Special relativity
LengthTimeEnergy SimultaneityVelocityLight
The 3D view needs WebGL. Every analysis below still works.
Lab frame, camera at rest
Motion along +x
Rest-frame outline

Drag to orbit. Click the view, then scroll or pinch to zoom.

Approaching c

Raise the speed and watch length, time, energy and light rearrange themselves. However far you push, you never arrive.

β0.0000000c
Lorentz factor γ relative to the lab
1.0000
Length along motion, L / L₀ in the lab frame
1.0000

Why never c: γ = 1/√(1 − β²) grows without bound as β → 1, so any object with mass would need infinite energy. The control stops at β = 0.9999999.

Object
Geometry

Where every point is at one instant of lab time. Lorentz-contracted along x only.

Length contraction

L = L₀ / γ

Measured in the lab, a moving object is shorter along its direction of travel by the factor γ. Its height and width do not change.

Rest length L₀ object’s rest frame
5.500 m
Length L lab frame
5.500 m
Height and width both frames
Unchanged
L / L₀ = 1/γ lab frame
1.0000

Time dilation

Δt = γ Δt₀

Seen from the lab, light in a moving clock follows a longer diagonal path at the same speed c, so each tick takes γ times longer. Every kind of clock slows in step.

Both clocks are compared in the lab frame. Display time runs at one lab second per second.

Tick of a clock at rest its own frame
1.000 s
Tick of the moving clock lab frame
1.000 s
One year aboard lasts lab frame
1.000 years

Energy and momentum

E = γmc²KE = (γ − 1)mc²p = γmv

Rest mass m is the same in every frame. What grows is energy and momentum, and they grow without limit as v approaches c.

Rest mass m invariant, all frames
1 kg
Total energy E / mc² lab frame
1.0000
Kinetic energy KE / mc² lab frame
0.0000
Momentum p / mc lab frame
0.0000
E for 1 kg lab frame
8.988 × 10¹⁶ J

Relativity of simultaneity

Δtlead = vL₀ / c²

Clocks synchronised aboard a train are not synchronised on the ground: there, the rear clock runs ahead by vL₀/c². A flash that reaches both ends together on the train reaches them at different times on the ground.

Train rest length L₀ train frame
1 light-µs ≈ 300 m
Train length ground frame
1.000 light-µs
Rear clock ahead by ground frame
0.000 µs
Flash reaches rear, front ground frame
0.500, 0.500 µs

Velocity addition

u = (v + w) / (1 + vw/c²)

Speeds do not simply add. Combine any two speeds below c and the result stays below c; it is rapidities, not velocities, that add.

0.5000000 c
Object v lab frame
0.0000000 c
Newtonian v + w lab frame, incorrect
0.0000000 c
Relativistic u lab frame
0.0000000 c

Doppler shift and aberration

f = f₀ / γ(1 − β cos θ′)cos θ′ = (cos θ + β) / (1 + β cos θ)

To a traveller, starlight ahead is blueshifted and crowded forward, while light from behind reddens and thins. The whole sky folds toward the direction of motion.

RedderBluer

Frequency ahead f / f₀ traveller’s frame
1.000
Frequency behind f / f₀ traveller’s frame
1.000
Frequency at 90° f / f₀ traveller’s frame
1.000
At the centre of your view f / f₀ traveller’s frame
1.000
Forward half of sky fits within traveller’s frame
90.00°

Notes on method

  • Units with c = 1 throughout. Speed is driven by rapidity φ (β = tanh φ, γ = cosh φ), which keeps values accurate near c.
  • Visual mode applies light-travel delay to vertex positions for a camera at rest in the lab, with observation time chosen so the object’s centre appears at the origin. Shading, Doppler colour and brightness of the object itself are not modelled; surface normals are estimated numerically.
  • The starfield uses an artistic colour mapping of the exact Doppler factor, with brightness compressed so dim, reddened stars stay visible; aberration and Doppler factors are exact.
  • Clock and simultaneity animations run at scaled display rates; every ratio shown is exact.
  • Objects are rigid and move inertially along x; acceleration and gravity are not modelled.
β 0.0000000 γ 1.0000
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