Geometrodynamics.

Geometrodynamics from wikipedia (short variant here) generally denotes a program of reformulation and unification which was enthusiastically promoted by John Archibald Wheeler in the 1960s and is today rather loosely used as a synonym for GR, and some authors use the phrase Einstein's geometrodynamics to denote the initial value formulation. Spacetimes are sliced up into spatial hyperslices, 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 is also what distinguishes TGD and GR, in the big structure, says Matti.

As described by Wheeler in the early 1960s, geometrodynamics attempts to realize three catchy slogans

• mass without mass,
• charge without charge,
• field without field.

Just LOOK. This is a pointing to thinking today?

"The vision of Clifford and Einstein can be summarized in a single phrase, 'a geometrodynamical universe': a world whose properties are described by geometry, and a geometry whose curvature changes with time – a dynamical geometry." 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. He searched the momentum constraint in geometry and wanted to show that GR is emergent, like a logical necessity; he talked of spacetime foam; requres the Einstein-Yang-Mills-Dirac System."

Scattering and virtual particles are similar modern notions? A dynamic metric.

"Geometrodynamics also attracted attention from philosophers intrigued by the suggestion that geometrodynamics might eventually realize mathematically some of the ideas of Descartes and Spinoza concerning the nature of space."

Is this a bad way to say that Matti Pitkänens work is too spiritual? Not even the name mentioned. Still he is 'famous'. See Topological Geometrodynamics: What Might Be the Basic Principles.

Modern geometrodynamics.
Christopher Isham, (he got the Dirac medal 2011), Jeremy Butterfield, (his homepage here) + students have continued to develop quantum geometrodynamics.

Addendum: About the Dirac medal, se all Dirac Medal winners here, note the many famous names:

Professor Christopher Isham

Imperial College London
For his major contributions to the search for a consistent quantum theory of gravity and to the foundations of quantum mechanics.

Chris Isham is a worldwide authority in the fields of quantum gravity and the foundations of quantum theory. Few corners of these subjects have escaped his penetrating mathematical investigations and few workers in these areas have escaped the influence of his fundamental contributions. Isham was one of the first to put quantum field theory on a curved background into a proper mathematical form and his work on anti-de Sitter space is now part of the subject’s standard toolkit.

His early work on conformal anomalies has similarly gone from “breakthrough to calibration”, as all good physics does. He invented the concept of twisted fields which encode topological aspects of the spacetime into quantum theory, and which have found wide application. He did pioneering work on global aspects of quantum theory, developing a group-theoretic approach to quantization, now widely regarded as the “gold standard” of sophisticated quantization techniques. This work laid some of the foundations for the subsequent development of loop-space quantum gravity of Ashtekar and collaborators (the only well-developed possible alternative to string theory). He has also made significant contributions to quantum cosmology and especially the notoriously conceptually difficult “problem of time”.

On the foundations of quantum theory, Isham has made many contributions to the decoherent histories approach to quantum theory (of Gell-Mann and Hartle, Griffiths, Omnes and others), a natural extension of Copenhagen quantum mechanics which lessens dependence on notions of classicality and measurement in the quantum formalism. In particular, using a novel temporal form of quantum logic, he established the axiomatic underpinnings of the decoherent histories approach, crucial to its generalization and application to the quantization of gravity and cosmology.

His recent work has been concerned with the very innovative application of topos theory, a generalization of set theory, into theoretical physics. He showed how it could be used to give a new logical interpretation of standard quantum theory, and also to extend the notion of quantization, giving a firm footing to ideas such as “quantum topology” or “quantum causal sets”. Isham’s contributions to all of these areas, and in particular his continual striving to expose the underlying mathematical and conceptual structures, form an essential part of almost all approaches to quantum gravity.

From Wikipedia cont. "Topological ideas in the realm of gravity date back to Riemann, Clifford and Weyl and found a more concrete realization in the wormholes of Wheeler characterized by the Euler-Poincare invariant. They result from attaching handles to black holes.
Observationally, 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. This dichotomy seems to be one of the main obstacles for quantizing gravity. Eddington suggested already 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 Pontrjagin 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 ghosts. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the Poincare lemma for the exterior derivative. 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 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-)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."

