Special Relativity Explorer

Drag the velocity slider to see how time, length, and mass transform at relativistic speeds.

Worked example — Special Relativity Explorer

Length contraction ratio	60%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Special Relativity Explorer

Special relativity is what follows from two statements held at once: the laws of physics look the same in every frame moving at constant velocity, and the speed of light is one of those laws. Neither mentions clocks or rulers. Everything on this page — the stretched time, the shortened length, the tilted axes — is a consequence of refusing to give either statement up.

What each symbol means

β: the speed, written as a fraction of light's. β = 0.8 is the value worth memorising: √(1 − 0.8²) = 0.6 comes out exact, a 3-4-5 triangle, so γ is exactly 5/3 and the ratio row reads a clean 60%.
γ: the Lorentz factor. It is a pure number — no units of its own — which is why the single value 1.6667 can act on seconds, metres and kilograms in the same panel without being converted between them.
t₀: the proper time: the interval read off one clock that is present at both events — the traveller's own watch. The subscript does not mean initial. It means measured where the thing is at rest, and reading it as "the time at the start" is the commonest way this formula gets applied upside down.
L₀: the proper length: measured in the frame where the object sits still. Measuring a moving object's length instead means marking both of its ends at the same moment — and simultaneity is exactly what two frames disagree about, which is where the contraction comes from.

Where the formula comes from

Build a clock out of light: two mirrors a distance d apart, a pulse bouncing between them. At rest, one tick is the round trip, t₀ = 2d/c.
Now let that same clock drift past you at speed v. The mirrors move sideways while the pulse is in flight, so the path you see is not the gap d but a longer diagonal — and the pulse still covers it at c, because the second postulate does not make exceptions for moving flashlights.
The diagonal, the gap and the sideways drift form a right triangle, so (c·Δt/2)² = d² + (v·Δt/2)². Substitute d = c·t₀/2 from step 1.
Collect the Δt² terms: Δt²(c² − v²) = c²t₀², so Δt = t₀/√(1 − β²) = γ·t₀. Nothing was assumed about what the clock is made of — which is why the same factor applies to a muon's decay as to a pulse between mirrors, and at β = 0.8 it is the 1.6667 s printed above.

How to read what you see

On the spacetime diagram, the orange dashed line at 45° is the path of light. Your axes are the blue pair; the traveller's x' and ct' are the green pair, and they lean towards that 45° line by the same angle from opposite sides — a scissor closing on the light cone but never crossing it. Watch what happens to the length: the rest length L₀ is drawn along the traveller's own x' axis, and the contracted L is its shadow on your x axis. Nothing is compressed; a segment is projected onto a differently angled ruler. The faint red hyperbola through the event is the calibration curve — every point on it is the same proper time from the origin, which is how you compare tick marks between two frames whose axes are scaled differently.

Assumes: Inertial frames only: one constant relative velocity, no acceleration, no gravity. There is a single β here and no way to turn around, so the twin paradox cannot actually be posed on this page — the return leg is the part that breaks the symmetry between the two observers, and there is no return leg.
Breaks when: The relativistic mass row is a survivor from an older way of writing this, and working physicists dropped it. Mass is treated as invariant now; the factor γ belongs to the momentum, p = γm₀v, not to the object. The reason is concrete: push the moving object along its direction of travel and its resistance goes as γ³m₀, push it sideways and it goes as γm₀. One object, two different "masses", depending on which way you shove — so the 1.6667 kg printed above is a bookkeeping device, not a heavier object.

Speeds refuse to add, but rapidities do 🖖

Set β = 0.5 and the Minkowski tab tilts the moving frame's axes by arctan β = 26.57°. Apply that same boost twice, though, and you do not arrive at β = 1: the speeds combine as (0.5 + 0.5)/(1 + 0.25) = 0.8 exactly, which you can check by setting β = 0.8 and reading γ = 1.6667. What does add is the rapidity ψ = artanh β, the quantity that diagram's tilt is built from — artanh 0.5 = 0.5493, doubled gives 1.0986, and tanh 1.0986 = 0.8. Rapidity is the hyperbolic angle of a boost, so boosts compose by plain addition the way rotations do, and no finite stack of them ever reaches c.

