# How can the D'Alembertian of a field be interpreted intuitively?

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.

• I think it was Rudy Rucker who wrote that in 15 years trying, he "saw 4 d" spacetime for a total of 15 mins, and he didn't like the experience. Maybe the scalar properties are easier for some people to imagine, personally I can't do it. – user81619 Jun 2 '15 at 10:30

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.