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The Michelson-Morley experiment splits coherent light into two perpendicular arms, reflects the beams and recombines them to observe interference fringes. Rotating the apparatus interchanges each arm's orientation relative to a proposed stationary luminiferous aether. The historical experiment used multiple reflections to increase the effective optical path and a rotatable support to compare orientations.^[ref]

For equal one-way effective arm lengths $\ell$, a classical stationary-aether model givesA right-angle rotation should reverse their difference, producing a fringe shift of approximately $2\ell v^2/(\lambda c^2)$. The observed orientation-dependent shift was much smaller than the stationary-aether prediction. In special relativity, the apparatus rest frame has the same vacuum speed of light along both arms, so no such orientation-dependent shift is expected. The result supports special relativity but does not, by itself, uniquely establish every postulate or exclude every modified aether theory.

For the rocket, take its tail at $x^{\prime }=0$ and nose at $x^{\prime }=L_0$, with coincident frame origins when $t=t^{\prime }=0$. Denote emission, reflection and return by $A,B,C$. In the rocket's inertial frame, their coordinates $(t^{\prime },x^{\prime })$ areThe inverse Lorentz transformation is $x=\Gamma (x^{\prime }+u{t}^{\prime })$, $t=\Gamma (t^{\prime }+u{x}^{\prime }/c^2)$, where $\Gamma =(1-u^2/c^2{)}^{-1/2}$. Thus the same spacetime events, in coordinates $(t,x)$, areThere is no requirement that emission have $t=0$ in this chosen laboratory coordinate system.

The nose-to-tail travel time is $\Delta t_{AB}=\Gamma L_0(1-u/c)/c$. If $L$ is the laboratory length measured at one laboratory time, a left-moving pulse and the right-moving tail close their initial separation at speed $c+u$. ThereforeThis derives length contraction from the pulse's travel. Comparing the positions of the two endpoints at different laboratory times would not measure the rocket's laboratory length.

Experimental evidence. Tests of special relativity check linked predictions about light propagation, clocks and four-momentum. Several independent kinds of experiment support those predictions.

The observed partial-drag coefficient agrees with this first-order prediction rather than simple Galilean addition of $u$. It tests a different consequence of the Lorentz transformation from a particle-lifetime measurement.^[ref]

Atmospheric muons and the two frame descriptions. Atmospheric cosmic rays create unstable particles whose decay chains produce muons. A stopped-muon experiment measures the time between a stopping signal and the subsequent decay electron, obtaining the proper mean lifetime ${\tau }_0\simeq 2.2\mu \mathrm{s}$. The associated light-travel distance is only $c{\tau }_0\simeq 660\mathrm{m}$, yet substantial numbers of energetic atmospheric muons reach detectors after travelling several kilometres.

For an ideal selected cohort produced height $h$ above a detector and moving downward at constant speed $v$, the flight time in the terrestrial frame is $h/v$. Time dilation predicts a laboratory mean lifetime $\gamma {\tau }_0$, hence survival fractionWithout time dilation, the predicted exponent would be $-h/(v{\tau }_0)$ and the surviving population much smaller. Altitude comparisons constitute a cosmic-ray muon time-dilation experiment: select comparable momenta or energies and account for detector acceptance, atmospheric energy loss and backgrounds before interpreting the count ratio.

For illustration, not as a claimed measurement, take $h=10\mathrm{km}$ and $v=0.998c$. Then $\gamma =15.82$, the laboratory flight time is $33.4\mu \mathrm{s}$, and the predicted surviving fraction is about $0.38$. Omitting time dilation would give only about $2.5\times 10^{-7}$ on the same idealized constant-speed model.

In the muon rest frame, its decay still follows $e^{-\Delta \tau /{\tau }_0}$. The atmosphere and ground move upward. Their separation, measured simultaneously in that frame, is shortened by length contraction:The same survival fraction follows. The terrestrial description uses a longer-lived moving particle; the muon-frame description uses a shorter atmospheric flight path. They describe the same production and detection events. This consistency illustrates both time dilation and length contraction, while the survival measurement is not two independent experimental tests merely because it can be described in two frames.

