Kennedy-Thorndike and Ives-Stilwell Experiments

A glass tube, a few tens of kilovolts, and a beam of hydrogen ions streaming down it fast enough to glow. Look along the beam with a spectrograph and the blue-green $H_\beta$ line appears twice: once blueshifted, from the ions coming at you, and once redshifted, from the light a mirror sends back the other way. Classical physics is entirely comfortable with that. It says the two lines sit at $\lambda_0(1+\beta)$ and $\lambda_0(1-\beta)$ — and therefore that their midpoint lands exactly on the undisplaced line, at $\lambda_0$ , no matter how fast the ions go. The first-order shifts are equal and opposite. They cancel. In 1938 Herbert Ives and G. R. Stilwell measured that midpoint. It had moved. By four hundredths of an ångström — toward the red — on a line of nearly five thousand ångströms. It was the first direct laboratory sighting of a moving clock running slow, and the ions were the clock.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive.
Michelson–Morley is not enough, and this is provable. FitzGerald–Lorentz contraction — a real aether, real length contraction, no time dilation — produces an exactly null Michelson–Morley result at every velocity. Two more experiments are needed, and they are these.
Kennedy–Thorndike (1932) deliberately makes the interferometer arms unequal and watches for months instead of rotating. Equal arms hide a common $\gamma(v)$ factor; unequal arms expose it, and the Earth's orbit sweeps $v$ for free. Their null forces velocity-independence of $c$ .
Ives–Stilwell (1938) looks along a fast ion beam at the forward- and backward-shifted lines of the same transition. The relativistic wavelengths are $\lambda_{\pm} = \lambda_{0}\gamma(1\pm\beta) = \lambda_{0}\sqrt{(1\pm\beta)/(1\mp\beta)}$ , so their arithmetic mean is $\gamma\lambda_0$ — displaced from $\lambda_0$ by $\gamma - 1 \approx \tfrac12\beta^2$ . Classically it would be $\lambda_0$ exactly. That displacement IS time dilation.
⚠ The base prints $\lambda_{\pm} = \lambda_{0}\gamma(1\mp\beta)^{\mp1}$ , and that is wrong. At $\beta = 0.9$ it gives $22.94\,\lambda_0$ where the correct value is $4.359\,\lambda_0$ — a factor of five. The base's own next equation, $\tfrac12(\lambda_++\lambda_-) = \gamma\lambda_0$ , only works with the correct form. The base refutes itself one line later.
Owned result, machine-proved: the two means swap. Classically the arithmetic mean of $\lambda_\pm$ is $\lambda_0$ and the geometric mean is $\lambda_0/\gamma$ . Relativistically the arithmetic mean is $\gamma\lambda_0$ and the geometric mean is exactly $\lambda_0$ , to all orders in $\beta$ . Relativity is precisely "the classical Doppler pair, multiplied by $\gamma$ " — and Ives–Stilwell is the measurement of that one factor.
Owned result, machine-proved: their headline number falls out of their own apparatus. At their top potential of $18{,}350\ \mathrm{V}$ , an $H_2^{+}$ ion has $\beta = 4.42\times10^{-3}$ , so $\gamma - 1 = 9.78\times10^{-6}$ , and on the $H_\beta$ line at $4861\ \text{Å}$ that is a mean displacement of $\mathbf{0.0475\ \text{Å}}$ . The largest shift they report is $0.047\ \text{Å}$ . No free parameters.
Modern versions: stored $^{7}\mathrm{Li}^{+}$ ions at $\beta = 0.338$ confirm $\gamma(\beta)$ to $\mathbf{2.3\times10^{-9}}$ and bound the test-theory parameter $|\hat\alpha| \le 2\times10^{-8}$ .
Key idea: Michelson–Morley measured a length, Kennedy–Thorndike measured a velocity dependence, Ives–Stilwell measured a clock. It takes all three to build the Lorentz transformation — and only the third one ever saw anything move.

The core insight

The three classic experiments are not three confirmations of the same thing. They are three independent measurements, and the Lorentz transformation is their unique common solution. Michelson–Morley fixes one combination of the test-theory parameters, Kennedy–Thorndike a second, Ives–Stilwell a third; three planes in a three-dimensional parameter space, meeting at exactly one point. Remove any one of them and you are left with a line of surviving theories — and one of the survivors is always a live aether.

Where this lives today

Every atomic clock on Earth is a running Ives–Stilwell experiment. The second-order Doppler shift is not a curiosity to be measured once; it is a systematic that every frequency standard must subtract. In an ion clock the trapped ion's residual thermal motion gives it a nonzero $\langle\beta^2\rangle$ , and its transition frequency is therefore pulled by $-\tfrac12\langle\beta^2\rangle$ — the same $\gamma$ , now an error budget entry called the "time-dilation shift." At the $10^{-18}$ accuracy of modern optical clocks it is one of the largest corrections applied.

