Understanding the Metric Tensor: A 4-Vector Perspective

[Maybe_Memorie]

Some subtleties of the metric tensor are just becoming clear to me now. If I take $g_{\mu\nu}=diag(+1,-1,-1,-1)$
and want to write $\partial_\mu\phi^\mu$, it would be $\partial_0\phi^0 -\partial_i\phi^i$, correct? $\phi$ is a 4-vector.

 

[dextercioby]

Not really. The minus only comes in once you you use the metric somewhere, but with $\partial_{\mu}\phi^{\mu}$ if you use the contravariant components of phi, then there's no use of the metric tensor.

 

[Maybe_Memorie]

Ah I see. So in that case, $\partial_0\phi^0 + \partial_i\phi^i=\partial_\mu\phi^{\mu}=g^{\mu\nu}\partial_\mu\phi_\nu=\partial_0\phi_0 - \partial_i\phi_i$ ?

 

[George Jones]

Expanding on what dextercioby wrote, and using the metric given in the original post:
$$
\begin{align}
\partial_\mu \phi^\mu &= \partial_0 \phi^0 + \partial_i \phi^i \\
&= g_{\mu \nu} \partial^\nu \phi^\mu \\
&= g_{0 0} \partial^0 \phi^0 + g_{1 1} \partial^1 \phi^1 + g_{2 2} \partial^2 \phi^2 + g_{3 3} \partial^3 \phi^3 \\
&= \partial^0 \phi^0 - \partial^1 \phi^1 - \partial^2 \phi^2 - \partial^3 \phi^3
\end{align}
$$

 

[George Jones]

Ah I see. So in that case, $\partial_0\phi^0 + \partial_i\phi^i=\partial_\mu\phi^{\mu}=g^{\mu\nu}\partial_\mu\phi_\nu=\partial_0\phi_0 - \partial_i\phi_i$ ?

A couple of comments:

1) Einstein summation convention often is used only for one up, one index down;

2) the components $\left\{ g^{\mu\nu} \right\}$ only equal the components $\left\{ g_{\mu\nu} \right\}$ for the metric given in the original post, not for general metrics.

 

[Maybe_Memorie]

A couple of comments:

1) Einstein summation convention often is used only for one up, one index down;

2) the components $\left\{ g^{\mu\nu} \right\}$ only equal the components $\left\{ g_{\mu\nu} \right\}$ for the metric given in the original post, not for general metrics.

Ah right, so it should be explicitly written as $g^{\mu\nu}\partial_\mu\phi_\nu=\partial_0\phi_0 - \partial_1\phi_1 -\partial_2\phi_2 -\partial_3\phi_3$

I knew about your second comment. Somehow I've survived a course on Jackson and a differential geometry course yet I'm only really thinking about this stuff now. Thanks for the help!