Addendum, Wheeler in wikipedia:

During the 1950s, Wheeler formulated geometrodynamics, a program of physical and ontological reduction of every physical phenomenon, such as gravitation and electromagnetism, to the geometrical properties of a curved space-time. Aiming at a systematical identification of matter with space, geometrodynamics was often characterized as a continuation of the philosophy of nature as conceived by Descartes and Spinoza. Wheeler's geometrodynamics, however, failed to explain some important physical phenomena, such as the existence of fermions (electrons, muons, etc.) or that of gravitational singularities. Wheeler therefore abandoned his theory as somewhat fruitless during the early 1970s.

Maybe he used the wrong concept for the unification? Why are forces making qubits?

Addendum: Wikipedia, the talk page for discussing improvements to the John Archibald Wheeler article.

information regarding geometrodynamics is not accurate

This is a good article on J.A. Wheeler. However, the information regarding geometrodynamics is not accurate, especially the following statement: "Wheeler abandoned it as fruitless in the 1970s".As a matter of fact, Wheeler kept using the term "geometrodynamics" to describe Einstein's theory of general relativity till his last days. For example, in Gravitation and Inertia, a book written with the Italian physicist I.Ciufolini in 1995(and which was missing from the bibliography), the authors keep referring to "Einstein Geometrodynamics"(the title of Chapter 2) throughout the the book: Chapter 3 is entitled " Tests of Einstein Geometrodynamics", Chapter 5 is "The Initial-Value Problem in Einstein Geometrodynamics" and Chapter 7:"Some Highlights of the past and a Summary of Geometrodynamics and Inertia".This proves that Wheeler did not abandon the concept at all in the 1970s! 

John A. Wheeler, 1990, "Information, physics, quantum: The search for links" in W. Zurek (ed.) Complexity, Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley.
Addendum about quantum geometrodynamics, hard linked to time and quantum gravity: Claus Kiefer, 2008. Quantum geometrodynamics: whence, whither? Total search here. Abstract:

Quantum geometrodynamics is canonical quantum gravity with the three-metric as the configuration variable. Its central equation is the Wheeler--DeWitt equation. Here I give an overview of the status of this approach. The issues discussed include the problem of time, the relation to the covariant theory, the semiclassical approximation as well as applications to black holes and cosmology. I conclude that quantum geometrodynamics is still a viable approach and provides insights into both the conceptual and technical aspects of quantum gravity.

And this is actually published; Gen.Rel.Grav.41:877-901, 2009 DOI:10.1007/s10714-008-0750-1
See also: Interpretation of the triad orientations in loop quantum cosmology
Scalar perturbations in cosmological models with dark energy - dark matter interaction

Look: Does time exist in quantum gravity?

Claus Kiefer

Comments: 10 pages, second prize of the FQXi "The Nature of Time" essay contest

Cosmological constant as result of decoherence. This means non-commutative geometry?

An earlier article (Adrian P. Gentle, Nathan D. George, Arkady Kheyfets, Warner A. Millerfrom 2004; Constraints in quantum geometrodynamics, http://arxiv.org/abs/gr-qc/0302044

And about time and geometrodynamics, by the same authors
http://arxiv.org/abs/gr-qc/0302051
http://arxiv.org/abs/gr-qc/0006001
http://arxiv.org/abs/gr-qc/9412037
http://arxiv.org/abs/gr-qc/9409058

A geometric construction of the Riemann scalar curvature in Regge calculus. Jonathan R. McDonald, Warner A. Miller http://arxiv.org/abs/0805.2411

and
A Discrete Representation of Einstein's Geometric Theory of Gravitation: The Fundamental Role of Dual Tessellations in Regge Calculus http://arxiv.org/abs/0804.0279

Quantum Geometrodynamics of the Bianchi IX cosmological model
Arkady Kheyfets, Warner A. Miller, Ruslan Vaulin 2006 http://arxiv.org/abs/gr-qc/0512040

and from 1995,
Quantum Geometrodynamics I: Quantum-Driven Many-Fingered Time
Arkady Kheyfets, Warner A. Miller http://arxiv.org/abs/gr-qc/9406031

All actually published.


References:

• Anderson, E. (2004). "Geometrodynamics: Spacetime or Space?". arXiv:gr-qc/0409123 [gr-qc]. This Ph.D. thesis offers a readable account of the long development of the notion of "geometrodynamics". University of London, Examined in June by Prof Chris Isham and Prof James Vickers. 226 pages including 21 figures. 396 cit.
  

