Timeline for Landau & Lifshitz "Classical Field theory" argument for invariance of $ds^2$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1, 2021 at 3:49 | comment | added | peek-a-boo | This is a purely infinitesimal assumption (i.e at the level of tangent spaces to the manifold). | |
| Dec 1, 2021 at 3:48 | comment | added | peek-a-boo | The only assumption made is about the equality of the light-cones (null-set) they generate (actually we only need a one-way inclusion, that the light cone generated by say the first quadratic form is contained in that of the second). From this, we already deduce that the two locally defined tensor fields $g,g'$ are related by a positive scalar multiple. (Showing the constant is $1$ then requires underlying Physical symmetry arguments.) At no point do we make any assumptions about how the coordinates are related to each other (i.e no assumption on the chart transitions). | |
| Dec 1, 2021 at 3:44 | comment | added | peek-a-boo | @WillG Wikipedia proves a certain pointwise theorem. Here, you're right that we are indeed supposing $ds^2$ (which is just classical notation) is the quadratic form associated to the pseudo-Riemannian metric (defined on some open set) $g=c^2dt\otimes dt-dx\otimes dx-dy\otimes dy-dz\otimes dz$. Likewise, $ds'^2$ is indeed the quadratic form associated to some $g'$. This makes no assumptions about the relationship between primed and unprimed coordinates. We're just taking two charts on the manifold, and constructing two tensor fields $g,g'$ on certain open sets, and taking their quadratic forms. | |
| Dec 1, 2021 at 3:13 | comment | added | WillG | The statement proven in the Wikipedia article assumes that $ds^2$ and $ds'^2$ are the quadratic forms associated with a pseudo–inner product and symmetric bilinear form. I think this amounts to assuming that the transformation from unprimed to primed coordinates is linear. It is worth mentioning this (and perhaps arguing for the validity of this assumption) since the OP asked for clarification on that. | |
| Jul 12, 2021 at 19:55 | history | answered | peek-a-boo | CC BY-SA 4.0 |