Friday, May 04, 2007

About Equivalence Principle and weaknesses of General Relativity

I wrote a lengthy email to Arkady Khodolenko explaining my views about Equivalence Principle (EP) and possible weaknesses of General Relativity. After having done all this work I realized that it would be a pity to not add the email to blog page after some necessary polishing of typos and addition of some juicy marketing details about the relationship to string models to the end (forgive me, I do my best but I cannot resist this temptation!;-)).

Dear Arkady,

the best manner to tell my own view about weaknesses of GRT is by comparing GRT and TGD based views about gravitational and inertial four momentum, about star models and black hole like singularities, and about cosmology.

1. Problems associated with the definition of four-momentum

The starting point of TGD was the well-known fact that energy-momentum cannot be defined as conserved Noether charges. This means that no general definition of even gravitational mass exists for general solutions of field equations. One can argue that the weakness of gravitation allowing perturbative approach around Minkowski space resolves the situation. In strong gravitational fields the situation is however very unsatisfactory.

The assumption that space-times are 4-surfaces in H=M4× CP2 cures this problem and exact Poincare invariance is realized at the level of H (M4). Space-time surface itself can have even Euclidian signature of metric.

2. The identification of Minkowski coordinates

This picture resolves the problem about physical identification of Minkowski coordinates which is highly non-trivial in GRT and makes the interpretation of the post-Newtonian approximation difficult. The reason is that Minkowski coordinates for M4 define unique choice of coordinates for space-time surface. Lorentz invariance of space-time surface defines unique cosmological time and cosmological principle reduces to the Lorentz invariance of space-time surface implying Robertson-Walker metric.

3. Equivalence Principle

I have been thinking a lot of about what form of Equivalence Principle is possible to realize in TGD framework.

Gravitational four-momentum and inertial four-momentum are well-defined in TGD framework. Inertial four-momentum is conserved but gravitational four-momentum is not. Therefore Einstein's equations cannot hold true generally except perhaps at some macroscopic structural equations analogous to D=epsilon E of electrodynamics under some additional conditions.

The recent formulation of Equivalence Principle (EP) in TGD framework at 3-D parton level is following. In quantum TGD light-like 3-surfaces ("partons") are fundamental objects and holography holds true in the sense that 4-D space-time surfaces associated with with these 3-D surfaces provide classical representation of quantum physics making possible quantum measurement theory among other things. EP holds true in the sense that inertial four-momentum for space-time sheet is temporal average of non-conserved gravitational four-momentum. This form of EP must have some counterpart at 4-D level but Einstein's equations cannot be it.

4. Zero energy ontology

A further distinction from GRT is what I call zero energy ontology which says that all conserved Noether charge type quantum numbers of physical states vanish. This means that physical states consist of positive and negative energy parts with opposite energies separated by some typical temporal distance T. Only in time scales short in the scale defined by TGD positive energy ontology applies. In longer time scales the state could be regarded as a quantum fluctuation: something pops up from vacuum and disappears. This resolves the problem about what where the initial values at the moment of big bang. Quantum jumps replaced the entire 4-D Universe (or superposition of them) with a new one all the "subjective" time. Entire 4-D Universe is created again and again. This resolves also the basic problem of quantum measurement theory.

Only this interpretation suggested very strongly by quantum TGD is consistent with the classical TGD which predicts that the imbeddings of Robertson-Walker cosmologies are always vacuum extremals with respect to inertial four-momentum but non-vacuum extremals with respect to gravitational four-momentum (in general not conserved). The interpretation would be that in the cosmological time resolution the positive and negative energy parts of states are not seen: one has quantum fluctuation below resolution. Gravitational four-momentum density which is not a conserved Noether charge can be however non-vanishing.

5. The problem of cosmological constant

Cosmological constant is basic problems of GRT based theories. Why I take TGD based cosmology very seriously is that it predicts that mass density for all globally imbeddable Robertson-Walker cosmologies is always sub-critical. Cosmologies with over-critical or critical density of gravitational mass make a transition to subcritical one after some time interval. Thus the large value of cosmological constant is not a problem in TGD.

6. Black-holes singularities and model for asymptotic state of star

In TGD framework the model for the asymptotic state of star is based on the assumption that inertial vacuum extremal is in question and gravitational four momentum densities are conserved locally. This gives a modification of minimal surface equation in which the contraction of contravariant metric with second fundamental form is replaced with the contraction of Einstein tensor (plus metric if cosmological constant is non-vanishing) with second fundamental form.

  • Schwartschild metric results as vacuum extremal solution fails below some radius because of non-imbeddability to 8-D space. Black-hole singularities are therefore not possible in TGD framework.

  • More complex vacuum extremals which are not gravitational vacua represent star interior. Solutions represent always rotating system carrying non-vanishing gauge charges. The special feature is dynamo-like structure meaning the existence of strong electric and magnetic fields orthogonal to each other. Solution is stationary and the collapse to a point like object is impossible.

  • Rotating black holes do not allow 8-D imbedding and for small vacuum extremals representing perturbations of the Schwartschild metric representing rotating system the predicted gravimagnetic field equals to that in GRT only at equator but becomes strong near poles. Unfortunately, the satellite tests are carried out at equator.

Here it was and it is actually essentially the same as the original email. I want however add some comments about the relationship of TGD to string models.

  • Light-like 3-surfaces as basic dynamical objects implies that super-conformal symmetries of string models generalize to larger symmetries by the metric 2-dimensionality of these surfaces. The reverse of quantum holography results in the sense that 4-D space-time surfaces provide classical representation of quantum dynamics at parton level (essential for quantum measurement theory). There is thus deep and completely unexpected connection between quantum measurement theory and quantum gravitational holography.

  • Fermions and gauge bosons are not string like objects in TGD framework. This is very important distinction from string models. A generalization of stringy mathematics of course applies inside partonic 3-surfaces by generalized super-conformal symmetries but this is different level.

  • Gravitons are necessarily parton-antiparton bound states that is superpositions of parton-antiparton states with parton and antiparton connected by gauge flux so that they and many many other similar states have a stringy character. Gravitational constant is proportional to the p-adic length scale squared so that an entire spectrum of gravitations with increasing value of G is possible. p-Adic length scale hypothesis suggests however Mersenne primes as preferred ones and M127=2127-1 corresponds to the largest Mersenne defining a not completely super-astrophysical p-adic length scale. The observed gravitation corresponds to this Mersenne. This p-adic length scale corresponds also to electron and color flux tubes associated with this Mersenne are crucial in the nuclear string model. This means a totally unexpected connection between gravitation and nuclear physics. A model of cold fusion, whose experimental verification was discussed in previous posting, is one of the implications of new nuclear physics.

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