Timeline for Euler-Lagrange equation in curved spacetime
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2022 at 14:22 | vote | accept | Rob Tan | ||
| Aug 20, 2022 at 14:21 | comment | added | Rob Tan | Ok, I'm really starting to see now, thank you very very much | |
| Aug 20, 2022 at 14:18 | comment | added | Michael Seifert | $$ \nabla_\mu \left( g^{\mu \nu} \nabla_\nu \phi\right) = 0$$and not $$\partial_\mu \left( g^{\mu \nu} \partial_\nu \phi\right) = 0$$as you seem to assume it should be; the left-hand sides of these equations are not equivalent. | |
| Aug 20, 2022 at 14:15 | comment | added | Michael Seifert | (2) From your comment on the other answer, I think you're implicitly assuming that for a scalar field, all covariant derivatives of scalar fields are equal to the corresponding expressions with partial derivatives. But that's not true; it's only true for the first derivative. So while we do have $\nabla_\mu \phi = \partial_\mu \phi$, this expression is a one-form field, and so $\nabla_\nu (\nabla_\mu \phi) \neq \partial_\nu (\nabla_\mu \phi) = \partial_\nu (\partial_\mu \phi)$. In particular, this means that the EOM for a massless Klein-Gordon field in curved spacetime is ... | |
| Aug 20, 2022 at 13:58 | comment | added | Michael Seifert | @RobTan: (1) The covariant derivative does satisfy the Liebniz product rule. It's constructed in such a way that for any tensors $A_{\dots}$ & $B_{\dots}$, we have $\nabla_a (A_{\dots} B_{\dots}) = (\nabla_a A_{\dots}) B_{\dots} + A_{\dots} \nabla_a (B_{\dots})$. See, for example, these notes. ... | |
| Aug 20, 2022 at 10:54 | comment | added | Rob Tan | Moreover for a scalar field $\nabla_\alpha=\partial_\alpha$, but this would give incorrect results | |
| Aug 20, 2022 at 10:01 | comment | added | Rob Tan | Very clear, excellent explanation, thank you. I still have a doubt, because seems to me that you implicitly state $\nabla_\alpha (AB) = \nabla_\alpha A\, B + A \nabla_\alpha B$, but that's not true. | |
| Aug 19, 2022 at 20:50 | history | edited | Michael Seifert | CC BY-SA 4.0 |
deleted 16 characters in body
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| Aug 19, 2022 at 17:23 | history | answered | Michael Seifert | CC BY-SA 4.0 |