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Intheoretical physics,geometrodynamics is an attempt to describespacetime and associated phenomena completely in terms ofgeometry. Technically, its goal is tounify thefundamental forces and reformulategeneral relativity as aconfiguration space of three-metrics, modulo three-dimensionaldiffeomorphisms. The origin of this idea can be found in an English mathematicianWilliam Kingdon Clifford's works.[1] This theory was enthusiastically promoted byJohn Wheeler in the 1960s, and work on it continues in the 21st century.
The term geometrodynamics is as a synonym forgeneral relativity. More properly, some authors use the phraseEinstein's geometrodynamics to denote theinitial value formulation of general relativity, introduced by Arnowitt, Deser, and Misner (ADM formalism) around 1960. In this reformulation,spacetimes are sliced up intospatial hyperslices in a rather arbitrary[citation needed] fashion, and the vacuumEinstein field equation is reformulated as anevolution equation describing how, given the geometry of an initial hyperslice (the "initial value"), the geometry evolves over "time". This requires givingconstraint equations which must be satisfied by the original hyperslice. It also involves some "choice of gauge"; specifically, choices about how thecoordinate system used to describe the hyperslice geometry evolves.
Wheeler[2] wanted to reduce physics to geometry in an even more fundamental way than the ADM reformulation of general relativity with a dynamic geometry whose curvature changes with time. It attempts to realize three concepts:
He wanted to lay the foundation forquantum gravity and unify gravitation with electromagnetism (the strong and weak interactions were not yet sufficiently well understood in 1960 to be included).
Wheeler introduced the notion ofgeons, gravitational wave packets confined to a compact region of spacetime and held together by the gravitational attraction of the (gravitational) field energy of the wave itself.[3] Wheeler was intrigued by the possibility that geons could affect test particles much like a massive object, hencemass without mass.
Wheeler was also much intrigued by the fact that the (nonspinning) point-mass solution of general relativity, theSchwarzschild vacuum, has the nature of awormhole. Similarly, in the case of a charged particle, the geometry of theReissner–Nordström electrovacuum solution suggests that the symmetry between electric (which "end" in charges) and magnetic field lines (which never end) could be restored if the electric field lines do not actually end but only go through a wormhole to some distant location or even another branch of the universe.George Rainich had shown decades earlier that one can obtain theelectromagnetic field tensor from the electromagnetic contribution to thestress–energy tensor, which in general relativity is directly coupled tospacetime curvature; Wheeler and Misner developed this into the so-calledalready-unified field theory which partially unifies gravitation and electromagnetism, yieldingcharge without charge.
In the ADM reformulation of general relativity, Wheeler argued that the full Einstein field equation can be recovered once themomentum constraint can be derived, and suggested that this might follow from geometrical considerations alone, making general relativity something like a logical necessity. Specifically, curvature (the gravitational field) might arise as a kind of "averaging" over very complicated topological phenomena at very small scales, the so-calledspacetime foam. This would realize geometrical intuition suggested by quantum gravity, orfield without field.
These ideas captured the imagination of many physicists, even though Wheeler himself quickly dashed some of the early hopes for his program. In particular, spin 1/2fermions proved difficult to handle. For this, one has to go to the Einsteinian Unified Field Theory of the Einstein–Maxwell–Dirac system, or more generally, the Einstein–Yang–Mills-Dirac-Higgs System.
Geometrodynamics also attracted attention from philosophers intrigued by the possibility of realizing some ofDescartes' andSpinoza's ideas about the nature of space.
More recently,Christopher Isham,Jeremy Butterfield, and their students have continued to developquantum geometrodynamics[4] to take account of recent work toward a quantum theory of gravity and further developments in the very extensive mathematical theory of initial value formulations of general relativity. Some of Wheeler's original goals remain important for this work, particularly the hope of laying a solid foundation for quantum gravity. The philosophical program also continues to motivate several prominent contributors.
Topological ideas in the realm of gravity date back toRiemann,Clifford, andWeyl and found a more concrete realization in the wormholes of Wheeler characterized by theEuler-Poincaré invariant. They result from attaching handles to black holes.
Observationally,Albert Einstein'sgeneral relativity (GR) is rather well established for theSolar System and double pulsars. However, in GR the metric plays a double role: Measuring distances in spacetime and serving as a gravitational potential for theChristoffel connection. This dichotomy seems to be one of the main obstacles for quantizing gravity.Arthur Stanley Eddington suggested already in 1924 in his bookThe Mathematical Theory of Relativity (2nd Edition) to regard the connection as the basic field and the metric merely as a derived concept.
Consequently, the primordial action in four dimensions should be constructed from a metric-free topological action such as thePontryagin invariant of the corresponding gauge connection. Similarly as in theYang–Mills theory, a quantization can be achieved by amending the definition of curvature and theBianchi identities viatopological ghosts. In such a gradedCartan formalism, the nilpotency of the ghost operators is on par with thePoincaré lemma for theexterior derivative. Using aBRSTantifield formalism with a duality gauge fixing, a consistent quantization in spaces of double dual curvature is obtained. The constraint imposesinstanton type solutions on the curvature-squared 'Yang-Mielke theory' of gravity,[5] proposed in its affine form already by Weyl 1919 and byYang in 1974. However, these exact solutions exhibit a 'vacuum degeneracy'. One needs to modify the double duality of the curvature via scale breaking terms, in order to retain Einstein's equations with an induced cosmological constant of partially topological origin as the unique macroscopic 'background'.
Such scale breaking terms arise more naturally in a constraint formalism, the so-calledBF scheme, in which thegauge curvature is denoted by F. In the case of gravity, it departs from the special linear groupSL(5,R) in four dimensions, thus generalizing (Anti-)de Sittergauge theories of gravity. After applying spontaneous symmetry breaking to the corresponding topological BF theory, again Einstein spaces emerge with a tiny cosmological constant related to the scale of symmetry breaking. Here the 'background' metric is induced via aHiggs-like mechanism. The finiteness of such a deformed topological scheme may convert into asymptotic safeness after quantization of the spontaneously broken model.[6]
The concepts of geometrodynamics have motivated work to use theKerr-Newman metric to create a relativistic but non-quantum model for the electron.[7]