Talk:PlanetPhysics/D'Alembertian

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: D'Alembertian %%% Primary Category Code: 40. %%% Filename: DAlembertian.tex %%% Version: 2 %%% Owner: invisiblerhino %%% Author(s): invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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The D'Alembertian is the equivalent of the \htmladdnormallink{Laplacian}{http://planetphysics.us/encyclopedia/LaplaceOperator.html} 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 \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} of the form $(+, +, +, -)$ as typically seen in \htmladdnormallink{special relativity}{http://planetphysics.us/encyclopedia/SR.html}. The connection between the Laplacian in Euclidean space and the D'Alembertian is clearer if we write both \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} 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 \htmladdnormallink{wave equation}{http://planetphysics.us/encyclopedia/WaveEquation.html} 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 \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} 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: \[ \Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} -\nabla^2 \]

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