Timeline for Invariance of $ds^2$ from invariance of all null intervals
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 12, 2019 at 23:26 | comment | added | Alex | Assumptions should be clear: 1. transformation is linear (i.e. no preferred origin of space and time). 2. something traveling at the speed of light in the first frame also travels at the speed of light in the other frame. Goal: prove that such a transformation preserves $\eta$ (maybe up to a scalar multiple). Can you explain why you are so sure this statement is false / give a counterexample? | |
| Mar 12, 2019 at 22:00 | comment | added | user4552 | I think the definition you give of Minkowski coordinates puts the cart before the horse in this case. I'm trying to show that the transformation Λ preserves the Minkowski metric η, not start from that point. Then you have to decide what premise you're going to start from. Your basic problem here is that you haven't clearly identified any particular set of assumptions that you want to start from, so you can't prove anything. There is more than one way to set up the logic here, but it won't work if you don't start with any set of assumptions at all. | |
| Mar 12, 2019 at 21:58 | comment | added | user4552 | Your counterexample is good and leads me to think that, in accordance with the wikipedia argument (originating with Landau & Lifshiftz?), that my statement needs to be changed to ΛTηΛ=constη No, that doesn't work. This counterexample is just a special case of the fact that all of these ideas only work for certain special transformations, not for all changes of coordinates. | |
| Mar 12, 2019 at 20:01 | comment | added | Alex | (1) Your counterexample is good and leads me to think that, in accordance with the wikipedia argument (originating with Landau & Lifshiftz?), that my statement needs to be changed to $\Lambda^T\eta\Lambda=\mathrm{const}\eta$. I'll edit the question. (2) I think the definition you give of Minkowski coordinates puts the cart before the horse in this case. I'm trying to show that the transformation $\Lambda$ preserves the Minkowski metric $\eta$, not start from that point. (3) There is no assumption that $\Lambda^T$ is the inverse transformation. $(\Lambda x)^T$ is a forward transformation of $x$ | |
| Mar 12, 2019 at 19:27 | history | edited | user4552 | CC BY-SA 4.0 |
added 478 characters in body
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| Mar 12, 2019 at 19:08 | history | answered | user4552 | CC BY-SA 4.0 |