Robertson-Mansouri-Sexl Test Theory

You cannot test a theory with an experiment whose analysis assumes the theory. Use the Lorentz transformation to convert your raw fringe counts into a statement about the speed of light, and you will find that the speed of light is invariant — and you will have found out nothing at all. You put it in. So how does anyone claim to test special relativity? By embedding it. Write the most general linear transformation between a hypothetical preferred frame and your laboratory, leave the coefficients as unknown functions of velocity, and let the data pin them down. Special relativity becomes one point in a three-dimensional space of theories: $\alpha = -\tfrac12$ , $\beta_{\!*} = +\tfrac12$ , $\delta = 0$ . Every null result becomes a constraint surface; every bound, a distance from that point. That is the Robertson–Mansouri–Sexl test theory, and it is what turns "Michelson and Morley saw nothing" into a number.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive.
The RMS transformation from a preferred frame $\Sigma$ to a lab $S$ moving at $v$ is $t = a(v)T + \varepsilon x$ , $x = b(v)(X - vT)$ , $y = d(v)Y$ — three unknown functions and a free synchronisation vector. SR is $a = 1/\gamma$ , $b = \gamma$ , $d = 1$ , $\varepsilon = -v/c^2$ .
⚠ The base prints the transformation as $x = (X - vT)/b$ — the $b$ is INVERTED, and it is the article's central equation. With the base's own $b = \gamma$ it neither reproduces the Lorentz transformation nor preserves the invariant interval: off by $4.67\%$ at $v = 0.6c$ . The base's own card claims to "verify the invariant interval" for it. Its LaTeX even prints `b = γ⁻¹?` — a question mark inside a displayed equation.
⚠ OWNED VERDICT — the cross-base sign contradiction, resolved. Two forms of the Michelson–Morley coefficient circulate, $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ and $\left(\beta_{\!*} - \delta - \tfrac12\right)$ : negatives of each other, and both vanish at the SR point, so the obvious check is blind. They are coefficients of reciprocal quantities — the first of $c(\theta)/c$ , the second of $c/c(\theta)$ , which is what the literature quotes.
The discriminator is the Galilean limit. At $\alpha = \beta_{\!*} = \delta = 0$ the derivation must return the exact aether result $c_\parallel = c(1-\beta^2)$ , $c_\perp = c\sqrt{1-\beta^2}$ . It does — and only with $+\tfrac12$ , i.e. only for $c(\theta)/c$ .
The synchronisation vector $\varepsilon$ cancels identically from every two-way quantity — which is why the one-way speed of light is a convention while the two-way speed is a fact.
Three experiments, three independent rows, one solution: $A_{\mathrm{IS}} = \alpha + \tfrac12$ , $A_{\mathrm{KT}} = \alpha - \beta_{\!*} + 1$ , $A_{\mathrm{MM}} = \beta_{\!*} - \delta - \tfrac12$ . The linear system has determinant $-1$ — nonsingular, so the Lorentz transformation is the unique common zero.
Key idea: a test theory is a coordinate system for ignorance. Its job is not to be true — it is to make "we saw nothing" into a number with error bars, and to say precisely which theories that number kills.

The core insight

The RMS framework is not an alternative to relativity; it is a ruler for relativity. It assumes only that the transformation is linear and that some preferred frame exists in which light is isotropic — and refuses to assume anything else. What makes it work looks like a technicality and is the whole foundation: the free synchronisation parameter drops out of every measurable quantity. You cannot measure how fast light goes one way, because you would have to synchronise the clocks first. You can measure how fast it goes there and back. Everything the framework says, it says about round trips.

Where this lives today

A common currency, and increasingly a historical one. Every modern isotropy experiment still quotes an RMS number, because it is the only way to compare a cryogenic sapphire resonator in 2015 with a sandstone slab floating on mercury in 1887. But nobody searches in RMS space any more, for a structural reason: three dials, and the assumption that every kind of matter responds to a preferred frame identically. It cannot express the possibility that electrons feel a background field and photons do not — which is precisely what a realistic Planck-scale scenario predicts.

The successor is a field theory, not a kinematics. `aq-sr-standard-model-extension-lorentz-violation` drops the universal-rods-and-clocks assumption and attaches independent coefficients to every field. Its price is complexity; its reward is that it can say which particle broke the symmetry.

But the framework's deepest lesson needs no successor. This is where physics worked out that the one-way speed of light is a convention and the two-way speed is a fact. Every experiment in this chapter is a round trip, and the reason is written into equation (3): $\varepsilon$ cancels. That is not a limitation of our instruments — it is a statement about what "the speed of light" can mean.

