nLab geometrodynamics

Idea

“This disappearance and regeneration of space in time and of time in space is motion;– a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter.” [PdN§261]

General

The term geometrodynamics has been coined, or at least promoted, by John Wheeler as a description for the dynamics of gravity according to general relativity. Since the field of gravity is entirely encoded in the spacetime metric geometry, there is some justification for thinking of the dynamics of the gravitational field as being “the dynamics of geometry” itself. Hence the term.

More specifically the term geometrodynamics is associated with Wheeler’s speculation that all of physics might fundamentally be described by configurations of gravity coupled to other fields, notably the field of electromagnetism, but without any matter: one can see that certain spacetimes without any matter content but with certain nontrivial topology may locally effectively look as if they contained massive and possibly charged bodies.

For instance on a spacetime that is obtained from two copies of Minkowski space connected by a thin (as measured by the metric) throat – often called a wormhole – an electric field configuration whose field lines all converge to the throat’s mouth in one of the two Minkowski sheets, pass through the throat and then emerge concentrically in the other Minkowski sheet may have no divergence anywhere, hence according to Maxwell's equations have no charge sources anywhere, and still effectively look to an observer constrained to one of the two Minkowski sheets but relatively far away from the throat’s mouth as if they were the field lines of a positively or negatively charged point source located where the mouth of the throat is.

These kinds of ideas Wheeler liked to describe by phrases such as charge without charge and mass without mass. Later these basic ideas have continued a life notably in the context of attempts to describe gravity by a topological quantum field theory, for instance in approaches to describe gravity as a BF-theory.

Similarity to KK-compactifications of pure super-gravity

A rather similar perspective arises in Kaluza-Klein compactification of systems of pure gravity to effective field theories. In the case of supergravity these reductions give rise to effective field theories which contain not just extra force field (such as the electromagnetic field in the original KK-mechanism) but also fermionic matter. For instance models such as the G2-MSSM consist entirely of pure 11d supergravity KK-compactified on G2-manifold fiber bundles and the resulting effective field theory contains all of the standard model of particle physics.

Geometric description of physical phenomena

Mass without Mass

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Momenta without Momenta

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Charge without Charge

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Spin without Spin

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Related concepts

• Wheeler superspace

References

General

Early speculation of similar nature is due to

• William Clifford, On the Space-Theory of Matter, Proceedings of the Cambridge Philosophical Society 2 (1864-1876 - Printed 1876) 157-158 reprinted in: The Concepts of Space and Time Boston Studies in the Philosophy of Science 22 Springer (1976) [doi:10.1007/978-94-010-1727-5_50]

(long before the formulation of Einstein gravity/general relativity).

The term “geometrodynamics” is due to:

• Charles W. Misner, John A. Wheeler, Classical physics as geometry, Annals of Physics 2 6 (1957) 525-603 [doi:10.1016/0003-4916(57)90049-0]

Review:

• Domenico Giulini, Matter from Space, in: Beyond Einstein Einstein Studies 14 Birkhäuser (2018) [doi:10.1007/978-1-4939-7708-6_12, arXiv:0910.2574]

See also:

• Wikipedia, Geometrodynamics

• The Algebraic Rainich Conditions pdf

• Return of the Wheeler wormhole pdf

• Spin 1/2 from gravity,

Matching KN black holes to elementary particles

On the similarity between Kerr-Newman black holes (charged and spinning “black 0-branes”) and elementary particles (sigma-model 0-branes) like electrons.

Historical precursor discussion on the possibility of geometrodynamics for fundamental particles:

• Albert Einstein, Leopold Infeld, B. Hoffmann, The gravitational equations and the problem of motion, Annals of Mathematics 39 1 (1938) [doi:10.2307/1968714, jstor:1968714]

Seminal observation that the gyromagnetic ratio of the Kerr-Newman black hole is 2, just as the for electron (cf. at anomalous magnetic moment):

• Brandon Carter, p. 1562 of: Global Structure of the Kerr Family of Gravitational Fields, Phys. Rev. 174 (1968) 1559 [doi:10.1103/PhysRev.174.1559]

• Werner Israel, Source of the Kerr Metric, Phys. Rev. D 2 641 (1970) [doi:10.1103/PhysRevD.2.641]

Further discussion:

• Carlos A. López, Extended model of the electron in general relativity, Phys. Rev. D 30 313 (1984) [doi:10.1103/PhysRevD.30.313]

• H. I. Arcos, J. G. Pereira: Kerr–Newman Solution as a Dirac Particle, General Relativity and Gravitation 36 (2004) 2441–2464 [doi:10.1023/B:GERG.0000046832.71368.a5, arXiv:hep-th/0210103]

• Alexander Burinskii, The Dirac-Kerr-Newman electron, Gravit. Cosmol. 14 (2008) 109–122 [doi:10.1134/S0202289308020011, arXiv:hep-th/0507109]

• Alexander Burinskii, Kerr–Newman electron as spinning soliton, International Journal of Modern Physics A 29 26, 1450133 (2014) [doi:10.1142/S0217751X14501334, arXiv:1410.2888]

See also:

• Wikipedia, Black hole electron