tisdag 24 januari 2012

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."

LinkAddendum, 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?
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.
GEOMETRODYNAMICS IS NOT DEAD YET!

23 kommentarer:

  1. Mielke with collaborators has a lot interesting published,142 papers on SPIRES. Much about topology and scaling.

    http://www.slac.stanford.edu/spires/find/hep/www?match=or&ea=%22Mielke%2C+E+%2C+%28Ed+%29%22&ea=%22Mielke%2C+E+W%22&ea=%22Mielke%2C+Eckehard+W%22

    Weak equivalence principle from a spontaneously broken gauge theory of gr avity. 2011

    Spontaneously broken topological SL (5, R) gauge theory with standard gravity emerging.

    Einsteinian gravity from a spontaneously broken topological BF theory. Phys.Lett.B688:273-277,2010. 5pp.2010

    Rotating black hole solution in a generalized topological 3-D gravity with torsion. 2009

    Topologically modified teleparallelism, passing through the Nieh-Yan functional.

    Limitations on the topological BF scheme in Riemann-Cartan spacetime with torsion.

    Axion condensate as a model for dark matter halos.

    Gravitational Stability of Boson Stars. 2008 http://arxiv.org/pdf/0810.0696.pdf

    Blue spectral inflation. arXiv:0809.4462

    Einsteinian gravity from a topological action. Gen.Rel.Grav.40:1311-1325,2008.

    Einsteinian gravity from BRST quantization of a topological action. arXiv:0707.3466

    Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy. arXiv:0704.1135

    Toroidal halos in a nontopological soliton model of dark matter. 2007

    S-duality in 3D gravity with torsion.

    Dark Matter Halos as Bose-Einstein Condensates. 2006 astro-ph/0608526

    Anomalies and gravity. hep-th/0605159

    Scalar field haloes as gravitational lenses. astro-ph/0603309

    Cosmological evolution of a torsion-induced quintaxion.

    Current status of Yang's theory of gravity. 2004

    Dark matter problem and effective curvature Lagrangians.

    Duality in Yang's theory of gravity.

    Flattened halos in a nontopological soliton model of dark matter.

    Consistent coupling to Dirac fields in teleparallelism: Comment on 'Metric-affine approach to teleparallel gravity'.

    Scaling behaviour of a scalar field model of dark matter halos. 2004 astro-ph/0401575

    Yang-Mills type BRST and co-BRST algebra for teleparallelism. 2002

    SvaraRadera
  2. Zbl 0695.58028 http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0695.58028&format=complete
    Pràstaro, A.
    Gauge geometrodynamics. (English)
    [J] Riv. Nuovo Cimento 5, No. 4, 1-122 (1982).

    This is an exposition of the geometrical framework for gauge theories, from the bundle viewpoint. In order to handle properly full covariance for physical fields under spacetime transformations the superbundle of geometric objects is introduced. This leads to an interpretation of a spinor field as a superfield of geometric objects, precisely ensuring full covariance for spinor fields. \par Contents: 1. Introduction. 2. Functors and fibre bundles. 3. Derivative spaces and differential equations. 4. Derivative spaces and variational calculus. 5. Connections and derivative spaces. 6. Geometrodynamics of gauge continuum systems and symmetry properties. 7. Classification of gauge continuum sytems. 8. Spinor superbundles of geometric objects and dynamics. 9. Conclusions. \par Appendix A: Categories, final and initial objects and related structures. Appendix B: Hamiltonian formulation of Noether theorem. Appendix C: 0- sequences, exact sequences and splittings. Appendix D: Homotopy and covering. Appendix E: Euclidean spaces and related isometry groups. Appendix F: Gauge geometrodynamics vs. physical language.
    [C.T.J.Dodson (MR 84e:83045)]

    MSC 2000:
    *58J90 Applications
    58C50 Analysis on supermanifolds, etc.
    53C27 Spin and Spin$^c$ geometry
    81R20 Covariant wave equations
    83E05 Geometrodynamics

    Keywords: conservation laws; covariance for spinor fields; gauge geometrodynamics

    Cited in: Zbl 0695.58029


    Pràstaro, A. has much more on the field. http://www.dmmm.uniroma1.it/~agostino.prastaro/PUBLICATIONS.HTM

    Geometrodynamics of non-relativistic continuous media.I: Space-time structures, II: Dynamic and constitutive structures,
    A geometric point of view for the quantization of non-linear field theories,
    http://www.dmmm.uniroma1.it/~agostino.prastaro/Prastaro-Firenze-1984.pdf
    On the geometric generalization of the Noether theorem, On the quantization of Newton equation, Quantum geometry of PDE's, Quantum gravity and group model gauge theory, quantum geometrodynamics. etc.