The physical picture

Imagine three dials on a black box labelled "the universe": how much a moving clock slows, how much a moving rod contracts along its motion, and whether the transverse scale changes at all. Galileo sets all three to zero. Einstein sets them to $(-\tfrac12, +\tfrac12, 0)$ . FitzGerald and Lorentz, trying to save the aether, set them to $(0, +\tfrac12, 0)$ — rods shrink, clocks do not. The RMS framework says you are allowed to turn the dials, and that experiment, not postulate, says where they stop. Take a preferred frame $\Sigma$ in which light really is isotropic at $c$ — aether or CMB rest frame, it does not matter — and write the most general linear map to a lab moving through it at $v$ :

t = a(v)\,T + \boldsymbol{\varepsilon}\cdot\mathbf{x}, \qquad x_\parallel = b(v)\,(X_\parallel - vT), \qquad \mathbf{x}_\perp = d(v)\,\mathbf{X}_\perp .

Linearity is the one assumption not under test: it encodes homogeneity of space and time. Everything else is up for grabs — and two things here are routinely got wrong.

First, $b$ multiplies. It does not divide. The base writes $x = (X - vT)/b$ , and that one misplaced $b$ makes its central equation wrong. Check it in four seconds: at the base's own SR point $b = \gamma$ , so the correct form gives $x = \gamma(X - vT)$ — Lorentz — and the base's gives $x = (X - vT)/\gamma$ , which is not. Its own derivation card hedges a page later, printing `b = γ⁻¹?`: a question mark inside a displayed equation. When a document typesets its own uncertainty, believe it.

Second, $\varepsilon$ is not a parameter of nature. It is the synchronisation convention the experimenter chooses, and different choices give different values of $\alpha$ , $\beta_{\!*}$ , $\delta$ for the same universe. That sounds fatal and is the point: $\varepsilon$ cancels identically from every two-way quantity, so everything measurable is convention-free. Hence every classic test is a round-trip experiment.

The core equations

The transformation, with $b$ on the correct side:

\boxed{\;t = a(v)\,T + \varepsilon\,x, \qquad x = b(v)\,(X - vT), \qquad y = d(v)\,Y, \quad z = d(v)\,Z\;}\qquad (1)

where: $a$ is the clock-rate factor, $b$ the longitudinal length factor, $d$ the transverse one, $\varepsilon$ the synchronisation vector. Physical reading: at $a = 1/\gamma$ , $b = \gamma$ , $d = 1$ , $\varepsilon = -v/c^2$ this reduces exactly to $t = \gamma(T - vX/c^2)$ , $x = \gamma(X - vT)$ . The base's $x = (X-vT)/b$ does not — see derivation 1, where it breaks the interval by $4.67\%$ . The three functions, expanded about the Galilean point:

a(v) = 1 + \alpha\frac{v^{2}}{c^{2}} + \dots, \qquad b(v) = 1 + \beta_{\!*}\frac{v^{2}}{c^{2}} + \dots, \qquad d(v) = 1 + \delta\frac{v^{2}}{c^{2}} + \dots \qquad (2)

where: $\alpha, \beta_{\!*}, \delta$ are the RMS test parameters ( $\beta_{\!*}$ written to keep it apart from $\beta = v/c$ ). Physical reading: special relativity is the single point $(-\tfrac12, +\tfrac12, 0)$ , because $1/\gamma = 1 - \tfrac12\beta^2 + \dots$ and $\gamma = 1 + \tfrac12\beta^2 + \dots$ There is no first-order term — every deviation from Galilean physics enters at $v^2/c^2$ , which is why every test of relativity is a second-order experiment, and why they are all so hard. The two-way speed of light — the only speed of light anyone can measure — with $\theta$ the angle between the light path and the lab's velocity through $\Sigma$ :

\boxed{\;\frac{c(\theta)}{c} \;=\; 1 \;+\; \left(\tfrac12 - \beta_{\!*} + \delta\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta \;+\; \left(\beta_{\!*} - \alpha - 1\right)\frac{v^{2}}{c^{2}}\;}\qquad (3)

and its reciprocal, which is the form the experimental literature reports bounds on:

\boxed{\;\frac{c}{c(\theta)} \;=\; 1 \;+\; \left(\beta_{\!*} - \delta - \tfrac12\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta \;+\; \left(\alpha - \beta_{\!*} + 1\right)\frac{v^{2}}{c^{2}}\;}\qquad (4)

where: the two coefficient sets are negatives, because the quantities are reciprocals to $O(v^2/c^2)$ . Physical reading: this, and not a sign error, is the source of the contradiction between the forms in circulation — and both vanish at the SR point, so "it goes to zero for SR" cannot tell them apart. The $\varepsilon$ -dependence cancels identically from both. Note also that $\theta$ enters only as $\sin^2\theta$ : the anisotropy is a quadrupole, which is why the turntable signal lives at $2\Omega$ and not at $\Omega$ . The experiment–parameter dictionary, read off (4):

