Gradient Four-Vector

The 4-dimensional version of the Gradient, encountered frequently in general relativity and special relativity, is

$$\nabla_\mu =\left[{\matrix{ {1\over c} {\partial\over\partia... ...tial\over\partial y}\cr {\partial\over\partial z}\cr}}\right],$$

which can be written

$$(\nabla^\mu)^2\equiv \vbox{\hrule height.6pt\hbox{\vrule wid... ...t height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}^2,$$

where $\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}^2$ is the d'Alembertian Operator.

See also d'Alembertian Operator, Gradient, Tensor, Vector

References

Morse, P. M. and Feshbach, H. ``The Differential Operator $\nabla$.'' §1.4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 31-44, 1953.