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## Main Question or Discussion Point

Why is Einstein's law of gravitation for empty space sometimes identified as Ricci tensor=0 instead of Einstein tensor=0. The first condition implies the second one, but not the other way around.

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Why is Einstein's law of gravitation for empty space sometimes identified as Ricci tensor=0 instead of Einstein tensor=0. The first condition implies the second one, but not the other way around.

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Not true. In vacuum the Einstein equations areWhy is Einstein's law of gravitation for empty space sometimes identified as Ricci tensor=0 instead of Einstein tensor=0.The first condition implies the second one, but not the other way around.

[tex]R_{ab} - \frac{1}{2}g_{ab}R = 0[/tex].

If you contract this equation using [itex]g^{ab}[/itex], you obtain [itex]R=0[/itex]; if you then substitute this back into the Einstein equations you'll find that [itex]R_{ab}=0[/itex] for flat space.

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For some reason I get that the Einstein tensor for the surface of a sphere is zero, while the Ricci tensor is not. This would be a case of the second condition not implying the first. Have I miscalculated?

Perhaps the equivalence of the two conditions is always true in 4-d space but not 2-d space..

Perhaps the equivalence of the two conditions is always true in 4-d space but not 2-d space..

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yes thats your problem I think. Its only in 4+ dimensions that you can have curvature in free space.Perhaps the equivalence of the two conditions is always true in 4-d space but not 2-d space..

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Well, what do you know. Thank you both!

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