- #1

cianfa72

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- Rules to raise or lower indices through metric tensor

I'm still confused about the notation used for operations involving tensors.

Consider the following simple example:

$$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$

Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get ##A^{\sigma}{}_{\nu}##. However if we work out explicitly the contraction employing the operator ##C_{\alpha}^{\mu} ()## we get:

$$C_{\alpha}^{\mu} (A_{\alpha \nu} \eta^{\mu \sigma} e^{\alpha} \otimes e^{\nu} \otimes e_{\mu} \otimes e_{\sigma}) = A_{\mu \nu} \eta^{\mu \sigma} e^{\mu} (e_{\mu}) e^{\nu} \otimes e_{\sigma} = A_{\mu \nu} \eta^{\mu \sigma} e^{\nu} \otimes e_{\sigma}$$

The latter is a tensor, say ##T = T_{\nu} {}^{\sigma} e^{\nu} \otimes e_{\sigma}##.

Is it the same as ##A^{\sigma}{}_{\nu} e_{\sigma} \otimes e^{\nu}## ?

Consider the following simple example:

$$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$

Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get ##A^{\sigma}{}_{\nu}##. However if we work out explicitly the contraction employing the operator ##C_{\alpha}^{\mu} ()## we get:

$$C_{\alpha}^{\mu} (A_{\alpha \nu} \eta^{\mu \sigma} e^{\alpha} \otimes e^{\nu} \otimes e_{\mu} \otimes e_{\sigma}) = A_{\mu \nu} \eta^{\mu \sigma} e^{\mu} (e_{\mu}) e^{\nu} \otimes e_{\sigma} = A_{\mu \nu} \eta^{\mu \sigma} e^{\nu} \otimes e_{\sigma}$$

The latter is a tensor, say ##T = T_{\nu} {}^{\sigma} e^{\nu} \otimes e_{\sigma}##.

Is it the same as ##A^{\sigma}{}_{\nu} e_{\sigma} \otimes e^{\nu}## ?

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