# D’Alembertian

The D’Alembertian is the equivalent^{} of the Laplacian in Minkowskian geometry. It is given by:

$$\mathrm{\square}={\nabla}^{2}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}$$ |

Here we assume a Minkowskian metric of the form $(+,+,+,-)$ as typically seen in special relativity. The connection^{} between the Laplacian in Euclidean space and the D’Alembertian is clearer if we write both operators and their corresponding metric.

### 0.1 Laplacian

$$\text{Metric:}d{s}^{2}=d{x}^{2}+d{y}^{2}+d{z}^{2}$$ |

$$\text{Operator:}{\nabla}^{2}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}$$ |

### 0.2 D’Alembertian

$$\text{Metric:}d{s}^{2}=d{x}^{2}+d{y}^{2}+d{z}^{2}-cd{t}^{2}$$ |

$$\text{Operator:}\mathrm{\square}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}$$ |

In both cases we simply differentiate twice with respect to each coordinate in the metric. The D’Alembertian is hence a special case of the generalised Laplacian.

## 1 Connection with the wave equation

The wave equation is given by:

$${\nabla}^{2}u=\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}u$$ |

Factorising in terms of operators, we obtain:

$$({\nabla}^{2}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}})u=0$$ |

or

$$\mathrm{\square}u=0$$ |

Hence the frequent appearance of the D’Alembertian in special relativity and electromagnetic theory.

## 2 Alternative notation

The symbols $\mathrm{\square}$ and ${\mathrm{\square}}^{2}$ are both used for the D’Alembertian. Since it is unheard of to square the D’Alembertian, this is not as confusing as it may appear. The symbol for the Laplacian, $\mathrm{\Delta}$ or ${\nabla}^{2}$, is often used when it is clear that a Minkowski space^{} is being referred to.

## 3 Alternative definition

It is common to define Minkowski space to have the metric $(-,+,+,+)$, in which case the D’Alembertian is simply the negative of that defined above:

$$\mathrm{\square}=\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}-{\nabla}^{2}$$ |

Title | D’Alembertian |
---|---|

Canonical name | DAlembertian |

Date of creation | 2013-03-22 17:55:18 |

Last modified on | 2013-03-22 17:55:18 |

Owner | invisiblerhino (19637) |

Last modified by | invisiblerhino (19637) |

Numerical id | 8 |

Author | invisiblerhino (19637) |

Entry type | Definition |

Classification | msc 31B15 |

Classification | msc 31B05 |

Classification | msc 26B12 |

Synonym | wave operator |

Synonym | D’Alembert operator |

Related topic | Laplacian |