratio t_ratio or the angle φ between light path orientation and velocity relative to the luminiferous ether. We are thereby moved to acknowledge the superiority of FitzGerald contraction factors over Lorentz contraction factors.

Experimental Setup.

Consider an experimental local frame with a non-zero velocity v = β⋅c with respect to the conjectured rest frame. The experimental frame's absolute velocity is to be determined. Suppose we fix two atomic clocks sufficiently far apart for significant timing of light travel and eliminate refraction vagaries of the light path between the clocks. Let a light signal be sent forward from clock one to clock two where it is reflected back to clock one, with times of these events noted according to the respective clocks. Several trials are executed with differing orientation of the line between the clocks. In the example below the orientation change of an evacuated light path is effected by Earth rotation. The clocks need not be synchronized, but are assumed to measure time at the same rate throughout all trials; if not, rate differences must be known and compensated for. Clock-one shall be regarded as the standard, so any synchronization error will be considered a clock-two error.

We shall be interested in:
--Unknowns to be found--

• beta, β = v/c, the unknown vector quantity to be found, velocity of the experimental frame ÷ speed of light,
• β, magnitude of β
• α_β, right ascension of β
• δ_β, declination of β
• phi, φ, the angle between the velocity, β, of the experimental frame and the directed line determined by the two clocks.
• t_err, clock-two synchronization error, (clock-one time = clock-two time − t_err)

• α_obs, the right ascension in radians of clock two relative to clock one.
• δ_obs, the declination in radians of clock two relative to clock one.
• t_A, the time on clock one when the light signal is emitted.
• t_B, the time on clock two when the light signal is reflected.
• t_C, the time on clock one when the returning light signal arrives.

• t_fwd = t_B − t_A, the nominal time taken by light to travel from clock one to clock two, (possibly inflated by t_err.)
• t_bck = t_C − t_B, the nominal time taken by light to travel from being reflected at clock two back to clock one, (possibly deflated by t_err.)
• t_dif = t_fwd − t_bck = 2t_B − t_A − t_C , (possibly inflated by a double synchronization error, 2 t_err.)
• t_sum = t_fwd + t_bck = t_C − t_A, (time for round trip, no synchronization error.)
• t_ratio = t_dif / t_sum = ( 2t_B − t_A − t_C ) / ( t_C − t_A ), (a ratio strictly between one and minus one if, relative to some frame of reference, the clocks are synchronized; possibly inflated by 2t_err / t_sum)

[Note: Boldface is used for the vector value, if not boldfaced, the scalar magnitude is meant.]

The measure of the angle φ is frame of reference dependant. Accordingly, subscript zero φ₀ will denote rest frame value, and subscript beta φ_β will denote experimental frame value. These will be equal at angles, 0, ½⋅π, and π, (radian measure.)

Derivation of the FitzGerald Contraction Factor.

Figure 1 idealizes a trial of the experiment in rest frame scale. The two clocks, synchronized in the rest frame, move to the right at speed β⋅c, the points A, B, and C are the fixed rest frame locations where three events occur: at A, clock one emits a light signal at time t_A as it passes through point A; at B, clock two reflects the light signal at t_B; at C, the light signal and clock one arrive together at t_C. The point D is the location of clock one at t_B. So we have time intervals t_fwd = t_B − t_A and t_bck = t_C − t_B. The angle between the directed line DB determined by the two clocks and the velocity v = β⋅c of the experimental frame is labeled φ₀.

The clocks are synchronized with respect to the rest frame, so for φ = ½⋅π the ratio t_ratio = t_dif / t_sum must be zero, because t_dif must be zero. However, for light path orientations at other angles φ from β non-zero ratios t_ratio are expected, and parallel to β the maximum t_ratio is expected.

A time dilation effect would act as a scaling factor independent of direction and would have no effect on the ratio t_ratio or the angle φ. We are thereby moved to acknowledge the superiority of FitzGerald contraction factors over Lorentz contraction factors.

Let us now derive equations for our variables of interest and the generalized transformations for any value of φ. Then, we will see the very factors derived by Michelson and Morley for φ = 0 and φ = ½⋅π.

The cosine law and quadratic formula yield solutions for t_fwd and t_bck in the triangles ADB and CDB respectively. First, t_fwd from triangle ADB in equations (1.1) through (1.3): 

	(1.1) cosine law for ΔADB
	(1.2) eq 1.1 in standard quadratic form
	(1.3) quadratic formula solution

Then, t_bck from triangle CDB in equations (1.4) through (1.6): 

	(1.4) cosine law for ΔCDB
	(1.5) eq 1.4 in standard quadratic form
	(1.6) quadratic formula solution

Now we see the time-difference, the time-sum, and the ratio t_ratio.

	(1.7) tdif
	(1.8) tsum
	(1.9) tratio

The experimental frame observer judges length by round-trip time of light, so d_β = ½c(t_fwd + t_bck). The equations for transformation of lengths between moving and ether frames are:

	(1.10) FitzGerald Contraction Factor,      [recall: γ = 1/(1 − β2)½.]
	(1.11) parallel, sin 0 = 0
	(1.12) perpendicular,        sin ½π = 1

The spatial distance appears greater in the moving frame because the local measuring sticks are contracted.

Applying length transformations to the cosine function, cos φ_β = cos φ₀(1 – β² sin²φ₀)^–½, it turns out that t_ratio is a quite simple function of φ_β independent of clock rate:

tratio = β cos φβ

(1.13) tratio(β,φβ)

The influence of unknown time dilation, i.e. clock rate change, must be isotropic, so it would enter these equations as a factor tau, τ; for example, d_β = τ γ²d₀ (1 − β² sin²φ₀)^½.

