View source for Geometro-dynamics

A variant of unified field theory which reduces all physical objects to geometric objects. Geometro-dynamics is constructed in several stages.

The first stage consists of the construction of a unified theory of gravitation and electromagnetism on the basis of general relativity theory. The principal problems of geometro-dynamics at this stage may be stated in a simplified manner as follows. Let there be given a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443101.png" /> of space-time which is a solution of Einstein's equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443102.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443103.png" /> is the Ricci tensor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443104.png" /> is the energy-momentum tensor of the electromagnetic field in vacuum, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443105.png" /> is the field-strength tensor of the electromagnetic field that satisfies the Maxwell equations. The task is to express <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443106.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443107.png" />. When put in this simplified manner, the problem has a solution in principle [[#References|[1]]], but its complete solution presents difficulties (such as allowance for non-electromagnetic fields) which have not yet (1988) been overcome.

The second stage consists of the construction of a theory of elementary particles. The model of a pair of interacting particles is the so-called  "handle" , the simplest form of which is one of the topological interpretations [[#References|[2]]] of the maximal analytic extension of the [[Schwarzschild field|Schwarzschild field]]. In this model the characteristics (e.g. the charge) of an elementary particle are given by certain integer invariants of the  "handle" . In geometro-dynamics space-time is multiply connected, while its first [[Betti number|Betti number]] is of the same order as the number of particles. The concept of a geon was introduced — a wave packet of some given radiation of a concentration which is sufficient for the corresponding distortion of space to make this wave packet metastable (i.e. existing for a long time). Geometro-dynamics predicts electro-magnetic, neutrino and gravitational geons. The concept of a geon is classical. It is believed that the quantum analogue of the concept of geometro-dynamics is a geometro-dynamic description of the mass of elementary particles (geons have not been experimentally observed).

The third stage consists in the construction of a theory of continuous media which yields, broadly speaking, the same results as does the general theory of relativity.

It is assumed that geometro-dynamics involves a violation of the law of conservation of baryon charge. A concrete example of this is the process of gravitational collapse and subsequent evaporation of black holes.

The fourth stage consists of attempts to subsequently construct a quantum geometro-dynamics. Quantum fluctuations of the metric are considered, and it is proved that at a distance of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443108.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g0443109.png" /> is the Planck constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g04431010.png" /> is Einstein's gravitational constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044310/g04431011.png" /> is the velocity of light) such fluctuations can substantially alter the topology of space-time and must correspond to elementary quantum particles.

At the time of writing (1970s) geometro-dynamics is not yet a fully developed theory. The interpretation of spin fields (as distinct from tensor fields), in particular of neutrino fields, is especially difficult. Many features of geometro-dynamics have no adequate mathematical foundation. The theory of superspace [[#References|[4]]] is one attempt to provide such a foundation.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.Y. Rainich,   "Electrodynamics in general relativity theory"  ''Trans. Amer. Math. Soc.'' , '''27'''  (1925)  pp. 106–136</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Wheeler,   "Geometrodynamics" , Acad. Press  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.K. Harrison,   K.S. Thorne,   M. Wakano,   J.A. Wheeler,   "Gravitational theory and gravitational collapse" , Univ. Chicago Press  (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Ya.B. Zel'dovich,   I.D. Novikov,   "Relativistic astrophysics" , '''2. Structure and evolution of the universe''' , Chicago  (1983)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Wheeler,   "Some implications of general relativity for the structure and evolution of the universe" , ''XI Conseil de Physique Solvay. Bruxelles''  (1958)  pp. 97–148</TD></TR></table>