Contravariant gradient operator

Reference: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Problem 12.55.

The gradient of a scalar function {\phi} is a covariant vector since it transforms as

\displaystyle  \frac{\partial\phi}{\partial\bar{x}{}^{a}}=\frac{\partial\phi}{\partial x^{i}}\frac{\partial x^{i}}{\partial\bar{x}{}^{a}} \ \ \ \ \ (1)

We can therefore regard the gradient operator {\partial_{a}} on its own as a covariant vector, so it should have a contravariant counterpart. In flat space, the only change in switching from covariant to contravariant is that the time component changes sign. Given that the Lorentz transformation for a contravariant four-vector is

\displaystyle   \bar{x}^{0} \displaystyle  = \displaystyle  \gamma\left(x^{0}-\beta x^{1}\right)\ \ \ \ \ (2)
\displaystyle  \bar{x}^{1} \displaystyle  = \displaystyle  \gamma\left(x^{1}-\beta x^{0}\right)\ \ \ \ \ (3)
\displaystyle  \bar{x}^{2} \displaystyle  = \displaystyle  x^{2}\ \ \ \ \ (4)
\displaystyle  \bar{x}^{3} \displaystyle  = \displaystyle  x^{3} \ \ \ \ \ (5)

the transformations for the covariant four-vector are obtained by lowering all indices and replacing the time components by their negatives:

\displaystyle   \bar{x}_{0} \displaystyle  = \displaystyle  \gamma\left(x_{0}+\beta x_{1}\right)\ \ \ \ \ (6)
\displaystyle  \bar{x}_{1} \displaystyle  = \displaystyle  \gamma\left(x_{1}+\beta x_{0}\right)\ \ \ \ \ (7)
\displaystyle  \bar{x}_{2} \displaystyle  = \displaystyle  x_{2}\ \ \ \ \ (8)
\displaystyle  \bar{x}_{3} \displaystyle  = \displaystyle  x_{3} \ \ \ \ \ (9)

where we multiplied the {\bar{x}_{0}} equation through by {-1}. The corresponding inverse transformations are obtained by replacing {\beta} by {-\beta}:

\displaystyle   x_{0} \displaystyle  = \displaystyle  \gamma\left(\bar{x}_{0}-\beta\bar{x}_{1}\right)\ \ \ \ \ (10)
\displaystyle  x_{1} \displaystyle  = \displaystyle  \gamma\left(\bar{x}_{1}-\beta\bar{x}_{0}\right)\ \ \ \ \ (11)
\displaystyle  x_{2} \displaystyle  = \displaystyle  \bar{x}_{2}\ \ \ \ \ (12)
\displaystyle  x_{3} \displaystyle  = \displaystyle  \bar{x}_{3} \ \ \ \ \ (13)

Thus the inverse covariant transformations are the same as the forward contravariant transformations.

The contravariant gradient is {\partial^{i}\phi=\frac{\partial\phi}{\partial x_{i}}} so

\displaystyle   \overline{\partial^{i}\phi} \displaystyle  = \displaystyle  \frac{\partial\phi}{\partial\bar{x}_{i}}\ \ \ \ \ (14)
\displaystyle  \displaystyle  = \displaystyle  \frac{\partial\phi}{\partial x_{k}}\frac{\partial x_{k}}{\partial\bar{x}_{i}}\ \ \ \ \ (15)
\displaystyle  \displaystyle  = \displaystyle  \partial^{k}\phi\frac{\partial x_{k}}{\partial\bar{x}_{i}} \ \ \ \ \ (16)

The transformations for each value of {i} are then

\displaystyle   \overline{\partial^{0}\phi} \displaystyle  = \displaystyle  \gamma\partial^{0}\phi-\beta\gamma\partial^{1}\phi\ \ \ \ \ (17)
\displaystyle  \overline{\partial^{1}\phi} \displaystyle  = \displaystyle  \gamma\partial^{1}\phi-\beta\gamma\partial^{0}\phi\ \ \ \ \ (18)
\displaystyle  \overline{\partial^{2}\phi} \displaystyle  = \displaystyle  \partial^{2}\phi\ \ \ \ \ (19)
\displaystyle  \overline{\partial^{3}\phi} \displaystyle  = \displaystyle  \partial^{3}\phi \ \ \ \ \ (20)

Thus {\partial^{i}\phi} transforms like a contravariant vector.

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