Why the Lorentz factor γ runs everything 🖖

The whole tool orbits one number: the Lorentz factor γ = 1/√(1−β²), where β = v/c. At walking speed γ ≈ 1, so relativity stays hidden; as v nears the speed of light c, γ climbs without limit. At v = 0.87c, γ = 2 — moving clocks tick half as fast and lengths halve. Because reaching c would demand γ = ∞, no massive object ever gets there: the price is infinite energy.

A speeding object looks rotated, not shrunk 🖖

Length contraction is what you measure, not what you see. Because light from an object's far side leaves earlier than light from its near side, a photograph of a body rushing past captures it apparently rotated rather than flattened — a sphere still shows a round outline. This Terrell–Penrose rotation went unnoticed for 54 years, until James Terrell and Roger Penrose spotted it independently in 1959.

Length contraction was invented to save the aether 🖖

In 1889 George FitzGerald sent a one-paragraph letter to Science proposing that a body moving through the aether physically shortens along its motion — by exactly the factor this page prints as L/L₀. Lorentz arrived at the same factor independently. Both were rescuing the aether from the Michelson–Morley experiment's null result: if the apparatus contracts by just the right amount, it cannot detect its own motion through the medium. Einstein's 1905 paper produced the identical number from two postulates and no aether at all. At β = 0.8 it is 0.6 either way — the arithmetic never changed, only what the arithmetic is about: not matter squeezed by a headwind, but two observers slicing spacetime along different angles. The factor still carries Lorentz's name.

The effect that matters most is the one you cannot see here 🖖

The muon preset is this page at its most convincing: β = 0.9992 gives γ = 25.005, stretching a muon's 2.2 µs lifetime to 5.501 × 10⁻⁵ s and thinning 650 m of atmosphere to 25.995 m, which is why particles that ought to decay far overhead reach the ground instead. That dilation has been measured directly, on muons circling a storage ring at CERN. The case with money attached is invisible here: load the GPS preset and the panel reads γ = 1 and Δt = 86400 s, because a satellite's β of about 10⁻⁵ puts the whole effect below five significant figures. It is real all the same — orbital speed loses a satellite clock roughly 7 µs a day, gravity gains it about 45 µs, and leaving the net uncorrected would walk a position fix off by some 11 km daily. Only the first of those three numbers is special relativity.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

At β = 0.8 the length contraction ratio prints as exactly 60%. Try β = 0.6, then β = 0.28, then β = 0.96 — and then try β = 0.5 to see why it behaves differently.

Show answer

0.6 gives 80%, 0.28 gives 96%, and 0.96 gives 28%: the pairs swap places. The ratio is 1/γ = √(1 − β²), so β² + (1/γ)² = 1 — the speed and the contraction are the two legs of a right triangle with hypotenuse 1, and that relation cannot tell which leg is which. Round answers therefore appear exactly where a Pythagorean triple does: (3, 4, 5) supplies the 0.6/0.8 pair and (7, 24, 25) supplies 0.28/0.96. β = 0.5 belongs to no triple, so it prints 86.603%, which is √3/2.
The GPS satellite preset runs at β = 0.0000133, a real orbital speed. γ reads 1, the dilated time reads 1 s, the contracted length reads 1 m, the ratio reads 100% — every relativistic effect has rounded away. Kinetic energy still reads 7.949×10⁶ J. Which readout is lying?

