---
tag: AYCCDQWCJF
kind: concept
name: Klein–Gordon equation
msc: [35Q40, 35L05, 35L71]
specializes: [92C3S03AQZ]
sources:
  - url: https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation
same_as:
  wikidata: Q868967
  wikipedia: Klein–Gordon equation
---

## Summary

The Klein–Gordon equation $u_{tt} - \Delta u + m^2 u = 0$ is the [[wave equation|5EHD2XBQDG]] with a mass term.
It is the relativistic wave equation of a free scalar particle of mass $m$, written in units with $c = \hbar = 1$.
It is hyperbolic, so signals travel no faster than light, and for $m > 0$ it is also dispersive: waves of different frequencies travel at different speeds.
Its nonlinear versions, such as $u_{tt} - \Delta u + m^2 u = \pm \abs{u}^{p-1}u$, are model problems in nonlinear wave theory.

## Explanations

### L1 human

Take a string that vibrates and is also held to its rest position by springs along its length.
The springs add the term $m^2 u$, which pulls the string back toward zero.
A plane wave $e^{i(kx - \omega t)}$ solves the equation exactly when $\omega^2 = k^2 + m^2$.
So short waves move almost at speed $1$, while long waves move more slowly, and a bump spreads out as it travels.