Two light clocks. The rest clock's pulse bounces straight up and down; the moving clock drifts right, so its pulse traces a longer zigzag — one full tick takes γ× as long. The bar below shows the ship contracting to 100 / γ metres against its 100 m rest length.
Special relativity says that time intervals and lengths themselves depend on how fast you are moving, and here two light-clocks race side by side so you can watch the moving one tick slow. Drag the speed slider (β = v/c) and watch the Lorentz factor γ stretch a 10-second trip and shrink a 100-metre ship, all from γ = 1/sqrt(1 - β2).
Special Relativity Lab
γ = 1 /
√1 − v²/c²
What Is the Special Relativity Simulator?
The special relativity simulator is a free interactive physics lab that runs in your browser — nothing to install and no sign-up. Adjust the speed β = v/c and watch the Lorentz factor γ stretch time and contract length.
| Control | Range | Step |
|---|---|---|
| Speed beta as a fraction of the speed of light | 0 – 0.99 | 0.01 |
How a Light Clock Reveals Time Dilation and Length Contraction
Special relativity stops being abstract the moment you watch a clock built from light. This simulator runs twin light-clocks: one at rest, one gliding past at a speed you set with the beta slider (β = v/c, from 0.00 to 0.99). Press Start to let the pulses bounce, and watch the moving clock's pulse trace a longer diagonal path. Because light travels at the same speed c for every observer, that longer path takes more time, so each tick stretches. That diagonal is the physical reason a moving clock runs slow.
Every readout follows from one quantity, the Lorentz factor γ = 1 / sqrt(1 - β2). It equals 1 at rest and climbs toward infinity as β approaches 1; at β = 0.6 it is exactly 1.25. The dilated time readout multiplies a 10.0 s proper-time interval on the moving clock by γ, so the stationary observer records γ times 10 = 12.5 s at β = 0.6. The contracted length readout divides a 100.0 m proper-length ship by γ, giving 80 m at that same speed. Only the dimension along the motion shrinks; transverse widths never change.
Notice that both effects vanish at everyday speeds and grow only as v nears c, which no massive object can ever reach. To put numbers on a specific journey, open the Time Dilation calculator; to explore why c is the universal limit, try the Speed of Light Simulator; or browse the full collection of interactive physics labs.
Where the ideal special relativity picture breaks down
The first thing to distrust is the labelling: one clock is called the rest clock and the other the moving clock, but no experiment can tell you which is which. Ride along with the gold clock and you would see the white one drift past at the same β, its ticks stretched by the same γ. Time dilation is reciprocal. Each observer measures the other's clock as the slow one, and neither is wrong; the simulator simply has to draw in some frame, and that choice makes the white clock look normal.
That reciprocity is why this page cannot settle the twin paradox. The relation γ = 1/sqrt(1 - β2) connects two frames in uniform motion, and the clock here glides at a fixed β forever. A twin who really leaves and returns has to turn around, and during that turn her frame is not inertial; the asymmetry that leaves her younger comes from the turnaround, not from γ. Apply the readout blindly to a round trip and you get a contradiction: a uniform-motion formula on a path that bends.
The contracting ship bar hides a second subtlety. Length is not a property a fast object carries; it is the distance between two marks made at the same instant, and simultaneity is what frames disagree about. Set β = 0.60 and the readout gives 80.0 m for a 100.0 m ship, but those 80 m are a measurement, not a photograph. A camera records something else: light from the far end left earlier than light from the near end, so a fast object photographs as sheared and rotated rather than flattened.
Finally, the formula is blind to gravity and the slider is blind at both ends. Speed is only half the story for a real clock, which also sits in a gravitational potential that shifts its rate; the satellite below needs two corrections of opposite sign. And at β = 0.01 the sim reports 2,998 km/s with γ = 1.000, when the true value is 1.00005: a real effect hiding below the third decimal. At the top end the growth turns violent, γ gaining just 1.294 from rest to β = 0.90 and a further 4.795 in the last nine steps of the slider.
Where this is actually used
Satellite navigation is the everyday case. A GPS satellite orbits at about 3.87 km/s, or β = 1.3 × 10-5, far below anything this slider resolves; through the same γ its clock loses 7.2 microseconds (µs) a day to motion. Altitude pushes the other way, adding about 45.7 µs a day, for a net gain near 38 µs. Since a receiver turns light-travel time into distance, an uncorrected 38 µs is roughly 11 km of position error a day, so the on-board clocks are built to run at an offset frequency before launch.
Cosmic-ray muons make the point with a stopwatch. A muon at rest lives 2.2 µs on average, so even at nearly light speed it should cover only about 660 m before decaying, and almost none created 15 km up should reach sea level. They arrive in quantity because they travel at roughly β = 0.995, where this expression gives γ = 10.0: the 2.2 µs becomes 22 µs in our frame and the flight stretches to about 6.6 km. In the muon's frame nothing lives longer; the atmosphere is what contracts, 15 km down to 1.5 km, crossed in about 2.3 lifetimes. Push the slider to its maximum, β = 0.99, for a comparable factor: γ = 7.089, the 10.0 s trip reads 70.9 s and the 100.0 m ship 14.1 m.
At the other extreme, the Large Hadron Collider accelerates protons to 6.8 TeV, which for a 938 MeV rest energy is γ = 7,247. Its 26,659 m ring measures 3.68 m in the proton's own frame, and a lap taking 88.9 µs in the laboratory occupies 12.3 nanoseconds of proton time. Every magnet ramp and radio-frequency kick is timed in the laboratory frame, because the beam's own clock runs over seven thousand times slow.
Frequently asked questions
What is time dilation?
A moving clock runs slow. A time interval measured on a clock moving at speed v is stretched by the Lorentz factor gamma when read by a stationary observer: dilated time = gamma times the proper time.
What is the Lorentz factor?
gamma = 1 / sqrt(1 - v^2/c^2). It equals 1 when at rest and grows without limit as the speed approaches the speed of light. At v = 0.6c it is exactly 1.25.
What is length contraction?
A moving object is measured shorter along its direction of motion by the factor gamma: contracted length = proper length / gamma. A 100 m ship at 0.6c measures 80 m. Lengths across the motion are unchanged.
Why can't anything reach the speed of light?
Because the Lorentz factor gamma heads to infinity as v approaches c, so it would take infinite energy to accelerate a massive object to light speed. Only massless things, like light itself, travel at c.
Does an object actually get heavier as it speeds up?
Its rest mass does not change. What grows with speed are momentum and energy, each carrying a factor of gamma: p = gamma m v and E = gamma m c^2. Calling the product gamma m a 'relativistic mass' is an older shortcut that most physicists have dropped, because the same factor does not work for a push applied across the direction of motion.
References & formula source
- Halliday, Resnick & Walker — Fundamentals of Physics, Chapter 37 (Relativity).
- Young & Freedman — University Physics with Modern Physics, §37.2–37.4 (Relativity of Time and Length).
- R. Nave — HyperPhysics, Georgia State University, "Time Dilation" / "Length Contraction" sections.
- Further reading: Special relativity — Wikipedia