    SvaraRadera
  3. The group structure of supergravity http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1986__44_1/AIHPA_1986__44_1_39_0/AIHPA_1986__44_1_39_0.pdf

    Geometrodynamics of some non-relativistic incompressible fluids.
    http://dmle.cindoc.csic.es/revistas/detalle.php?numero=1597

    In some previous papers [1, 2] we proposed a geometric formulation of continuum mechanics, where a continuous body is seen as a suitable differentiable fiber bundle C on the Galilean space-time M, beside a differential equation of order k, Ek(C), on C and the assignement of a frame Psi on M. This approach allowed us to treat continuum mechanics as a unitary field theory and to consider constitutive and dynamical properties in a more natural way. Further, the particular intrinsic geometrical framework allowed to utilize directly the formal theory of differential equations in order to obtain criteria of existence of solutions.

    In the present paper we apply this general theory to some incompressible fluids. The scope is to demostrate that also for these more simple materials our theory is a suitable tool in order to understand better the fundamental principles of continuum mechanics.

    SvaraRadera
  4. Penrose 'On the origins of twistor theory'. Gravitation and Geometry, a volume in honour of I. Robinson, Biblipolis, Naples 1987
    http://users.ox.ac.uk/~tweb/00001/

    This 'new approch' is actually very old,e ven older than GR? Introduction, somewhat shorted:
    Felix Klein put forth his correspondence between the lines in complex projective 3-space and a general quadric in projective 5-space as long ago as 1870 (Klein 1870, 1926), this correspondence being based on the coordinates of Julius Plücker (1865, 1868/9) and Arthur Cayley (1860, 1869), (or of Hermann Grassmann, even earlier). Sophus Lie had noted essentially the key "twistor" geometric fact that oriented spheres in complex Euclidean 3-space (including various degenerate cases) could be represented as lines in complex projective 3-space (contact between spheres represented as meeting of lines) already in 1869 (cf. Lie & Scheffers 1896) as was pointed out to me by Helmuth Urbantke some years ago. The spheres may be thought of as the t = 0 representation of the light cones of events in Minkowski space, so the Lie correspondence in effect represents the points of (complexified compactified) Minkowski space by lines in complex projective 3-space, where meeting lines describe null-separated Minkowski points - the twistor correspondence! The local isomorphism between the "twistor group" SU(2,2) and the connected component of the group 0(2,4) was explicitly part of the Cartan's (1914) general study and classification of Lie groups. The physical relevance of 0(2,4) in relation to the conformal motions of (compactified) Minkowski space-time had been exploited by Paul Dirac (1936 b) and the objects which I call twistors (namely the spinors for 0(2,4)) had been explicitly studied by Murai (1953, 1954, 1958) and by Hepner (1962). (See also Gindikin 1983 for a discussion of these matters.)

    Moreover, as Ivor Robinson pointed out to me some ten or so years ago, a certain line-integral expression for representing the general (analytic solution of the wave equation in terms of holomorphic functions of three complex variables was known to Bateman in 1904 (see Bateman 1904 and 1944, p. 96), this having arisen from a similar expression due to Whittaker (1903) for solving the three-dimensional Laplace equation, and Bateman also gave a similar line-integral expression for solving the free Maxwell equations (Bateman 1944, p. 100). By a simple transformation of variables, these become the helicities zero and one cases of the basic contour integral formula (Penrose 1968, 1969a) giving the linear field case of the so-called "Penrose transform" of twistor theory. The Radon transform (Radon 1917, Gel'fand Graev & Vilenkin 1966) and its generalizations may also, from a different angle, be regarded as providing models for (and generalizations of) this twistor expression. In addition, the classic Weierstrass (1866) construction (cf. Darboux 1914) (which was known to me!) had provided a paradigm for the explicit solution, in terms of free holomorphic data, of an important non-linear problem (Plateau's problem). This may be regarded as a direct antecedent of the later non-linear twistor constructions for (anti-) self dual gravitational (Penrose 1976; cf. also Hitchin 1979) and Yang-Mills fields (Ward 1977, Atiyah & Ward 19771 Aityah, Hitchin, Drinfeld & Manin 1978).

