Timeline for Why can we put these conditions on coordinates of worldsheet?
Current License: CC BY-SA 4.0
6 events
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| Dec 22, 2022 at 5:19 | comment | added | octonion | Actually my answer was intended to be a sketch of a proof of that by explicitly constructing the coordinates. I rewrote it (using components rather than differential forms too). | |
| Dec 22, 2022 at 5:17 | history | edited | octonion | CC BY-SA 4.0 |
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| Dec 21, 2022 at 21:17 | comment | added | particle-not good at english | ＞"I'll put up an alternate derivation in an edit" --- Thank you very much!! I also want to know about your sentence "All 2d manifolds are conformally flat, i.e. they can always be put in this form at least in a coordinate patch". Can we put parameters which gives expression of metric, $g^{ind}_{ab}(x)=h(x)η_{ab}$ to whole a 2d manifold which is diffeomorphic to$I\times I$? And I'd like to also know the proof of the fact "All 2d manifolds are conformally flat" Is there any website show the proof? | |
| Dec 21, 2022 at 20:23 | comment | added | octonion | You can't confirm $d(*d\tau)=0$ for an arbitrary function, the idea is that you have to choose a function $\tau$ that solves this equation. I'll put up an alternate derivation in an edit. | |
| Dec 21, 2022 at 16:30 | comment | added | particle-not good at english | Thanks for answering!! >>"All 2d manifolds are conformally flat" Then, for each 2d surface which is given metric and homeomorphic to $I\times I$, we can choose coordinates in which the metric is expressed as h(x)η_{ab}, right? >> "timelike gradient dτ" Does that mean g^{ττ} is negative? >>"By construction ∗dτ is orthogonal to dτ..." Yes, I just checked. >>"This is just Laplace's equation associated to the metric g^{ind}..." Could you explain it in more detail or introduce any website in which I can confirm d( ∗dτ)=0? | |
| Dec 21, 2022 at 9:53 | history | answered | octonion | CC BY-SA 4.0 |