Thursday, December 14, 2017

Does instability of Einstein's equations explain accelerated expansion?

The article Doing without dark energy: Mathematicians propose alternative explanation for cosmic acceleration (thanks for link to Steven Crowley) makes a very interesting claim. The instability of Einstein's equations requires accelerated expansion usually assumed to be due to a small value of cosmological constant. I understood that the instability of Einstein' cosmological solutions around cosmological solutions is however not proven.

What is remarkable that the exact analog of this instability occurs in TGD.

  1. I believed for a long time that Kähler action alone determines the dynamics. The special feature of Kähler action was its huge vacuum degeneracy, which I interpreted in terms of 4-D spin glass nature of theory and did not regard it as a problem. This instability however destroyed all hopes about path integral approach to the quantum TGD and led to the notion of "world of classical worlds". It became soon clear that it really works and leads to a beautiful physical interpretation. Mistakes sometimes force to see a really big problem and even solve it!

    In particular, for the canonically imbedded empty Minkowski space, which should give rise to an analog of massless field theory, the field equations give 0=0! The propagator defined by the kinetic term vanishes identically. The situation if of course mathematically very singular and the interpretation is as instability since the infinite-D matrix defined by the second order functional derivatives has not only some vanishing eigenvalues but vanishes identically. One expects a hierarchy of singularities with increasing number of non-vanishing eigenvalues.

  2. Few years ago my views changed as I started to think about twistorialization of TGD at the level of space-time dynamics. The basic idea is simple: in TGD framework particles are massless in 8-D sense but can be massive in 4-D sense. 8-D twistorialization could solve the basic problem of ordinary twistorialization due to massive particles.

    Twistor lift of space-time surfaces is needed. This requires the introduction of 6-D twistor space of space-time surface as a dynamical entity: a sphere bundle of space-time surface receiving twistor structure by induction from the product of twistor spaces for M4 and CP2.

    What puts bells ringing is that M4 and CP2 are the unique 4-D manifolds with twistor space possessing Kähler structure. This makes possible the generalization of Kähler action to its 6-D variant in the 12-D twistor space of M4×CP2. The existence of twistor lift of TGD would lead to standard model symmetries and fix TGD completely!

    The allowed solutions of field equations/preferred externals should be twistor spaces of spacetime surfaces. The natural guess is that they are also extremals of 6-D Kähler action, which is a generalization of Maxwell action. Dimensional reduction for 6-D Kähler action to sum of Kähler action and volume term serving as analog of cosmological term is proposed to determine the 6-D preferred extremals, which should correspond to 6-D twistor spaces of 4-D space-time surfaces (sphere bundles with space-time surface as base space). It however seems that cosmological term at GRT limit of TGD corresponds to the sum of Kähler action and volume term rather than only volume term: here I must be however cautious.

    Twistor space property for 6-surface should determine the dynamics completely. This generalizes Penrose's approach reducing Maxwell's equations to holomorphy. A generalization of the approach of Penrose from gauge fields to induced gauge fields would be in question.

Preferred extremals for Kähler action plus volume term come in two classes.
  1. Minimal surface solutions for which field equations reduce to holomorphy conditions. One has minimal surfaces and the equations for Kähler action and volume term are separately satisfied and reduce to holomorphy conditions. These solutions are analogs of light-like geodesic lines: massless particles but in 8-D sense allowing massive particles in 4-D sense so that the basic problem of ordinary twistor approach is solved.

  2. Second type of solution corresponds to an analog for the motion of massless particle in Abelian gauge field, which is however internally determined and vanishes in equiibrium situations: that is for external particles entering causal diamond. Physically these solutions describe interaction regions and minimal surfaces to external particles with interactions coupled off.

The addition of volume term solves the instability/degeneracy problem and the analog of volume term is indeed cosmological constant term in GRT! The spin glass degeneracy could quite well remain as approximate property since the cosmological constant in recent cosmology is extremely small and just takes care that the degeneracy disappears. At the limit of infinitely long p-adic length scale cosmological constant vanishes and one has vacuum degeneracy. The reason is that cosmological constant depends on p-adic length scale going like 1/p-adic length scale squared propto 1/p. This predicts that cosmological constant evolves and approaches its maximal and very large value in the early Universe. Its value is not "un-natural" anymore. One should however be critical and check for whether two strong assumptions are made. Everything is guesswork after all.
  1. The preferred extremals in the interaction regions correspond at the level of M8 (recall M8-H duality) to space-time surfaces, which are not associative/quantum critical (having quaternionic (associative) tangent or normal space) so that they cannot be mapped to surfaces in M4 ×CP2 (requires quaternionicity of tangent space). It is however enough to do this only for external particles which are associative surfaces and correspond to quantum criticality. Kähler action +volume term take care of the time evolution inside CDs once 3-surfaces at boundaries of CD are known.

  2. For minimal surfaces defining external particles dynamics is determined by holomorphy and holomorphy is the key feature of twistorialization. Could it be that twistor bundle property for the 6-D sphere bundle is true only for external particles? This would correspond to associativity of external particles at the level of M8. Could it be that twistor space property (characterizing also for standard twistors free gauge fields) fails in interaction regions - CDs - and is true only for external particles? Just asking;-).

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


Stephen A. Crowley said...

What is so hard about determining whether Einstein's equations have an instability? I must confess, I still do not understand tensors. A matrix is just a "2-tensor" ? The inverted pendulum is a classical example used in control theory to test stabilization methods.. why can't the same type of analysis be done?

Matti Pitkänen said...

Good question.

Certainly Einstein's equations are highly non-linear. Infinite number of variables is certainly part of answer. Any metric with Minkowski signature will due if one takes the source term given by energy momentum tensor T as free input. If one expresses it in terms of other fields satisfying action principle, not all metrics are possible. This does not help much but makes things more complicated. Besides this one must make fermionic fields Grassmann algebra valued and the classical field equations make no sense anymore and one must use path integral approach.

In TGD frame one does not have path integral: allowed space-time surfaces are like Bohr orbits, preferred extremals and there are very few of them. The situation is also very simple locally: just 4 coordinates of M^4xCP_2 appear as "field like" variables. Strong form of holography reducing the boundary data to 2-D data implies preferred extremal property. Preferred extremal property reduces to a huge generalization of already infinite-D conformal symmetries in H=M^4xCP_2 picture and to a reduction to algebraic equations for algebraic surfaces in M^8 picture.

This allows simple interpretation of solutions in terms of external particles entering into interaction region defined by causal diamond (there is of course infinite hierarchy of CDs but fractal reductionism makes life still very simple). It is amusing that the phenomenological picture of particle physics translates to the structure of space-time surfaces very precisely.