  This thesis concerns the split of Einstein's field equations (EFE's) with respect to nowhere null hypersurfaces. Areas covered include A) the foundations of relativity, deriving geometrodynamics from relational first principles and showing that this form accommodates a sufficient set of fundamental matter fields to be classically realistic, alternative theories of gravity that arise from similar use of conformal mathematics. B) GR Initial value problem (IVP) methods, the badness of timelike splits of the EFE's and studying braneworlds under guidance from GR IVP and Cauchy problem methods.

Abstract
The work in this thesis concerns the split of Einstein field equations (EFE’s) with respect to nowhere-null hypersurfaces, the GR Cauchy and Initial Value problems (CP and IVP), the Canonical formulation of GR and its interpretation, and the Foundations of Relativity. I address Wheeler’s question about the why of the form of the GR Hamiltonian constraint “from plausible first principles”. I consider Hojman–Kuchar–Teitelboim’s spacetime-based first principles, and especially the new 3-space approach (TSA) first principles studied by Barbour, Foster, ´O Murchadha and myself. The latter are relational, and assume less structure, but from these Dirac’s procedure picks out GR as one of a few consistent possibilities. The alternative possibilities are Strong gravity theories and some new Conformal theories. The latter have privileged slicings similar to the maximal and constant mean curvature slicings of the Conformal IVP method.
The plausibility of the TSA first principles are tested by coupling to fundamental matter. Yang–Mills theory works. I criticize the original form of the TSA since I find that tacit assumptions remain and Dirac fields are not permitted. However, comparison with Kuchaˇr’s hypersurface formalism allows me to argue that all the known fundamental matter fields can be incorporated into the TSA. The spacetime picture appears to possess more kinematics than strictly necessary for building Lagrangians for physically-realized fundamental matter fields. I debate whether space may be regarded as primary rather than spacetime. The emergence (or not) of the Special Relativity Principles and 4-d General Covariance in the various TSA alternatives is investigated, as is the Equivalence Principle, and the Problem of Time in Quantum Gravity.
Further results concern Elimination versus Conformal IVP methods, the badness of the timelike split of the EFE’s, and reinterpreting Embeddings and Braneworlds guided by CP and IVP knowledge.

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

Topological ideas in the realm of gravity date back to Riemann, Clifford and Weyl and found a more concrete realization in the wormholes of Wheeler characterized by the Euler-Poincare invariant. They result from attaching handles to black holes.

Observationally, 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. This dichotomy seems to be one of the main obstacles for quantizing gravity. Eddington suggested already 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 Pontrjagin 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 ghosts. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the Poincare lemma for the exterior derivative. 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 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-)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.

Although many details remain to be seen, topological actions are prospective in being renormalizable and, after symmetry breaking, are inducing general relativity as an "emergent phenomenon" for macroscopic spacetime.

Older ref.

• Butterfield, Jeremy (1999). The Arguments of Time. Oxford: Oxford University Press. ISBN 0-19-726207-4. This book focuses on the philosophical motivations and implications of the modern geometrodynamics program.
• Prastaro, Agostino (1985). Geometrodynamics: Proceedings, 1985. Philadelphia: World Scientific. ISBN 9971-978-63-6.
• Misner, Charles W; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0716703440. See chapter 43 for superspace and chapter 44 for spacetime foam.
• Wheeler, John Archibald (1963). Geometrodynamics. New York: Academic Press. LCCN 62-013645.
• Misner, C.; and Wheeler, J. A. (1957). "Classical physics as geometry". Ann. Phys. 2 (6): 525. Bibcode 1957AnPhy...2..525M. doi:10.1016/0003-4916(57)90049-0. online version (subscription required)
• 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.
• J. Wheeler (1961). "Geometrodynamics and the Problem of Motion". Rev. Mod. Physics 44 (1): 63. Bibcode 1961RvMP...33...63W. doi:10.1103/RevModPhys.33.63. online version (subscription required)
• J. Wheeler (1957). "On the nature of quantum geometrodynamics". Ann. Phys. 2 (6): 604–614. Bibcode 1957AnPhy...2..604W. doi:10.1016/0003-4916(57)90050-7. online version (subscription required)

GEOMETRODYNAMICS IS NOT DEAD YET!