Timeline for Why can integrals be written as $I=\int \phi(x) \epsilon$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2022 at 7:27 | vote | accept | Ryder Rude | ||
| Aug 15, 2022 at 21:46 | answer | added | peek-a-boo | timeline score: 8 | |
| Aug 15, 2022 at 18:37 | comment | added | Kurt G. | The point is probably that on an $\color{red}{n}$ dimensional manifold $dx^{\mu_1}\wedge...\wedge dx^{\mu_\color{red}{n}}$ (what you call $n$-form field) is the volume form. The Graviation book by Misner, Thorne & Wheeler spends a lot of time explaining this very pedagogically. | |
| Aug 15, 2022 at 12:17 | comment | added | Ryder Rude | @KurtG. I don't understand how its interpretation as a volume integral and as a n-form field can co-exise. If it's an n-form field, then what does it mean to apply $\int$ on it? | |
| Aug 15, 2022 at 9:52 | comment | added | Kurt G. | Carroll writes on the previous page : that "the naive volume element $d^nx$ transforms as a density, not as a tensor...". Furthermore, his $\epsilon$ on the LHS of $\equiv$ in (2.96) is not the Levi-Civita tensor. The integral is a volume integral of a scalar function $\phi$. That's it. | |
| Aug 15, 2022 at 9:48 | comment | added | Ryder Rude | @KurtG. $d^n x$ is still a tensor | |
| Aug 15, 2022 at 9:46 | comment | added | Ryder Rude | @KurtG. Yes, but that's still the same tensor field, even when expressed in terms of $d^n x$. Integrals are supposed to be numbers. | |
| Aug 15, 2022 at 9:42 | comment | added | Kurt G. | ... and Carroll showed $$\tag{2.96} \epsilon\equiv\epsilon_{\mu_1...\mu_n}dx^{\mu_1}\otimes...\otimes dx^{\mu_n} =...=\sqrt{|g|}d^nx. $$ | |
| Aug 15, 2022 at 9:30 | comment | added | Kurt G. | Carroll writes explicitly that (2.98) is an abstract notation for $$\tag{2.97} \boxed{I=\int \phi(x)\sqrt{|g|}d^nx}. $$ Imho: we can do it because nothing stops us from doing it. | |
| Aug 15, 2022 at 8:08 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
added 10 characters in body; edited tags
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| Aug 15, 2022 at 7:17 | history | asked | Ryder Rude | CC BY-SA 4.0 |