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The D'Alembertian operator is defined as $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu $$ For the Minkowski metric in Cartesian coordinates that is $$ \Box=\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} $$

Can it be intuitively described, just as a gradient or divergence, curl or Laplacian may be?

I'm looking for something similar to the interpretation of a Laplacian given in this question and answer.

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The operator is just $\partial_t^2-\nabla^2$. So it is the difference between a "temporal laplacian" and a "spatial laplacian". Since laplacian measures curvature, this is basically telling you the difference in curvature between the spatial and temporal variation of the field.

One reason this comes up in physics is in describing elastic sheets under tension. In an elastic sheet, if there is (spatial) curvature at a point, the tension in the sheet will pull the point in order to flatten out the curvature. Thus the point feels a force in the same direction as the curvature. So by newtons second law, the point on the sheet will accelerate, that is, have a second time derivative, in the direction of the curvature. This is why you would expect the difference in the "temporal laplacian" and "spatial laplacian" to be zero.

If this operator is non-zero, then it means the temporal and spatial variations are inconsistent with each other, and it looks like there is an external force acting on the point in your elastic sheet.

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