Conditions for flatness in general relativity

Abstract

In the absence of sources, the Einstein field equations, R /sub mu nu / = 0, may be thought of as defining a special class of Riemann spaces whose twenty curvature components R/sub mu nu alpha BETA / have ten vanishing linear combinations; R/sub mu nu / = 0. The necessary and sufficient criterion for the absence of a gravitational field is that space-time be flat, namely, that the curvature tensor vanish: R /sub mu nu alpha BETA / = 0. The conditions for such a flatness in general relativity are discussed primarily and several flatness theorems are derived. (R.E.U.)