Lorentz Transformation

Equivalent forms

ct' = \gamma(ct - \beta x)

\begin{pmatrix}ct'\\x'\end{pmatrix} = \begin{pmatrix}\gamma & -\gamma\beta\\-\gamma\beta & \gamma\end{pmatrix}\begin{pmatrix}ct\\x\end{pmatrix}

Four scalar equations encode the entire kinematics of special relativity — including time dilation and length contraction as special cases.

Symbol	Name	SI unit	Dimension	Range
$x_prime$	Transformed Positionoutput Position of event in moving frame S'	m	L	-100000000 – 100000000
$t_prime$	Transformed Timeoutput Time of event in moving frame S'	s	T	-1 – 1
$x$	Position Position of event in frame S	m	L	-100000000 – 100000000
$t$	Time Time of event in frame S	s	T	-1 – 1
$v$	Boost Velocity Velocity of S' relative to S	m/s	LT^-1	-299792457 – 299792457
$c$	Speed of Light Speed of light in vacuum	m/s	LT^-1	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$x_prime$	m	cm	ft
$t_prime$	s	s	s
$x$	m	cm	ft
$t$	s	s	s
$v$	m/s	cm/s	ft/s
$c$	m/s	cm/s	ft/s

Where it holds

Valid for inertial frames in flat spacetime with relative velocity |v| < c.

Dimensional analysis

x' \to [L]

,

t' \to [T]

[1] * ([L] - [LT^-1][T]) = [L]

;

[1] * ([T] - [LT^-1][L]/[L^2T^-2]) = [T]

Both space and time transformations dimensionally consistent.

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Discovery

Hendrik Lorentz / Henri Poincaré · 1904

Lorentz derived these transformations to preserve Maxwell's equations; Einstein (1905) showed they were the true kinematics of spacetime, not just mathematical tricks.

Try this

If a train moves at 0.6c and a passenger walks forward, what does the platform see?

Lorentz transformation mixes space and time: x' = gamma(x - vt), t' = gamma(t - vx/c^2). Solve for the platform-frame coordinates of any event aboard the train.

Research status: stable

Real-world applications

GPS timing corrections
Particle accelerator kinematics
Astronomical observations of relativistic jets
Electromagnetic field transformations

Common misconceptions

Time is not absolute — t' generally differs from t even at the origin
Lorentz transformations are not rotations in space, but hyperbolic rotations in spacetime
They reduce to Galilean transformations only at v ≪ c, not exactly

Experimental verification

Tests include the Hafele-Keating clock experiment (1971), GPS clock corrections, particle accelerator kinematics, and the Ives-Stilwell transverse Doppler experiment.

Derivation

Demand (i) linearity, (ii) invariance of c, and (iii) symmetry between S and S'.

Postulate

x' = A(x - vt)

.

Apply to a light pulse

x = ct

,

x' = ct

'; substitute and require consistency to find

A = \gamma

.

Time transformation follows from the inverse relation and yields

t' = \gamma (t - vx/c^2)

.

Limiting cases

v ≪ c⟶

x' \to x - vt

,

t' \to t

Recovers Galilean transformation — time is universal at low speeds.

v \to c

⟶

\gamma \to \infty

Transformation diverges; massive frames cannot reach c.

v = 0

⟶

x' = x

,

t' = t

Identity transformation — same frame.

What if…

What if c were infinite?

Then $\gamma \to 1$ and $t' = t$ ; spacetime collapses to Newton's absolute time.

What if v > c were allowed?

gamma becomes imaginary — coordinates lose physical meaning; tachyons (if they existed) violate causality.

What if you boost twice in a row?

The composition is another Lorentz boost with velocity given by the relativistic addition formula.

1

Train clock at the front

Given ·

x:: 100
t:: 0
v:: 100000000

Find · t_prime

Steps

$\beta = 1e_{8} / 3e_{8} = 0.333$
$\gamma = 1/\sqrt{1 - 0.111} \approx 1.061$
$t' = 1.061 * (0 - 1e_{8} * 100 / (3e_{8})^2) \approx -1.18e-7\,\mathrm{s}$

Result ·

t' = \gamma * (0 - 1e_{8} * 100 / 8.99e16) \approx -1.18e-7\,\mathrm{s}

— the front clock reads earlier by 0.118 microseconds.

2

Spaceship's own clock

Given ·

x:: 150000000
t:: 0.5
v:: 150000000

Find · x_prime

Steps

$\beta = 0.5$ , $\gamma \approx 1.155$
$x - vt = 1.5e_{8} - 1.5e_{8} * 0.5 = 7.5e_{7}$
$x' \approx 1.155 * 7.5e_{7} \approx 8.66e_{7} m$

Result ·

x' = \gamma (x - vt) = \gamma * (1.5e_{8} - 7.5e_{7}) \approx 8.66e_{7} m

.

Lorentz Factor→Time Dilation→Length Contraction→Relativistic Velocity Addition→