\boxed{\;A_{\mathrm{MM}} = \beta_{\!*} - \delta - \tfrac12, \qquad A_{\mathrm{KT}} = \alpha - \beta_{\!*} + 1, \qquad A_{\mathrm{IS}} = \alpha + \tfrac12\;}\qquad (5)

where: each is zero in special relativity, and each is bounded by one experiment. Physical reading: the coefficient matrix of $(A_{\mathrm{MM}}, A_{\mathrm{KT}}, A_{\mathrm{IS}}) = 0$ in $(\alpha, \beta_{\!*}, \delta)$ has determinant $-1$ — nonsingular, so the Lorentz values are the unique common zero, and dropping any row leaves a line of survivors.

Limiting cases

Case	Result	Meaning
$a=1/\gamma,\ b=\gamma,\ d=1,\ \varepsilon=-v/c^2$	the Lorentz transformation, exactly	special relativity. All three of (5) vanish; $c(\theta) = c$ at every angle
$a=b=d=1,\ \varepsilon=0$	$c_\parallel = c(1-\beta^2)$ , $c_\perp = c\sqrt{1-\beta^2}$	Galilean lab in a real aether. The discriminator — (3) must reproduce this exactly
$a=1,\ b=\gamma,\ d=1$	$A_{\mathrm{MM}} = 0$ but $A_{\mathrm{KT}} = A_{\mathrm{IS}} = +\tfrac12$	FitzGerald–Lorentz. Passes Michelson–Morley exactly; fails the other two
$v \to 0$	all corrections $\to 0$ as $v^2/c^2$	no first-order effect exists, whatever the parameters. Every test is second order
any $\varepsilon$	$c(\theta)$ unchanged	synchronisation is a convention. Two-way quantities are convention-free
$\delta$ alone	affects only $A_{\mathrm{MM}}$	the transverse scale is invisible to Kennedy–Thorndike and Ives–Stilwell

Deriving the framework, and settling the sign

Derivation AWhich side of the fraction b belongs on and why the base's transformation loses the interval

The Mansouri–Sexl transformation has one job: to contain the Lorentz transformation as a special case, so that "special relativity is true" is a statement inside the framework rather than an assumption underneath it. If it does not reduce to Lorentz at its own declared special-relativistic point, it is not a generalisation of relativity — it is a different theory wearing relativity's parameter names, and every bound extracted from it is meaningless. The check fails for the form the base prints.

Step 1. Write the candidate transformation with $b$ multiplying: $t = aT + \varepsilon x$ , $x = b(X - vT)$ , $y = dY$ . Substitute the SR values $a = 1/\gamma$ , $b = \gamma$ , $\varepsilon = -v/c^2$ (set $c = 1$ ):

x = \gamma(X - vT), \qquad t = \tfrac{1}{\gamma}T - v\,\gamma(X - vT) = \gamma T\!\left(\tfrac{1}{\gamma^{2}} + v^{2}\right) - \gamma v X = \gamma(T - vX) .

Why: the collapse in the middle is the whole check, and it is not a coincidence. $\tfrac{1}{\gamma^2} + v^2 = (1 - v^2) + v^2 = 1$ exactly — the definition of $\gamma$ doing the only work it ever does. Both Lorentz equations come out, from the correct $b$ and nothing else.

Step 2. Now try the base's form, $x = (X - vT)/b$ , with the base's own $b = \gamma$ :

x = \frac{X - vT}{\gamma} .

Why: no choice of $\varepsilon$ can rescue this, because $\varepsilon$ appears only in $t$ ; the spatial equation is simply wrong. Note what the base would need to make its form work: $b = 1/\gamma$ , hence $\beta_{\!*} = -\tfrac12$ — while its own parameter table declares $\beta_{\!*} = +\tfrac12$ . It is internally inconsistent, and it knows it.

Step 3. Kill it properly — test the invariant interval, which is what the base's own card claims to do. Take $v = 0.6$ , $\gamma = 1.25$ , event $(T, X) = (1, 0)$ . Correct form: $x = 1.25(0 - 0.6) = -0.75$ , $t = 1.25$ , so $t^2 - x^2 = 1.5625 - 0.5625 = \mathbf{1.000} = T^2 - X^2$ . Base's form: $x = -0.6/1.25 = -0.48$ , $t = 0.8 - (0.6)(-0.48) = 1.088$ , so

t^{2} - x^{2} = 1.18374 - 0.23040 = \mathbf{0.95334} \;\ne\; 1 .

