REL-01

Lorentz factor

γ = 1 / √(1 − v²/c²).

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Special relativityLorentz / Einstein 1905

Governing equation

\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}

$v/c \in \left[0, 0.999\right]$

where

v/c: Speed in c (—)
\gamma: Lorentz factor (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $\gamma = 1$

Along the curve

②

Domain of the variable — blue

v/c_{\min} = 0,\quad v/c_{\max} = 0.999

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $v/c = 0$

Along the curve

③

Response curve — green

\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $\gamma = 1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\gamma}{\mathrm{d}v/c}

$v/c \in \left[0, 0.999\right]$

$v/c = 0.006$ $\gamma^{\prime} = 0.006$

Along the curve

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-01 — Lorentz factor) is the form associated with Lorentz / Einstein 1905. Working symbols: $v/c$ $\rightarrow$ $\gamma$ . The Lorentz transformation that leaves Maxwell's equations invariant mixes t and x with the factor γ, later identified by Einstein as time dilation.

Purpose

Purpose: compute

\gamma

from

v/c

in Relativity via

\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}

γ = 1 / √(1 − v²/c²). Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

v/c = 0.600\,\mathrm{—}

, the governing relation

\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}

yields

\gamma = 1.2500\,\mathrm{—}

. Inertial boost, v < c. Enter v/c. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Speed in c v/c0.6 —

Outputs

Lorentz factor \gamma1.2500 —

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Narration of this film

Inertial boost, v < c. Enter v/c.

The Lorentz transformation that leaves Maxwell's equations invariant mixes t and x with the factor γ, later identified by Einstein as time dilation.

Reading speed

Watch on YouTube