Well it sort of bothered me that there were these two definitions floating around and no demonstration that they were equivalent. Later I realized that this task was assigned as an exercise. I looked at it. No hope. Came back again later. No hope. Drat. Google. No help?
But then I took my copy of Gravitation and held it very close to my eyes defocusing just a bit. As I moved the copy away from my face slowly a few terms came into view. It was like magic! So anyhow I was satisfied that the two expressions are the same. Yay!
Well. One other thing. It follows from 1.4 that the Einstein Tensor, as defined by 1.2, is symmetric since both the Ricci Tensor and the metric are.
This is kind of a key point. Why take the double dual of Riemann? Taking just a single dual on the last two indices also gives a result with vanishing divergence. (This can be shown from the Bianchi Identities and follows the development of Electrodynamics given in MTW.) So why take the double dual? Well, because it is the double dual which has vanishing divergence and also the correct symmetry when contracted.
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