What Equivalence Principle means?

In GRT Equivalence Principle can be coded typically by Einstein-YM action. What Equivalence Principle means in TGD framework is not at all obvious since this action does not make sense in TGD framework the fundamental action being so called Kähler action.

Basically the problem is about the relationship between inertial and gravitational four-momenta. The real achievement at the level of formal rigor is that inertial and gravitational four-momenta and their generalizations color quantum numbers are well-defined as Noether charges associated with curvature scalar. The most general option is that cosmological constant and gravitational constant are by definition chosen so that gravitational and inertial four-momenta are identical. It however seems un-necssary to introduce cosmological constant and also it seems that one must accept failure of Equivalence Principle although it does not occur for ordinary matter. For instance, string like objects which are vacuum extremals but have gigantic gravitational mass are possible.

Here one must however be very cautious. It is asymptotic behavior of gravitational field created by topologically condensed space-time sheet which matters experimentally, and it is not at all clear under what conditions the mass parameter characterizing the gravitational field equals to the gravitational mass of the space-time sheet defined by Einstein tensor. The reason is that one does not anymore have Einstein's field equations which in linear approximation identify energy momentum tensor as the source of gravitational field.

On the other hand, the covariant divergence of Einstein tensor vanishes and the components of Einstein tensor are essentially what one obtains by applying d'Alembert type operator on components of metric. Hence it is natural to regard topologically condensed space-time sheets as sources of the gravitational field defined by the metric. If these sources corresponds to gravitational charges of the topologically condensed space-time sheets then there are good hopes of obtaining Equivalence Principle at the level of asymptotic behavior of the metric. That pseudo-Riemannian geometry codes for the dynamics of gravitational field without any variational principle is something which is highly non-trivial and means that Einstein's equations derived from EYM action are only a manner to state Equivalence Principle.

The are good reasons to expect that small deformations of vacuum extremals, which are extremals of curvature scalar define in the stationary situation exterior metrics. Field equations state in this kind of situation the conservation of various kinds of gravitational charges. For the simplest exterior metrics one can indeed cast the conservation laws in a form in which one has Einstein tensor at the left hand side of field equations and a term depending on geometric data at the right hand side. The optimistic conjecture is that this is the case more generally but - as noticed - this is not necessary for the realization of Equivalence Principle in the sense of asymptotic behavior.

For details see the updated chapter The Relationship Between TGD and GRT of "Classical Physics in Many-Sheeted Space-Time".