    Much of this previously existing material was not known to me twenty years ago. But the Klein correspondence was something I had been well acquainted with since my undergraduate days. So some might argue that there was not a great deal left to be original about in the basic twistor scheme. Nevertheless I do feel that I have a good claim to some sort of originality! This - if we discount a fair number of (non-trivial) later mathematical developments - lies primarily in the essential "physical idea" that the actual space-time we inhabit might be significantly regarded as a secondary structure arising from a deeper twistor-holomorphic reality.

    SvaraRadera
  5. Spinors and space-time: Spinor and twistor methods in space-time ... - Google böcker, resultat
    books.google.fi/books?isbn=0521347866...Roger Penrose, Wolfgang Rindler - 1988 - Mathematics - 512 sidor

    SvaraRadera
  6. TWISTOR DIAGRAMS AND GAUGE-THEORETIC SCATTERING
    AMPLITUDES
    Twistor String Workshop, Oxford, 13 January 2005
    http://www.twistordiagrams.org.uk/papers/jan05.pdf

    twistor representations for free z.r.m. fields, thought of as in-states (positive frequency) and out-states (negative frequency). There are multilinear functionals of these fields yielding (Feynman) scattering amplitudes in Minkowski space.

    We would like to do this in way that expresses:
    ▼ manifest gauge invariance
    ▼ manifest finiteness
    ▼ manifest decomposition into atomic elements like the propagators in Feynman diagrams, with the hope of finding a new dynamical principle in twistor space analogous to the Lagrangian in space-time.

    All the basic elements in the theory were identified by Roger Penrose in about 1970, and scattering amplitudes for massless QED were studied as helicity amplitudes.
    Perhaps no-one guessed how important helicity amplitudes would be...

    Comment: in particular it would hardly have been expected in 1970 that Feynman diagrams for the strong interactions would be needed for interpreting actual collision experiments, and that the helicity amplitude approach to simplifying the Feynman predictions for SU(3) scattering would be vital in practical computation.

    Comment: by manifest finiteness I mean finiteness of the S-matrix for all finitenormed
    states in the Hilbert space. This is a much more stringent criterion than the usual practice of computing functions of momenta, without worrying much about the singularities exhibited by these functions for special external momentum values.

    Momentum states are obviously very important for comparison with collision experiments, but for real consistency with the principles of quantum mechanics, the S-matrix should be completely well-defined for all states.

    Conformal geometry and geometric gravity are the links?

    Maybe this is enough for a short introduction? The geometric aspect of spacetime, and the emergence of geometry, even quantum geometry can be found linking TGD to this. The use of YangMills equations for the complemental side of the equations is also here.

    Prastaro has a link to geometric algebra. http://dmle.cindoc.csic.es/revistas/listado.php?clas=120101

    Lots and lots of useful links...

    I will go on to the foundations of gravity now.

    SvaraRadera
  7. There was a comment on Sarfattis Fb that said Geometrodynamics of Wheeler and GR was the same thing.

    Mattis comment on it:

    At 10:00 PM, matpitka@luukku.com said...

    To Ulla:

    Wheeler's Geometrodynamics cannot be identified with General Relativity. The infinite-D space of 3-metrics is the basic object and the dream is to quantize gravitation in this geometric framework generalizing Einstein's geometrization program. This is one of the deep ideas of Wheeler.

    At least following two basic problems plague this approach.

    The first problem is that one looses time: space-times are what we want. How to make 3-D of geometrodynamics to 4-D of general relativity? Semiclassical approximation to postulated path integral over 4-geometries is the obvious approach but has formidable mathematical difficulties.

    [The mathematical non-existence of the path integral is quite general problem. QFT colleagues have done their best to forget this. Pretend that their is no problem when problem is too difficult: this has been the strategy of modern mainstream theoretical physics and guaranteed that nothing new has emerged for four decades;-).]

