Invitation to discussion

 

Mach principle in multidimensional dynamics

U.N.Zakirov

Institute of Mechanics and Mechanical Engineering of RAS

2/31, Lobachevsky str., Kazan, 420111, Russia

 

In five-dimensional standard cosmological solution of Ross on the basis of standard cosmological closed model and the induced energy-momentum tensor, the function influencing the inertial properties of a trial particle and permitting to point to the connection with mass distribution including dark matter in the Universe and in its nearest environment is postulated. This can be an argument to prove Mach principle.

Ernst Mach realized the historical and critical analysis of mechanics and paid attention to its feeble aspects; in particular, to Newtonian concept of absolute Space and absolute Motion. In 1921 at the Royal college of London A. Einstein said on this point: "In Newton theory if one considers motion not from the causative but only form descriptive point of view, it exists only as a relative movement of subjects in relation to each other. However from this point of view the acceleration appearing in Newton equations is not clear. Newton had to invent physical space, in relation to which acceleration must exist. Mach principle is reduced to a motion not in relation to absolute space, but in respect to plurality of all the rest of ponderable bodies, which determine the properties of space, and Universe in particular": A.Einstein's speculations brought him to more concrete formulations of this principle :

1.      The inertia of a body should increase in accordance with aggregation of ponderable masses in its vicinity.

2.      The body should undergo accelerating force, when the neighboring masses accelerate; directions of this force and acceleration should coincide.

3.      Rotating hollow body should initialize "Coriolis field of forces" inside itself which seeks to diverge moving bodies in moving direction, and also radial centrifugal field of forces.

A.Pais wrote that he tried to follow this principle in his theory:

as g_μν quantities (gravitational and metric potentials) cause inertia activity, they must be, in turn, completely determined by mass distribution in the Universe. A. Einstein assumed Space to be closed three-dimensional spherical Universe without Newtonian infinity; further, for stationary solution he introduced the term Λ, but with appearing of Friedmann's theory of nonstationary Universe he abandoned it; realization of Mach principle still remained a problem. As A.Pais wrote: "In my opinion, the problem of inertia origin was and remains the most obscure problem in theory of particles and fields". R.Tolman in his book wrote: "In addition to principles of covariance and equivalence we should, apparently, enter some complementary element into gravitation theory": and further "now we should enter the third principle of relativistic gravitation theory, namely: the precise formulation of law, reflecting dependence of metric and gravitational fields on space-time state, the knowledge of which will allow one to compute the gravitational effects at given substance and energy distributions": J.A.Smorodinsky, Editor of translation of R.Tolman's work, noted: "At present time (the seventies) : there is no evidence for any connection between mass distribution in the Universe and mass of any body or particle". Authors of workbook on relativity and gravitation theory wrote that nobody till that time had managed to achieve more or less noticeable success in derivation of Mach principle from equation of halfgeneral relativity theory.

J.Weber studying this problem wrote: "Einstein and Wheeler studied a problem of opportunity to explain Mach principle not as a consequence of field equation, but as a requirement imposed on boundary conditions. If a system is "isolated", then it is necessary to introduce the requirement, that over large distances the metrics should properly turn into the metrics of remaining Universe. It would mean the interaction of a substance of remaining part of the Universe with a mass of "isolated" system. The realization of such a program could give a relation linking the inert properties with substance configuration in remote areas of the Universe. At last, it is necessary to note F.Hoyle's work (middle seventies), where he suggested the modification, based on Mach principle (partitioning the Universe into "compartments": the deeper one looks the Universe, the less mass he can observe; in this case the size of particles also decreases, resulting in "compartment" extremity.

The present work deals with attempt of partial solution of this problem on the basis of five- dimensional theory of Kaluza-Klein, as the parameters and functional capability of constructing the induced energy-momentum tensor using these parameters are built in it. Such parameters are metric tensor component at the fifth coordinate γ₅₅ and coupling coefficient of five-dimensional and four-dimensional metric σ.

zakirov@kfti.knc.ru