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A variant of unified field theory which reduces all physical objects to geometric objects. Geometro-dynamics is constructed in several stages.
 
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
+
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 g _ {ij} $
 +
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>
+
$$
 +
R _ {ik} - {
 +
\frac{1}{2}
 +
} g _ {ik} R  = \
 +
T _ {ik} ( f _ {\mu \sigma }  , g),
 +
$$
  
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.
+
where $  R _ {ik} $
 +
is the Ricci tensor, $  T _ {ik} $
 +
is the energy-momentum tensor of the electromagnetic field in vacuum, and $  f _ {\mu \sigma }  $
 +
is the field-strength tensor of the electromagnetic field that satisfies the Maxwell equations. The task is to express $  f _ {\mu \sigma }  $
 +
in terms of g _ {ik} $.  
 +
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 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).
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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.
 
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.
+
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 $  ( \hbar \kappa /c  ^ {3} )  ^ {1/2} \approx 10 ^ {- 33 }  \mathop{\rm cm} $(
 +
where $  \hbar $
 +
is the Planck constant, $  \kappa $
 +
is Einstein's gravitational constant and $  c $
 +
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.
 
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.
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====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:41, 5 June 2020


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 $ g _ {ij} $ of space-time which is a solution of Einstein's equations

$$ R _ {ik} - { \frac{1}{2} } g _ {ik} R = \ T _ {ik} ( f _ {\mu \sigma } , g), $$

where $ R _ {ik} $ is the Ricci tensor, $ T _ {ik} $ is the energy-momentum tensor of the electromagnetic field in vacuum, and $ f _ {\mu \sigma } $ is the field-strength tensor of the electromagnetic field that satisfies the Maxwell equations. The task is to express $ f _ {\mu \sigma } $ in terms of $ g _ {ik} $. When put in this simplified manner, the problem has a solution in principle [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 [2] of the maximal analytic extension of the 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 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 $ ( \hbar \kappa /c ^ {3} ) ^ {1/2} \approx 10 ^ {- 33 } \mathop{\rm cm} $( where $ \hbar $ is the Planck constant, $ \kappa $ is Einstein's gravitational constant and $ c $ 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 [4] is one attempt to provide such a foundation.

References

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

Comments

References

[a1] 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
How to Cite This Entry:
Geometro-dynamics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometro-dynamics&oldid=47091
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article