    Second problem is that one does not obtain fermions. Fermions are the problem of classical general relativity too: space-time need not allow spin structure at all so that one cannot talk about spinor fields. Also this problem is much more general and plagues also string models and M-theory, would have been excellent hint that space-times must be replaced with 4-surfaces and spinor structure with induced spinor structure, has been put under the rug.

    3-metrics are replaced with 3-surfaces in TGD framework. This solves both basic problems of geometrodynamics. WCW assigns to 3-surfaces space-time surfaces as analogs of Bohr orbits and classical physics becomes part of quantum physics. The problems with fermions and spin are circumvented via induced spinor structure. One fruit of labor is geometrization of fermionic statistics in terms of spinors of WCW.


    Observer participancy is another very deep idea if Wheeler. Delayed choice experiment in which one changes geometric past is inspired by this idea.

    Skeptic would react by saying that before we can talk about observer participancy we must have a physical definition for observer: we do not. Optimistic skeptic might try to imagine what this definition might be on basis of existing and maybe some new ideas. The notion of self is TGD inspired attempt to meet the challenge.

    Evolution as a sequence of quantum jumps recreating the Universe repeatedly would realize observer-participancy in TGD Universe.

    Most of us speak about cognitive, social, and cultural developments as something self evident. But theoretical physicist does not use these words. Very many words of biology and neuroscience are absent from his vocabulary: behavior, function, goal, homeostasis, punishments and rewards, evolution...: everything relating to intentionality, goal directedness, values is absent .., The brutal reason is that the existing mathematical tools do not allow even attempt to define these notions. New mathematics and new concepts are needed.

    There is of course the easy way out: self-deception which is as easy as cheating the innocent laymen. Just say that all that is is nothing but dance of quarks and consciousness is illusion and life is nothing but complexity!

    This is why I am talking about physics as generalized number theory, p-adic physics, infinite primes, hierarchy of Planck constants, etc.... I am observer wanting to participate the expansion of our understanding about the white regions of the map;-).

    SvaraRadera
  8. http://www.matpitka.blogspot.com/2012/02/progress-in-number-theoretic-vision.html#c7049639570994917306 The link left out.

    SvaraRadera
  9. Spacetime and the Philosophical Challenge of Quantum Gravity
    J.Butterfield, C.J.Isham
    1999 http://arxiv.org/abs/gr--qc/9903072

    We survey some philosophical aspects of the search for a quantum theory of gravity, emphasising how quantum gravity throws into doubt the treatment of spacetime common to the two `ingredient theories' (quantum theory and general relativity), as a 4-dimensional manifold equipped with a Lorentzian metric. After an introduction, we briefly review the conceptual problems of the ingredient theories and introduce the enterprise of quantum gravity We then describe how three main research programmes in quantum gravity treat four topics of particular importance: the scope of standard quantum theory; the nature of spacetime; spacetime diffeomorphisms, and the so-called problem of time. By and large, these programmes accept most of the ingredient theories' treatment of spacetime, albeit with a metric with some type of quantum nature; but they also suggest that the treatment has fundamental limitations. This prompts the idea of going further: either by quantizing structures other than the metric, such as the topology; or by regarding such structures as phenomenological.

    SvaraRadera
  10. Hamed asked at TGD blog, this is Mattis answer:

    http://matpitka.blogspot.com/2012/03/icarus-measures-light-velocity-for.html#c8574462032044329126

    Your first question was following:

    "I learned the basic notions of Riemannian geometry like geodesics, Riemann tensor and … but I am skeptic about the basic notions of the manifold. For example one can assigns to each point of a manifold a tangent and a cotangent vector space but I don’t think in TGD a 3-surface composes of points? Does in TGD Classical space-time points would be replaced by regions of the space-time as like the viewpoint of C. J. Isham? Then it should be reconsideration in basic notions of manifold."


    I am not quite sure what you mean! 3-surface is locally a manifold and decomposes to points: in other words it is chartable and therefore can be represented by 3-D maps with each page of the map 3-D Euclidian space E^3 and there are chart maps identifying the images of same point at different pages. Chart book about the surface of our planet is an example about what I mean! There are diffeomorphisms between different pages of the book.

