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9.6 The geodesic hypothesis

In elementary textbooks on general relativity we read that the Einstein equations imply that small bodies move on geodesics of the spacetime metric. It is very hard to make this into a mathematically precise statement which refers to actual solutions of the Einstein equations (and not just to some formal approximations). Recently a theorem relating to this question was proved by Stuart [329]. He considers a nonlinear wave equation which possesses soliton solutions in flat space. He studies families of solutions of the equations obtained by coupling a nonlinear wave equation of this kind to the Einstein equations. Initial data are chosen in such a way that as the parameter labelling the family tends to a limiting value the support of the data contracts to a point p. He shows that if the family is chosen appropriately then the solutions exist on a common time interval (although the data are becoming singular), that the geometry converges to a regular limit and that the support of the solutions converges to a timelike geodesic passing through p.


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