How can we determine if a spacetime is flat?

Abstract

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A spacetime is locally flat if and only if no geodesical deviation exists for congruences of all kinds of geodesics. However, while for causal geodesics the deviation can be measured observing the motion of (infinitesimal) falling bodies, it does not seem possible to evaluate the geodesical deviation of spacelike geodesic. So a physical problem may arise. To tackle it we analyze the interplay of local flatness and geodesic deviation measured for causal geodesics. We establish that a generic spacetime is (locally) flat if and only if there is no geodesic deviation for timelike geodesics or, equivalently, there is no geodesic deviation for null geodesics.

Keywords

• general relativity
• Geodesical deviation
• Local flatness
• Causal geodesic
• Riemann tensor