    One can assign to it tangent and co-tangent spaces and their tensor products and powers and this is necessary in order to define forms and vector fields. For instance, induced metric can be interpreted in terms of induced tangent bundle.

    There are of course some delicacies. Local manifold property can fail for surfaces and it does so for the "lines" of generalized Feynman diagrams at vertices in the same manner as it fails for ordinary Feynman diagrams at vertices due to the fact that the topology of "Y" is not topology of "I".

    There is also the notion of finite measurement resolution which could be also understand as discretization obtained b y replacing space-time regions with points. Consider as an concrete example all space-time points for which coordinates have same first N decimal digits. This defines certain space-time region replaced with single point.

    About Isham's view point I do not know enough to say anything.

    SvaraRadera
  11. Endre Szemeredi (Abel Prize winner 2012)http://gowers.files.wordpress.com/2012/03/talktalk2.pdf

    is a towering gure in the area of mathematics known as combinatorics, with particularly important contributions to the subarea called extremal combinatorics. I will explain what these terms mean in a moment, but fi rst here are a few bald facts about his extraordinary mathematical output. The achievement for which he is best known is his proof, in 1975, of what is now called Szemeredi's theorem but which at the time was a notorious and decades-old conjecture of Erdös and Turan. This theorem is one of the highlights of twentieth-century mathematics, but it also lies at the heart of a great deal of very recent research. He also gave us Szemeredi's regularity lemma, a result that originated in the proof of Szemeredi's theorem but went on to become a major tool in extremal combinatorics. As well as these results, he has published over 200 papers, many of them representing important advances. I shall pick one or two, but it should be understood that they are just a small sample from a huge output that has profoundly influenced many areas of mathematical thought.
    What, then, is combinatorics? One possible de nition is that it is the study of discrete structures. And what are they? Well, the word "discrete" is typically contrasted with the word "continuous": a structure is continuous if you can move smoothly from one part to another, whereas it is discrete if you have to jump . For example, if you are modelling the flow of a fluid, then the mathematical structures you study will be continuous, since you will specify things like velocities and pressures at various points, and these vary smoothly. By contrast, if you are modelling the inside of a computer, then you will be interested in sequences of 0s and 1s, which is an example of a discrete structure, since to get from one such sequence to another you have to cause at least one 0 to jump to a 1 or vice versa.
    Another discrete structure, and perhaps the single most important in combinatorics, is the graph(net). The points are called vertices and the lines are called edges. You might think that a graph is continuous, because you can move continuously along its edges. However, it is just the picture that is continuous rather than the graph itself.

    SvaraRadera
  12. http://morfometriagraceli.blogspot.com.br/

    Some scattered thoughts.

    Graceli cosmic theory.
    Phases, flows, mutabilidades planning, desplanificação, stretching and shortening, rotations and translations of galaxies and the cosmos itself. By the action of phenomenality graceli.

    This causes the galaxies go through phases of stretches and shortenings, schedules, rotations and translations.

    That is, the galaxies within a cluster do not follow a sync between all cluster galaxies and therefore depends on the stage you are on.

    The recession itself dependent on this energy graceli and phenomenality graceli.

    Relativity and the revolutions of unicidades graceli.
    1 - Rotations and translations of electrons, stars and galaxies happen around centers, and in the case of rotation of its own center, and develop plans around magnetic power tracks graceli that exist within matter and space. That is, happens around centers of stars and power plans graceli.

    2 - The moves follow stages, mutuals, mutabilidades, alternancidades following the system and fenomenalidades graceli alquimifísica.

    3 - Where the movements are part of the nature of energy and matter, ie, it's the nature and transformation.

    4 - It can be uneven as its variability, desmorfolizado, twisting, flipping continuing stretches flows, flows bamboleios [varied forms], vibrating, oscillating, variational [see graceli theories in other articles].

    The tracks and plan graceli stabilize the orbits and rotations, layers graceli slows the recession of the stars and galaxies.

    Graceli with its una astrocosmicafísica [micro, medium and macro] phase, layers, and energy bands, one graceli field and other phenomena, and its relationship with the unifying alquimifísica and physical, in your system includes all the movements and directions of movements regarding the chirality of the translation of the secondary relative to the primary rotation [systems on systems], so all science leads to new coverage and perspectives, including philosophy and theology.