Why: the base's transformation does not preserve the Minkowski interval — it is off by $\mathbf{4.67\%}$ at $v = 0.6c$ . And the base's own derivation card, at its final step, states: "Verify invariant interval: $c^2t^2 - x^2 = c^2T^2 - X^2$ ." Its own claimed verification fails on its own equation. This is the trap-27 pattern in its purest form: a check that was asserted, never run.

Step 4. Read off the correct expansions. With $a = 1/\gamma = 1 - \tfrac12\beta^2 + O(\beta^4)$ and $b = \gamma = 1 + \tfrac12\beta^2 + O(\beta^4)$ and $d = 1$ , comparison with (2) gives

\alpha = -\tfrac12, \qquad \beta_{\!*} = +\tfrac12, \qquad \delta = 0 .

Why: these are the SR values every source quotes, and they are consistent only with $b$ multiplying. The base quotes them too — which is precisely how you know its transformation, and not its parameter table, is the thing that is wrong. *Verified in verify_aq-sr-robertson-mansouri-sexl-framework.py: sympy confirms that $t = aT + \varepsilon x$ , $x = b(X-vT)$ with $(a,b,\varepsilon) = (1/\gamma, \gamma, -v/c^2)$ reduces identically to the Lorentz transformation and preserves $c^2t^2 - x^2$ ; that the base's $x = (X-vT)/b$ does not; and that at $v = 0.6c$ it returns an interval of $0.95334$ instead of $1$ .*

Go deeper · Derivation B · 2 minThe two-way speed of light and the sign that has been wrong in three places

Derivation BThe two-way speed of light and the sign that has been wrong in three places

The framework's central prediction, and this article's owned result. Three articles in this corpus print the Michelson–Morley coefficient and do not agree: two give $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ , one gives its negative $\left(\beta_{\!*} - \delta - \tfrac12\right)$ — and both vanish at the special-relativity point, so the first check anyone would reach for cannot discriminate them. This card derives the quantity from scratch, in one stated convention, and settles it in a limit where the answer is known independently.

Step 1. Invert (1). From $x = b(X - vT)$ and $t = aT + \varepsilon x$ : $T = (t - \varepsilon x)/a$ and $X = x/b + vT$ , with $Y = y/d$ . Differentiate, and impose that light is null in $\Sigma$ (where it is isotropic at $c = 1$ by hypothesis):

\mathrm{d}X^{2} + \mathrm{d}Y^{2} = \mathrm{d}T^{2} .

Why: this is the only physical input in the entire derivation. Everything downstream is algebra. Note that the lab's light cone is now a quadratic form with cross terms, not a cone — which is exactly what "the speed of light might depend on direction" means.

Step 2. Put a ray down a lab direction $\hat n = (\cos\theta, \sin\theta)$ at lab speed $u$ , and solve the quadratic order by order in $v$ with $a = 1 + \alpha v^2$ , $b = 1 + \beta_{\!*} v^2$ , $d = 1 + \delta v^2$ , $\varepsilon = e_1 v$ . Repeat for the return ray, whose one-way speed $u'$ is different. Why: the two one-way speeds are not equal — that is the entire content of a preferred frame — and each depends on $\varepsilon$ . Neither is measurable.

Step 3. Form the two-way speed, which is what a round trip measures:

c(\theta) \;=\; \frac{2}{\dfrac{1}{u(\theta)} + \dfrac{1}{u(\theta + \pi)}} \qquad\Longrightarrow\qquad \frac{c(\theta)}{c} = 1 + \left(\tfrac12 - \beta_{\!*} + \delta\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta + \left(\beta_{\!*} - \alpha - 1\right)\frac{v^{2}}{c^{2}} .

Why: the synchronisation parameter has vanished. It appears in both one-way speeds and cancels in the harmonic mean identically — the fact that makes the framework testable rather than a parametrisation of arbitrary conventions. And it must be the harmonic mean: a round trip of total length $2L$ takes $L/u + L/u'$ .

Step 4. Discriminate the sign in the Galilean limit, where the answer is known independently of any framework. Set $\alpha = \beta_{\!*} = \delta = 0$ : the lab is now just an aether frame with Galilean coordinates, and the round-trip speeds are elementary — $c_\parallel = c(1-\beta^2)$ from $L/(c-v) + L/(c+v)$ , and $c_\perp = c\sqrt{1-\beta^2}$ from the transverse hypotenuse. Equation (3) at those parameters gives

\frac{c(\theta)}{c} = 1 + \tfrac12\beta^{2}\sin^{2}\theta - \beta^{2} \qquad\Longrightarrow\qquad \frac{c(0)}{c} = 1 - \beta^{2}, \qquad \frac{c(\pi/2)}{c} = 1 - \tfrac12\beta^{2} . \;\checkmark

Why: this is the verdict, and it is airtight. The derived expression reproduces both exact aether round-trip speeds, with anisotropy coefficient $+\tfrac12$ , so $c(\theta)$ increases with $\sin^2\theta$ — the transverse round trip is the fast one, as it must be. The other convention gives $-\tfrac12$ here and would call the transverse direction slow, which is false. $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ is the coefficient of $c(\theta)/c$ . Full stop.

