Is General Relativity a simplified theory?

Martín Rivas

doi:10.1088/1742-6596/437/1/012008

Abstract

Gravity is understood as a geometrization of spacetime. But spacetime is also the manifold of the boundary values of the spinless point particle in a variational approach. Since all known matter, baryons, leptons and gauge bosons are spinning objects, it means that the manifold, which we call the kinematical space, where we play the game of the variational formalism of an elementary particle is greater than spacetime. This manifold for any mechanical system is a Finsler metric space such that the variational formalism can always be interpreted as a geodesic problem on this space. This manifold is just the flat Minkowski space for the free spinless particle. Any interaction modifies its flat Finsler metric as gravitation does. The same thing happens for the spinning objects but now the Finsler metric space has more dimensions and its metric is modified by any interaction, so that to reduce gravity to the modification only of the spacetime metric is to make a simpler theory, the gravitational theory of spinless matter. Even the usual assumption that the modification of the metric only involves dependence on the metric coefficients of the spacetime variables is also a restriction because in general these coefficients are dependent on the velocities. In the spirit of unification of all forces, gravity cannot produce, in principle, a different and simpler geometrization than any other interaction.

Highlights

Things should be made simple, but not simpler
We consider that General Relativity is a constrained, and a simpler formalism for describing gravity for two reasons: One is that the geometrization of spacetime has to be enlarged to consider Finsler metrics instead of pseudo-Riemannian metrics, and another that the manifold which describes the boundary states of spinning matter is larger than spacetime
The manifold X of the boundary variables of any Lagrangian dynamical system is always a Finsler metric space, so that any variational approach is equivalent to a geodesic statement on this metric manifold

Summary

Introduction

Things should be made simple, but not simpler. From this sentence attributed to Albert Einstein is where we take the title of this work to show that if the spin concept of elementary particles had been known to physics before General Relativity was born most probably the geometrization of spacetime proposed by its creator should be changed by the geometrization of a different manifold, larger than spacetime, so that today’s General Relativity would be considered as a theory of gravitation of simpler and spinless matter. The variational approach of classical mechanics can always be interpreted as a geodesic statement on the space X of the boundary variables of the variational formalism [1] This metric manifold X, is not a pseudo-Riemannian space but rather a Finsler space [2], [3], where the symmetric metric gij(x, x ) is a function of the point x ∈ X, and of its velocity x , where the overdot means derivative with respect to some arbitrary evolution parameter. The relativistic point particle of mass m and spin 0 has a kinematical space span√ned by time t and the position of the point r, so that the free Lagrangian L0 = −mc c2t2 − r2, is clearly a homogeneous function of first degree of the derivatives tand r.

Example

Conclusions