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It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see previous question). The standard derivation of this is through Koszul's formulae either in coordinates (for example wikipedia), or in abstract index notation (for example, in Wald's General Relativty), or in coordinate-free notation (for example, as pointed out by Thomas Richard in Besse - Einstein manifold). This approach is mainly algebraic by using the definition in terms of Koszul's formulae and then calculus in various notations. Essentially the derivation is a direct calculation without the need to even mention the manifold.

I am wondering if there is a way to derive/interprete the statement refered to at the beginning using an alternative method which is more geometric, ie. using parallel transport or alike. The criterion for "geometric" being (a) a direct reference to the manifold is necessary; or (b) a picture, at least in principle a mental picture, can be drawn to at least carry the main idea of the derivation (of course, pictures of formulae don't count).

flag	asked Sep 10 at 21:33 Lizao Li 261●2●8

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This seems very hard to do. Can you cite an example of what you're looking for for a simpler variational problem? Also, you could take a look at the book "Gravitation" by Misner, Thorne, and Wheeler. They try very hard to explain everything using pictures. – Deane Yang Sep 10 at 22:36

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For example, can the equation for a geodesic in a Riemannian manifold be derived geometrically? I'd love to know about this, if it can. – Deane Yang Sep 10 at 22:37

It is also not clear to me what you mean by "direct reference to the manifold". To even begin to contemplate Einstein's equation you have to have a smooth manifold with its associated tangent bundle. If the fact that the manifold itself is required for whatever you are computing to be even defined is not sufficient as a reference, can you give an example of an argument where "direct reference to the manifold is necessary"? Also, I don't quite understand the distinction you are trying to draw between parallel transport and connection, can you explain more? – Willie Wong Sep 11 at 10:54

Thank you for the comments. @Deane Yang - I am not that optimistic about it as well. But the situation for the geodesic equation is somewhat different. The equation itself has the Christoffel symbol in it. So I don't think there is much you can do about it. The meaning of the various curvatures appeared here are beyond mere calculation tools. Hilbert action is the total curvature (integration of the 2nd Lipschitz-Killing curvature), which at least for simple objects have a very clear interpretation other than the formulae itself. The Einstein tensor has geometric characterizations too. – Lizao Li Sep 11 at 15:17

But I don't really see how things fit in here. Also, it seems even in MTW - Gravitation they just give a direct derivation. – Lizao Li Sep 11 at 15:18

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Geometric derivation of the Einstein’s field equation from the Hilbert action.

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