(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given a symmetric tensor [itex]T_{\mu\nu}[/itex] on the flat Euclidean plane ([itex]g_{\mu\nu}=\delta_{\mu\nu}[/itex]), we want to change to complex coordinates [itex]z=x+iy, \,\overline{z}=x-iy[/itex].

Show, that the components of the tensor in this basis are given by:

[itex]T_{zz}=\frac{1}{4}(T_{00}-2iT_{10}-T_{11}),\, T_{\overline{z}\overline{z}}=\frac{1}{4}(T_{00}+2iT_{10}-T_{11}),\, T_{\overline{z}z}=T_{z\overline{z}}=\frac{1}{4}(T_{00}+T_{11})[/itex]

2. Relevant equations

[itex]T_{\mu\nu}=T_{\nu\mu}[/itex]

[itex]T'_{\alpha\beta} = U_{\alpha}{}^{\mu}U_{\beta}{}^{\nu}T_{\mu\nu}[/itex]

3. The attempt at a solution

In priciple, this should be an easy linear algebra problem. I just don't get the right result.

I use [itex]T'[/itex]for the tensor in the new coordinates. Then, there should be a transformation matrix [itex]U[/itex], s.t.

[itex]T'_{\alpha\beta} = U_{\alpha}{}^{\mu}U_{\beta}{}^{\nu}T_{\mu\nu}[/itex]

But what is [itex]U[/itex]? Neiter with the transformation matrix from the new to the old coordinates [itex]\begin{pmatrix}1 & 1 \\ i & -i\end{pmatrix}[/itex] nor with the transformation matrix from the old to the new coordinates [itex]\begin{pmatrix}\frac{1}{2} & \frac{-i}{2} \\ \frac{1}{2} & \frac{i}{2} \end{pmatrix}[/itex] it is working. What am I doing wrong?

physicus

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# Homework Help: Coordinate transformation of a tensor in 2 dimensions

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