🚀 Lorentz Transformation Calculator
Transform spacetime coordinates between inertial reference frames using the Lorentz transformation. Enter an event's coordinates (x, t) in frame S and the relative velocity, and get the coordinates (x′, t′) in frame S′. Includes forward and inverse transforms with step-by-step solutions.
What Is the Lorentz Transformation?
The Lorentz transformation is the mathematical foundation of Albert Einstein's special theory of relativity. It describes how the space and time coordinates of an event change when observed from two different inertial reference frames moving at a constant relative velocity.
Unlike the classical Galilean transformation (which assumes time is absolute), the Lorentz transformation mixes space and time coordinates. This mixing is what gives rise to relativistic effects like time dilation and length contraction.
Key properties of the Lorentz transformation:
- Linear: The transformation is a linear map between coordinate systems.
- Preserves the spacetime interval: The quantity Δs² = c²Δt² − Δx² is invariant under Lorentz transformations.
- Reduces to Galilean: At velocities much less than c (β ≪ 1), the Lorentz transformation reduces to the Galilean transformation x′ ≈ x − vt, t′ ≈ t.
- Speed of light is invariant: The speed of light c is the same in all inertial frames.
The Lorentz Transformation Equations
For a boost along the x-axis with relative velocity v, the forward transformation (from frame S to S′) is:
x′ = γ(x − vt)
t′ = γ(t − vx/c²)
y′ = y, z′ = z
where the Lorentz factor is: γ = 1 / √(1 − v²/c²) and β = v/c
The inverse transformation (from S′ back to S) is obtained by replacing v with −v:
x = γ(x′ + vt′)
t = γ(t′ + vx′/c²)
Worked Example
Consider an event at x = 100 m, t = 1 μs in frame S. Frame S′ moves at v = 0.6c relative to S. Calculate the coordinates in S′.
- Lorentz factor: γ = 1/√(1 − 0.6²) = 1/√0.64 = 1/0.8 = 1.25
- x′: x′ = 1.25 × (100 − 0.6 × 3×10⁸ × 1×10⁻⁶) = 1.25 × (100 − 180) = 1.25 × (−80) = −100 m
- t′: t′ = 1.25 × (1×10⁻⁶ − (0.6 × 100)/(3×10⁸)) = 1.25 × (1×10⁻⁶ − 2×10⁻⁷) = 1.25 × 8×10⁻⁷ = 1×10⁻⁶ s
Notice that the x-coordinate changed sign — this is the relativity of simultaneity in action. The event is at a different position in the moving frame.
Why Use This Calculator?
- Forward and inverse: Calculate both S → S′ and S′ → S transformations.
- Step-by-step solution: See exactly how the calculation is performed.
- Spacetime interval check: Verify that the interval Δs² is preserved.
- Visual coordinate mapping: See how coordinates map between frames.
- Free & private: No registration, no data storage — all calculations are done in your browser.
❓ Lorentz Transformation Calculator FAQ
What is the Lorentz transformation?
The Lorentz transformation is a set of equations that describes how spacetime coordinates (x, t) change between two inertial reference frames moving at a constant relative velocity. It is the mathematical core of special relativity.
What is the difference between forward and inverse Lorentz transformation?
The forward transformation converts coordinates from frame S to frame S′ (S → S′). The inverse transformation converts from S′ back to S (S′ → S) and is obtained by replacing v with −v.
Why is time mixed with space in the Lorentz transformation?
In special relativity, space and time are not separate — they form a unified spacetime. The Lorentz transformation mixes the x and t coordinates because measurements of space and time depend on the observer's relative motion. This is why simultaneity is relative.
What is the spacetime interval?
The spacetime interval is the quantity Δs² = c²Δt² − Δx². It is invariant under Lorentz transformations — all inertial observers agree on its value. This is analogous to how distance is invariant under rotations in ordinary space.
What happens at low velocities?
At velocities much less than c (β ≪ 1), the Lorentz factor γ ≈ 1, and the Lorentz transformation reduces to the Galilean transformation: x′ ≈ x − vt, t′ ≈ t. This is why we don't notice relativistic effects in everyday life.
Can the Lorentz transformation be used for accelerations?
No. The Lorentz transformation applies only to inertial frames (frames moving at constant velocity relative to each other). For accelerating frames, general relativity is required.
What is a Lorentz boost?
A Lorentz boost is a Lorentz transformation that does not involve spatial rotation — just a change in velocity. The transformation along the x-axis (as in this calculator) is the simplest example of a Lorentz boost.
How accurate is this calculator?
This calculator uses the exact Lorentz transformation equations. Accuracy depends on the precision of your input values. You can adjust the decimal precision in the output to suit your needs.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required.