    SvaraRadera
  13. http://xxx.arxiv.org/abs/gr-qc/0106075 . GEOMETRODYNAMICS, INERTIA AND THE QUANTUM VACUUM, Bernard Haisch & Alfonso Rueda, AIAA paper 2001-3360, presented at AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Salt Lake City, July 8-12, (2001).

    http://xxx.arxiv.org/abs/gr-qc/0009036 INERTIAL MASS AND THE QUANTUM VACUUM FIELDS, B. Haisch, A. Rueda & Y. Dobyns, Annalen der Physik, 10, 393-414 (2001).

    We do not enter into the problems associated with attempts to explain inertia via Mach’s Principle, since we have discussed this at length in a recent paper [48]: a detailed discussion on intrinsic vs. extrinsic inertia and on the inability of the geometrodynamics of general relativity to generate inertia reaction forces may be found therein. It had already been shown by Rindler [49] and others that Mach’s Principle is inconsistent with general relativity, and Dobyns et al. [48] further elaborate on a crucial point in general relativity that is not much appreciated: Geometrodynamics merely defines the geodesic that a freely moving object will follow. But if an object is constrained to follow some different path, geometrodynamics has no mechanism for creating a reaction force. Geometrodynamics has nothing more to say about inertia than does classical Newtonian physics. Geometrodynamics leaves it to whatever processes generate inertia to generate such a force upon deviation from a geodesic path, but this becomes an obvious tautology if an explanation of inertia is sought in geometrodynamics.

    From http://www.calphysics.org/haisch/publications.html

    Mattis comment: I have considered in what sense Mach's principle could make sense in TGD framework. I do not believe Mach's principle in the standard sense: http://en.wikipedia.org/wiki/Mach%27s_principle .
    That is distant stars would explain centrifugal acceleration.

    I have considered just for fun alternative interpretations in the section "Miscellaneous topics" http://tgdtheory.com/public_html/tgdclass/tgdclass.html#tgdgrt . Usually the idea ending in a section with this title is near the end of its life cycle;-).

    SvaraRadera
  14. Why does F = m a in Newton's equation of motion? How does a gravitational field produce a force? Why are inertial mass and gravitational mass the same? It appears that all three of these seemingly axiomatic foundational questions have an answer involving an identical physical process: interaction between the electromagnetic quantum vacuum and the fundamental charged particles (quarks and electrons) constituting matter. All three of these effects and equalities can be traced back to the appearance of a specific asymmetry in the otherwise uniform and isotropic electromagnetic quantum vacuum. This asymmetry gives rise to a non-zero Poynting vector from the perspective of an accelerating object. We call the resulting energy-momentum flux the {\it Rindler flux}. The key insight is that the asymmetry in an accelerating reference frame in flat spacetime is identical to that in a stationary reference frame (one that is not falling) in curved spacetime. Therefore the same Rindler flux that creates inertial reaction forces also creates weight. All of this is consistent with the conceptualizaton and formalism of general relativity. What this view adds to physics is insight into a specific physical process creating identical inertial and gravitational forces from which springs the weak principle of equivalence. What this view hints at in terms of advanced propulsion technology is the possibility that by locally modifying either the electromagnetic quantum vacuum and/or its interaction with matter, inertial and gravitational forces could be modified. http://arxiv.org/abs/gr-qc/0106075

    SvaraRadera
  15. http://www.worldscientific.com/worldscibooks/10.1142/8095

    Series on Knots and Everything: Volume 47
    Introduction to the Anisotropic Geometrodynamics
    By (author): Sergey Siparov (State University of Civil Aviation, Russia)

    The well-known observations of the flat rotation curves of spiral galaxies and of the gravitational lensing effect greatly exceeding the expectations based on the classical GRT can be explained without bringing in the notion of dark matter. The Tully-Fisher law and the unusual features of globular clusters' motion then become clearer. It also turns out that new features appear in the cosmological picture that involves the Universe expansion and acceleration.

    The theory and the first observational results of the specific galactic scale experiment based on the optical-metrical parametric resonance are also discussed in the book. Instead of the direct measurements of the extremely small gravitational waves, it appears sufficient just to register their action on the radiation of the space masers for special cases when the source of the gravitational wave is strictly periodic and presents a close binary system. When the amount of data obtained in such observations is large enough, it would be possible to judge upon the geometrical properties of the space-time region enveloping our galaxy, the Milky Way.

