- #1
WWCY
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- TL;DR Summary
- What are the rules?
If I have a metric of the form ##g_{\mu \nu} = f_{\mu \nu} + h_{\mu \nu}## where ##f_{\mu \nu}## is the background metric and ##h_{\mu \nu}## the perturbation, how do I raise and lower indices of tensors?
For instance, I was told that ##G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }##. But shouldn't ##G_{ \ \nu}^{\mu} = g^{\mu \nu '} G_{\nu ' \nu }## be true as well? What exactly are the "rules" behind these operations?
PS If the answer is related to differential geometry, could I be also be pointed to a (introductory) source that explains it?
Cheers.
For instance, I was told that ##G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }##. But shouldn't ##G_{ \ \nu}^{\mu} = g^{\mu \nu '} G_{\nu ' \nu }## be true as well? What exactly are the "rules" behind these operations?
PS If the answer is related to differential geometry, could I be also be pointed to a (introductory) source that explains it?
Cheers.