(adsbygoogle = window.adsbygoogle || []).push({}); Kronicker Delta, Levi-Civita, Christoffel ... and "tensors"

For quick reference in grabbing latex equations:

http://en.wikipedia.org/wiki/Levi-Civita_symbol

http://en.wikipedia.org/wiki/Kronecker_delta

http://en.wikipedia.org/wiki/Christoffel_symbols

Wiki warns that the Christoffel symbols are not actually tensors. But they appears to be defined purely in terms of the metric tensor:

[tex]\Gamma^i_{k\ell}=\frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} \right) = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m})[/tex]

so why isn't this a tensor?

And if that is not a tensor, how do we show that the curvature is a tensor if we were forced to start with:

[tex]{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}

- \partial_\nu\Gamma^\rho_{\mu\sigma}

+ \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}

- \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}[/tex]

I'm starting to realize that I've been making a lot of assumptions that basically amount to: if it is in a tensor equation, it must transform like a tensor. So now I'm wondering, are the Kronicker Delta, and Levi-Civita symbols actually tensors ... or should they only be considered a convenient way of writing out constants that appear in front of terms?

For instance, starting with (ct,x) in an inertial frame as my coordinates, the kronicker delta is:

[tex]{\delta^\mu}_\nu = \left(\begin{matrix}

1 & 0 \\

0 & 1

\end{matrix}\right)[/tex]

Then any coordinate transformation S, in matrix form would be a similarity transformation [itex] S^{-1} \delta S[/itex], and so should stay the same since delta is just the identity matrix. So can I consider that as a tensor (that just happens to be the same in all coordinate systems)?

What about the Levi-Civita symbol?

In EM when I go to the dual field tensor

[tex]G^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}[/tex]

how can I know if the dual is really a tensor or not? Are the components of the Levi-Civita symbol the same in all coordinate systems? (if so, is there some quick/obvious way to see this without writing out all the transformations?) ... wikipedia claims they can only scale by an overall constant, but that is not intuitive to me. Even if that is true, it seems that the tensor would then depend on which coordinate system you originally defined it in ... so is this a tensor at all?

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# Kronicker Delta, Levi-Civita, Christoffel and tensors

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