Energy-momentum invariant. For a particle with rest mass $m$,Since $E=m{c}^2+K$, substitution and cancellation of $m^2{c}^4$ giveThis relativistic energy-momentum relation reduces to $K\simeq p^2/(2m)$ when $K\ll m{c}^2$.

Laboratory beam energies. If both final muons are at rest in the laboratory, their total momentum is zero and their total energy is $2m_{\mu }{c}^2$. Four-momentum conservation therefore requires equal and opposite incoming momenta. Because the electron and Positron have equal rest masses, they then have equal total energies:Subtract each particle's own rest energy:Using the usual handbook rest energies gives approximately $105.1\mathrm{MeV}$ for each beam. These are the ideal kinematic threshold values. Just above threshold, the pair has small, equal and opposite momenta in this laboratory frame.

Positron momentum. Apply the relativistic energy-momentum relation to $E_{+}=m_{\mu }{c}^2$ and use its stipulated negative-$x$ direction:The electron has the opposite momentum, $p_{e,x}=+p_0$ with $p_0=c\sqrt{m_{\mu }^2-m_e^2}$.

Boost to the electron rest frame. The electron's velocity is $v=p_{e,x}{c}^2/E_{-}=p_0/m_{\mu }>0$. Thus $S^{\prime }$ moves in the positive-$x$ direction, and its Lorentz factor satisfiesThe positron has the same laboratory total energy as the electron, so $v{E}_{+}/c^2=v{m}_{\mu }=p_0$. The supplied Lorentz transformation then givesAs a sign check, the same boost gives $p_{e,x}^{\prime }=\gamma (p_0-v{E}_{-}/c^2)=0$, as required for the electron rest frame.

In the double-slit experiment, each photon is detected as a localized event carrying energy $h\nu$, while repeated events build the interference pattern of coherent probability amplitudes from the two slits. Acquiring information that distinguishes the paths suppresses the interference; splitting the classical intensity into two incoherent alternatives would not reproduce the fringes.

Put one plate before each slit at normal incidence, with the same known refractive index $n$ and illumination of vacuum wavelength $\lambda$. Relative to the plate-free reference, their thickness difference $\Delta d=d_1-d_2$ produces an optical path difference $(n-1)\Delta d$ and a phase shiftObserve the displacement of the full interference fringe pattern, or scan a controllable reference phase and estimate $\Delta \varphi$. This compares the plates directly; a detectable phase displacement establishes unequal thickness under the common-material assumption. For fringe spacing $\lambda D/b$, the displacement is $-D(n-1)\Delta d/b$, where $b$ is slit separation and $D$ the screen distance, with sign set by the slit labeling.

A differential phase measurement can resolve a small difference sensitively and avoid determining two large absolute thicknesses. It does not guarantee a universally faster or more photon-efficient test: noise, coherence, alignment, and required accuracy matter. Moreover a null phase shift at one wavelength only impliesso equality requires a prior bound, a broadband or multiple-wavelength measurement, or continuously counting interference fringes as the plates are introduced. If the refractive indices differ, this apparatus measures optical thickness difference rather than geometrical thickness difference.

A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index $n$ along the moving arm, and fixed vacuum wavelength ${\lambda }_0$. Write $\lambda ={\lambda }_0/n$ for the wavelength in that medium and take $\Delta L$ positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by $\Delta L$ adds $\Delta L$ before reflection and $\Delta L$ afterward. One bright-to-bright cycle means mirror travel $\lambda /2$; bright-to-dark means $\lambda /4$, half a cycle. All lengths must use compatible units, and $\Delta N$ is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.

