Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Metric tensor

  1. Dec 22, 2014 #1
    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.
     
  2. jcsd
  3. Dec 22, 2014 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  4. Dec 22, 2014 #3
    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## ?
     
  5. Dec 22, 2014 #4

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Expanding on what dextercioby wrote, and using the metric given in the original post:
    $$
    \begin{align}
    \partial_\mu \phi^\mu &= \partial_0 \phi^0 + \partial_i \phi^i \\
    &= g_{\mu \nu} \partial^\nu \phi^\mu \\
    &= g_{0 0} \partial^0 \phi^0 + g_{1 1} \partial^1 \phi^1 + g_{2 2} \partial^2 \phi^2 + g_{3 3} \partial^3 \phi^3 \\
    &= \partial^0 \phi^0 - \partial^1 \phi^1 - \partial^2 \phi^2 - \partial^3 \phi^3
    \end{align}
    $$
     
  6. Dec 22, 2014 #5

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  7. Dec 22, 2014 #6
    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Metric tensor
  1. The metric tensor (Replies: 14)

  2. Metric tensor (Replies: 10)

  3. Metric tensor (Replies: 1)

  4. Metric Tensors (Replies: 1)

  5. Metric tensor (Replies: 10)

Loading...