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The binary-pulsar Einstein delay is the periodic clock-rate contribution to pulse arrival times from a pulsar moving in an eccentric binary. Near closest approach, the pulsar's orbital speed is larger and the companion's gravitational potential is deeper. Time dilation and gravitational redshift therefore make its clock run at a different rate relative to a distant time coordinate.

At leading order the variable timing term is ${\Delta }_E=\gamma \sin E$, with eccentric anomaly $E$ and time amplitude $\gamma$. Constant clock-rate offsets are absorbed into the fitted pulsar spin. This term is distinct from the geometric binary-pulsar Roemer delay and the propagation Shapiro time delay. Fitting it with other orbital observables constrains the component masses and tests general relativity.^[ref]

Bruno Rossi used penetrating cosmic-ray particles to test relativistic particle lifetimes. His experiment with David B. Hall compared particles selected by momentum and examined how many survived propagation through the atmosphere.

The decay probability depends on the proper time elapsed along the particle's path. Faster particles can travel farther in the laboratory because their rest-frame clock accumulates less proper time over a given laboratory interval. The measured momentum dependence connected an observed particle population to time dilation.^[ref]

Rossi also coauthored the early rocket detection of extrasolar X-rays, extending astronomical measurement beyond wavelengths accessible through the atmosphere.^[ref]

Atmospheric cosmic rays generate muons whose mean lifetime at rest is about $2.2\mu \mathrm{s}$. For a selected speed $v$ and production-to-detection distance $h$ in the terrestrial frame, special relativity predicts survival fraction $e^{-h/(\gamma v{\tau }_0)}$, where $\gamma$ is the Lorentz factor. Without time dilation, the exponent would instead be $-h/(v{\tau }_0)$. Comparing altitude-dependent counts at controlled energies, allowing for acceptance and energy loss, tests these alternatives. In the muon rest frame the terrestrial distance is shortened to $h/\gamma$ by length contraction, so the proper flight time and survival fraction agree with the terrestrial calculation. This frame comparison does not turn the same decay measurement into two independent tests.

An experimental particle physicist who, with Bruno Rossi, compared the decay of cosmic-ray muons selected by their penetrating power. The less energetic group decayed more rapidly than the more penetrating group, consistent with time dilation. Using absorbers to separate momentum ranges turned naturally occurring particles into a test of the rate of moving clocks.^[ref]

In the 1923 note Waves and Quanta, Louis de Broglie proposed a periodic process associated with rest energy, remaining in phase with a wave along the moving particle's path. He distinguished the internal frequency slowed by time dilation from the wave frequency based on total energy.^[ref] The proposal developed into his 1924 thesis.^[ref] In the 1929 lecture The wave nature of the electron, he reconstructed the argument using a Lorentz transformation.^[ref]

Here is that quantum phase argument in modern notation. Set ${\omega }_0=m{c}^2/\hslash$ in the rest frame. A Lorentz transformation to a frame where the particle has velocity $v$ givesAlong $x=vt$, proper time is $\tau =t/\gamma$, soThis is the harmony: evaluating the spatial wave at the moving particle gives the slower clock phase. At fixed $x$, its angular frequency is instead $E/\hslash =\gamma {\omega }_0$. Since $k=p/\hslash$, the de Broglie wavelength follows as $\lambda =2\pi /k=h/p$. The proposed clock was a theoretical construction, not an observed mechanical vibration.

An experimental contributor to measurements linking the lifetime of fast muons with relativistic time dilation.^[ref]

An experimental researcher whose team tested time dilation by measuring muon lifetimes in a circular orbit.^[ref]

An experimental researcher who, with Richard E. Keating, tested relativistic clock predictions using four portable atomic clocks flown around the world. Comparing their readings with a reference timescale after eastward and westward journeys gave different accumulated times. The comparison brought time dilation into a transportable laboratory instrument: the prediction depends on both the aircraft’s motion and its altitude through gravitational time dilation.^[ref]

A contributor to testing relativistic time dilation with muons, using particle decay as a measurable clock.^[ref]

A light clock measures time by counting round trips of a light pulse between mirrors. In its rest frame, mirrors separated by $h$ give a tick duration $2h/c$, where $c$ is the speed of light. Comparing this round trip in another inertial frame provides the geometric time dilation argument above.