Step 5. Reconcile with the literature, which reports $\left(\beta_{\!*} - \delta - \tfrac12\right)$ and is also right. Invert (3): to $O(v^2/c^2)$ , $c/c(\theta) = 1 - \left(\tfrac12 - \beta_{\!*} + \delta\right)\beta^2\sin^2\theta - \left(\beta_{\!*}-\alpha-1\right)\beta^2$ , which is (4). Why: the two forms are coefficients of reciprocal quantities, not rival sign conventions. The experimental literature quotes bounds on $\left|\beta_{\!*} - \delta - \tfrac12\right|$ — a magnitude — so the sign never bites anyone in practice, which is exactly why the confusion survives in print. It bites the moment you ask which way an anisotropy would actually go. *Verified in verify_aq-sr-robertson-mansouri-sexl-framework.py: sympy derives $c(\theta)/c$ perturbatively from (1), confirms the coefficients $\left(\tfrac12-\beta_{\!*}+\delta\right)$ and $\left(\beta_{\!*}-\alpha-1\right)$ ; confirms the reciprocal carries their negatives; confirms $\varepsilon$ is absent from both; confirms BOTH vanish at the SR point (so the naive check is blind); and confirms the Galilean limit returns $1-\beta^2$ and $\sqrt{1-\beta^2}$ exactly.*

Figure

PUNCHLINE — the sign verdict, and why the obvious check cannot make it. The plot is the fractional deviation of the TWO-WAY speed of light from c, as a function of sin²θ, for the two rival conventions. At the SPECIAL-RELATIVITY point (centre panel) BOTH lines are flat at zero — which is why "it goes to zero for SR" proves nothing and why this defect survived in print. The discriminator is the GALILEAN limit (left panel), where the aether answer is known independently: the round trip is SLOWEST along the motion (c(1−β²)) and FASTEST across it (c√(1−β²)). Only the coefficient (½ − β⋆ + δ) rises with sin²θ and reproduces it. The other convention predicts the transverse direction is slow — and it is the coefficient of the RECIPROCAL, c/c(θ), which the experimental literature quotes because it reports magnitudes

Go deeper · Derivation C · 2 minThree experiments three rows one theory

Derivation CThree experiments three rows one theory

It is often said, loosely, that the three classic experiments "together imply the Lorentz transformation". That is a claim about a linear system, and a claim about a linear system is either provable or false. This card writes the system down, computes its determinant, and shows the implication is exact — and, just as importantly, exactly what is lost when any one row is removed.

Step 1. Read the three constraints off (4) and off the Ives–Stilwell second-order Doppler shift:

A_{\mathrm{MM}} = \beta_{\!*} - \delta - \tfrac12, \qquad A_{\mathrm{KT}} = \alpha - \beta_{\!*} + 1, \qquad A_{\mathrm{IS}} = \alpha + \tfrac12 .

Why: Michelson–Morley bounds the $\sin^2\theta$ coefficient (anisotropy), Kennedy–Thorndike the isotropic one (velocity dependence), Ives–Stilwell the clock-rate function $a(v)$ directly. Three experiments, three different physical quantities.

Step 2. Write it as a matrix on $(\alpha, \beta_{\!*}, \delta)$ and set all three to zero:

\begin{pmatrix} 0 & 1 & -1 \\ 1 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta_{\!*} \\ \delta \end{pmatrix} = \begin{pmatrix} \tfrac12 \\ -1 \\ -\tfrac12 \end{pmatrix} .

Why: the third row is the simplest — Ives–Stilwell measures $\alpha$ and nothing else. The system can be solved by eye, but the determinant is the thing that matters.

Step 3. Expanding along the bottom row, $\det = 1 \cdot (1\cdot 0 - (-1)(-1)) = \mathbf{-1}$ . Nonsingular, so the solution exists and is unique:

\alpha = -\tfrac12, \qquad \beta_{\!*} = +\tfrac12, \qquad \delta = 0 .

Why: this is the theorem the three experiments actually prove. Not "relativity is consistent with the data" — that is weak — but "within this framework, relativity is the only theory consistent with the data". The determinant is what upgrades a consistency check into a uniqueness proof.

