Calculating the metric tensor.

  • #1
MathematicalPhysicist
Gold Member
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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.
 

Answers and Replies

  • #2
WannabeNewton
Science Advisor
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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
HallsofIvy
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Homework Helper
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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]

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.
[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]
 

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