# How is d'Alembertian operator is defined in differential geometry?

Which general formula for the box operator is correct, $$\Box=g^{ij}\partial_i\partial_j$$ or $$\Box=\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)$$? I have seen both the definition being used when both produce different results, e.g. for 2-dimensional polar coordinate, former yields $$\Box=\frac{\partial^2}{\partial r^2}+ \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}$$ while the latter yields $$\Box=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial}{\partial r})+ \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}$$.

## 1 Answer

The second one. The definition is the double covariant derivative. The first covariant derivative acts on a scalar field, and so it is just a derivative. The second is a covariant derivative on the vector field $$\partial_i \phi$$, which can be shown to equal the second expression you gave.

• Thanks Rd Basha, finally understood the difference. Box or D'Alembertian operator is special case of general Laplace-Beltrami operator for flat spacetime, i.e. spacetime with $g^{ij}$ being constant or independent of coordinates. This converts latter definition into former one. Right?? – rim Sep 11 at 17:54