For an original hypothetical example, suppose $\lambda =600\mathrm{nm}$ in the actual surrounding medium and a continuous scan gives $40.25$ cycles, including the final fractional cycle. Then $\Delta L=(600\mathrm{nm})(40.25)/2=12075\mathrm{nm}=12.075\mu \mathrm{m}$. With the wavelength treated as exact, a fringe-reading measurement uncertainty of $0.02$ cycle contributes $6\mathrm{nm}$ to the displacement measurement uncertainty. An uncertainty of $0.01\mathrm{rad}$ in a local phase difference instead corresponds to $600(0.01)/(4\pi )\mathrm{nm}\approx 0.477\mathrm{nm}$. This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.

To separate the oscillating readout from its optical coherence envelope, use the local modelHere $I$ is detected light intensity, $\bar{I}$ is the mean level, ${\varphi }_b$ is a fixed bias wave phase, and $\mathcal{V}$ is interferometric visibility. A fine scan of $D$ measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through $\mathcal{V}(D)$. The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known $\mathcal{V}$, one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.

The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving $\mathcal{V}(D)=\exp (-|D|/L_c)$ with $L_c=1\mathrm{mm}$. In vacuum, arm mismatches $|L_2-L_1|=0.60\mathrm{mm}$ and $0.10\mathrm{mm}$ give $\mathcal{V}=e^{-1.2}\approx 0.301$ and $e^{-0.2}\approx 0.819$. Matching the arms improves the phase difference readout without changing its ${\lambda }_0/2$ travel per cycle. A fine mirror scan of one cycle changes $D$ by only $600\mathrm{nm}$ for the stipulated $600\mathrm{nm}$ source, much less than $L_c$. These are calculated values, not historical data.

A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where $|dI/d\varphi |=\bar{I}\mathcal{V}$ is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.

Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed ${\lambda }_0$, the first-order change of the moving arm's round-trip path is $\Delta D\approx 2n\Delta L+2L\Delta n$. Thus an unrecognized $\Delta n$ mimics $\Delta L_{\mathrm{app}}=L\Delta n/n$. A hypothetical $L=0.10\mathrm{m}$, $n\approx 1$, and $\Delta n=10^{-7}$ give $10\mathrm{nm}$ apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.

The Sagnac effect is a difference in return times, or an associated interference fringe displacement, for light sent in opposite directions around a closed optical path in a rotating apparatus. The light leaves one beam splitter, travels around the same loop in opposite senses, and returns to the rotating beam splitter and readout. For a slowly rotating apparatus in flat spacetime, the leading laboratory arrival-time difference isHere $c$ is the vacuum speed of light, $\mathbf{\Omega }$ is the apparatus's angular velocity relative to an inertial frame, and $\mathbf{A}$ is the vector area oriented by the chosen positive direction around the closed curve $C$. With positive rotation about the upward normal vector, the counterclockwise beam is labelled $+$ and arrives later. Reversing rotation, or reversing the orientation convention alone, changes the sign of $\Delta t$.

The surprise is that a small loop can reveal its rotation by comparing two beams locally, even though each local vacuum speed of light measurement gives $c$. A rotating reference frame is not a single inertial frame: a beam must catch the moving receiver after a full circuit. The Sagnac area law makes this difference a practical way to measure angular velocity, while the Relativistic Sagnac delay on a circular ring distinguishes laboratory coordinate time from the receiver's proper time.

For a stationary rotating apparatus driven by a source with optical coherence and proper frequency ${\nu }_0$, define ${\lambda }_0=c/{\nu }_0$. The source wave phase advances by $2\pi {\nu }_0\Delta \tau$ between the emission times of the two contributions that meet at one readout spacetime event. Define the signed phase lag $\Delta \varphi$ of the $+$ contribution and the corresponding displacement $N$ in units of one interference fringe byThe sign of a photographed spatial displacement also depends on the imposed fringe orientation; this definition fixes the theoretical sign through the delayed beam. In an equal-amplitude output, a representative light intensity is $I(x)=I_{\max }[1+\cos (Kx-\Delta \varphi )]/2$, where $K$ sets the chosen spatial interference fringe spacing $2\pi /K$. Moving $\Delta \varphi$ by $2\pi$ shifts the pattern by one fringe. A fixed output exactly at a bright or dark fringe has zero first-order intensity slope, so a known phase bias or a spatial fringe readout is useful for determining a small signed delay.

