DAlembertian 2008-03-20 09:25:49 2008-03-20 09:25:49 Definition % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The D'Alembertian is the equivalent of the Laplacian in Minkowskian geometry. It is given by: \[ \Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \] 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. \subsection{Laplacian} \[ \mbox{Metric: } ds^2 = dx^2 + dy^2 + dz^2 \] \[ \mbox{Operator: } \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \] \subsection{D'Alembertian} \[ \mbox{Metric: } ds^2 = dx^2 + dy^2 + dz^2 -cdt^2 \] \[ \mbox{Operator: } \Box = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \] 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. \section{Connection with the wave equation} The wave equation is given by: \[ \nabla^2 u = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} \] Factorising in terms of operators, we obtain: \[ (\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2})u = 0 \] or \[ \Box u = 0 \] Hence the frequent appearance of the D'Alembertian in special relativity and electromagnetic theory. \section{Alternative notation} The symbols $\Box$ and $\Box^2$ 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, $\Delta$ or $\nabla^2$, is often used when it is clear that a Minkowski space is being referred to. \section{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: \[ \Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} -\nabla^2 \]