Monday, December 24, 2007

Is the description of accelerated expansion in terms of cosmological constant possible in TGD framework?

The introduction of cosmological constant seems to be the only manner to explain accelerated expansion and related effects in the framework of General Relativity. As summarized in the previous posting, TGD allows different explanation of these effects. I will not however go to this here but represent comments about the notion of vacuum energy and the possibility to describe accelerated expansion in terms of cosmological constant in TGD framework.

The term vacuum energy density is bad use of language since De Sitter space is a solution of field equations with cosmological constant at the limit of vanishing energy momentum tensor carries vacuum curvature rather than vacuum energy. Thus theories with non-vanishing cosmological constant represent a family of gravitational theories for which vacuum solution is not flat so that Einstein's basic identification matter = curvature is given up. No wonder, Einstein regarded the introduction of cosmological constant as the biggest blunder of his life.

De Sitter space is representable as a hyperboloid a2-u2= -R2, where one has a2=t2-r2 and r2=x2+y2+z2. The symmetries of de Sitter space are maximal but Poincare group is replaced with Lorentz group of 5-D Minkowski space and translations are not symmetries. The value of cosmological constant is Λ= 3/R2. The presence of non-vanishing dimensional constant is from the point of view of conformal invariance a feature raising strong suspicions about the correctness of the underlying physics.

1. Imbedding of De Sitter space as a vacuum extremal

De Sitter Space is possible as a vacuum extremal in TGD framework. There exists infinite number of imbeddings as a vacuum extremal into M4×CP2. Take any infinitely long curve X in CP2 not intersecting itself (one might argue that infinitely long curve is somewhat pathological) and introduce a coordinate u for it such that its induced metric is ds2=du2. De Sitter space allows the standard imbedding to M4×X as a vacuum extremal. The imbedding can be written as u= ±[a2+R2]1/2 so that one has r2< t2+R2. The curve in question must fill at least 2-D submanifold of CP2 densely. An example is torus densely filled by the curve φ = αψ where α is irrational number. Note that even a slightest local deformation of this object induces an infinite number of self-intersections. Space-time sheet fills densely 5-D set in this case. One can ask whether this kind of objects might be analogs of branes in TGD framework. As a matter fact, CP2 projections of 1-D vacuum extremals could give rise to both the analogs of branes and strings connecting them if space-time surface contains both regular and "brany" pieces. It is not clear whether the 2-D Lagrangian manifolds can fill densely D> 3-dimensional submanifold of CP2.

It might be that the 2-D Lagrangian manifolds representing CP2 projection of the most general vacuum extremal, can fill densely D> 3-dimensional sub-manifold of CP2. One can imagine construction of very complex Lagrange manifolds by gluing together pieces of 2-D Lagrangian sub-manifolds by arbitrary 1-D curves. One could also rotate 2-Lagrangian manifold along a 2-torus - just like one rotates point along 2-torus in the above example - to get a dense filling of 4-D volume of CP2.

The M4 projection of the imbedding corresponds to the region a2>-R2 containing future and past lightcones. If u varies only in range [0,u0] only hyperboloids with a2 in the range [-R2,-R2+u02] are present in the foliation. In zero energy ontology the space-like boundaries of this piece of De Sitter space, which must have u02>R2, would be carriers of positive and negative energy states. The boundary corresponding to u0=0 is space-like and analogous to the orbit of partonic 2-surface. For u02<R2 there are no space-like boundaries and the interpretation as zero energy state is not possible. Note that the restriction u02>R2 plus the choice of the branch of the imbedding corresponding to future or past directed lightcone is natural in TGD framework.

2. Could negative cosmological constant make sense in TGD framework?

The questionable feature of slightly deformed De Sitter metric as a model for the accelerated expansion is that the value of R would be same order of magnitude as the recent age of the Universe. Why should just this value of cosmic time be so special? In TGD framework one could of course consider space-time sheets having De Sitter cosmology characterized by a varying value of R. Also the replacement of one spatial coordinate with CP2 coordinate implies very strong breaking of translational invariance. Hence the explanation relying on quantization of gravitational Planck constant looks more attractive to me.

It is however always useful to make an exercise in challenging the cherished beliefs.

  1. Could the complete failure of the perturbation theory around canonically imbedded M4 make De Sitter cosmology natural vacuum extremal. De Sitter space appears also in the models of inflation and long range correlations might have something to do with the intersections between distant points of 3-space resulting from very small local deformations. Could both the slightly deformed De Sitter space and quantum critical cosmology represent cosmological epochs in TGD Universe?

  2. Gravitational energy defined as a non-conserved Noether charge in terms of Einstein tensor TGD is infinite for De Sitter cosmology (Λ as characterizer of vacuum energy). If one includes to the gravitational momentum also metric tensor gravitational four-momentum density vanishes (Λ as characterizer of vacuum curvature). TGD does not involve Einstein-Hilbert action as fundamental action and gravitational energy momentum tensor should be dictated by finiteness condition so that negative cosmological constant might make sense in TGD.

  3. The imbedding of De Sitter cosmology involves the choice of a preferred lightcone as does also quantization of Planck constant. Quantization of Planck constant involves the replacement of the lightcones of M4× CP2 by its finite coverings and orbifolds glued to together along quantum critical sub-manifold. Finite pieces of De Sitter space are obtained for rational values of α and there is a covering of lightcone by CP2 points. How can I be sure that there does not exist a deeper connection between the descriptions based on cosmological constant and on phase transitions changing the value Planck constant?

Note that Anti de Sitter space having similar imbedding to 5-D Minkowski space with two time like dimensions does not possess this kind of imbedding. Very probably no imbeddings exist so that TGD would allow only imbeddings of cosmologies with correct sign of Λ whereas M-theory predicts a wrong sign for it. Note also that Anti de Sitter space appearing in AdS-CFT dualities contains closed time-like loops and is therefore also physically questionable.

For details see the chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time".


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