Four-gradient

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

$\partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)$

and is sometimes also represented as D.

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

$D\cdot D=\partial _{\alpha }\partial ^{\alpha }=-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}$.

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