Kennedy–Thorndike lives on as the clock-versus-cavity comparison. The modern version replaces the unequal arms with an unequal pair of oscillators: an ultra-stable optical cavity, whose frequency depends on the speed of light, against an atomic clock, whose frequency does not. Their beat is watched as the Earth orbits. The best such experiments bound any velocity dependence of $c$ at the $10^{-8}$ level — a factor of roughly a million beyond 1932, and still, obstinately, zero.

And the tripod is now a scaffold with far more legs. The RMS parameters buy exactly three numbers, and they assume that every kind of matter responds to a preferred frame in the same way. Modern searches drop that assumption and let electrons, protons, neutrons and photons each carry their own coefficients — `aq-sr-standard-model-extension-lorentz-violation`. But the experiments have not changed. A cryogenic cavity on a turntable is still Michelson–Morley; a stored ion beam probed by two lasers is still Ives–Stilwell. Only the bookkeeping got richer.

The physical picture

Refuse to assume relativity. Allow a general theory in which moving clocks tick at some unknown rate and moving rods take some unknown length: three numbers describe the deviation from Galilean physics to second order — $\alpha$ (how much clocks slow), $\beta_{\!*}$ (how much rods contract) and $\delta$ (how the transverse scale changes). Special relativity is one point in that space: $\alpha = -\tfrac12$ , $\beta_{\!*} = +\tfrac12$ , $\delta = 0$ . The three experiments are three cuts through it.

Michelson–Morley cuts once. Its null result kills the combination $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ . But look what survives: the FitzGerald–Lorentz aether, $\alpha = 0,\ \beta_{\!*} = \tfrac12,\ \delta = 0$ — rods contract, clocks do not — sits exactly on that cut. A rotating interferometer can never tell it from relativity. `aq-sr-michelson-morley-isotropy-tests` carries the modern form of that test; what it cannot do is finish the job.

Kennedy–Thorndike cuts a second time, and the trick is the arms. In a symmetric interferometer both round-trip times carry a common factor $\gamma(v)$ , which cancels in the difference — all a rotation can show you — so the apparatus is blind to it. Make the arms unequal and the cancellation fails: the fringe position now depends on $v$ itself. Then wait. The Earth's orbital velocity swings by $\pm 30\ \mathrm{km/s}$ over six months, so any velocity dependence of $c$ must appear as a seasonal drift. Kennedy and Thorndike used an arm difference of about $16\ \mathrm{cm}$ — set by the coherence length of their mercury lamp, not by ambition — and watched for months. Nothing drifted.

Ives–Stilwell cuts the third time, and it is the only one of the three that ever measured a nonzero effect. It goes straight at the clock. Classically the forward- and backward-emitted wavelengths are $\lambda_0(1 \pm \beta)$ , symmetric about $\lambda_0$ , so their midpoint is pinned to the undisplaced line. Relativistically both pick up an extra factor of $\gamma$ , because the atom's own clock is running slow — and $\gamma$ multiplies the midpoint too. The midpoint moves. That is the whole experiment.

Figure

The tripod. Each classic experiment is a NULL result on one combination of the RMS test-theory parameters, and each null is a PLANE in the (alpha, beta, delta) space of possible theories. Michelson-Morley alone leaves a whole LINE of surviving theories — and the FitzGerald-Lorentz aether (contraction without time dilation, marked in red) lies on it. Kennedy-Thorndike cuts the line down. Ives-Stilwell fixes the last coordinate. The three planes meet at exactly ONE point: alpha = −1/2, beta = +1/2, delta = 0, which is the Lorentz transformation. No two of the three suffice

The core equations

The two longitudinal wavelengths from a source moving at $\beta$ , observed along the beam axis:

\boxed{\;\lambda_{\pm} \;=\; \lambda_{0}\,\gamma\,(1 \pm \beta) \;=\; \lambda_{0}\sqrt{\frac{1 \pm \beta}{1 \mp \beta}}\;}\qquad (1)

where: $\lambda_-$ is the light emitted toward the observer (blueshifted) and $\lambda_+$ the light emitted away and returned by a mirror (redshifted). Physical reading: the two forms are identically equal for every $\beta$ . The base prints $\gamma(1\mp\beta)^{\mp1}$ instead, a different and wrong function (see the warning). The $\gamma$ multiplying both is the moving atom's slow clock; the $(1\pm\beta)$ is ordinary Doppler. The two means, which is where the physics is:

\boxed{\;\bar\lambda \equiv \tfrac12(\lambda_+ + \lambda_-) = \gamma\,\lambda_{0}\;} \qquad\qquad \boxed{\;\sqrt{\lambda_+\lambda_-} = \lambda_{0} \;\;\text{(exactly, all orders)}\;} \qquad (2)

where: the arithmetic mean carries the signal; the geometric mean is an exact identity. Physical reading: the first-order shifts cancel in the arithmetic mean, and what is left is $\gamma$ and nothing else. The geometric mean is $\lambda_0$ for any $\beta$ — a free calibration line telling the experimenter their beam energy and spectrograph are honest. The observable, and its size:

\boxed{\;\frac{\bar\lambda - \lambda_{0}}{\lambda_{0}} \;=\; \gamma - 1 \;\approx\; \tfrac12\beta^{2}\;}\qquad (3)

where: this is the Ives–Stilwell signature. Physical reading: classical physics predicts exactly zero here, at every order in $\beta$ — the classical pair $\lambda_0(1\pm\beta)$ has arithmetic mean $\lambda_0$ identically. A nonzero $\bar\lambda - \lambda_0$ is therefore not a small correction to a classical effect; it is an effect with no classical counterpart at all. That is what makes it a clean test. The beam speed, from the accelerating potential:

\beta \;=\; \sqrt{\frac{2qU}{mc^{2}}} \qquad (4)

where: $U$ is the accelerating voltage, $m$ the ion mass. Physical reading: the only input the experiment needs beyond the spectrograph. At $U = 18{,}350\ \mathrm{V}$ for $H_2^{+}$ it gives $\beta = 4.42\times10^{-3}$ — and (3) then predicts the observed shift with no adjustable parameter. The test-theory dictionary, which is what makes the three experiments a system rather than a list:

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

where: each combination is zero in special relativity and each is bounded by one experiment. Physical reading: the three are linearly independent, so the system $A_{\mathrm{MM}} = A_{\mathrm{KT}} = A_{\mathrm{IS}} = 0$ has the unique solution $\alpha = -\tfrac12,\ \beta_{\!*} = \tfrac12,\ \delta = 0$ . Drop any row and the solution set becomes a line. `aq-sr-robertson-mansouri-sexl-framework` carries the transformation and the derivation of (5); this article carries the experiments.

Limiting cases

Case	Result	Meaning
$\beta \to 0$	$\lambda_\pm \to \lambda_0$ , $\bar\lambda \to \lambda_0$	no beam, no signal. The effect is second order — you need real speed
classical Doppler, $\lambda_\pm = \lambda_0(1\pm\beta)$	$\bar\lambda = \lambda_0$ exactly	the null prediction Ives–Stilwell falsified. No classical counterpart
$\theta = 90^\circ$ (transverse)	$\lambda_\perp = \gamma\lambda_0$	pure time dilation, no first-order term. Same $\gamma$ , harder geometry — see the warning
$\beta \to 1$	$\lambda_+ \to \infty$ , $\lambda_- \to 0$ , $\sqrt{\lambda_+\lambda_-} = \lambda_0$	the geometric-mean identity survives even here
$\alpha = -\tfrac12$	$A_{\mathrm{IS}} = 0$	special relativity. The clock rate is exactly $1/\gamma$
equal arms, $L_1 = L_2$	Kennedy–Thorndike signal $\to 0$	the common $\gamma(v)$ cancels. Unequal arms are the entire idea

The two means, and the number Ives and Stilwell actually saw

Derivation AWhy the arithmetic mean carries the signal and the geometric mean is exactly one

The Ives–Stilwell experiment is usually explained as "the second-order Doppler shift" — true, and completely uninformative. It does not say why anyone would think to average two lines, nor why the average is the right thing to average. This card derives both wavelengths, forms both natural means, and shows that relativity and classical physics swap which mean is trivial. That swap is the article's owned structural result, and it explains the apparatus: the geometric mean is the calibration, the arithmetic mean is the signal.

Step 1. Write the relativistic longitudinal Doppler shift. For a source receding at $\beta$ the received wavelength is $\lambda_{+} = \lambda_{0}\sqrt{(1+\beta)/(1-\beta)}$ , and for one approaching, $\lambda_{-} = \lambda_{0}\sqrt{(1-\beta)/(1+\beta)}$ . Rationalise: since $\sqrt{(1+\beta)/(1-\beta)} = (1+\beta)/\sqrt{(1-\beta)(1+\beta)} = \gamma(1+\beta)$ ,

\lambda_{\pm} = \lambda_{0}\,\gamma\,(1 \pm \beta) .

Why: the second form is the transparent one — it factorises the shift into an ordinary Doppler piece $(1\pm\beta)$ , which classical physics also has, and an overall $\gamma$ , which it does not. The base writes the middle expression as $\gamma(1\mp\beta)^{\mp1}$ — $\gamma/(1-\beta)$ for the upper sign — a different function entirely: at $\beta = 0.9$ it gives $22.94\lambda_0$ against the correct $4.359\lambda_0$ .

Step 2. Take the arithmetic mean.

\bar\lambda = \tfrac12(\lambda_+ + \lambda_-) = \tfrac12\lambda_{0}\gamma\big[(1+\beta) + (1-\beta)\big] = \gamma\,\lambda_{0} .

Why: the $\pm\beta$ terms are equal and opposite and vanish identically, removing the first-order shift (parts in $10^{3}$ for canal rays) that would otherwise swamp everything. What survives is a clean, unadorned $\gamma$ . This is also the internal check the base fails: its own next equation is $\tfrac12(\lambda_++\lambda_-) = \gamma\lambda_0$ , which is false for the form it printed one line earlier.

Step 3. Take the geometric mean.

\sqrt{\lambda_+\lambda_-} = \lambda_{0}\,\gamma\sqrt{(1+\beta)(1-\beta)} = \lambda_{0}\,\gamma\sqrt{1-\beta^{2}} = \lambda_{0} .

Why: exactly $\lambda_0$ , for every $\beta$ , to all orders. Not approximately, not to second order — identically. The $\gamma$ that the arithmetic mean exposes is precisely the $\gamma$ that the geometric mean cancels, because $\gamma\sqrt{1-\beta^2} = 1$ is the definition of $\gamma$ .

Step 4. Now do the same to the classical pair, $\lambda_{\pm} = \lambda_{0}(1\pm\beta)$ :

\tfrac12(\lambda_++\lambda_-) = \lambda_{0}, \qquad \sqrt{\lambda_+\lambda_-} = \lambda_{0}\sqrt{1-\beta^{2}} = \frac{\lambda_{0}}{\gamma} .

Why: the two means have swapped roles. Classically the arithmetic mean is trivial and the geometric mean carries a $1/\gamma$ ; relativistically the geometric mean is trivial and the arithmetic mean carries a $\gamma$ . And the reason is a single sentence: relativity multiplies the entire classical Doppler pair by $\gamma$ , and multiplying both entries by $\gamma$ multiplies both means by $\gamma$ . The experiment is the detection of that one factor, in the one mean where classical physics predicts a flat zero.

Step 5. Read off the observable. From Step 2, $(\bar\lambda - \lambda_0)/\lambda_0 = \gamma - 1 = \tfrac12\beta^2 + \tfrac38\beta^4 + \dots$ Why: a positive displacement — the midpoint moves to the red — and it is quadratic, so it is immune to the sign of the beam velocity and cannot be faked by a spectrograph miscalibration that is linear in $\beta$ . It also cannot be faked by getting the beam energy slightly wrong, because the geometric mean of Step 3 independently pins $\lambda_0$ . *Verified in verify_aq-sr-kennedy-thorndike-ives-stilwell.py: sympy confirms $\gamma(1\pm\beta) \equiv \sqrt{(1\pm\beta)/(1\mp\beta)}$ identically; that $\tfrac12(\lambda_++\lambda_-) = \gamma\lambda_0$ and $\sqrt{\lambda_+\lambda_-} = \lambda_0$ exactly; that the classical pair gives $\lambda_0$ and $\lambda_0/\gamma$ ; and that the base's $\gamma(1-\beta)^{-1}$ is a different function, disagreeing at $\beta = 0.9$ by $22.94$ against $4.359$ .*

Go deeper · Derivation B · 2 minReconstructing Ives and Stilwell's 0.047 angstroms from their own voltmeter

Derivation BReconstructing Ives and Stilwell's 0.047 angstroms from their own voltmeter

A derivation that stops at $\gamma - 1 \approx \tfrac12\beta^2$ has not tested anything: it has produced a formula. The experiment is a test only if the formula, fed with the apparatus's own settings and nothing else, reproduces the number on the plate. This card does that. The inputs are the accelerating potential Ives and Stilwell used, the ion they used and the line they watched — all three published — and the output is the largest displacement they reported. There is no adjustable parameter anywhere in it.

Step 1. Get the beam speed from the voltage. An ion of charge $q = e$ and mass $m$ falling through a potential $U$ acquires kinetic energy $qU$ , so for $qU \ll mc^2$ , $\beta = \sqrt{2qU/mc^{2}}$ . Ives and Stilwell ran canal rays of $H_2^{+}$ and $H_3^{+}$ over $6{,}788$ to $18{,}350\ \mathrm{V}$ . Take the top potential and the lighter ion, $H_2^{+}$ , whose rest energy is $mc^2 = 2m_pc^2 + m_ec^2 = 1877.06\ \mathrm{MeV}$ :

\beta = \sqrt{\frac{2 \times 18{,}350\ \mathrm{eV}}{1.87706\times10^{9}\ \mathrm{eV}}} = \mathbf{4.42\times10^{-3}} .

Why: the lighter ion is the faster one at fixed voltage, so $H_2^{+}$ sets the largest shift — the number an experimenter would quote as their headline. Note $\beta$ is only one part in $230$ : the first-order Doppler shift is a thousand times bigger than the effect being hunted, which is why the averaging trick of the previous card is not a convenience but the entire method.

Step 2. Convert to a clock rate.

\gamma - 1 = \frac{1}{\sqrt{1 - \beta^{2}}} - 1 = \mathbf{9.78\times10^{-6}} .

Why: ten parts per million. Form $\gamma$ and subtract $1$ in single precision and you lose most of the significant figures — $\gamma$ differs from $1$ in the sixth decimal place. Use $\gamma - 1 = \beta^2/(1-\beta^2+\sqrt{1-\beta^2})$ , or high precision, or the series $\tfrac12\beta^2 + \tfrac38\beta^4$ .

Step 3. Convert to ångströms on their line. They watched the $H_\beta$ Balmer line, $\lambda_0 = 4861\ \text{Å}$ . From equation (3),

\Delta\bar\lambda = (\gamma - 1)\lambda_{0} = 9.78\times10^{-6} \times 4861\ \text{Å} = \mathbf{0.0475\ \text{Å}} .

Why: this is the prediction, and it is made of nothing but a voltmeter reading, a proton mass and a wavelength. No fitting, no normalisation, no free scale.

Step 4. Compare with the plate. The largest displacement Ives and Stilwell report is $0.047\ \text{Å}$ . Why: agreement to about one percent, on a shift that classical physics says must be identically zero. It is worth pausing on how small the target was: $0.0475\ \text{Å}$ out of $4861\ \text{Å}$ is one part in $10^{5}$ — they were measuring the position of a spectral line to a hundredth of its own width, on a photographic plate, in 1938.

Step 5. Note what could not have faked it. A first-order Doppler error is odd in $\beta$ and cancels in the arithmetic mean. A wavelength-scale error is killed by the geometric-mean identity $\sqrt{\lambda_+\lambda_-} = \lambda_0$ , which is exact and independent of the beam energy. And the sign is fixed: $\gamma > 1$ , so the midpoint must move to the red, and it did. Why: this is why the experiment persuaded people. It is not that it found a small number where a small number was expected. It is that classical physics predicted a hard zero, the apparatus had a built-in calibration that made a scale error impossible, and the effect appeared with the right size and the right sign. *Verified in verify_aq-sr-kennedy-thorndike-ives-stilwell.py at mp.dps = 50: $\beta = 4.4218\times10^{-3}$ from $U = 18{,}350$ V and $mc^2 = 1877.06$ MeV; $\gamma - 1 = 9.776\times10^{-6}$ computed in a cancellation-safe closed form and cross-checked against the series; $\Delta\bar\lambda = 0.04752$ Å on $\lambda_0 = 4861$ Å; and the whole chain artefact-guarded against perturbations of $U$ , $m$ and $\lambda_0$ .*

Figure

PUNCHLINE — the spectrum Ives and Stilwell photographed, and the one classical physics predicted. BOTH theories put a blueshifted line to the left of λ₀ and a redshifted line to the right. They differ in ONE thing: where the MIDPOINT of the pair falls. Classically the two first-order shifts are exactly equal and opposite, so the midpoint is pinned to the undisplaced line at λ₀ — for any beam speed, to all orders. Relativity multiplies BOTH lines by γ, so the midpoint moves to γλ₀ — to the RED, by (γ − 1)λ₀. At their top voltage of 18,350 V that is 0.0475 Å on the 4861 Å Hβ line, and the largest shift they reported was 0.047 Å. The moving clock is the ion, and it is running slow

Common trapThe base's Doppler exponent is wrong and its own next line proves it

The base prints $\lambda_{\pm} = \lambda_{0}\gamma(1\mp\beta)^{\mp 1} = \lambda_{0}\sqrt{(1\pm\beta)/(1\mp\beta)}$ . The right-hand expression is correct; the middle one is not. For the upper sign it reads $\lambda_{0}\gamma/(1-\beta)$ , and $\gamma/(1-\beta) \ne \gamma(1+\beta)$ unless $\beta = 0$ . At $\beta = 0.9$ the correct value is $4.359\,\lambda_0$ and the base's is $\mathbf{22.94\,\lambda_0}$ — wrong by a factor of five. The tell is free and instant: the base's own next equation is $\tfrac12(\lambda_++\lambda_-) = \gamma\lambda_0$ , and that identity holds only for $\lambda_\pm = \lambda_0\gamma(1\pm\beta)$ . A defect that the next line of the same document refutes is the signature of transcription, not of physics — and it is exactly the class of error that survives every check except substituting the formula back into its own consequences.