    The foundations of the new approach stem from the equivalence principle which is the basics of the classical GRT. In order to make the presentation self-contained, the roots of century-old ideas are discussed again. This makes the book interesting not only to the specialists in the field but also to graduates and ambitious undergraduate students.

    http://www.worldscientific.com/doi/suppl/10.1142/8095/suppl_file/8095_chap01.pdf

    SvaraRadera
  16. Time in quantum geometrodynamics, 2000 by Arkady Kehyfets, arxiv gr-qc/0006001
    Klaus Kiefer 2008, Quantum geometrodynamics: whence, wither? arxiv gr-qc/08120295

    SvaraRadera
  17. http://arxiv.org/pdf/gr-qc/0409123v1.pdf GEOMETRODYNAMICS:
    SPACETIME OR SPACE? by Edward Anderson, Ph.D.thesis 2004, 231 pp.

    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 a
    nd 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 m
    ethods, the badness of the timelike split of the EFE’s, and reinterpreting Embeddings and Braneworlds guided by CP and IVP knowledge.

    SvaraRadera
  18. More of Andersson (55 total).
    http://arxiv.org/abs/gr-qc/0211022 Scale-Invariant Gravity: Geometrodynamics

    http://arxiv.org/abs/gr-qc/0312037 Spacetime or Space and the Problem of Time

    http://arxiv.org/abs/gr-qc/0511068 Relational Particle Models. I. Reconciliation with standard classical and quantum theory http://arxiv.org/abs/gr-qc/0511069 Relational Particle Models. II. Use as toy models for quantum geometrodynamics

    http://arxiv.org/abs/gr-qc/0511070 On the recovery of geometrodynamics from two different sets of first principles

    http://arxiv.org/abs/gr-qc/0611007 Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models http://arxiv.org/abs/gr-qc/0611008 Emergent Semiclassical Time in Quantum Gravity. Full Geometrodynamics and Minisuperspace Examples

    http://arxiv.org/abs/gr-qc/0702083 Classical dynamics on triangleland

    http://arxiv.org/abs/0711.0285 Does relationalism alone control geometrodynamics with sources?

    http://arxiv.org/abs/0809.1168 Triangleland. I. Classical dynamics with exchange of relative angular momentum http://arxiv.org/abs/0809.3523 Triangleland. II. Quantum Mechanics of Pure Shape

    http://arxiv.org/abs/0909.2439 Shape Space Methods for Quantum Cosmological Triangleland
    http://arxiv.org/abs/1005.2507 Scaled Triangleland Model of Quantum Cosmology

    http://arxiv.org/abs/1009.2157 The Problem of Time in Quantum Gravity http://arxiv.org/abs/1111.1472 The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena http://arxiv.org/abs/1206.2403 Problem of Time in Quantum Gravity http://arxiv.org/abs/1306.5816 Problem of Time: Facets and Machian Strategy
    http://arxiv.org/abs/1209.1266 Machian Time Is To Be Abstracted From What Change? http://arxiv.org/abs/1305.4685 Machian Classical and Semiclassical Emergent Time
    http://arxiv.org/abs/1307.1916 Minisuperspace model of Machian Resolution of Problem of Time. I. Isotropic Case

    SvaraRadera
  19. Now 2017. Kip Thorne also has started to look to Geometrodynamcs to explain the gravitational Waves (he got Nobel Prize 2017 for it). https://arxiv.org/pdf/1706.09078.pdf

    SvaraRadera
  20. 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

    (old URL)

    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.

    Mielke, E.W. (1982): “Ueber die Hypothesen, welche der Geometrodynamik

    zugrund liegen”, Habilitation-Thesis, Universidad de Kiel, 1982, 273 pag.




    Mielke, E.W. (1987): Geometrodynamics of Gauge Fields --- On

    the geometry of Yang--Mills and gravitational gauge theories, (Akademie--Verlag, Berlin), 242 pages.

    Mielke, E.W. (1977): “Knot wormholes in geometrodynamics?”, Gen. Rel. Grav. 8, 175 -- 196. [reprinted in Knots and Applications, L.H. Kauffman, ed. (World Scientific, Singapore 1995), p. 229 -- 250].

