Empty space

snoopies622 · Apr 11, 2008

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.

shoehorn · Apr 11, 2008

snoopies622 said: ↑

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.

Not true. In vacuum the Einstein equations are

[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.

snoopies622 · Apr 11, 2008

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..

michael879 · Apr 11, 2008

snoopies622 said: ↑

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

yes thats your problem I think. Its only in 4+ dimensions that you can have curvature in free space.

snoopies622 · Apr 11, 2008

OK, now I'm getting that the equivalence does hold in any number of dimensions except two, in which case the Einstein tensor is always zero.

Well, what do you know. Thank you both!

snoopies622 · Apr 11, 2008

Oops, that's the contraction of the Einstein tensor=0 in two dimensional space, not the tensor itself.

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