⏱️ Time Dilation & Length ContractionRelativity🇺🇦 Українська
Velocity
Speed v / c
0.6000 c
Presets
Options
Show Minkowski diagram
Animate clocks
Show proper time
Relativistic Values
Lorentz γ1.250
v / c0.600
Time ratio τ/t0.800
Length ratio L/L₀0.800
Relativistic p0.750 mc
Kinetic E0.250 mc²
γ = 1/√(1 − v²/c²)
Time dilation: t = γ·τ — moving clocks run slow.
Length contraction: L = L₀/γ — moving rulers
shrink.
The Minkowski diagram shows world lines: the tilted axis belongs
to the moving frame. Events simultaneous in one frame are not in
another.
Clock comparison — rest frame (left) vs moving frame (right)
Length contraction — rest ruler (top) vs contracted ruler
Minkowski spacetime diagram — ct vs x
About Time Dilation & Length Contraction
This simulation visualises the kinematic effects of Einstein's special relativity. As you raise the velocity of a moving observer, two synchronised clocks drift apart, a ruler shrinks along the direction of motion, and a Minkowski spacetime diagram tilts the moving frame's axes. Everything is driven by the Lorentz factor γ = 1/√(1 − v²/c²), with proper time τ = t/γ and contracted length L = L₀/γ.
The single velocity slider sets v/c from 0 up to 0.9999c, and preset buttons jump to physically meaningful cases such as the ISS, GPS satellites, muon decay, an LHC proton and the γ = 2 point at 0.866c. Toggles reveal the Minkowski diagram, animate the ticking clocks and show proper time. The live readout reports γ, time and length ratios, relativistic momentum p = γβmc and kinetic energy (γ−1)mc².
Frequently Asked Questions
What is time dilation?
Time dilation is the slowing of a moving clock as measured from a frame in which it is in motion. The simulation shows it through Bob's clock, which advances by the proper time tau = t divided by gamma while Alice's rest clock advances by the full coordinate time t. The faster Bob moves, the larger the gap between the two clock faces.
What is length contraction?
Length contraction is the shortening of an object along its direction of motion, given by L = L0 divided by gamma. The lower ruler in the simulation is the contracted one: at 0.866c it shrinks to half its rest length because gamma equals 2. Lengths perpendicular to the motion are unaffected.
What is the Lorentz factor gamma?
Gamma is defined as 1 divided by the square root of (1 minus v squared over c squared). It equals 1 when the object is at rest and grows without bound as the speed approaches c. It is the single number that scales both time dilation and length contraction, and the simulation displays it to four decimal places.
What does the velocity slider control?
The slider sets the speed as a fraction of the speed of light, v/c, from 0 to 0.9999 in steps of 0.0001. Every quantity in the panel, the clocks, the rulers and the Minkowski diagram updates instantly as you drag it. A value of exactly zero is nudged to 0.0001 to avoid dividing by zero.
What do the preset buttons do?
The presets load real or instructive velocities: the ISS at about 0.0000266c, a GPS satellite at roughly 0.0000039c, a cosmic-ray muon at 0.9941c, an LHC proton at 0.9999c, a twin paradox example at 0.5c, and 0.866c where gamma is exactly 2. They let you compare everyday and extreme regimes quickly.
Why do moving muons reach the ground?
Muons created high in the atmosphere have a half-life of only about 2.2 microseconds, too short to reach the surface at nearly light speed in a naive calculation. At 0.9941c, gamma is roughly 9.2, so their clocks run slow enough that many survive the trip. From the muon's own frame the atmosphere is length contracted instead, giving the same result.
What is the Minkowski diagram showing?
It plots time (ct) vertically against space (x) horizontally. Alice's rest axes are drawn at right angles, while Bob's primed axes tilt inward by an angle theta = arctan(beta). The dashed 45-degree lines are the light cone, and the moving dots mark each observer's current event on their world line. The tilt illustrates why simultaneity is frame dependent.
Is this simulation physically accurate?
The core formulae are exact special relativity: gamma, the time and length ratios, momentum p = gamma times beta, and kinetic energy gamma minus one (in units of mc squared) are all computed directly. The clock and ruler animations are simplified illustrations rather than a full ray-traced view, but the numerical readouts are correct for any speed below c.
Why can the speed never reach exactly c?
As v approaches c, the term under the square root in gamma tends to zero, so gamma and the relativistic energy diverge to infinity. Accelerating a massive object to c would require infinite energy, which is why the slider caps at 0.9999c. Only massless particles such as photons travel at exactly c.
Does each observer see the other's clock slow down?
Yes. Time dilation is symmetric: each inertial observer measures the other's clock as running slow, because neither frame is privileged. The apparent paradox is resolved only when one observer accelerates or turns around, as in the twin paradox, which breaks the symmetry and produces a genuine difference in elapsed proper time.
How does relativistic momentum differ from the classical value?
Classical momentum is just mass times velocity, but the relativistic version is p = gamma times beta times mc, shown in the readout. Because gamma rises steeply near c, the momentum grows far faster than velocity alone would suggest, which is why a particle's speed barely increases even as accelerators pump in huge amounts of energy.
Where does time dilation matter in everyday technology?
The GPS system is the classic example. Satellite clocks run slightly slow due to their orbital speed and slightly fast due to weaker gravity, and both effects must be corrected or positions would drift by kilometres each day. The simulation's GPS preset highlights just how tiny, yet practically essential, these velocity-based corrections are.