Abstract
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.
Highlights
The presence of tidal forces, i.e., geodesic deviation for causal geodesics, can be adopted to give a notion of gravitation valid in the general relativistic context, as the geodesic deviation is not affected by the equivalence principle and it cannot be canceled out by an appropriate choice of the reference frame
From a physical viewpoint, the geodesic deviation can be measured for causal geodesic, observing the stories of falling bodies, but it can hardly be measured for spacelike geodesics
The popular slogan “gravitation = curvature,” that is “absence of gravitation ⇔ flatness,” seems to encounter an obstruction on the physical ground to be rigorously proved. This is not the case because we establish by Theorem 2.1 that, in a generic spacetime, the absence of geodesic deviation for timelike geodesics – or, equivalently, for null geodesics – is equivalent to the local flatness
Summary
Dipartimento di Matematica, Università di Trento and Istituto Nazionale di Fisica Nucleare - Gruppo Collegato di Trento, Povo, Italy. A spacetime is locally flat if and only if no geodesical deviation exists for congruences of all kinds of geodesics. 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. To tackle this problem 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
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
