This lecture will start by a discussion of the (weak) principle of equivalence in Newtonian theory and in General Relativity theory and it will shown how, in the latter case, this leads to Einstein?s idea of geodesic motion for ?free? particles. Then a mathematical problem will be raised; suppose two metrics on a space-time manifold give rise, through their Levi-Civita connections, to the same (unparametrised) geodesic paths (that is to the same geodesic paths, ignoring their parameters and whether they are affine or not). How are these metrics and their associated Levi-Civita connections related?
Perhaps the most natural place to start is by considering vacuum (Ricci flat) space-times and this will be done. It will be shown that, under such circumstances, the metric is essentially uniquely determined (up to the choice of units) by the knowledge of the geodesics. (In fact, more will be shown). This work will then be extended to include more general situations than vacuum space-times. During the course of the lecture, a review will be given of the recent contributions to such a study. Much of this lecture is based on joint work with Dr David Lonie in Aberdeen. |