The weak field limit


Einstein’ theory, deformed products and the covariant Lorentz force at the weak field limit - see the document below.

I have started this document in remarking that the equation of geodesic deviation (at the weak field limit) can be interpreted within a mathematical context involving the extrinsic decomposition of deformed products ⊗A(..., x).

Depending on the meaning which is given to A and to ..., that remark can be applied either to a new concept of deformed angular momentum (A is anti-symmetric and ... = u, the four-speed of some particle) or to a pre-generator of the gravitational term (A = Γ(2) and ... = u).

Although E. Cartan' considerations on Einstein' theory would give arguments to choose the first interpretation for (A, ...), I prefer the second one because Einstein' work is rooted in Christoffel' one which is involving symmetric symbols.

Because I am aware of recent works involving Clifford algebras, I do not abandon the idea to analyze the first option for ever. In opposition, I remark by the way that tensor products deformed by antisymmetric cubes are equivalent to Lie bracket and equip any space with a Lie algebra structure. But I leave that path for later and I focus my energy on the second option.

One of my preoccupations is the search for a rationalistic way justifying the covariant versus of the Lorentz force. I remark that situations such that the connection and the four-speed are invariant (i) allow the recovery of the gravitational part of that force and (ii) give a simplified formalism of it. A simple ordinary derivation of ⊗Γ(x, u) is enough.

Because I am not certain that that simplified formalism corresponds to some reality, I examine a concrete example; namely, the one of an accelerator of particles. I come to a strange qualitative conclusion. If that formalism describes a real phenomenon, then the particles would no longer run at invariant speed in a circular ring because they should feel a Lorentz force at least twice per period.

Despite of my doubts, I continue my theoretical exploration in rewriting the Lagrange equations. I do it at the weak field limit. This allows me to propose the existence of some Newton-like acceleration. This hypothesis transforms the equations in such a manner suggesting to interpret them in a context taking the elasticity of the tiny geometry into account (nota bene: it has been proved via the results of Gravity Probe B and LIGO experiments). Concretely, I introduce the gradient of a Lagrange function into my theory and that function is related to the components of the Riemann' curvature tensor.

The last paragraph of the document opens the door for an extrapolation of this first approach.

© Thierry PERIAT, 10 August 2020.

The document – 23 pages - GB.


Bibliography (abstract).

M. T. W.: Gravitation; © 1973 by W.H. Freeman and Company.

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event Date de dernière mise à jour : 09/09/2020