What is the physical meaning to vanishing Ricci scalar $R=0$ of a metric in general relativity? Note that this is not the same questions as the geometric meaning of $R_{\mu\nu}=0$ which has been asked before.
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1Related: http://physics.stackexchange.com/q/2447/2451 and links therein. – Qmechanic Nov 23 '16 at 10:53
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The scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. So zero Ricci tensor means zero deviation to euclidean space.
Josh Hulks
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First sentence correct, but your last sentence is only true for two dimensions, when the scalar curvature wholly defines the Ricci tensor. But in higher dimensions a nonzero Ricci tensor can have zero trace (and the OP was at pains to state he/she wasn't asking about $R_{\mu,\nu}=0$). Admittedly, the case of two dimensions is quite an important one historically and conceptually and fun to think about too. You could add a little detail to your first sentence, which, although perfectly true, is a little dense for someone coming across this notion for the first time. – Selene Routley Nov 23 '16 at 11:37
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1The sentence was supposed to be ricci scalar opposed to ricci tensor. – Josh Hulks Nov 24 '16 at 09:22