theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, geometrodynamics is an attempt to describe

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

and associated phenomena completely in terms of

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

. Technically, its goal is to unify the

fundamental forces In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist: * gravity * electromagnetism * weak int ...

and reformulate

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

as a configuration space of three-metrics, modulo three-dimensional

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

s. The origin of this idea can be found in an English mathematician

William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was a British 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. 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

. 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 Arnowitt–Deser–Misner (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 nume ...

) around 1960. In this reformulation,

s 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 v ...

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, has the nature of a

wormhole A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are base ...

. 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. T ...

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 subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

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 Baruch (de) Spinoza (24 November 163221 February 1677), also known under his Latinized pen name Benedictus de Spinoza, was a philosopher of Portuguese-Jewish origin, who was born in the Dutch Republic. A forerunner of the Age of Enlightenmen ...

'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, and

Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

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 Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

(GR) is rather well established for the

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

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. 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 Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...

, a quantization can be achieved by amending the definition of curvature and the

Bianchi identities In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra ...

via topological ghosts. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the

Poincaré lemma In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R''n'' is exact for ''p'' ...

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 re ...

. 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 An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...

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 is denoted by F. In the case of gravity, it departs from the special linear group SL(5, R) in four dimensions, thus generalizing ( Anti-) 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. Richard J. Petti believes that cosmological models with torsion but no rotating particles based on Einstein–Cartan theory illustrate a situation of "a (nonpropagating) field without a field".

References

Works cited

General 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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