What are the counterparts of Einstein's equations in TGD?
Whether TGD is consistent with Einstein's equations or not has been a continual source of worry and my interpretations have fluctuated. The working with a quite different problem forced however to ask the basic questions again and led to challenge the assumption inspired by the construction of preferred extremals that the local conservation of energy momentum tensor of Kähler action always forces Einstein-Maxwell action with cosmological term. As a matter, fact much milder conditions are enough to achieve this and imply the existence of several cosmological "constants" (not constants actually) multiplying mutually orthogonal projectors, whose sum replaces the ordinary cosmological constant term. This view unifies the two quite different interpretations of cosmological constant.
The first - "Einsteinian" - interpretation is as a fundamental constant not characterizing real energy density. According to the second interpretation cosmological constant characterizes genuine contributions to energy assigned to inflaton type fields. In the recent case these contributions would correspond to topologically condensed matter such as K\"ahler magnetic flux tubes and particles. As a matter fact, this was the original interpretation, which I challenged after the discovery of preferred extremals. I glue below the abstract of the new chapter What are the counterparts of Einstein's equations in TGD?.
The original motivation of this work was related to Platonic solids. The playing with Einstein's equations and the attempts to interpret them physically forced the return to an old interpretational problem of TGD. TGD allows enormous vacuum degeneracy for Kähler action but the vacuum extremals are not gravitational vacua. Could this mean that TGD forces to modify Einstein's equations? Could space-time surfaces carrying energy and momentum in GRT frameword be vacua in TGD context? Of course, also in GRT context cosmological constant means just this and an experimental fact, is that cosmological constant is non-vanishing albeit extremely small.
Trying to understand what is involved led to the realization that the hypothesis that preferred extremals correspond to the solutions of Einstein-Maxwell equations with cosmological constant is too restricted in the case of vacuum extremals and also in the case of standard cosmologies imbedded as vacuum extremals. What one must achieve is the vanishing of the divergence of energy momentum tensor of Kähler action expressing the local conservation of energy momentum currents. The most general analog of Einstein's equations and Equivalence Principle would be just this condition giving in GRT framework rise to the Einstein-Maxwell equations with cosmological constant.
One can however wonder whether it could be possible to find some general ansätze allowing to satisfy this condition. This kind of ansätze can be indeed found and can be written as kG+∑ΛiPi=T, where Λi are cosmological "constants" and Pi are mutually orthogonal projectors such that each projector contribution has a vanishing divergence. One can interpret the projector contribution in terms of topologically condensed matter, whose energy momentum tensor the projectors code in the representation kG=-∑ΛiPi+T. Therefore Einstein's equations with cosmological constant are generalized. This generalization is not possible in General Relativity, where Einstein's equations follow from a variational principle.
The suggested quaternionic preferred extremals and the preferred extremals involving Hamilton-Jacobi structure might allow identification as different families characterized by the little group of particles involved and assignable to time-like/light-like local direction. One should prove that this ansatz works also for all vacuum extremals. If not, the local conservation of K¨hler energy momentum tensor would be the general formulation for the counterpart of Equivalence Principle in TGD framework. This progress - if it really is progress - provides a more refined view about how TGD Universe differs from the Universe according to General Relativity and leads also to a model for how the cosmic honeycomb structure with basic unit cells having size scale 108 ly could be modelled in TGD framework.
For details see the article What are the counterparts of Einstein's equations in TGD? or the new chapter of "Physics in many-Sheeted space-time".