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:
“Distinctions drawn by the mind are not necessarily equivalent to distinctions in reality.”
—Thomas Aquinas (c. 1225–1274)
“I can’t say that the college-bred woman is the most contented woman. The broader her mind the more she understands the unequal conditions between men and women, the more she chafes under a government that tolerates it.”
—Susan B. Anthony (1820–1906)