Monday, December 07, 2009

Long length scale limit of TGD as General Relativity with sub-manifold constraint

What is the precise relationship of the long length scale limit of TGD to General Relativity as a description of gravitational interactions? On basis of physical intuition it is clear that Einstein's equations hold true for the matter topologically condensed around vacuum extremals of Kähler action and that energy momentum tensor can be described as average description for small deformations of vacuum extremals. The question is what happens in the case of non-vacuum extremals. Does a simple variational principle leading to Einstein's equations at long length scale limit exist and allow to identify the solutions as extremals of Kähler action?

The answer to the question is affirmative. It has been clear from the beginning that TGD in long length scales as a theory of gravitational interactions is General Relativity with a sub-manifold constraint. The problem is to formulate this statement so that extremals of Kähler action are consistent with Einstein's equations. This requires basic wisdom about sub-manifold geometry and about variational principles and boils dow to two and half pages of formulas difficult to transform to html. Interested reader can click this to see the details.

This could be also a good exercise in the noble art of debunking: the average debunker does not understand anything about contents but it seems that he cannot avoid saying something about it. How to debunk convincingly in this kind of situation? This is a real challenge in bad rhetorics. But only for an advanced debunker. Novices should develop their skills with simpler targets such as entire life work. After all, a life work consisting of about 15.000 pages is much easier to debunk than two and half pages of text because one can make "general statements" and avoid comments about content.

The relationship between TGD and GRT is described in the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".


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