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

Calculating the metric tensor.

  1. Mar 12, 2013 #1

    MathematicalPhysicist

    User Avatar
    Gold Member

    Suppose, I have the next metric:
    [tex]g = du^1 \otimes du^1 - du^2 \otimes du^2[/tex]

    And I want to calculate [tex]g(W,W)[/tex], where for example [tex]W=\partial_1 + \partial_2[/tex]

    How would I calculate it?

    Thanks.
     
  2. jcsd
  3. Mar 12, 2013 #2

    WannabeNewton

    User Avatar
    Science Advisor

    The metric tensor is bilinear so [itex]g_p(\partial _{1} + \partial _{2}, \partial _{1} + \partial _{2}) = g_p(\partial _{1} + \partial _{2},\partial _{1}) + g_p(\partial _{1} + \partial _{2},\partial _{2}) = \\g_p(\partial _{1},\partial _{1}) + g_p(\partial _{2},\partial _{1}) + g_p(\partial _{1},\partial _{2}) + g_p(\partial _{2},\partial _{2}) = g_{11}(p) + 2g_{12}(p) + g_{22}(p) = 1 +0 - 1 = 0 [/itex].

    Assuming by [itex]\partial _{i}|_{p}[/itex] you are talking about the coordinate basis vectors, [itex]g_p(\partial _{i},\partial _{j}) = g_{ij}[/itex].
     
  4. Mar 12, 2013 #3

    MathematicalPhysicist

    User Avatar
    Gold Member

    ok, thanks.
     
  5. Mar 12, 2013 #4

    HallsofIvy

    User Avatar
    Science Advisor

    In matrix terms we can representn [itex]g = du^1 \otimes du^1+ 0 du^1\otimes du^2+ 0 du^2\otimes du^1 - du^2 \otimes du^2[/itex] as
    [tex]\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}[/tex]

    [tex]\begin{pmatrix}1 & 1\end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}1 \\ 1 \end{pmatrix}= \begin{pmatrix}1 & -1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}= 1+ (-1)= 0[/tex]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook