Understanding Einstein's Law of Gravitation in Empty Space

[snoopies622]

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]

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

$$R_{ab} - \frac{1}{2}g_{ab}R = 0$$.

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

 

[snoopies622]

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]

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

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

 

[snoopies622]

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]

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