Compute relativistic coordinate transformations between inertial frames. Input relative velocity β = v/c and event coordinates (t, x, y, z) to obtain transformed coordinates (t′, x′, y′, z′), Lorentz factor γ, and visualize the γ‑β relation.
In 1905, Albert Einstein revolutionized physics by postulating the constancy of the speed of light and the principle of relativity. The Lorentz transformation mathematically relates the coordinates of an event as measured in two inertial frames moving at constant relative velocity. Unlike Galilean transformations, Lorentz transformations preserve the spacetime interval and unify space and time into a four‑dimensional Minkowski spacetime.
Standard Lorentz boost (along x‑axis):
t′ = γ (t − β x) , x′ = γ (x − β t) , y′ = y , z′ = z
where β = v/c, γ = 1 / √(1 − β²).
The inverse transformation (S′ → S) is obtained by replacing β with −β. This symmetry ensures that physical laws remain identical in all inertial frames. The Lorentz factor γ grows rapidly as β approaches 1, leading to time dilation, length contraction, and the relativity of simultaneity — effects that become dominant at speeds close to light.
The Lorentz transformation can be derived from two postulates: the relativity principle and the invariance of the speed of light. Consider two inertial frames S and S′ with S′ moving at velocity v along the x‑axis. The linear transformation that keeps the wavefront equation c²t² − x² = c²t′² − x′² leads uniquely to the Lorentz formulas. Important consequences:
Our calculator uses natural units (c=1) for simplicity; you can treat t in seconds and x in light‑seconds, or any consistent unit system.
| Scenario | β | Event (t, x) in S | Transformed (t′, x′) in S′ | Physical meaning |
|---|---|---|---|---|
| Simultaneity test | 0.8 | (1, 2) & (1, 3) | Δt′ ≠ 0 | Events simultaneous in S are not simultaneous in S′ |
| Time dilation | 0.866 | (1, 0) | t′ = γ = 2.0 | Moving clock runs slower |
| Length contraction | 0.9 | (0, 1) | x′ = γ(1) ≈ 2.294 | Length in S′ appears contracted (measuring simultaneously) |
| Null interval (light) | any | (1,1,0,0) | t′ = x′ | Light cone preserved |
Global Positioning System satellites orbit Earth at about 14,000 km/h (β ≈ 1.3×10⁻⁵). Although small, the combined effects of special relativity (time dilation) and general relativity (gravitational blueshift) cause clock offsets of roughly 38 microseconds per day. Without Lorentz transformations and relativistic adjustments, GPS would accumulate errors of several kilometers within hours. This calculator demonstrates the basic Lorentz factor γ = 1/√(1−β²) — for β = 1.3e‑5, γ is extremely close to 1, but integrated over time the effect is significant.
The Lorentz transformation was initially proposed by Hendrik Lorentz (1899, 1904) to explain the null result of the Michelson‑Morley experiment. However, it was Einstein who gave it a physical interpretation based on the relativity of simultaneity and the abandonment of absolute time. Today, Lorentz invariance is a cornerstone of the Standard Model of particle physics, and every high‑energy physics experiment (like those at CERN) uses Lorentz boosts to compute decay products and collision events.