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Lorentz Transformation (Space-Time) Calculator

Coordinate transformations between two inertial frames moving at constant relative velocity.

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Transformed Position

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Overview

The Lorentz Transformation equations describe how space and time coordinates change when moving between inertial frames at constant relative velocities. They were developed to ensure the speed of light remains constant across all observers, effectively replacing Galilean relativity for high-speed motion.

Symbols

Variables

x' = Transformed Position, t' = Transformed Time, x = Stationary Position, t = Stationary Time, v = Relative Velocity

x'
Transformed Position
t'
Transformed Time
Stationary Position
Stationary Time
Relative Velocity
m/s
Speed of Light
m/s

Apply it well

When To Use

When to use: Use these formulas when calculating the space and time coordinates of events in relativistic scenarios where relative velocities are a significant fraction of the speed of light. They are essential for resolving paradoxes in special relativity and for high-precision synchronization in moving systems. Do not use them for accelerating frames, which require General Relativity.

Why it matters: These transformations are critical for the operation of modern satellite technology like GPS, which would lose accuracy without relativistic corrections. They also allow physicists to accurately predict particle lifespans in accelerators and explain how high-energy cosmic rays interact with our atmosphere. Essentially, they reveal that space and time are not absolute, but are interwoven into a single four-dimensional continuum.

Avoid these traps

Common Mistakes

  • Applying Galilean transforms (x' = x - vt) to systems where velocity is a significant fraction of c.
  • Forgetting to transform the time coordinate, assuming time is absolute.
  • Confusing the primed and unprimed frames during the algebraic inversion.

One free problem

Practice Problem

A spaceship travels past Earth at 0.6c (v = 1.8 × 10⁸ m/s). An observer on Earth measures a signal flash at a distance of 3.0 × 10⁹ meters from the origin at a time t = 10.0 seconds. Calculate the spatial coordinate (x') of this flash in the spaceship's frame. Use c = 3.0 × 10⁸ m/s.

Stationary Position3000000000 m
Stationary Time10 s
Relative Velocity180000000 m/s
Speed of Light300000000 m/s

Solve for:

Hint: First find the Lorentz factor γ using the ratio of v to c.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Halliday, Resnick, Walker, Fundamentals of Physics
  2. Wikipedia: Lorentz transformation
  3. Britannica: Lorentz transformations
  4. NIST CODATA
  5. Halliday, Resnick, and Walker, Fundamentals of Physics, 10th ed.
  6. Halliday, Resnick, and Walker, Fundamentals of Physics
  7. John R. Taylor, Edwin F. Taylor, and Wolfgang Rindler, Spacetime Physics
  8. Britannica, Lorentz transformations