I returned to the topic later again with a boost given by one of the few people in the finnish academic establishment who have regarded me as a life form with some indications about genuine intelligence. What demonstrates the power of a good idea is that just posing some naturally occurring questions led rapidly to a TGD inspired phenomenology of EG allowing to see what is good and what is bad in EG hypothesis and also to see possible far reaching connections with apparently completely unrelated basic problems of recent day physics.
Consider first the phenomenology of EG in TGD framework.
- Gravitating bodies can be seen as sources of virtual and real gravitons propagating along flux tubes. The gravitons at flux tubes are thermalized and thus characterized by temperature and entorpy when the wavelength is much shorter than the distance between the source and receiver. One can say that massive object serves as a heat source. One could also say that the pair of bodies connected by flux tubes serves as a heat source for the flux tubes with temperature determined by reduced mass so that their is a complete symmetry between the two bodies.
- The expression for the gravitonic entropy of the flux tube is naturally proportional to the length of flux tube at a given "holographic screen" - and for the gravitonic temperature-naturally proportional to the inverse of distance squared in absence of other heat sources from standard Laplace equation- are consistent with their forms at the non-relativistic limit discussed by Sabine Hossenfelder in very transparent manner. In general case, the stringy slicing for the preferred extremals of Kähler action provide the preferred coordinates in which gravitational potential and the counterpart of the radial coordinate can be identified.
- EG generalizes to all interactions but negative temperatures mean a severe problem. This in turn suggests a direct connection with matter-antimatter asymmetry. Could thermally stable matter and antimatter correspond in zero energy ontology to different arrows of geometric time and appear therefore in different space-time regions? I have made this question also earlier but with a motivation coming directly from the formalism of quantum TGD.
This approach leads to the question whether the mathematical formalism of quantum TGD could make sense also in General Relativity when appropriately modified. In particular, do the notions of zero energy ontology and causal diamond and the identification of generalized Feynman diagrams as space-time regions of Euclidian signature of the metric make sense? Does the Kähler geometry for world of classical worlds realizing holography in strong sense lead to a formulation of GRT as almost topological QFT characterized by Chern-Simons action with a constraint depending on metric?
- Einstein-Maxwell theory generalizes Kähler action and the conditions guaranteing reduction of action to
3-D "boundary term" are realized automatically by Einstein-Maxwell equations and the weak form of electric-magnetic duality leads to Chern-Simons action.
- One distinction beween GRT and TGD is the possibility of space-time regions of Euclidian signature of the induced metric in TGD representing the lines of generalized Feynman diagrams. The deformations of CP2 type vacuum extremals with Euclidian signature of the induced metric represent these lines replace black holes in TGD Universe. Black hole horizons are big particles and are suggested to possess gigantic effective value of Planck constant for which Schwartshild radius is essentially the Compton length for gravitational Planck constant so that black hole becomes indeed a particle in quantum sense. Blackholes represent dark matter in TGD sense.
- CP2 type vacuum extremals are solutions of Einstein's equations with a unique value of cosmological constant fixing CP2 radius and this constant can be non-vanishing only in regions of Euclidian signature. The average value of the cosmological constant would be proportional to the ratio of the three-volume of Euclidian regions to the whole volue of 3-space and therefore very small. Could this be equivalent with the smallness of the actual cosmological constant? To answer the question one should understand the interaction between Euclidian and Minkowskian regions. I have proposed alternative manners to understand apparent cosmological constant in TGD Universe. Negative pressure could be understood in terms of the magnetic energy of magnetic flux tubes. On the other hand, quantum critical cosmology replacing inflation in TGD framework characterized by single parameter - its duration- corresponds to "negative pressure". These explanations need not be mutually exclusive.
At the formal level the formalism for WCW Kähler geometry generalizes as such to almost topological quantum field theory but the conditions of mathematical existence are extremely powerful and the conjecture is that this requires sub-manifold property.
- The number of physically allowed space-times is much larger in GRT than in TGD framework and this leads to space-time with over-critical and arbitrarily large mass density and other problems plaguing GRT. M-theory exponentiates the problem and leads to landscape misery. The natural conjecture is that one cannot do without assuming that physically acceptable metrics are representable as surfaces in M4× CP2.
- CP2 type regions give rise to electroweak quantum numbers and Minkowskian regions to four-momentum and spin. This almost gives standard model quantum numbers just from Einstei-Maxwell system! It is however far from clear whether one obtains both of them at the wormhole throats between the Minkowskian and Euclidian regions (perhaps from the representations of super-conformal algebras associated with light-like 3-surfaces by their geometric 2-dimensionality). Since both are needed it seems that one must replace geometry with sub-manifold geometry. Also electroweak spin is obtained naturally only if spinors are induced spinors of the 8-D imbedding space rather than 4-D spinors for which also the existence of spinor structure poses problems in the general case.