Equivalent Conditions
It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the trace reverse of each other:
where the traces are .
A third equivalent condition follows from the Ricci decomposition of the Riemann curvature tensor as a sum of the Weyl curvature tensor plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree, in some region if and only if it is a vacuum region.
Read more about this topic: Vacuum Solution (general Relativity)
Famous quotes containing the words equivalent and/or conditions:
“Inter-railers are the ambulatory equivalent of McDonalds, walking testimony to the erosion of French culture.”
—Alice Thompson (b. 1963)
“One cannot divine nor forecast the conditions that will make happiness; one only stumbles upon them by chance, in a lucky hour, at the world’s end somewhere, and hold fast to the days, as to fortune or fame.”
—Willa Cather (1876–1947)