For the Population of all Possible Angles φ_β.

Two-dimensional polar coordinate plots with radial coordinate t_ratio as a function of angular coordinate φ_β for constant t_sum and β appear in figure 2.

For clocks absolutely synchronized, the plot of t_ratio for all values of φ_β is a circle that intersects the origin at φ_β = ½⋅π, and the diameter from the origin will equal β. The plot for φ_β ∈ [−½⋅π, ½⋅π] coincides with the plot for φ_β ∈ ([−π, −½⋅π]∪[½⋅π, π]).

A synchronization error would make the plot a limaçon. Depending on the magnitude of the synchronization error, the limaçon could have inner and outer loops, be heart-shaped with a cusp, or be oval.

The three-dimensional celestial-coordinate plot of t_ratio for all values of right ascension and declination is a sphere when there is no synchronization error, and β, (β, α_β, δ_β), is the point of the sphere farthest from the origin.

With synchronization error, the 3-D plot of t_ratio becomes a limaçon rotated about its symmetry axis. The values 0 and π of φ yeild maximum and minimum of t_ratio, respectively, and the maximum less the minimum equals twice β, and the velocity direction is that of the maximum. The sum of maximum and minimum t_ratio equals four times t_err / t_sum.

For a Sample Including All Observations of a Single Declination.

In the practical case, observations limited to a single declination, the values 0 and π of φ are no longer expected to be included. Values of t_ratio will span an interval from a maximal t_ratio at minimal φ to a minimal t_ratio at maximal φ.

As Earth turns, observations will sample a conic surface intersecting the spherical or limaçonic surface. Positive values of t_ratio will lie in one nappe of the cone while negative values lie in the other nappe.

A plot of t_ratio (vertical axis) against φ (horizontal axis) is a cosine wave with amplitude determined by β shifted up or down according to t_err. The width of the sample is twice the conic half angle, with possible redundancy. For different sets of parameter values, the curves within the span of the sample are distinct but may have up to three points in common, not more.

Two javascript routines are appended to this document: (1) a data simulator, (2) a data solver to find β from a set of actual or simulated data points. The data simulator generates simulated data based on parameter input by the user. It also generates plots with optional animation. The animated plots will assist in illustrating how a data plot reflects values of the unknown parameters.

The solver accepts input data in the same CSV format generated by the simulator, consisting of five observed quantities, α_obs, δ_obs, t_A, t_B, t_C, for each trial; (the optional linefeeds between trials are for display only.) Independent of plots, the javascript data solver employs matrix determinants to evaluate the four unknown coeficients, t_err, x_β, y_β, and z_β, of the general equation for the limaçon: 0 = t_err ( 2 sqrt(x² + y² + z²) / t_sum ) + x_β x + y_β y + z_β z − (x² + y² + z²)
However, for this or any other method of solution to work, the solution for a set of data must be unique. This we now test with simulations in which t_err and β are chosen so as to pin maximal and minimal t_ratio to fixed values to test whether varying δ_β alters the plot as it must.

A 200px × 200px space for animated plot of simulated data was to appear here, but apparently either it is not supported by your browser or JavaScript is disabled.

The symmetry plane of the intersection will contain the earth/cone axis and β and therefore also α_β. A lack of synchronization does not alter the orientation of the symmetry plane provided the synchronization error is constant. A light path perpendicular to the symmetry plane will reveal t_err = ½ t_dif. If that light path is not sampled, untangling the remaining unknowns, t_err, β, and δ_β, is a more daunting task, best explored with the aid of animated plots from simulations.

The distance between extrema of t_ratio is a function of β, and the values of t_ratio can be uniformly shifted by adjustment of t_err, so we can pin the zero point midway between extrema and maintain constant extrema to reveal the effect of varying δ_β. ** insert button to show animation **

Possibilities for a Terrestrial Test.

Possible sites for a long level course for carrying out such an earth-bound test might include a salt flat or a frozen lake. Sacrificing the ideal of an evacuated light path, a sheltered valley might serve. Specifically, a test site situated at the Bonneville Salt Flats (Lat. 40.736556, Long. -113.411537) with path up to ninety kilometer long and forward direction about 19° east of true north would give a constant 45° declination, but temperature fluctuations might be a problem. Under the ice of Green Bay at the border between Wisconsin and Michigan, (Lat. 45.1, Long. -87.6) with path up to seventy kilometer long directed 26.5° east of true north to give a constant 19° declination, would be temperature stable and possibly sufficiently free of disturbing current. At certain seasons the air may be sufficiently clear and stable across a sheltered valley or caldera such as La Garita Caldera in Colorado or Crater Lake in Oregon to run the test in air. I am skeptical of experiments in refractive media, such as fiber optics, because additional questions about the unknown effect of motion on the medium's optical properties are involved.

How accurate must clocks be to make a decisive test? Let us suppose a 90 km test course situated as suggested at the Bonneville Salt Flats. The relative velocity of the endpoints of the path would be less than 5 meter per second. The value of t_sum is 0.6 millisecond. Further, suppose β = (β, α_β, δ_β) = (0.001, 10°, −34°). The expected 0.00117 range of t_ratio, 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.

For highest precision, an extraterrestrial site, possibly the GPS system, would surely be best but would require more complex control and analysis.