Four-gradient

The four-gradient is the four-vector generalization of the gradient:

:$partial\_alpha\; stackrel\{mathrm\{def\{=\}\; left(frac\{1\}\{c\}\; frac\{partial\}\{partial\; t\},\; abla\; ight)$

and is sometimes also represented as "D".

The square of "D" is the four-Laplacian, which is called the d'Alembert operator:

:$Dcdot\; D\; =\; partial\_alpha\; partial^alpha\; =\; -\; frac\{1\}\{c^2\}frac\{partial^2\}\{partial\; t^2\}\; +\; abla^2$.

As it is the dot product of two four-vectors, the d'Alembertian is a Lorentz invariant scalar.

It is also written $Box$

References

* S. Hildebrandt, "Analysis II" (Calculus II), ISBN 3-540-43970-6, 2003
* L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
* J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley ISBN 0-471-30932-X