Spacetime gradient

We define the spacetime gradient as an operator that consumes a scalar field and produces a 4-vector. Conceptually it is similar to a regular gradient, but adapted to fit the math we use for relativity.

\begin{align*} \partial_\alpha = \frac{\partial}{\partial x^\alpha}. \end{align*}

Consider the gradient in two reference frames, $\O$ and $\O'$ . We will relate the gradient in the primed frame to the gradient in the unprimed frame.

\begin{align*} \partial_{\mu'} &= \frac{\partial}{\partial x^{\mu'}} = \frac{\partial x^\alpha}{\partial x^{\mu'}} \frac{\partial}{\partial x^{\mu'}} = \frac{\partial x^\alpha}{\partial x^{\mu'}} \partial_{\mu'} \\ x^\alpha &= \Lambda^\alpha{}_{\mu'} x^{\mu'} \\ \frac{\partial}{\partial x^{\mu'}} \big[ x^\alpha \big] &= \frac{\partial}{\partial x^{\mu'}} \big[ \Lambda^\alpha{}_{\mu'} x^{\mu'} \big] = \Lambda^\alpha{}_{\mu'}, \\ \partial_{\mu'} &= \Lambda^\alpha{}_{\mu'} \partial_\alpha. \end{align*}

We see the gradient transforms with the Lorentz transform.

We can equally well define an upstairs gradient

\begin{align*} \partial^\alpha = \eta^{\alpha\beta} \partial_\beta = \frac{\partial}{\partial\, \eta_{\alpha\beta} x^\beta}. \end{align*}

Following a similar analysis to above, we can show that the up gradient transforms with the Lorentz transform. We can use both these gradients to define a new operator

\begin{align*} \square := \partial^\alpha \partial_\alpha = -\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. \end{align*}

We call the box $\square$ the wave operator.