Show answer

None of them. The kinetic energy row is the only one multiplied by c², so it is the only one where the excess survives: γ − 1 is 8.845×10⁻¹¹, nine parts in a hundred billion, and the row substitutes exactly that rather than the rounded γ. Times c² = 8.99×10¹⁶ it comes to 7.9 megajoules per kilogram. Check it against Newton: the preset’s orbital speed is 3987.2 m/s, and ½mv² = ½ × 1 × 3987.2² = 7.949×10⁶ J, the same number. At orbital speeds relativity is unmeasurable in the lengths and unmissable in the energies, and the whole difference is how large c² is.

Problem solved in full

A metre stick measuring 60 cm at 0.8 c 5 steps
At 0.8c a metre stick measures 60 cm. Derive that, then find how badly Newtonian kinetic energy underestimates the same object.
1. Everything comes from one factor. At 0.8c the square root evaluates to 0.6 exactly — one of the few speeds where the arithmetic is clean, which is why textbooks use it.
  
  $γ = \frac{1}{1 - β ^{2}} = \frac{1}{1 - 0.64} = \frac{1}{0.6} = 1.6667$
2. Length contracts by the reciprocal of that factor, so 60% is not an approximation; it is 3/5 exactly.
  
  $\frac{L}{L _{0}} = \frac{1}{γ} = 0.60 = 60%$
3. The factor is worth tabulating, because its shape is the whole subject. It barely moves up to half light speed, then turns sharply: 2.29 at 0.9c, 7.09 at 0.99c, 22.4 at 0.999c.
  
  $β = 0.9 : γ = 2.294; β = 0.99 : γ = 7.089; β = 0.999 : γ = 22.37$
4. Kinetic energy is where the practical consequence sits. The relativistic expression is not ½mv² with a correction bolted on; it is the excess of the total energy over the rest energy.
  
  $K = (γ - 1) m c^{2} = 0.6667 m c^{2}$
5. At 0.8c that is 0.667mc², against the Newtonian 0.32mc².
  
  $\frac{1}{2} m v^{2} = 0.32 m c^{2} ⟹ Newton is short by a factor of 2.08$
Answer

The tool prints a length contraction ratio of 60%. The number the panel does not print is the energy one: Newton is short by a factor of 2.08 at this speed, and the shortfall diverges as the factor does. That is why particle accelerators are described by energy rather than speed — a machine can double a proton's energy and move its speed from 0.99c to 0.997c, which sounds like nothing and is the entire upgrade. Push the slider to 0.999c and watch the length ratio fall to 4.471% while the speed changes in the third decimal.

$γ = (1 - β^{2})^{- 1/2} : L / L_{0} = 60% at 0.8 c, K = 0.667 m c^{2}$

References (7)

Rapidity as the additive hyperbolic angle of a boost: J. A. Rhodes and M. D. Semon, "Relativistic velocity space, Wigner rotation, and Thomas precession." American Journal of Physics 72(7), 943–960, 2004.
On stating time dilation and contraction without the usual sleights: N. D. Mermin, "Lapses in relativistic pedagogy." American Journal of Physics 62(1), 11, 1994.
The contraction proposed to reconcile the aether with the Michelson–Morley null result: G. F. FitzGerald, "The Ether and the Earth's Atmosphere." Science ns-13(328), 390, 1889.
The same factor derived from two postulates, with no aether: A. Einstein, "Zur Elektrodynamik bewegter Körper." Annalen der Physik 322(10), 891–921, 1905.
Why the relativistic-mass row is a historical convention rather than current usage: L. B. Okun, "The Concept of Mass." Physics Today 42(6), 31–36, 1989.
Time dilation measured directly on muons in a storage ring: J. Bailey et al., "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit." Nature 268, 301–305, 1977.
The satellite-clock corrections, both signs of them: N. Ashby, "Relativity in the Global Positioning System." Living Reviews in Relativity 6, 1, 2003.

By the 42Math team · Updated 2026-06-14

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Muon (0.9c) - Muon at 0.9c: γ=2.29, time slows
GPS satellite - GPS satellite: tiny but measurable dilation
Half c - Half light speed: γ=1.155
0.99c - 0.99c: γ=7.09, dramatic effects