A Lorentz transformation preserves the Minkowski metric. For a Lorentz boost, let $S^{\prime }$ move at constant velocity $+v$ along the $x$-axis of $S$, with parallel spatial axes and coincident origins at $t=t^{\prime }=0$. With $\beta =v/c$ and the Lorentz factor $\gamma$, the coordinates of the same spacetime event obey

The inverse Lorentz transformation replaces $v$ by $-v$. In particular, $x^{\prime }=0$ traces the moving origin $x=vt$, fixing the sign convention. The time formula is $t^{\prime }=\gamma (t-vx/c^2)$; dropping its position term would lose the relativity of simultaneity. ^[ref].

For the moving-clock example in time dilation, the second tick has $(t,x)=(5\mu \mathrm{s},3c\mu \mathrm{s})$. At $v=0.6c$, substitution gives $t^{\prime }=1.25(5-0.6\times 3)\mu \mathrm{s}=4\mu \mathrm{s}$ and $x^{\prime }=1.25(3-0.6\times 5)c\mu \mathrm{s}=0$. The clock stays at its own origin, and its coordinate time there equals its proper time.

Accelerator-produced muons circulating in a magnetic storage ring allow a direct comparison between their measured laboratory mean lifetime and $\gamma$ times the rest lifetime. The 1977 experiment at $\gamma =29.33$ measured positive- and negative-muon lifetimes near $64.4\mu \mathrm{s}$, consistent with time dilation. This is a controlled complement to atmospheric survival measurements, with different systematic effects.^[ref]

Particle decay is a spontaneous transformation of an unstable particle into allowed final particles. Conservation of energy, momentum, electric charge and the relevant interaction's other conserved quantities constrain the possibilities. A decay can be energetically allowed yet absent or weak because its transition amplitude is suppressed.

For an isolated state with an approximately constant rest-frame decay rate, the surviving population follows an exponential in proper time. The mean lifetime $\tau$ and energy decay width $\Gamma$ obey $\Gamma =\hslash /\tau$. A moving particle's laboratory flight-time distribution also includes time dilation.

Experimentally, detectors reconstruct decay products and sometimes a separated decay vertex. Correcting for acceptance and reconstruction efficiency is necessary before interpreting their counts as a lifetime or branching fraction. Different decay channels compete to make the total width.

A particle decay cascade follows successive unstable daughters to longer-lived or stable particles. Probabilities of specified independent decay paths multiply their branching fractions. Waiting times are governed by each particle's mean lifetime, with time dilation relating proper and laboratory times. A particle which appears stable on one observation timescale can be an intermediate on another.

For a timelike circular Schwarzschild geodesic, the coordinate orbital period and onboard proper time per orbit areUsing $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the Schwarzschild radius and orbital areal radius are $R_s=2GM/c^2\simeq 1.21789\times 10^{10}\mathrm{m}$ and $R=200R_s\simeq 2.43578\times 10^{12}\mathrm{m}$. ThereforeThe crew accumulates about $3836$ fewer seconds, or $63.9$ fewer minutes, per orbit. Both gravitational time dilation and kinematic time dilation contribute: a static clock at $R$ has $d{T}_{\mathrm{static}}=\sqrt{1-R_s/R}dt$, while the orbiting clock additionally has $d\tau =d{T}_{\mathrm{static}}/\gamma$, where $\gamma =\sqrt{398/397}$. A distant observer measures the coordinate period; comparing arrivals emitted at the same orbital phase also removes the repeated light propagation delay.

