Lorentz Transformation (Space-Time) Equation, Derivation & Rearrangements

Core idea

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.

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.

Symbols

Variables

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

x'

Transformed Position

m

t'

Transformed Time

s

x

Stationary Position

m

t

Stationary Time

s

v

Relative Velocity

m/s

c

Speed of Light

m/s

Walkthrough

Derivation

Derivation: Lorentz Transformations

The Lorentz transformations replace the Galilean transformations when relative velocities are comparable to c, preserving the invariant spacetime interval.

Frames S and S' are inertial; S' moves at speed v along x relative to S.
Lorentz factor γ = 1/√(1 − v²/c²).
Origins coincide at t = t' = 0.

1

Require invariance of the spacetime interval:

Any valid transformation must preserve s² (the invariant interval) between inertial frames.

s^{2} = c^{2} t^{2} - x^{2} = c^{2} t^{'2} - x^{'2}

2

Derive the transformation for position:

Time dilation and length contraction are built into this single equation.

x^{'} = γ (x - v t)

3

Derive the transformation for time:

Simultaneity is relative: two events with the same t but different x have different t' — the relativity of simultaneity.

t^{'} = γ (t - \frac{v x}{c ^{2}})

Result

t^{'} = γ (t - \frac{v x}{c ^{2}})

Source: University Physics — Special Relativity

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line because the variable t prime relates to x through a linear transformation where the slope remains constant. This linear relationship means that for a fixed velocity, a uniform change in position x results in a proportional shift in the transformed time t prime. The most important feature is that the slope is determined by the constant gamma, which dictates how significantly the spatial coordinate influences the time measurement between frames. Large values of x increase the impact of the velocity term on the transformed time, while small values of x show that the time coordinate remains closer to the original time value.

Graph type: linear

Why it behaves this way

Intuition

The Lorentz transformation can be visualized as a 'hyperbolic rotation' in a four-dimensional spacetime, where the relative velocity determines the 'angle' of rotation, causing space and time coordinates to mix.

x'

Spatial coordinate of an event in the 'primed' (moving) inertial reference frame.

Where an event occurs according to an observer moving at velocity v.

t'

Time coordinate of an event in the 'primed' (moving) inertial reference frame.

When an event occurs according to an observer moving at velocity v.

x

Spatial coordinate of an event in the 'unprimed' (stationary) inertial reference frame.

Where an event occurs according to a stationary observer.

t

Time coordinate of an event in the 'unprimed' (stationary) inertial reference frame.

When an event occurs according to a stationary observer.

v

Constant relative velocity of the 'primed' frame with respect to the 'unprimed' frame, typically along the x-axis.

How fast one observer is moving relative to another.

c

The speed of light in a vacuum, a universal constant.

The ultimate speed limit in the universe, linking space and time.

γ

The Lorentz factor, quantifying the magnitude of relativistic effects like time dilation and length contraction.

A scaling factor that grows larger as relative velocity approaches the speed of light, indicating increasingly dramatic relativistic changes.

Signs and relationships

x - vt: The -vt term accounts for the relative motion between frames. If the 'primed' frame moves in the positive x-direction, its origin shifts by vt relative to the 'unprimed' frame, making an event's x' coordinate smaller
t - vx/c^2: The -vx/ $c^{2}$ term is responsible for the relativity of simultaneity. It shows that two events simultaneous in the 'unprimed' frame but separated in space (x)
1 - v^2/c^2: This term in the denominator of the Lorentz factor (γ) ensures that γ becomes very large as v approaches c. This mathematically prevents objects from exceeding the speed of light and causes extreme relativistic effects

Free study cues

Insight

Canonical usage

All spatial and temporal quantities must be expressed in a consistent system of units to ensure dimensional homogeneity.

Common confusion

A common mistake is using inconsistent units for velocity (v) and the speed of light (c), or for position (x) and time (t) across different terms.

Dimension note

The Lorentz factor ( $γ$ ) is inherently dimensionless, as it is derived from the ratio of velocities (v/c). This ensures that the transformation equations correctly relate quantities of the same dimension.

Unit systems

x, x'meter (m) · Spatial coordinates in the respective inertial frames.

t, t'second (s) · Temporal coordinates in the respective inertial frames.

$v$ meter per second (m/s) · Relative velocity between the two inertial frames.

$c$ meter per second (m/s) · Speed of light in vacuum.

$γ$ dimensionless · The Lorentz factor, \gamma = 1 / \sqrt{1 - (v/c)^2}, is always dimensionless.

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: $t_{p} r im e$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In precise coordinate tracking of particles in high-energy accelerators like the LHC, Lorentz Transformation (Space-Time) is used to calculate Transformed Position from Transformed Time, Stationary Position, and Stationary Time. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Calculate the Lorentz factor γ = 1 / √(1 - v²/c²) as your very first step.
Always verify that velocity v is expressed in the same units as the speed of light c.
Check the sign of the relative velocity based on whether the frames are moving apart or together.
Remember that x and vt must both have units of distance before subtraction.

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.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Lorentz transformations replace the Galilean transformations when relative velocities are comparable to c, preserving the invariant spacetime interval.

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.

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.

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.