Step 4. Now delete a row and see what survives. Drop Ives–Stilwell and the remaining two equations leave a one-parameter family, $\alpha = \beta_{\!*} - 1$ , $\delta = \beta_{\!*} - \tfrac12$ : at $\beta_{\!*} = \tfrac12$ that is relativity, but at $\beta_{\!*} = 1$ it is $(0, 1, \tfrac12)$ , which is not. Drop Michelson–Morley and $\delta$ becomes completely unconstrained — every $(-\tfrac12, \tfrac12, \delta)$ survives. Why: a line of theories, not a point. The FitzGerald–Lorentz aether $(0, \tfrac12, 0)$ is the sharpest illustration of what this buys: it passes Michelson–Morley exactly — $A_{\mathrm{MM}} = \tfrac12 - 0 - \tfrac12 = 0$ — while failing both of the others, $A_{\mathrm{KT}} = A_{\mathrm{IS}} = +\tfrac12$ . It is the theory the first experiment cannot see and the other two annihilate. Every one of the three rows is load-bearing, and the framework is what lets you prove it.

Step 5. Put in the numbers: $|A_{\mathrm{IS}}| \le 2\times10^{-8}$ , $|A_{\mathrm{KT}}| \lesssim 10^{-8}$ , $|A_{\mathrm{MM}}| = |(4 \pm 8)\times10^{-12}|$ . Why: the three are pinned at very different levels, because $A_{\mathrm{MM}}$ is the only one whose signal is a quadrupole and can therefore be modulated by simply spinning the apparatus. The parameters are boxed into a region around the Lorentz point, and nothing has ever been found outside it. *Verified in verify_aq-sr-robertson-mansouri-sexl-framework.py: sympy builds the $3\times3$ system, computes $\det = -1$ , solves it and returns $(-\tfrac12, \tfrac12, 0)$ exactly; and it confirms that deleting any single row leaves a one-parameter family containing at least one non-Lorentzian theory.*

Common trapBoth sign conventions vanish for special relativity — so that check is worthless

The anisotropy coefficient appears in print as $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ and as $\left(\beta_{\!*} - \delta - \tfrac12\right)$ — negatives of each other, and both zero at $\alpha = -\tfrac12,\ \beta_{\!*} = \tfrac12,\ \delta = 0$ . So the natural sanity check, "does it vanish for special relativity?", passes for both and therefore proves nothing. That is the mechanism by which the error propagates: the cheap test is blind to it. Two tests are not: (i) the Galilean limit, where the aether answer is known exactly and independently, and only $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ reproduces it; and (ii) asking which quantity the coefficient belongs to — $c(\theta)/c$ or its reciprocal. The literature reports magnitudes, $\left|\beta_{\!*} - \delta - \tfrac12\right|$ , so it has never had to care. An article that predicts the direction of an anisotropy does.

Figure

The base's central equation, tested on one event. The hyperbola is the invariant set c²t² − x² = 1 — every observer must agree that the event (T,X) = (1,0) lies ON it, because the interval is what a Lorentz transformation preserves. Transform that event into a lab moving at v = 0.6c. The CORRECT Mansouri-Sexl form, x = b(X − vT) with b = γ, lands exactly on the hyperbola at (t,x) = (1.25, −0.75) — and it IS the Lorentz transformation. The base's form, x = (X − vT)/b, lands at (1.088, −0.48), which is OFF the hyperbola: it returns an interval of 0.95334 where 1 is required, an error of 4.67%. And the base's own derivation card ends with the step "Verify invariant interval". It asserted the check. It never ran it

Worked examples

1 · What do the experiments actually box in?

Given $|A_{\mathrm{IS}}| \le 2\times10^{-8}$ , $|A_{\mathrm{KT}}| \lesssim 10^{-8}$ and $|A_{\mathrm{MM}}| \approx 8\times10^{-12}$ , how tightly is each parameter pinned to its Lorentz value?

Step 1 — identify. Invert the system of derivation 3, which is nonsingular ( $\det = -1$ ).

Step 2 — solve. From $A_{\mathrm{IS}} = \alpha + \tfrac12$ : $\ \alpha + \tfrac12 = A_{\mathrm{IS}}$ , so $\alpha$ is pinned directly, to $2\times10^{-8}$ . From $A_{\mathrm{KT}} = \alpha - \beta_{\!*} + 1$ : $\ \beta_{\!*} - \tfrac12 = (\alpha + \tfrac12) - A_{\mathrm{KT}}$ , so $\beta_{\!*}$ inherits both errors, $\lesssim 3\times10^{-8}$ . From $A_{\mathrm{MM}} = \beta_{\!*} - \delta - \tfrac12$ : $\ \delta = (\beta_{\!*} - \tfrac12) - A_{\mathrm{MM}}$ , so $\delta$ inherits all three, $\lesssim 3\times10^{-8}$ .

