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Conclusion

The main intermediaries allowing this generalization are the concept of potential and the idea of space-time continuum. This last idea, introduced by Einstein through the notion of space-time metric, asked for the generalization of that metric. This generalization is considered to represent the gravitational field. However the generalization led subsequently to some contradictions unsolved even today. One of these is the problem of gravitational waves the other is the impossibility of quantization of the gravitational field (Loinger, 1998, 1999, ). We daresay these contradictions have the source in the primordial contradiction between Geometry and Physics: the dichotomy of time as illustrated in the first equation of geodesics. Indeed, on one hand the time is a parameter along geodesics of the new Geometry - which is what Physics requires - while, on the other hand it is itself a coordinate - which is what Geometry requires. The dichotomy is made possible by the indeterminacy in the affine parameter of geodesics: being defined up to a linear transformation this parameter is taken as solution of a second order differential equation which then is interpreted as equation of a geodesic component. This is an example where the meaning of a concept is abusively extended: the indeterminacy is translated directly into a dynamical property!

On the other hand, one can ask: did we really need the Poisson equation? Newtonian force is a vacuum force, a solution of Poisson equation for zero density (Laplace equation). Come to think of it: has it been discovered as anything else? Isn’t it a force between two material points in vacuum? One must say that, on the contrary, Newton’s law has both empirical and theoretical foundation, and this last one is given by the Laplace equation. Now, one can reply that we are sure that the density of the Universe is not zero, so that the Laplace equation itself has no foundation, but this cannot be held against what we just said. Indeed, the (however small) density of the Universe considered today is not the Newtonian density, but a figure obtained from a density in the sense of Hertz (Hertz, 2003) and referred to a quite arbitrary volume.

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