Choose a square of side $0.30\mathrm{m}$, a rectangle $0.40\mathrm{m}$ by $0.20\mathrm{m}$, ${\lambda }_0=633\mathrm{nm}$ and $\Omega =+10\mathrm{rad}{\mathrm{s}}^{-1}$, with angular velocity parallel to the positive normal vectors. Both perimeters are $1.20\mathrm{m}$, but the areas are $0.090{\mathrm{m}}^2$ and $0.080{\mathrm{m}}^2$. Using $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the leading predictions areThese are original model choices and predictions, not measured points. The square has $9/8$ times the response despite the equal path lengths. Reversing from $+\Omega$ to $-\Omega$ gives $N(-\Omega )-N(+\Omega )=-2N(+\Omega )$: the displacement magnitude between opposite rotations is twice the displacement magnitude from rest. Confusing these two comparisons introduces a factor-of-two error.

A Sagnac interferometer uses a beam splitter to send light with optical coherence around a common closed optical path in opposite directions, then recombines the beams to read their phase difference. Mirrors or a guiding path define the loop; the source, beam splitter, loop and readout rotate together in the basic experiment. A small relative beam tilt makes spatial interference fringes whose displacement reveals a change of phase difference. At one fixed output, a photodetector can instead record the corresponding change in light intensity.

In 1913 Georges Sagnac mounted a lamp, collimator, beam splitter, four mirrors and photographic plate on a horizontal rotating platform. The mirrors defined a closed polygon; the two beams traversed it in opposite senses. A slight beam splitter adjustment produced visible interference fringes. A motor brought the platform to steady rotation, and switching the lamp exposed the plate. Comparing opposite senses of rotation revealed a reversing displacement. Changing the beam splitter adjustment supplied a further check against optical deformation. The report compared measured and calculated displacements. Georges Sagnac interpreted this as evidence for a luminiferous aether; the relativistic derivation below reproduces the delay without that hypothesis.^[ref]

The readout measures a relative wave phase, not two stopwatch readings separated by an optical transit time. Reversal isolates a contribution odd in angular velocity: if a signed interference fringe position is $q(\Omega )=q_0+k\Omega$ to leading order, then $[q(\Omega )-q(-\Omega )]/2=k\Omega$ cancels the fixed offset $q_0$. Changes of alignment or temperature correlated with reversal can survive this subtraction. Thus the experimental inference concerns a reproducible rotating-loop phase difference, with controls for disturbances; it is not by itself a unique test of an underlying propagation hypothesis.

An actual teaching experiment in LM-0300 Fabry Perot Spectrum Analyser by Walter Luhs, §§3.1.1–3.1.2, prepares a green laser beam by aligning it near and far along a rail, then centring back-reflected light from adjustable curved mirrors. An actuator using piezoelectricity moves one mirror. A screen first displays interference fringes; a focusing optical lens and photodiode then replace the viewing arrangement. The oscilloscope records the photodiode signal and the piezo drive reference on separate channels, using the latter as its trigger. Alignment and mirror separation are adjusted to suppress extra transverse normal modes; source temperature and drive electric current are controlled. Repeated peaks calibrate the scan using the appropriate free spectral range, and their spacing divided by their full width at half maximum estimates finesse.^[ref]

This experiment uses a confocal curved-mirror cavity, whose repeated peak families can have spacing $c/(4d)$ in air; it must not be calibrated blindly with the parallel-mirror $c/(2d)$. Extra transverse normal modes can mimic source features. A finite measured peak width bounds source sharpness only after instrumental broadening is considered; it does not determine the source's intrinsic width by itself.^[ref]