Step 3 — the boundary number, and the structure. $\alpha + \tfrac12$ , $\beta_{\!*} - \tfrac12$ and $\delta$ are all confined to about $\mathbf{3\times10^{-8}}$ of their Lorentz values. And note the lesson: the tightest experiment does not give the tightest parameter. Michelson–Morley pins its combination to $10^{-12}$ , four orders better than the others — but $\delta$ is not that combination, and solving for it drags in the loose $\alpha$ and $\beta_{\!*}$ . The bound on an individual parameter is set by the worst experiment in its dependency chain, not the best. The dials on the front of the universe are set to Einstein's numbers, and they have been checked to eight decimal places.

Physics of the portrait

Origins · 2 minCoordinate transformation and how this was discovered

Coordinate transformation

1949–1977 · Robertson embeds relativity in a family · Mansouri and Sexl add synchronisation · the framework becomes the language of every test

Why a transformation and not a person. The object at the centre of this article is a map between coordinate systems, and it carries physics — but not all of it, and not unambiguously. Part ( $a$ , $b$ , $d$ ) is physical: clocks really do run slow, rods really do contract. Part ( $\varepsilon$ ) is pure convention. Separating those two parts is the entire discipline of the subject. Robertson, 1949. H. P. Robertson asked, in Rev. Mod. Phys. 21, 378, a question that had gone unasked for forty years: is the Lorentz transformation forced by experiment, or merely consistent with it? He wrote a family of transformations from a preferred frame, showed that the three classic experiments each constrain a different coefficient, and proved that together they force the Lorentz values. It is the first time relativity was shown to be experimentally unique rather than merely successful. Mansouri and Sexl, 1977. Reza Mansouri and Roman Sexl rebuilt the framework around what Robertson left implicit — clock synchronisation — in a three-part paper in Gen. Relativ. Gravit. 8; the first part is about nothing else. Their insight: the convention is free, it changes the values of the test parameters, and it cancels from every two-way observable. That is why the RMS bounds in a 2015 cavity paper mean anything at all.

History

1887

— Michelson and Morley find the speed of light isotropic. The result is compatible with a live aether plus length contraction, so it settles less than it appears to.

1932

— Kennedy and Thorndike show, with an unequal-arm interferometer, that the speed of light does not depend on the laboratory's velocity.

1938

— Ives and Stilwell measure the second-order Doppler shift of a fast ion beam and see a moving clock run slow.

1949

— H. P. Robertson (Rev. Mod. Phys. 21, 378) asks whether the Lorentz transformation is FORCED by these three results rather than merely consistent with them, embeds relativity in a parameterised family, and proves that it is.

1977

— Mansouri and Sexl (Gen. Relativ. Gravit. 8, 497; 515; 809) rebuild the framework around clock synchronisation, in three papers, and give the test parameters alpha, beta and delta the form every modern experiment now reports.

1979

— Brillet and Hall run the first laser-cavity isotropy test and quote their result as a bound on the Michelson-Morley combination, establishing the framework as the standard currency.

1993

— Kostelecky and Potting open the Standard-Model Extension programme, which replaces the RMS framework's three universal dials with independent coefficients for every field.

2015

— Rotating cryogenic cavities bound the Michelson-Morley combination at the parts-in-10-to-the-12 level; the RMS parameters remain pinned to Einstein's values to about eight decimal places.

Going further

Primary sources. H. P. Robertson, "Postulate versus Observation in the Special Theory of Relativity", Rev. Mod. Phys. 21, 378 (1949) · R. Mansouri & R. U. Sexl, "A test theory of special relativity", Gen. Relativ. Gravit. 8: I. Simultaneity and clock synchronization, 497; II. First order tests, 515; III. Second-order tests, 809 (1977) · Herrmann et al., Phys. Rev. D 80, 105011 (2009). Texts: C. M. Will, Theory and Experiment in Gravitational Physics, §2.4 — the standard modern account.

Provenance. Signature $(+,-,-,-)$ . Every equation is re-derived and every number recomputed in verify_aq-sr-robertson-mansouri-sexl-framework.py, which reads this file, regexes the numbers out of the prose and asserts them. Base defects repaired: (i) the base prints $x = (X - vT)/b$ — the $b$ is inverted, and with the base's own $b = \gamma$ it neither reduces to Lorentz nor preserves the invariant interval ( $0.95334$ instead of $1$ at $v = 0.6c$ ), while the base's own card claims to verify that very interval; (ii) its LaTeX for that step reads `b = \gamma^{-1}?` — a question mark inside a displayed equation; (iii) it labels $\left(\beta_{\!*} - \delta - \tfrac12\right)$ as the coefficient of $c(\theta)/c$ , when it is the coefficient of the reciprocal $c/c(\theta)$ .

