D'Alembert operator

From Wikipedia, the free encyclopedia

In special relativity, electromagnetism and wave theory, the d'Alembert operator $Δ$ , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space and other solutions of the Einstein equation. In Minkowski space in standard coordinates (t, x, y, z) it has the form

$\Delta_{\mathbf{M}} = \partial^\mu \partial_\mu = \eta^{\nu\mu} \partial_\nu \partial_\mu = \pm(\partial_0^2 - \delta^{i j} \partial_i \partial_j = \partial_t^2 - \sum_{i=1}^3 \partial_i^2 = {\partial^2 \over \partial t^2} - \Delta_{\mathbf{R}^3})$

where $\Delta_{\mathbf{R}^3} = \nabla_{\mathbf{R}^3}^2$ is the three dimensional Laplacian, $η 00 = 1$ , $η 0 i = 0$ and $η i j = - δ i j$ for i,j = 1 to 3; η being the Minkowski metric, and δ being the Kronecker delta. Note that μ and ν range from 0 to 3, whereas i and j range from 1 to 3: see Einstein notation. The sign of these expressions depends on the sign convention used for the Minkowski metric.

Lorentz transformations leave the metric invariant, thus the above coordinate expressions remain valid for the standard coordinates in every inertial frame.

[edit] Alternate notations

In physics the symbol $\Box$ or $\Box^2$ is usually used for the d'Alembertian: the four sides of the box representing the four dimensions of space-time. Sometimes $\Box$ is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol $\nabla$ is then used to represent the space derivatives, but this is coordinate chart dependent. In such case, the three sides of the triangular nabla may be taken to represent the three dimensions of space.

Another way to write the d'Alembertian in flat standard coordinates is $\partial^2$ . The notation $\partial^2$ is useful in quantum field theory where partial derivatives are usually indexed: so the lack of an index with the squared partial derivative signals the presence of the D'Alembertian.