Experimental Relations and Tests

Apply the general directional factors to the two longitudinal legs separately, then add:

For each transverse leg, set $\phi=\pi/2$:

The ideal equal-arm model therefore gives $T'_\parallel-T'_\perp=0$. The common factor $\gamma$ cancels from this difference. A null differential result alone does not measure the transverse true scale.

Original observation: Michelson & Morley (1887), Chapter 1 reference [10].

Using this section's convention for the direction of object motion, first write the counter- and co-directional elapsed times:

Then use the rest radius, the radius associated with motion and the area in sequence:

The substitution is $\Delta t=(2\pi R/c)(2R'\omega/c)\gamma$. With $\gamma R=R'$, it yields the area expression. The manuscript retains $\gamma$ here and absorbs it into $R'$. Agreement in form with the standard low-speed area relation does not establish an all-orders comparison between the two timing definitions.

The observation reference follows Chapter 1 [16]: Sagnac (1913), Comptes Rendus 157, 708–710. This website has not refitted the original fringe data.

In the chapter's transverse case, $\phi=\pi/2$ gives $f'/f=1/\gamma$ and a wavelength ratio of $\gamma$. The forward/backward spectral-line combination in Ives–Stilwell is not the same apparatus as a directly drawn ninety-degree light path. Its actual spectral combination must be mapped to the manuscript's angle convention.

Original paper: Ives & Stilwell (1938), Chapter 1 reference [9]. The general wavelength–frequency derivation is on the physical-meaning page.

Under the manuscript's simplified balancing condition $L_{\mathrm{vac}}=nL_{\mathrm{med}}$, the vacuum-segment expression $2\gamma L_{\mathrm{vac}}/c$ equals the medium-segment expression $2n\gamma L_{\mathrm{med}}/c$. This calculation is conditional. Its correspondence to the closed optical path, segment directions and phases in Hoek's original apparatus requires a segment-by-segment check against the apparatus diagram.

The observation reference follows Chapter 1 [14]: Hoek (1868), Archives Néerlandaises 3, 180–185. The chapter does not provide a complete apparatus data table for direct recalculation.

Chapter 1, Section 3.4, interprets the experiment using effective optical path in a medium. Section 3.5 applies Eq. (1.38) to a linearly moving medium and proposes $t=nL/c$. Direct substitution into the manuscript's $\Delta t=2t\beta\gamma$ gives:

This is the website's explicit expansion of the substitution described in the chapter. It has no separate manuscript equation number. It does not yet account for all propagation segments, boundaries and phases of the same apparatus. Define $L$, medium speed $w$ and the number of traversals first, then check refractive-index and dispersion dependence. A complete reproduction of Fizeau fringes cannot yet be claimed from this substitution.

The original observation follows Chapter 1 reference [15]: Fizeau (1851), Comptes Rendus 33, 349–355. The standard velocity-composition comparison uses Einstein 1905, Section 5. It is an independent comparison and does not enter the ASD derivation.

The final manuscript’s standard-theory comparison in §3.4 prints a velocity-addition denominator that is not dimensionless when $w$ is a speed. This source issue is recorded explicitly; the page does not use that typography as a valid comparison formula or change ASD Eq. (1.38) on its basis. The Fizeau apparatus model and refractive-index dependence remain the specific tasks stated above.

The manuscript inverts the relation using $L'=643\,\mathrm m$ and $\beta=0.998$:

Here $L$ is the path from production to reception in the ground reference, $L'$ is the effective path assigned to the moving process in Chapter 1, and $\beta=v/c$. The website expands manuscript Eqs. (1.45)–(1.46) for the stated example inputs as

Rounding to one decimal place gives $20323.3\,\mathrm m$, consistent with the manuscript’s approximate $20323\,\mathrm m$. The intermediate radicals and extra digits are a website arithmetic check. The inputs $643\,\mathrm m$ and $0.998$ remain approximate illustrative values, not measurements with this many significant digits. ASD links process duration here to a directional effective path rather than introducing a separate change in a flow of time.

This algebraic calculation must be distinguished from an experimental measurement. Eq. (1.46) provides no particle populations at different energies, initial flux, energy losses, detector efficiency or survival fit. The fact that some particles reach the ground is insufficient to test a particular factor for a mean distance.

As a website consistency check, inversion of Eq. (1.45) gives $L/L'=\gamma(1+\beta)$. This differs from the SR factor $\gamma$ when the same $L'$ is interpreted as a constant-speed path associated with proper lifetime. The author must specify the statistic and endpoints represented by $L'$. The two factors cannot yet be treated as completed predictions for the same experiment.

Continue with OP-LVE-01: muon survival under the LVE path interpretation → · Chinese original

Original observations: Rossi & Hall (1941), Chapter 1 [17]; Frisch & Smith (1963), American Journal of Physics 31, 342–355, reference [18]. These studies measured relations between decay or survival and motion, not the illustrative number in Eq. (1.46) directly.

To identify a possible difference, the website expands the manuscript expression at low speed:

For constant speed in the same inertial background with gravity omitted, the standard proper-time relation is $d\tau/dt=\sqrt{1-\beta^2}$, whose first correction is second order. This is a formal comparison of two expressions. The speed and process path of the manuscript's equatorial circuit have not yet been fully aligned with standard coordinate time.

Actual clock transport requires flight trajectories, Earth rotation, gravitational potential, stop durations and the clock-comparison procedure. Chapter 1 supplies neither these inputs nor a total clock difference. Hafele–Keating observations are therefore not labeled as verification of Eq. (1.47). Precision requirements should follow matched-trajectory predictions and an error budget.

Original report: Hafele & Keating, Observed Relativistic Time Gains (1972), Chapter 1 reference [20]; the accompanying prediction is reference [19]. For the standard constant-speed timing relation, see Einstein 1905, Section 4.