Timeline for Variation of induced metric in Nambu-Goto action
Current License: CC BY-SA 4.0
10 events
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| Feb 5, 2022 at 20:55 | comment | added | user3517167 | I appreciate the help. It's clear that there's a gap in my knowledge. I remember talking a lot about variations in the formulation of the Schwinger-Dyson equations - maybe I'll look back at those notes. | |
| Feb 5, 2022 at 20:52 | vote | accept | user3517167 | ||
| Feb 5, 2022 at 20:26 | history | edited | Prahar | CC BY-SA 4.0 |
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| Feb 5, 2022 at 20:22 | comment | added | Prahar | I will modify my answer to explain variations, but I seriously suggest you go back to absolute basics - QFT 101. Trust me, it'll be exceedingly difficult for you to follow anything if you do not first strengthen your fundamentals. Variations occur so often in physics that you need to be able to do it while asleep to continue onwards. | |
| Feb 5, 2022 at 20:15 | comment | added | user3517167 | 🤦♂️I'm looking at a few treatments of the material, so of course there's nothing about $g_{\mu\nu}$ here - that's not how the pull-back is defined in Tong's notes. So $\delta\gamma_{ab}=2\eta_{\mu\nu}\delta(\frac{dX^{\mu}}{d\zeta^a})\frac{dX^{\nu}}{d\zeta^b}$. The variation of the derivative term is the limit as $\varepsilon\rightarrow0$ of $\frac{d(X^{\mu}+\varepsilon Y^{\mu})}{d\zeta^a}$. The second term vanishes in the limit of little $\varepsilon$ so the variation of $X^{\mu}_{,~a}$ is $X^{\mu}_{,~a}$. Is this on the right track here? | |
| Feb 5, 2022 at 13:34 | comment | added | Prahar | OK, then you need to learn that first. No point in trying to study string theory without learning the basics. | |
| Feb 5, 2022 at 5:27 | comment | added | user3517167 | No, it's been a long time since I've seen this stuff. I'm not comfortable with it anymore. I guess I naïvely am assuming that since a variation is like a derivative that $\delta\gamma_{ab}$ should look something like $\delta g_{\mu\nu}\partial_aX^{\mu}\partial_bX^{\nu}+2g_{\mu\nu}\delta\partial_aX^{\mu}\partial_bX^{\nu}$ but even if that's correct, I'm not to sure where to go from there. It seems like we should get a nasty term with Christoffels in it from varying $g_{\mu\nu}$ with respect to $X$, but that isn't the case here. | |
| Feb 5, 2022 at 3:00 | comment | added | Prahar | I defined gamma in the first line. Then I just literally varied it. Do you know how to vary fields to derive equations of motion? | |
| Feb 5, 2022 at 2:59 | comment | added | user3517167 | In your second line you write an expression for $\gamma^{ab}\delta\gamma_{ab}$. How do you find $\delta\gamma_{ab}$? | |
| Feb 5, 2022 at 1:20 | history | answered | Prahar | CC BY-SA 4.0 |