Lorentz Transformation Calculator
Transforms the spacetime coordinates (x, t) of an event from one inertial reference frame to another moving at relativistic velocity. Used in special relativity to reconcile observations made by different observers.
About this calculator
The Lorentz transformation gives the position and time of an event as seen from a moving inertial frame S′ relative to a rest frame S. The key formulas are: γ = 1 / √(1 − v²/c²), x′ = γ(x − vt), and t′ = γ(t − vx/c²), where v is the relative velocity between frames, c = 299,792,458 m/s, x is the position in S, and t is the time in S. The Lorentz factor γ is always ≥ 1 and grows without bound as v → c. These equations encode both length contraction (Δx′ = Δx/γ for simultaneous events) and time dilation (Δt′ = γΔt for co-located events). They replace the classical Galilean transformation x′ = x − vt, which assumed absolute time.
How to use
An event occurs at x = 3×10⁸ m and t = 2 s in frame S. A second frame S′ moves at v = 0.6c = 1.799×10⁸ m/s. First compute γ = 1/√(1 − 0.6²) = 1/√0.64 = 1.25. Then: x′ = 1.25 × (3×10⁸ − 1.799×10⁸ × 2) = 1.25 × (3×10⁸ − 3.598×10⁸) = 1.25 × (−5.98×10⁷) ≈ −7.48×10⁷ m. And: t′ = 1.25 × (2 − 1.799×10⁸ × 3×10⁸ / (3×10⁸)²) = 1.25 × (2 − 0.5997) ≈ 1.75 s.
Frequently asked questions
What does the Lorentz factor gamma tell you physically?
The Lorentz factor γ = 1/√(1 − v²/c²) quantifies how much time dilation and length contraction a moving observer experiences relative to a stationary one. At v = 0, γ = 1 and there is no relativistic effect. At v = 0.87c, γ ≈ 2, meaning a moving clock ticks at half the rate of a stationary clock and lengths contract to half their rest value. As v approaches c, γ diverges to infinity, which is why no massive object can reach the speed of light — it would require infinite energy.
How do Lorentz transformations prove that simultaneity is relative?
Two events that are simultaneous in frame S (same t, different x) generally have different t′ values in frame S′, because the transformation for time is t′ = γ(t − vx/c²). The extra term −vx/c² means the time assigned to an event depends on its spatial location relative to the moving frame. This shatters the Newtonian notion of absolute, universal time. Einstein used this insight to argue that there is no single correct answer to whether two spatially separated events happen 'at the same time' — it depends on the observer's state of motion.
When should I use Lorentz transformations instead of classical Galilean transformations?
Use Lorentz transformations whenever the relative velocity between frames is a significant fraction of c (roughly v > 0.01c, or 3,000 km/s). In particle physics accelerators, muon decay experiments, and GPS systems, the differences are physically meaningful and must be accounted for. The Galilean transformation x′ = x − vt is the low-velocity limit of the Lorentz transformation — it is recovered when v/c → 0 and is perfectly adequate for everyday engineering problems involving cars, aircraft, or even rockets at chemical propulsion speeds.