Cross-corpus consistency. The coefficient $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ printed in `aq-sr-michelson-morley-isotropy-tests` and `aq-sr-kennedy-thorndike-ives-stilwell` is correct for $c(\theta)/c$ and agrees with equation (3) here. The three articles use one convention and state the reciprocal relation explicitly. Scope. The synchronisation question is `aq-sr-conventionality-of-simultaneity`'s; the field-theoretic successor is `aq-sr-standard-model-extension-lorentz-violation`.

Builds on. The Lorentz Transformation · Time Dilation · Length Contraction · Conventionality of Simultaneity. Leads to. The Standard-Model Extension · Tests of the Photon Mass.

If the RMS parameters depend on the synchronisation convention, in what sense are they measured?

They are not, strictly — the combinations are. Change your synchronisation convention and the values of $\alpha$ , $\beta_{\!*}$ , $\delta$ change with it; nothing observable does. What is measured is always a two-way quantity, from which $\varepsilon$ has cancelled identically, and $A_{\mathrm{MM}}$ , $A_{\mathrm{KT}}$ , $A_{\mathrm{IS}}$ are exactly the convention-free combinations. A paper quoting " $\beta_{\!*} - \delta - \tfrac12 = (4 \pm 8)\times10^{-12}$ " has adopted the standard Einstein synchronisation to turn that combination into parameter values. The bound is real; the parameter values are that bound reported in a chosen chart.

Is the RMS framework a rival to special relativity?

No, and treating it as one is the commonest misreading. It is a superset: relativity sits inside it at a point, and the framework's only purpose is to measure how far the world is from that point. Nobody believes the dials are set anywhere but Einstein's values; the framework exists so that "nobody has found otherwise" can be given a number. The move is standard — the PPN formalism does the same job for general relativity. A test theory is a measuring instrument for theories, and like any instrument it is more useful for the nulls it delivers than for the possibility it entertains.

Why does the framework assume a preferred frame at all, when relativity says there is none?

Because you cannot look for something you have assumed away. A test theory's job is to give Lorentz violation somewhere to live, so that the data can then say it is not there. The preferred frame is a hypothesis under test, and the experiments detect the laboratory's motion through it — which is why the analysis needs a candidate frame and a value for $v$ . Historically that was the Sun's rest frame ( $v \approx 30\ \mathrm{km/s}$ ); today it is the frame in which the microwave background has no dipole ( $v = 369.8\ \mathrm{km/s}$ ), which has the great advantage of being observationally defined rather than postulated. The nulls say: whichever frame you nominate, we are not moving through it in any way light can feel.

The framework is purely kinematic. What does that cost?

It costs the ability to distinguish particles. RMS assumes rods and clocks — and therefore all matter, and light too — transform in one universal way. That is an extremely strong assumption, and not one any realistic Lorentz-violating theory would make: a Planck-scale background tensor field would couple to electrons, quarks and photons with independent strengths, and could easily be invisible to an optical cavity while showing up in a nuclear spin. RMS has no vocabulary for that, nor for violations acting on dynamics rather than kinematics — a modified dispersion relation, say. Everything the framework cannot see is exactly what the Standard-Model Extension was built to catalogue.

Why is the two-way speed of light the only measurable one?

Because measuring a one-way speed requires two clocks in different places to agree on what "the same time" means, and establishing that agreement requires sending a signal between them and assuming how long it took. Whatever you assume is what you will then "measure." That is not a technological limitation to be beaten with better clocks — it is a genuine circularity, and it is why $\varepsilon$ appears in the RMS transformation as a free parameter at all. A round trip uses one clock, in one place, and needs no convention. The isotropy of the one-way speed of light is a definition. The isotropy of the two-way speed is an experimental result — and it is the result this whole chapter is about.

What would it look like if one of these parameters were nonzero?

Take $\delta \ne 0$ , the transverse-scale parameter. Then $A_{\mathrm{MM}} \ne 0$ and the two-way speed of light acquires a $\sin^2\theta$ dependence: light would take measurably different times to go out and back along two perpendicular rods of the same proper length. Put them on a turntable and the difference would oscillate at twice the rotation frequency — a quadrupole, not a dipole — with sidebands at one sidereal day and one year as the Earth carried the apparatus through the preferred frame, and amplitude $|A_{\mathrm{MM}}|\,v^2/c^2 \approx |A_{\mathrm{MM}}| \times 1.5\times10^{-6}$ . We know exactly what it would look like. It is not there.