Use signed velocities in Skaro's inertial frame; the returning velocity $w$ must be negative. For the first two rockets, the Lorentz transformation of infinitesimal separations is $dx={\gamma }_u(d{x}^{\prime }+ud{t}^{\prime })$, $dt={\gamma }_u(d{t}^{\prime }+ud{x}^{\prime }/c^2)$. Dividing with $d{x}^{\prime }/d{t}^{\prime }=v^{\prime }$ gives the relativistic velocity-addition formulaFor the third rocket, its signed velocity in the second frame is $-w^{\prime \prime }$, so another application givesSince $0<v<c$ and $0<w^{\prime \prime }<c$, the denominator $1-v{w}^{\prime \prime }/c^2$ is positive. The third rocket returns from positive $x$ to Skaro if and only if $w<0$, henceEquality leaves it stationary in Skaro's frame, so it never returns from $x=2L$; a smaller $w^{\prime \prime }$ leaves it travelling outward.

Let $O$ denote departure, $A$ boarding the second rocket, $B$ boarding the third, and $C$ reunion with Skaro. In coordinates $(t,x)$, their positions on the worldline areIndeed, the first two coordinate durations are $L/u$ and $L/v$, and the returning leg has duration $-2L/w>0$. Equivalently, their $(ct,x)$ coordinates are obtained by multiplying each displayed time by $c$; these are the coordinates used for the spacetime diagram.

The diagram uses representative velocities satisfying the return condition; its event labels apply to the general formulas above. Skaro's elapsed time isOn a constant-velocity leg the traveller's proper time is $\Delta t/{\gamma }_V=\Delta t\sqrt{1-V^2/c^2}$, by time dilation. Neglecting the specified brief accelerations, Davros therefore ages byEvery contribution is positive because $w<0$. Each square-root factor is smaller than one, so $\Delta \tau <T$: Davros returns younger than a companion who remained on Skaro. The elapsed times compare the same departure and reunion events; they cannot be obtained by applying one time-dilation factor to the whole journey, whose worldline has three different velocities.

In the Earth inertial frame, the return journey takes $2d/v$. Bob is at rest in this frame, while Alice's proper time is reduced by the Lorentz factor $\gamma =(1-v^2/c^2{)}^{-1/2}$. Neglecting the short turnaround interval, their increases in age are

The twin paradox is the apparent contradiction from applying reciprocal time dilation to claim that each twin must be younger than the other. Alice changes inertial frames at the turnaround, so a single comoving inertial frame cannot describe her whole journey; Bob uses one inertial frame throughout. The twins follow different world lines, and integrating proper time along those world lines gives the displayed, consistent ages. The change in Alice's notion of simultaneity at turnaround removes the apparent contradiction. Acceleration distinguishes the paths; its magnitude is not an additional local rule for aging.

At time $t_0$, the accelerated proton has momentum $p=qE{t}_0$ and, by the relativistic energy-momentum relation, energyThe incident cosmic microwave background photon adds energy $E_{\gamma }$. Since the final state contains only the Delta-plus baryon, conservation of relativistic energy gives its laboratory-frame Lorentz factorIts rest-frame decay time $t_{\Delta }$ is a proper time, so time dilation gives the laboratory decay time

In the parent's rest frame, momentum conservation makes the massive daughter and photon momenta equal and opposite, of magnitude $p$. The relativistic two-body decay formulas giveSince $v=p{c}^2/E_R$ and $r=m/M$,The daughter's Lorentz factor isIf it decays after proper time $\tau$, time dilation gives $t=\gamma \tau$, while uniform motion gives $x=v\gamma \tau$. Therefore the event coordinates in the parent frame are

Since $r>3M$,Relative to the distant observer, the static clock $O_2$ suffers gravitational time dilation. The orbiting clock $O_1$ suffers the same gravitational effect together with the kinematic time dilation caused by its orbital motion, so it accumulates the least proper time.