For an independent ideal scan calculation, hold a narrow laser wavelength fixed and vary the parallel-mirror gap $d$ at normal incidence. Advancing one resonance order requires $\Delta d=\lambda /(2n)$. If $d$ changes uniformly over the measured interval, the time between repeated peaks divided by their full width at half maximum in time equals finesse. The cancellation of scan rate follows from $d(t)=d_0+vt$: both peak spacing and peak width are divided by the same $|v|$. It fails if the scan rate changes appreciably over that interval. A practical control in the University of Washington's motor-driven flat-mirror experiment is to check that successive peaks are equally spaced before using the time axis this way; its handout also warns that table strain and aggressive low-pass filtering alter the trace.^[ref]

Spatial optical coherence compares field samples at different positions, usually across a transverse plane at equal time. Two small openings separated by $d$ sample this correlation: recombine their light with nearly matched travel times and measure interferometric visibility near the central interference fringes. A narrow opening in front of a broad-spectrum lamp can improve transverse spatial optical coherence while leaving its short temporal optical coherence. These are separate controls: narrow spectral selection by itself need not make an extended source spatially coherent. MIT's lecture notes illustrate the two-opening test and these contrasting source preparations. ^[ref].

To isolate source width, stipulate a distant, narrowband source with uniform angular brightness on $-\Theta /2\le \alpha \le \Theta /2$, and mutually uncorrelated emission from distinct source directions. Here $\Theta \ll 1$ is the full angular width in radians, $\lambda$ the common wavelength, and the two openings are separated by $d$ along the same transverse direction as the source extent. Neglect amplitude differences across the openings. Each source direction contributes a phase difference approximately $2\pi d\alpha /\lambda$. Averaging its complex cross term givesThe sinc function is extended continuously to one at $q=0$. The sign convention in the exponential depends on which opening is numbered first; the symmetric model gives the same real result either way. Equal detected single-opening light intensities make the magnitude equal to interferometric visibility. A displaced source center adds a deterministic wave phase, shifting the interference fringes without changing their contrast.

The strips show $I/(I_1+I_2)=1+g_{12}^{(1)}(0)\cos \varphi$ with equal single-opening light intensities, a common phase difference coordinate $\varphi$ and no single-slit diffraction envelope. At $q=1/2$, contrast is $2/\pi \approx 0.636620$; at $q=1$, it vanishes. At $q=3/2$, the correlation is $-2/(3\pi )\approx -0.212207$: weak fringes return with bright and dark positions exchanged. For $\lambda =500\mathrm{nm}$ and $d=0.5\mathrm{mm}$, the first zero occurs at $\Theta =1\mathrm{mrad}$. This is a calculated example, not an experimental measurement or a universal source-width law.

A proposed source-width control follows directly from this model: illuminate two fixed openings with a distant adjustable luminous slit of full width $w$ at distance $R$, so $\Theta \approx w/R$ under the small-angle approximation, and hold $d$ and the spectral selection fixed. Increasing $w$ overlays more mutually incoherent, displaced interference fringe patterns. Record profiles over several near-central fringes, measure each opening separately and correct intensity imbalance as in first-order optical coherence. Restrict the recorded region so the path difference stays small compared with the optical coherence length, and resolve the fringes with the detector. Real source nonuniformity, finite opening widths and spectral spread modify the stipulated sinc function law. This protocol is a model-based extension of the verified two-opening method, not a claim that the plotted curves were measured.

Thus an extended incoherent source can give locally excellent wave interference when the sampled separation is small, and almost none when it is larger. The useful scale $d\Theta \sim \lambda$ expresses how source extent limits the aperture over which coherent addition is available. The same angular averaging explains why narrowing a source opening can recover visible interference fringes, even without narrowing its optical spectrum.

A two-element radio interferometer coherently combines signals from two receiving antennas. With equal single-arm wave intensity $I_0$, projected baseline $b$, free-space wavelength $\lambda$ and imposed phase difference $\delta$, its directional response is the displayed formula. It follows by adding the two complex amplitudes and taking the squared modulus. Near the normal to the baseline, the first null is $\lambda /(2b)$ from a maximum. This narrow response repeats in interference fringes, so a long baseline alone does not give an unambiguous image.