Undergrad Using 4-derivative

[deuteron]

Hi! I am trying to understand the object $\partial^\mu$, and wanted to check if the result I am getting below is true.

The definition of $\partial_\mu$ is:

$$\partial_\mu = ( \frac \partial {\partial x^0} , \frac \partial {\partial x^1}, \frac \partial {\partial x^2},\frac \partial {\partial x^3})$$

We define $\partial^\mu$ as:

$$\partial^\mu = \eta^{\mu\nu}\partial_\nu = (\frac \partial {\partial x^0} , -\frac\partial {\partial x^1} , -\frac \partial {\partial x^2}, -\frac \partial {\partial x^3})$$

In this case, if we were to apply $\partial^\mu$ to $x_\nu$, we would get:

$$\partial^\mu x_\nu = (\eta^{\mu\rho}\partial_\rho)x_\nu=\eta^{\mu\rho} \partial_\rho (\eta_{\nu\sigma} x^\sigma)$$

Since $\eta$'s are just number, I rearrange the equation to be:

$$=\eta^{\mu\rho} \eta_{\nu\sigma} \partial_\rho x^\sigma$$

Here, my knowledge of indices and contraction ends and I try to think of the objects like matrices, in which case I write:

$$= \begin{bmatrix} 1&0&0&0\\ 0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\begin{bmatrix} 1&0&0&0\\ 0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix} \begin{bmatrix} \frac {\partial x^0}{\partial x^0} & \frac {\partial x^1}{\partial x^0} & \frac {\partial x^2}{\partial x^0} & \frac {\partial x^3}{\partial x^0} \\ \frac {\partial x^0}{\partial x^1} & \frac {\partial x^1}{\partial x^1} & \frac {\partial x^2}{\partial x^1} & \frac {\partial x^3}{\partial x^1}\\ \frac {\partial x^0}{\partial x^2} & \frac {\partial x^1}{\partial x^2} & \frac {\partial x^2}{\partial x^2} & \frac {\partial x^3}{\partial x^2} \\ \frac {\partial x^0}{\partial x^3} & \frac {\partial x^1}{\partial x^3} & \frac {\partial x^2}{\partial x^3} & \frac {\partial x^3}{\partial x^3}\end{bmatrix} $$

Here, one can observe the last matrix reduces to the identity matrix, and the multiplication of two $\eta$'s also reduces to the identity matrix, thus I get:

$$=\mathbb I$$

Are these steps correct?

 

[anuttarasammyak]

It seems OK.
\partial^\mu x_\nu = \delta_\nu^\mu

 

[Orodruin]

It looks ok. I would only comment that both $\partial^\mu$ and $x_\mu$ are somewhat artificial objects compared to the more natural $\partial_\mu$ and $x^\mu$.

I much prefer actually writing out the metric.