Timeline for Varying the Einstein-Hilbert action without reference to a chart
Current License: CC BY-SA 4.0
9 events
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| Nov 5, 2018 at 11:44 | comment | added | Bence Racskó | @BobKnighton 2) As far as I recall, all physically reasonable spacetimes are paralellizable, so using local orthonormal frames is in fact global. 2+1) You can likely use index-based tensor calculus on the frame bundle, which is formally basis-free and global, despite the use of indices. See Gauge Theory and Variational Principle by David Bleecker for such an approach. | |
| Nov 5, 2018 at 11:43 | comment | added | Bence Racskó | @BobKnighton I don't think you can calculate the variation of the canonical volume element without coordinates or frames. Aside from that, there are two considerations: 1) You can derive the variation in a "purely covariant" manner, eg. you never need to use something like $\partial_\mu$ or $\gamma^\rho_{\mu\nu}$. If you then declare that you use abstract index notation, your derivation will be formally coordinate free, aside from the pesky volume element. You can also try to do that without using indices whatsoever, but you will run into notational issues with contractions. | |
| Nov 4, 2018 at 11:53 | comment | added | Arnold Neumaier | @BobKnighton: Thirring only uses local orthogonal frames, which always exist, and only to do the computations in a simple way - ''it is easiest to perform the variation of L in an orthogonal basis'' (Remark 4.3.5.1). For a local Lagrangian density, the variation of the action always reduces to a local computation. Thus I do not understand why you want to avoid that. Of course you could avoid it by first proving (in an orthogonal frame - like all basic properties for differential geometric objects) the required properties for variation of the trace. | |
| Nov 3, 2018 at 19:48 | comment | added | Bob Knighton | While this derivation doesn't use a chart, it still requires the existence of an orthogonal frame on the base manifold, which is not guaranteed to exist. Thus, this derivation still only makes sense locally. I was hoping for a derivation that can be applied globally, without referring to a basis for the calculations. | |
| Nov 2, 2018 at 16:57 | history | edited | Arnold Neumaier | CC BY-SA 4.0 |
added some requested details
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| Nov 2, 2018 at 16:57 | comment | added | Arnold Neumaier | @user7777777: The reference is to a well-known textbook, and remains valid even when the link (provided just for convenience) dies. | |
| Nov 2, 2018 at 16:42 | comment | added | Vincent Thacker | While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review | |
| Nov 2, 2018 at 14:55 | review | Low quality answers | |||
| Nov 3, 2018 at 1:01 | |||||
| Nov 2, 2018 at 14:22 | history | answered | Arnold Neumaier | CC BY-SA 4.0 |