    Mielke, E.W. (1977): “Outline of a new geometrodynamical model of extended baryons”, Phys. Rev. Lett. 39, 530 -- 533; 851 (E).

    Mielke, E.W. (1980): “The eightfold way to color geometrodynamics”, Int. J.

    Theor. Phys. 19, 189 -- 209.

    Mielke, E.W. (1985): “Bemerkungen zur Geometrisierung fundamentaler

    Wechselwirkungen der Physik”, Naturwissenschaften 72, 118 -- 124.

    Mielke, E.~W.: ``Einsteinian gravity from a spontaneously broken topological BF theory",Phys. Letters B 688, 273 -277 (2010).

    Mielke, E.W. (1974a): “Quantumstatistics of knot wormholes in geometrodynamics”, Bull. Am. Phys. Soc. 19, 508.

    SvaraRadera
  21. Den här kommentaren har tagits bort av skribenten.

    SvaraRadera
  22. I copy here Mattis comment on Thornes article.

    Matti Pitkänen Geometrodydynamics was originally a creation of Wheeler and was my inspiration during the Odysseia eventually leading to the discovery of TGD forty years ago.

    Wheeler introduced the space of 3-geometries - superspace as he called it - in order to quantize gravitation using the analog of Schroedinger equation. It did not work. Canonical quantization does not generalize to infinite-D contex except for systems with very mild non-linearities and also in this case one has infinities. Fermions are not obtained. Time is lost. It is difficult to understand how gravitational waves emerge using only super-space. The geometry of superspace did not exist mathematically: at that time it was not yet realized that infinite-D geometries are extremely tricky objects and the mere existence of this geometry determines it to high degree. as the case of loop spaces show. This could be actually a excellent news to physicist but string theorists were too busy to stop thinking for the few minutes neeeded to realize this.

    The deepest problem is the loss of classical conservation laws plaguing already classical GRT: no unique definition of translations and Lorentz transformations except when space-time has them as isometries. As general coordinate transformations they are just gauge transformations and one does not obtain the conservation laws.

    The solution of the energy problem is simple: add just single word "Topological" to "Geometrodynamics" and there it is! More precisely: identify space-times as 4-surfaces in H=M^4xCP_2 fixed uniquely the existence of twistor lift of the theory. Superspace is replaced with "world of classical worlds" (WCW).

    One obtains conservation laws: energy and momentum are those assignable to symmetries of H and also color quantum numbers and electroweak quantum numbers are obtained.Geometrization of fermions with many particle states of second quantized ordinary fermions becoming WCW spinors and gamma matrices expressible as superpositions of fermionic oscillator operators. Time are not lost but corresponds to that lM^4. The infinite-D geometry is unique by the infinite-dimensionality of aWCW replacing super-space and has maximal isometries being generalization of loop space Kaehler geometry. Also a unification of standard model and gravity is obtained as an additional bonus.

    Ulla Mattfolk But the 'topological' alsoallow for a variational c, a ftl situation? If you look into the BIG BOOK, there are always bigger sides, or smaller ones. ?

    Matti Pitkänen You probably mean maximal signal velocity with c. It is fixed. The effective propagation velocity can be reduced at space-time surfaces since the road along them is bumpier than along light-like geodesics of M^4xCP_2. Variation of Hubble constant and the strange time lag between neutrinos and gamma rays from SN1987A (for instance) could be interpreted in terms of different space-time sheets for the arrival of neutrinos and gamma rays
    Ulla Mattfolk This is good. http://www.ams.org/journals/tran/1925-027-01/S0002-9947-1925-1501302-6/S0002-9947-1925-1501302-6.pdf

    SvaraRadera
  23. Matti Pitkänen Simple argument based on conformal invariance shows that Einstein tensorand therefore by Einstein's equations energy momentum tensor vanishes for the solutions representing single gravitational wave. This is additional problem besides the ill-definedness of energy and momentum. Also redshift is physically is not understood if one requries conservation of energy and momentum. For some reason these problems have been put under the rug. I thought that already the solution of these problems would mean immediate breakthrough for TGD but I was wrong. Colleagues have not time to think since they must do calculations on existing theory to increase their CV.

    SvaraRadera