Quantum Gravity and String Theory

   

What Are the Counterparts of Einstein's Equations in TGD?

Authors: Matti Pitkänen

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 framework 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. The vanishing or light-likeness of Kähler current guarantees the vanishing of the divergence for the known extremals.

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. This kind of ansätze can be indeed found and involve the analogs of cosmological constant, which are however not genuine constants anymore. 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 preferred extremals involving Hamilton-Jacobi structure could be identified 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. 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.

Comments: 29 Pages.

Download: PDF

Submission history

[v1] 2013-09-09 06:26:21

Unique-IP document downloads: 141 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus