Experiment: Time Dilation and Relativistic Kinematics

Lorentz Transformations derived time dilation (Phenomenon 38.13) and Relativistic Dynamics the energy–momentum relation. This chapter reports the measurements. They span five orders of magnitude in \(\gamma-1\), from canal-ray ions at \(\beta\approx4\times 10^{-3}\) to storage-ring muons at \(\gamma=29.33\), and they agree with the same one-parameter formula throughout.

Ives and Stilwell: the moving atomic clock (1938)

Tests Phenomenon 38.13. Assuming Definition 38.12.

The cleanest laboratory test uses light itself as the clock. A moving atom radiating at a known rest frequency is a clock whose ticking is read optically; the first-order Doppler shift is a classical effect, but the second-order part is pure time dilation and survives when the first-order part is cancelled.

Apparatus

A canal-ray tube producing fast hydrogen ions (\(\mathrm{H}_{2}^{+}\) and \(\mathrm{H}_{3}^{+}\)) at accelerating potentials from about \(6788\,\mathrm{V}\) to \(18350\,\mathrm{V}\) — the tube sparked above roughly \(20\,\mathrm{kV}\). A spectrograph viewed the beam at \(7\,^\circ\) to its axis, recording simultaneously the direct beam and its image in a mirror behind the tube, so that light emitted forward and backward reached the plate side by side. The line studied was the blue-green H\(\beta\) Balmer line at \(\lambda_{0}=486.1\,\mathrm{nm}\).

Procedure

Viewing the same emitters both ways gives two lines displaced by the first-order Doppler effect, one to the red and one to the blue. Their mean position is what matters: the first-order terms cancel in the average, leaving the second-order shift

From Equation (38.27) (written for a moving light source). The experimental art is that the first-order displacement is some \(2\,\mathrm{mm}\) on the plate while the second-order shift to be extracted is about \(0.005\,\mathrm{mm}\).

Observations and data

Ion speeds ran from \(\beta=2.184736\times 10^{-3}\) to \(\beta=4.433244\times 10^{-3}\), that is \(v\) from about \(6.55\times 10^{5}\,\mathrm{m}/\mathrm{s}\) to \(1.33\times 10^{6}\,\mathrm{m}/\mathrm{s}\); the beam velocities inferred from the first-order shifts agreed with \(eV=\tfrac{1}{2}Mv^{2}\) to within \(1\,\mathrm{\%}\). The measured second-order shifts for the eight tabulated cases, against the values predicted by Equation (41.1), were

Interpretation

The mean displacement is nonzero, of the predicted sign, and of the predicted size — a classical clock would show none at all, since the first-order effects cancel exactly. Ives and Stilwell quote no single headline ratio with an uncertainty, presenting instead graphs of \(\Delta\lambda\) against \(\tfrac{1}{2}\beta^{2}\lambda_{0}\) and concluding that the results are a satisfactory confirmation. A later assessment by Mandelberg and Witten puts the experimental uncertainty of this generation of measurements nearer \(10\text{–}15\,\mathrm{\%}\), and their own repeat found the exponent \(n\) in \(\left(1-\beta^{2}\right)^{n}\) to be \(0.498(25)\) where special relativity requires exactly \(\tfrac{1}{2}\).

Primary references

[Ives:1938]. The \(10\text{–}15\,\mathrm{\%}\) uncertainty estimate and the exponent \(0.498(25)\) are from Mandelberg and Witten, J. Opt. Soc. Am. 52, 529 (1962), and are quoted here at second hand.

A muon is a clock with a single tick: it decays, and the distribution of decay times is exponential with a mean proper lifetime \(\tau_{0}\approx2.2\,\mu\mathrm{s}\). Muons are created by cosmic-ray primaries in the upper atmosphere and arrive at the ground in numbers that Galilean kinematics cannot account for.

Rossi and Hall: muon survival with altitude (1941)

Tests Phenomena 38.13 and 115.55. Assuming Definition 38.12.

Apparatus

A telescope of self-quenching Geiger–Müller tubes of \(4\,\mathrm{cm}\) internal diameter, in coincidence and anticoincidence, with a permanent absorber equivalent to \(186\,\mathrm{g}/\mathrm{cm}^{2}\) of lead above and between the counters and \(11.5\,\mathrm{cm}\) lead side walls. The assembly, except the variable absorber, sat in a thermostatic box and was carried by truck between stations.

Procedure

Coincidences with anticoincidence in the lower counters select particles that stop in the absorber; inserting \(115\,\mathrm{g}/\mathrm{cm}^{2}\) of lead and differencing isolates muons in a chosen residual-range interval, and hence a chosen momentum band. Comparing rates at two altitudes measures the fraction surviving the flight between them. A \(200\,\mathrm{g}/\mathrm{cm}^{2}\) iron compensator at the higher station equalizes the total material traversed, so that the difference in counts reflects decay rather than absorption.

Observations and data

Stations: Echo Lake, Colorado, at \(3240\,\mathrm{m}\) altitude and \(709\,\mathrm{g}/\mathrm{cm}^{2}\) atmospheric depth, and Denver at \(1616\,\mathrm{m}\) and \(856\,\mathrm{g}/\mathrm{cm}^{2}\) — a difference of \(1624\,\mathrm{m}\), equivalent to \(147\,\mathrm{g}/\mathrm{cm}^{2}\) of air, at practically the same geomagnetic latitude. Survival ratios between the two stations:

The softer group, of effective momentum \(5.0\times 10^{8}\,\mathrm{eV}/c\), disintegrates about three times faster than the penetrating group, of effective momentum \(1.5\times 10^{9}\,\mathrm{eV}/c\). Converting the survival fractions to a mean range before decay, \(L=4.5(6)\times 10^{5}\,\mathrm{cm}\), and using a muon mass of \(8\times 10^{7}\,\mathrm{eV}/c^{2}\), the authors obtain the proper lifetime

From Equations (41.2) and (41.3) (the two survival fractions converted to a mean range before decay).

Interpretation

The two momentum groups traverse the same distance, so under Galilean kinematics they would decay in the same proportion; they do not. The faster group survives better, by the amount Equation (38.27) requires for its larger \(\gamma\). The abstract states the conclusion plainly: the rates are “in agreement with the theoretical predictions based on the relativity change in rate of a moving clock”.

Primary references

[Rossi:1941]. “Mesotron” in the original is the modern muon.

Frisch and Smith: Mount Washington to sea level (1963)

Tests Phenomena 38.13 and 115.55. Assuming Definition 38.12.

Apparatus

A doped-polystyrene scintillator cylinder \(28\,\mathrm{cm}\) high and \(28\,\mathrm{cm}\) in diameter (about \(30\,\mathrm{g}/\mathrm{cm}^{2}\)), viewed by a \(13\,\mathrm{cm}\) photomultiplier, with the pulse train displayed on a Tektronix 581 oscilloscope at \(8.5\,\mu\mathrm{s}\) sweep and photographed. Iron absorber above the scintillator: \(600\,\mathrm{g}/\mathrm{cm}^{2}\) on the mountain and \(359\,\mathrm{g}/\mathrm{cm}^{2}\) at sea level (the paper reports the apparatus in inches and feet; SI equivalents are given here per the unit axiom of Measurement, SI Units, and the Theory of Errors), the difference compensating the \(218\,\mathrm{g}/\mathrm{cm}^{2}\) of air between the stations.

Procedure

The absorber and geometry select muons in a narrow speed band, between \(0.9950c\) and \(0.9954c\). A muon that stops in the scintillator gives a first pulse; its decay electron gives a second. Counting stopped muons per hour at the two altitudes measures the survival fraction over the descent, and the distribution of intervals between the pulse pairs measures the lifetime in the same apparatus.

Observations and data

Six one-hour runs on the summit of Mount Washington at \(1909\,\mathrm{m}\) gave 568, 554, 582, 527, 588 and 559 counts; five at Cambridge, Massachusetts, \(3\,\mathrm{m}\) above sea level, gave 412, 403, 436, 395 and 393. The averages are

The vertical descent is \(1907\,\mathrm{m}\), covered in \(6.4\,\mu\mathrm{s}\) of earth-frame time at \(0.9952c\). With backgrounds of \(13\pm2\) and \(11\pm2\) subtracted, the survival ratio is

and equating this to \(\exp\left[-1907/\left(\gamma c\tau_{0}\right)\right]\) with \(\tau_{0}=2.211(3)\times 10^{-6}\,\mathrm{s}\) gives the observed dilation factor

against the predicted effective value \(8.4(20)\) for this geometry.

Interpretation

Without dilation the expected sea-level rate would be about \(27\) per hour rather than \(408\): the muons survive some fifteen times better than Galilean kinematics permits. The measured factor Equation (41.7) agrees with the prediction within the uncertainties, which the authors are careful to describe as statistical only, the quoted \(\pm2\) on the prediction being “essentially only guessed at”. The experiment is included here because it is the most direct possible reading of Phenomenon 38.13: two numbers of counts, one distance, one lifetime.

Primary references

[Frisch:1963].

The CERN muon storage ring (1977)

Tests Equations (38.26) and (38.27).

Apparatus

The CERN Muon Storage Ring, holding polarized positive and negative muons in a circular orbit at Lorentz factor \(\gamma=29.33\), with detectors counting the decay electrons as a function of time.

Procedure

Muons circulate and decay; the decay-electron rate falls exponentially with a time constant that is the dilated lifetime. Measuring it for \(\mu^{+}\) and \(\mu^{-}\) separately tests the dilation factor and, by comparison with the at-rest lifetime, also tests CPT.

Observations and data

The dilation factor agrees with the special-relativistic prediction with a fractional error of \(2\times 10^{-3}\) at \(95\,\mathrm{\%}\) confidence. Dividing, \(\tau_{-}/\gamma=2.1948\,\mu\mathrm{s}\), consistent with the free-muon lifetime.

Interpretation

This is the most precise direct test of Equation (38.27), and it is performed at a \(\gamma\) thirteen times larger than Frisch and Smith's. Note also what it does not disturb: the muons are in circular orbit and therefore permanently accelerated, at some \(10^{18}\) times the acceleration of gravity, and the dilation depends only on their speed. That is the experimental content of the clock hypothesis — proper time along a worldline depends on the instantaneous velocity and not on the acceleration — which Equation (38.26) assumes and which the twin-paradox analysis of Section 40.6 needs.

Primary references

[Bailey:1977]. The derived rest lifetime \(\tau_{0}=2.1948(10)\,\mu\mathrm{s}\) is quoted from the abstract record rather than from the article body, which was not consulted; the storage-ring beam momentum and radius are deliberately not quoted here for the same reason. The 1979 final report of the same collaboration states the special-relativistic time transformation valid to \(8(7)\times 10^{-4}\).

Hafele and Keating: caesium clocks flown around the world (1972)

Tests Phenomenon 38.13. Assuming Phenomenon 42.5.

Apparatus

Four caesium-beam atomic clocks, flown as ordinary airline passengers eastward and then westward around the world, and compared before and after with the reference ensemble of the United States Naval Observatory.

Procedure

Both flights circumnavigate the globe; the eastward one moves with the Earth's rotation and the westward against it, so the two have different speeds relative to the non-rotating frame in which the analysis is done. The prediction [Hafele:1972a] was published before the observation [Hafele:1972b].

Observations and data

Predicted and observed time differences relative to the ground clocks, in nanoseconds, are reported for each direction, with the eastward clocks losing time and the westward clocks gaining it — opposite signs, which is the striking part.

Interpretation

This experiment is included with a caveat that matters for the architecture of this treatise: it is not a clean test of special relativity. The total effect is the sum of the kinematic dilation of this part, which always slows the flying clock, and the gravitational blueshift of The Equivalence Principle and Classical Tests, which speeds it up at altitude. The two are comparable in size at airline altitudes and have opposite signs, which is why the eastward and westward results differ in sign. The experiment therefore tests the sum of two predictions from two different theories; it is quoted here because it is the most familiar demonstration that dilation applies to ordinary macroscopic clocks, and its gravitational half is properly treated in Part V.

Primary references

[Hafele:1972a], the advance prediction; [Hafele:1972b], the observation.

Bertozzi: the speed limit (1964)

Tests Equations (40.4) and (40.5).

Apparatus

A Van de Graaff accelerator and a linear accelerator delivering electrons of kinetic energy from about \(0.5\,\mathrm{MeV}\) to \(15\,\mathrm{MeV}\), with a time-of-flight baseline of several metres and a calorimeter to measure the beam energy independently of the applied potential.

Procedure

Electron bunches are timed over the baseline, giving the speed directly; the kinetic energy is known from the accelerating potential and confirmed calorimetrically. Plotting \(v^{2}\) against \(T\) separates the two predictions.

Observations and data

As \(T\) rises through the megavolt range, \(v\) approaches \(c\) and saturates: the measured \(v^{2}\) flattens while \(T\) continues to climb in proportion to the applied potential. The calorimetric energies confirm that the energy really is being delivered to the beam and not lost.

Interpretation

Newtonian mechanics predicts \(T=\tfrac{1}{2}mv^{2}\), hence \(v^{2}\) linear in \(T\) without bound and \(v>c\) above \(0.25\,\mathrm{MeV}\). The data instead follow \(T=\left(\gamma-1\right)mc^{2}\) from Equation (40.7), with \(v\to c\) asymptotically. This is Corollary 40.10 measured directly, and it is the most economical demonstration that the relativistic energy–momentum relation Equation (40.5) and not the Newtonian one governs fast particles.

Primary references

[Bertozzi:1964].

Summary of the evidence

No result in Table 41.2 is compatible with Galilean kinematics, and all of them are described by the single factor \(\gamma\) with no adjustable parameter. Together with the isotropy tests of Experiments: Light, the Aether, and Time, they close the experimental case for the transformations of Lorentz Transformations.