Timeline for Calculating the Riemann Christoffel tensor for a diagonal metric
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2015 at 19:40 | vote | accept | imranal | ||
| Oct 2, 2015 at 17:55 | comment | added | Deane Yang | I also don't know what you mean by "trivial terms", but a natural one is the terms that can be eliminated by using a particular set of co-ordinates. One can verify that one can choose co-ordinates in the neighborhood of a point such that all first order terms in the formula for curvature vanish at that single point (but not necessarily anywhere else). This is equivalent to saying that the Christoffel symbols vanish. The only surviving terms involve 2 derivatives of the metric. | |
| Oct 2, 2015 at 17:43 | comment | added | Deane Yang | I don't understand what you mean by "visualize". How will you be able to visualize the curvature from a rather messy formula? Also, it might be easier to do this first with special cases, say metrics with constant sectional curvature, and then a slightly more general one such as conformally flat ones, as suggested by Thomas Richard. | |
| Oct 2, 2015 at 16:04 | answer | added | rpanai | timeline score: 0 | |
| Oct 2, 2015 at 15:11 | comment | added | Liviu Nicolaescu | The fastest way to do this is to use Cartan's moving frame method that is ideally suited to the situation at hand. For details about this method see section 4.2.3 of the notes www3.nd.edu/~lnicolae/Lectures.pdf | |
| Oct 2, 2015 at 13:47 | comment | added | imranal | I want to visualize the scalar curvature. I have a metric for a non trivial case. I want to find a correlation between the space the metric describes and the scalar field that I visualize from this metric. | |
| Oct 2, 2015 at 13:41 | history | edited | imranal | CC BY-SA 3.0 |
added more specific case in title
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| Oct 2, 2015 at 13:38 | answer | added | imranal | timeline score: 1 | |
| Oct 2, 2015 at 13:32 | comment | added | Thomas Richard | You want to compute the scalar curvature of your metric $g$ if I understand correctly. Still the question stands, what use would you have of the formula you will get for the scalar curvature ? | |
| Oct 2, 2015 at 13:07 | comment | added | imranal | I intend to contract the fourth order tensor and find the Ricci tensor. After that I intend to perform another contraction on the Ricci tensor. I want to visualize the contracted quantity for a given metric. | |
| Oct 2, 2015 at 12:58 | comment | added | Thomas Richard | I agree with Deane, except if the metric is conformal to the euclidean one, in which case the formulas get much simpler. | |
| Oct 2, 2015 at 12:58 | comment | added | Deane Yang | Second, why are you doing this? | |
| Oct 2, 2015 at 12:57 | comment | added | Deane Yang | First, ignore the assumption that the metric is diagonal. It doesn't help. | |
| Oct 2, 2015 at 12:50 | review | First posts | |||
| Oct 2, 2015 at 13:40 | |||||
| Oct 2, 2015 at 12:46 | history | asked | imranal | CC BY-SA 3.0 |