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:\Delta_{\mathbf{M}} = \partial^\mu \partial_\mu = \eta^{
u\mu} \partial_
u \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} =
abla_{\mathbf{R}^3}^2 is the three dimensional Laplacian, \eta^{00} = 1 , \eta^{0i} = 0 and \eta^{ij} = -\delta^{ij} for ''i'',''j'' = 1 to 3; η being the Minkowski metric, and δ being the . The sign of these expressions depends on the Sign Convention used for the Minkowski metric.

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


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
abla 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.


APPLICATIONS

The Continuity Equation for the Four-current ''J'' = (''ρc'', j)
:
abla_{\mathbf{R}^3} \cdot \mathbf{j} = - rac{\partial ho}{\partial t}
can be written
:\Delta^{1/2} \cdot J = 0.

The Klein-Gordon Equation would look like
: (\Delta \pm m^2) \psi = 0 .

A Wave Equation for the electromagnetic field is
: \Delta \mathbf{A} = 0
where A is the Vector Potential .