Spacetime Algebra - Spacetime Gradient

Spacetime Gradient

The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied

$a \cdot \nabla F(x)= \lim_{\tau \rightarrow 0} \frac{F(x + a\tau) - F(x)}{\tau}$

One can show that this requires the definition of the gradient to be

$\begin{align}\nabla &= \frac{1}{{\gamma_\mu}} \frac{\partial {}}{\partial {x^\mu}} \equiv \gamma^\mu \partial_\mu \\ &= \frac{1}{{\gamma^\mu}} \frac{\partial {}}{\partial {x_\mu}} \equiv \gamma_\mu \partial^\mu.\end{align}$

Written out explicitly with, these partials are

$\begin{align}\partial_0 &= \partial^0 = \frac{1}{{c}} \frac{\partial {}}{\partial {t}} \\ \partial_k &= \frac{\partial {}}{\partial {x^k}} \\ \partial^k &= \frac{\partial {}}{\partial {x_k}} = -\partial_k \\ \end{align}$

Read more about this topic: Spacetime Algebra