Timeline for Do contravariant and covariant partial derivatives commute in GR?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 21, 2017 at 11:15 | comment | added | AccidentalFourierTransform | @bianchira fair enough. | |
| Jul 21, 2017 at 2:57 | comment | added | tparker | The same ambiguity can come up even if you only have one derivative. Suppose you have a one-form $A_\mu$. If we were to interpret $\partial_\mu A^\mu$ as the four-divergence of the corresponding four-vector $A^\mu$, then we would have $\partial_\mu A^\mu \neq \partial^\mu A_\mu$, because the former would equal $\partial_\mu (g^{\mu \nu} A_\nu) = A_\nu \partial_\mu g^{\mu \nu} + g^{\mu \nu} \partial_\mu A_\nu = A_\nu \partial_\mu g^{\mu \nu} + \partial^\nu A_\nu$. In practice, we usually adopt the notational convention that all derivatives are taken before raising or lowering any indices. | |
| Jul 20, 2017 at 17:43 | comment | added | zzz | @AccidentalFourierTransform no this should stay on physics SE because it's a notation issue, and the notation is one invented and popularized and abused by physicists. | |
| Jul 20, 2017 at 16:28 | comment | added | tparker | Your same objection also applies to two contravariant partial derivatives, which look even more like they "should" commute. | |
| Jul 20, 2017 at 16:24 | comment | added | Prahar | No. Your feeling is wrong. You cannot commute raised partial derivatives. | |
| Jul 20, 2017 at 16:12 | answer | added | zzz | timeline score: 3 | |
| Jul 19, 2017 at 21:53 | comment | added | Prof. Legolasov | What is your definition of $\partial^{\mu}$? | |
| Jul 18, 2017 at 18:46 | comment | added | Kyle Kanos | Probably related: physics.stackexchange.com/q/187590/25301 | |
| Jul 18, 2017 at 18:26 | answer | added | J.G. | timeline score: 9 | |
| Jul 18, 2017 at 18:23 | review | Close votes | |||
| Jul 19, 2017 at 19:02 | |||||
| Jul 18, 2017 at 18:15 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
deleted 21 characters in body; edited tags
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| Jul 18, 2017 at 17:55 | review | First posts | |||
| Jul 18, 2017 at 18:00 | |||||
| Jul 18, 2017 at 17:55 | history | asked | user163020 | CC BY-SA 3.0 |