Common trapIves-Stilwell is not a transverse Doppler measurement — and the distinction is the experiment

It is very often said that Ives and Stilwell "measured the transverse Doppler effect." They did not, and could not: the transverse configuration requires you to know the emission angle to a precision that a diffuse canal-ray beam cannot deliver, and a $1^\circ$ error in a $90^\circ$ geometry leaks a first-order shift of $\beta\sin(1^\circ) \approx 1.7\times10^{-2}\beta$ — comparable to the entire second-order signal. Ives and Stilwell looked along the beam, where the geometry is robust, and recovered $\gamma$ from the average of two longitudinal shifts instead. The quantity measured — the factor $\gamma$ in the moving atom's clock rate — is the same one the transverse effect exposes, which is why the loose name stuck. The method is the opposite: they leaned into the first-order shift and cancelled it, rather than trying to geometrically avoid it. `aq-sr-transverse-doppler-effect` owns the $90^\circ$ configuration.

Worked examples

1 · The shift Ives and Stilwell should have seen, from their own dials

Their canal rays were $H_2^{+}$ ions accelerated through $18{,}350\ \mathrm{V}$ ; they watched the $H_\beta$ Balmer line at $4861\ \text{Å}$ . What displacement of the mean of the two Doppler lines does relativity predict, and what does classical physics predict?

Step 1 — identify. Equation (4) for $\beta$ , then equation (3) for the fractional shift, then multiply by $\lambda_0$ . Classical prediction: exactly zero, from the limiting-cases table.

Step 2 — the beam. $mc^2(H_2^{+}) = 2 \times 938.272 + 0.511 = 1877.06\ \mathrm{MeV}$ , so

\beta = \sqrt{\frac{2 \times 18{,}350}{1.87706\times10^{9}}} = \mathbf{4.42\times10^{-3}} .

Step 3 — the clock. $\gamma - 1 = 9.78\times10^{-6}$ — and compute this properly. Naively forming $1/\sqrt{1-\beta^2}$ and subtracting $1$ in single precision destroys the answer; $\gamma$ and $1$ agree to five decimal places.

Step 4 — the boundary number. $\Delta\bar\lambda = (\gamma-1)\lambda_0 = 9.78\times10^{-6} \times 4861\ \text{Å} = \mathbf{0.0475\ \text{Å}}$ , to the red. Ives and Stilwell's largest reported displacement is $0.047\ \text{Å}$ . Classical physics predicts $0$ . One part in $10^{5}$ , measured on a photographic plate, and it decided the question.

2 · A modern Ives-Stilwell: lithium ions at a third of the speed of light

The ESR storage ring at Darmstadt holds $^{7}\mathrm{Li}^{+}$ ions at $\beta = 0.338$ , probed by two counter-propagating lasers. How much bigger is the signal, and what does the null buy?

Step 1 — identify. Same equation (3), vastly larger $\beta$ .

Step 2 — the clock. $\gamma = 1/\sqrt{1 - 0.338^{2}} = 1.0625$ , so $\gamma - 1 = \mathbf{6.25\times10^{-2}}$ . Against the $9.78\times10^{-6}$ of 1938 that is a signal six thousand times larger — the effect is no longer a whisper at the edge of a photographic plate but a six-percent shift.

Step 3 — the leverage. This is why the modern experiment wins twice over. It has a bigger signal and a better ruler: laser saturation spectroscopy locks to the resonance rather than estimating a line centroid. Botermann and co-workers confirm the relation between $\gamma$ and $\beta$ to $\mathbf{2.3\times10^{-9}}$ at this speed.

Step 4 — the boundary number. Translated into the test theory, that is $|\hat\alpha| \le \mathbf{2\times10^{-8}}$ — the tightest direct bound on the time-dilation parameter, and the third leg of the tripod nailed down. Time dilation is now confirmed to better than one part in $10^{8}$ , and the confirming apparatus is a ring of ions that would cross Australia in a fifth of a second.

Physics of the portrait

Origins · 2 minRelativistic Doppler effect and how this was discovered

Relativistic Doppler effect

1842–1938 · Doppler proposes the shift · Einstein predicts the extra gamma · Ives and Stilwell find it

The classical effect, and the gap in it. Christian Doppler proposed in 1842 that the pitch of a wave depends on the motion of its source. For light the classical prediction is $\lambda = \lambda_0(1 \pm \beta)$ , first order in $\beta$ , and for a century that was the whole story. The gap is invisible until you ask what happens to the average of an approaching and a receding shift: classically it is $\lambda_0$ , exactly, forever, and no amount of precision will move it. What relativity adds. Einstein's 1905 paper derives the shift from the two postulates and gets an extra factor of $\gamma$ multiplying the classical expression — because the source's own clock, which sets $\lambda_0$ , is running slow in the observer's frame. That single factor is the whole of relativistic kinematics visible in one measurement: it is why light emitted at exactly $90^\circ$ is redshifted at all (there is no classical transverse effect), and it is why the average of two longitudinal lines refuses to sit where it should. Why the effect, not a person, is the portrait. Herbert Ives spent his career at Bell Labs and was a lifelong sceptic of special relativity; he framed his own experiment as a test of a Lorentzian ether theory and never accepted that it confirmed Einstein. It is one of the cleaner ironies in the history of physics: the most direct laboratory confirmation of time dilation was performed, with great care and great skill, by a man who did not believe in it. The measurement outlived the interpretation, which is the correct order of events.

History

1842

— Christian Doppler proposes that the observed frequency of a wave depends on the motion of its source.

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 seems to.

1905

— Einstein derives the relativistic Doppler formula from the two postulates; it carries a factor of gamma that classical theory has no room for.

1932

— Kennedy and Thorndike (Phys. Rev. 42, 400) build a deliberately UNEQUAL-arm interferometer, arm difference about 16 cm, and watch it for months. No seasonal drift: the null limits any preferred-frame velocity to about 10 plus or minus 10 km/s and forces time dilation.

1938

— Ives and Stilwell (J. Opt. Soc. Am. 28, 215) measure the mean of the forward- and backward-shifted H-beta lines of hydrogen canal rays at 6,788 to 18,350 volts. The midpoint is displaced to the red by up to 0.047 angstroms. The first direct laboratory sighting of a moving clock running slow.

1941

— Ives and Stilwell repeat the measurement with improved technique and confirm the second-order displacement.

2007

— Reinhardt and co-workers use laser spectroscopy on stored lithium ions at beta = 0.03 and 0.064, bounding the time-dilation test parameter below 8 times 10 to the minus 8.

2014

— Botermann and co-workers (Phys. Rev. Lett. 113, 120405) reach beta = 0.338 in the ESR storage ring and confirm the relation between gamma and beta to 2.3 parts per billion, bounding the test parameter below 2 times 10 to the minus 8.

Going further

Primary sources. Kennedy & Thorndike, "Experimental Establishment of the Relativity of Time", Phys. Rev. 42, 400 (1932) · Ives & Stilwell, "An Experimental Study of the Rate of a Moving Atomic Clock", J. Opt. Soc. Am. 28, 215 (1938) · Botermann et al., Phys. Rev. Lett. 113, 120405 (2014), erratum Phys. Rev. Lett. 114, 239902 (2015) · Reinhardt et al., Nature Physics 3, 861 (2007). Texts: Rindler, Relativity, §4 · Will, Theory and Experiment in Gravitational Physics, §2.4.

Provenance. Signature $(+,-,-,-)$ . Every number is recomputed at 50 decimal digits in verify_aq-sr-kennedy-thorndike-ives-stilwell.py, which reads this file, regexes the numbers out of the prose and asserts them against what it computed. Base defect repaired: the base's eq[2] prints $\lambda_{\pm} = \lambda_{0}\gamma(1\mp\beta)^{\mp1}$ ; the correct middle form is $\lambda_{0}\gamma(1\pm\beta)$ , and the two differ by a factor of five at $\beta = 0.9$ ( $22.94$ vs $4.359$ ). The base's own eq[3] — $\tfrac12(\lambda_++\lambda_-) = \gamma\lambda_0$ — is right and is inconsistent with the form it printed one line earlier. The mean-wavelength result itself, $(\bar\lambda - \lambda_0)/\lambda_0 = \gamma - 1$ , is correct in the base and is retained.

Scope. The $90^\circ$ configuration is owned by `aq-sr-transverse-doppler-effect`; the longitudinal formula by `aq-sr-longitudinal-doppler-effect`. This article owns the experiments — the unequal-arm design, the two-means structure, and the reconstruction of the 1938 number. The isotropy leg is `aq-sr-michelson-morley-isotropy-tests`; the transformation behind equation (5) is `aq-sr-robertson-mansouri-sexl-framework`'s; the field-theoretic successor is `aq-sr-standard-model-extension-lorentz-violation`.

Builds on. Time Dilation · The Longitudinal Doppler Effect · The Lorentz Transformation. Leads to. The Robertson–Mansouri–Sexl Test Theory · The Standard-Model Extension.

If the first-order Doppler shift is a thousand times larger than the effect, how could anyone see the effect at all?

By arranging for the large term to cancel identically rather than trying to measure through it. The two lines sit at $\lambda_0\gamma(1+\beta)$ and $\lambda_0\gamma(1-\beta)$ ; add them and the $\pm\beta$ terms are exactly equal and opposite and vanish, leaving $2\gamma\lambda_0$ with no first-order residue at all. This is a structural cancellation, not a subtraction of two big measured numbers: it does not degrade as $\beta$ grows, it does not depend on knowing $\beta$ accurately, and it is immune to any error that is odd in the beam direction. The design lesson generalises far beyond 1938 — when a small signal sits on a large background, look for a symmetry that annihilates the background exactly.

Does Ives-Stilwell measure time dilation, or does it measure the Doppler effect?

Both, and separating them is the point. The Doppler effect is what carries the information out of the beam; time dilation is what the information is about. Classically the beam's own clock rate does not enter — the emitted wavelength in the atom's frame is $\lambda_0$ and stays $\lambda_0$ — so the classical Doppler pair is symmetric about $\lambda_0$ . Relativistically the atom's clock runs slow by $\gamma$ , so it emits at $\gamma\lambda_0$ in the lab before any Doppler shifting at all, and the Doppler factors $(1\pm\beta)$ then multiply that. The average strips the Doppler factors off and leaves the clock rate naked. The Doppler effect is the microscope; time dilation is the specimen.

Why does Kennedy-Thorndike need a whole year, when Michelson-Morley needed only a rotation?

Because they interrogate different things, and only one of them can be changed on a laboratory bench. Michelson–Morley asks whether $c$ depends on direction, and you can change direction by turning a table. Kennedy–Thorndike asks whether $c$ depends on the observer's speed — and you cannot change the laboratory's speed relative to a hypothetical preferred frame by any means available in a basement. The only velocity modulation on offer is the one the Earth supplies for free: its orbital velocity reverses direction over six months, changing its speed relative to any candidate frame by up to $60\ \mathrm{km/s}$ . The experiment has to be slow because the only lever available is the calendar.

Could a length change of the interferometer have faked the Kennedy-Thorndike null?

This is exactly the loophole, and it is why the experiment is not simply "Michelson–Morley with different arms." A null result can arise either because $c$ is velocity-independent, or because the arms happen to change length seasonally in just the way that cancels a real velocity dependence. Kennedy and Thorndike could not fully close this by themselves — their quartz frame was chosen precisely to be dimensionally stable, and the argument was that no thermal or elastic mechanism plausibly mimics a $v^2/c^2$ dependence with the right phase. The genuinely airtight statement is the joint one: the three experiments together are inconsistent with every set of $(\alpha, \beta_{\!*}, \delta)$ except the Lorentz values. No single one of them, alone, closes every door.

Herbert Ives did not believe in special relativity. Does that matter?

Not to the result, and that is the point worth taking away. Ives interpreted his own experiment as support for a Lorentzian ether theory — a real preferred frame, in which moving clocks physically slow down and moving rods physically shrink — and he maintained that position for the rest of his life. He was not being unreasonable: as a kinematic matter, Lorentzian ether theory and special relativity make identical predictions for every experiment in this article, because they sit at the same point $(\alpha, \beta_{\!*}, \delta) = (-\tfrac12, \tfrac12, 0)$ . The experiments cannot distinguish them, and no experiment can. What distinguishes them is that one posits an undetectable preferred frame and the other does not — a question of parsimony, not of measurement. Ives measured the right number and drew the metaphysics he preferred. The number was still right.

Why is the mean displacement a redshift rather than a blueshift?

Because $\gamma > 1$ , always, and $\gamma$ multiplies both wavelengths. A moving clock runs slow, so a moving atom emits at a lower frequency — a longer wavelength — than it would at rest, before any directional Doppler shift is applied. The directional shifts then push one line blue and the other red, symmetrically about the already-reddened centre. So the midpoint sits at $\gamma\lambda_0 > \lambda_0$ : to the red, always, whichever way the beam is pointed. That the sign is fixed and independent of the beam direction is itself a check — reverse the beam and the two lines swap places, but the midpoint stays put on the red side. A systematic that tracked the beam direction could not do that.