In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to unify the fundamental forces and reformulate general relativity as a configuration space of three-metrics, modulo three-dimensional diffeomorphisms. The origin of this idea can be found in a English mathematician

William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

's works. Wheeler, John Archibald. 1962 960 "Curved empty space as the building material of the physical world: an assessment." In ''Logic, Methodology, and Philosophy of Science'', edited by

E. Nagel Ernest Nagel (November 16, 1901 – September 20, 1985) was an American philosopher of science. Suppes, Patrick (1999)Biographical memoir of Ernest Nagel In '' American National Biograph''y (Vol. 16, pp. 216-218). New York: Oxford University P ...

Stanford University Press Stanford University Press (SUP) is the publishing house of Stanford University. It is one of the oldest academic presses in the United States and the first university press to be established on the West Coast. It was among the presses officially ...

. This theory was enthusiastically promoted by John Wheeler in the 1960s, and work on it continues in the 21st century.

Einstein's geometrodynamics

The term geometrodynamics is as a synonym for general relativity. More properly, some authors use the phrase ''Einstein's geometrodynamics'' to denote the initial value formulation of general relativity, introduced by Arnowitt, Deser, and Misner (

ADM formalism The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was fir ...

) around 1960. In this reformulation, spacetimes are sliced up into ''spatial hyperslices'' in a rather arbitrary fashion, and the vacuum Einstein field equation is reformulated as an ''evolution equation'' describing how, given the geometry of an initial hyperslice (the "initial value"), the geometry evolves over "time". This requires giving ''constraint equations'' which must be satisfied by the original hyperslice. It also involves some "choice of gauge"; specifically, choices about how the ''coordinate system'' used to describe the hyperslice geometry evolves.

Wheeler's geometrodynamics

Wheeler 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: *mass without mass *charge without charge *field without field He wanted to lay the foundation for

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

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 of geons, 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. Wheeler was intrigued by the possibility that geons could affect test particles much like a massive object, hence ''mass without mass''. Wheeler was also much intrigued by the fact that the (nonspinning) point-mass solution of general relativity, the

Schwarzschild vacuum In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...

, has the nature of a wormhole. Similarly, in the case of a charged particle, the geometry of the Reissner–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 the

electromagnetic field tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...

from the electromagnetic contribution to the

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...

, which in general relativity is directly coupled to spacetime curvature; Wheeler and Misner developed this into the so-called ''already-unified field theory'' which partially unifies gravitation and electromagnetism, yielding ''charge without charge''. In the ADM reformulation of general relativity, Wheeler argued that the full Einstein field equation can be recovered once the ''momentum 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-called spacetime foam. This would realize geometrical intuition suggested by quantum gravity, or ''field 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 1/2

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...

s 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 of Descartes' and Spinoza's ideas about the nature of space.

Modern notions of geometrodynamics

More recently, Christopher Isham, Jeremy Butterfield, and their students have continued to develop '' quantum geometrodynamics'' 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 to Riemann,

Clifford Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford *Baron Clifford *Baron Clifford of Chudleigh *Baron de Clifford *Clifford baronets *Clifford fami ...

, and Weyl and found a more concrete realization in the wormholes of Wheeler characterized by the Euler-Poincaré invariant. They result from attaching handles to black holes. Observationally, Albert Einstein's general relativity (GR) is rather well established for the solar system and double pulsars. However, in GR the metric plays a double role: Measuring distances in spacetime and serving as a gravitational potential for the

Christoffel connection In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distan ...

. This dichotomy seems to be one of the main obstacles for quantizing gravity. Arthur Stanley Eddington suggested already in 1924 in his book ''The 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 the Pontryagin invariant of the corresponding gauge connection. Similarly as in the Yang–Mills theory, a quantization can be achieved by amending the definition of curvature and the Bianchi identities via

topological ghost In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

s. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the Poincaré lemma for the

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...

. Using a BRST antifield formalism with a duality gauge fixing, a consistent quantization in spaces of double dual curvature is obtained. The constraint imposes instanton type solutions on the curvature-squared 'Yang- Mielke theory' of gravity, proposed in its affine form already by Weyl 1919 and by Yang 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-called BF scheme, in which the

gauge curvature In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...

is denoted by F. In the case of gravity, it departs from the meta-linear group SL(5, R) in four dimensions, thus generalizing (

Anti- Anti- is an American record label founded in 1999 as a sister label to Epitaph Records. While Epitaph's focus has mostly been on punk rock, Anti-'s roster includes gospel (Mavis Staples), country (Merle Haggard), hip hop (Sage Francis, The Cou ...

) de Sitter gauge 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 a Higgs-like mechanism. The finiteness of such a deformed topological scheme may convert into asymptotic safeness after quantization of the spontaneously broken model.

References

* This Ph.D. thesis offers a readable account of the long development of the notion of "geometrodynamics". * This book focuses on the philosophical motivations and implications of the modern geometrodynamics program. * * See ''chapter 43'' for superspace and ''chapter 44'' for spacetime foam. *
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* J. Wheeler (1960) "Curved empty space as the building material of the physical world: an assessment", in Ernest Nagel (1962) ''Logic, Methodology, and Philosophy of Science'', Stanford University Press.
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* Mielke, Eckehard W. (2010, July 15). Einsteinian gravity from a topological action. SciTopics. Retrieved January 17, 2012, from http://www.scitopics.com/Einsteinian_gravity_from_a_topological_action.html *Wang, Charles H.-T. (2005). "Conformal geometrodynamics: True degrees of freedom in a truly canonical structure". ''Phys. Rev.'' D 71, 124026. .

Einstein's geometrodynamics

Wheeler's geometrodynamics

Modern notions of geometrodynamics

See also

References

Further reading