WARNING:
JavaScript is turned OFF. None of the links on this concept map will
work until it is reactivated.
If you need help turning JavaScript On, click here.
This Concept Map, created with IHMC CmapTools, has information related to: Equivalence Principle, EQUIVALENCE PRINCIPLE (EP) 2. EP need not be problem of principle in TGD. a) Gravitational interaction couples to inertial four-mo- mentum which is well-defi- ned as classical Noether charge associated with Kähler action. b) Very close analogy of TGD with string models suggest the same. c) Only if one assumes that gravitational and inertial exist separately and are identical, one ends up with potential problems in TGD. d) In cosmology mass is not conserved. This does not mean breaking of Poinare invariance in Zero Energy Ontology (ZEO). Four-momentum depends on the scale of causal dia- mond (CD)., EQUIVALENCE PRINCIPLE (EP) 4. The understanding of EP at classical level has been a long standing head- ache in TGD framework. The even- tual solution looks disappointingly tri- vial. in the sense that its discovery requires only common sense. The most elegant understanding of EP at classical level relies on following argument sug- gesting how GRT space-time emerges from TGD as an effective notion. a) Particle experiences the sum of the effects caused by gravitational forces. The linear superposition for gravitatio- nal fields is replaced with sum of ef- fects describable in terms of effective metric in GRT framework. The effective space-time metric is identified as the sum of M^4 metric and the deviations of the metrics of various space-time sheets from M^4 metric to which par- ticle has topological sum contacts. b) The effective metric is not in gene- ral imbeddable to M^4xCP_2. Under- lying Poincare invariance is not lost but global conservation laws cannot hold true for the effective metric. This suggests that energy-momentum con- servation translates to the vanishing of covariant divergence of energy mo- mentum tensor. c) By standard argument this implies Einstein's equations with cosmological constant Λ: this at least in statistical sense. Λ would parametrize the pre- sence of topologically condensed mag- netic flux tubes. d) Gravitational constant and cosmo- logical constant would come out as predictions. More precisely: dimensi- onless constant n characterizing Planck length is predicted: L_P^2=R^2/n, R CP_2 radius., EQUIVALENCE PRINCIPLE (EP) 1. Several interpretati- ons: a) Global form: inertial mass = gravitational mass (more generally four-momentum). Does not make sense in general relativity since four-momentum is not well-defined. b) Local form: Einstein's equations. Gravitational mass a parameter ap- pearing in asymptotic expression of solutions of Einstein's equations., EQUIVALENCE PRINCIPLE (EP) 3. EP in quantum TGD. a) Inertial momentum is de- fined as Noether charge for Kähler action. b) One can assign to Kähler- Dirac action quantal four-mo- mentum. Its conservation is however not at all all trivial since imbedding space coordi- nates appear in KD action like external fields. c) It however seems that at least for the modes localized at string world sheets the four-momentum conservation could be guaranteed by an assumption motivated by holomorphy. d) Quantum Classical Corre- spondence (QCC) suggests that the eigenvalues of quan- tal four-momentum are equal to those of Kähler four-mo- mentum. QCC would imply EP! e) This generalizes to Car- tan sub-algebra of symmet- ries and would give a corre- lation between geometry of space-time sheet and con- served quantum numbers, EQUIVALENCE PRINCIPLE (EP) 5. To sum up: TGD is microco- pic theory of gravitation and GRT its statistical limit obtained by replacing M^4 with metric with effective metric. The ex- perimental challenge is to make many-sheetedness visible. One can of course ask whether EP or something akin to it could be realized for preferred extre- mals of Kähler action. a) In cosmological and astro- physical models vacuum extre- mals play key role. Could small deformations of them provide realistic enough models for astrophysical and cosmological scales in statistical sense. b) Could preferred extremals satisfy something akin to Einstein's equations? Maybe! The mere condition that the covariant divergence of energy momentum tensor for Kähler action vanishes, is sa- tisfied if Einsteins equations with cosmological terms are satisfied. One can however consider also argue that this condition can be satisfied also in other manners. c) For instance, four-momen- tum currents associated with them be given by Einstein's equations involving several cosmological "constants". The vanishing of covariant diver- gence would however give a justification for why energy- momentum tensor is locally conserved for the effective metric and thus gives rise to Einstein's equations.
