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 a^{2}-u^{2}= -R^{2}, where one has a^{2}=t^{2}-r^{2} and r^{2}=x^{2}+y^{2}+z^{2}. 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/R^{2}. 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 M^{4}×CP_{2}. Take any infinitely long curve X in CP_{2} 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 ds^{2}=du^{2}. De Sitter space allows the standard imbedding to M^{4}×X as a vacuum extremal. The imbedding can be written as u= ±[a^{2}+R^{2}]^{1/2} so that one has r^{2}< t^{2}+R^{2}. The curve in question must fill at least 2-D submanifold of CP_{2} 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, CP_{2} 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 CP_{2}.

It might be that the 2-D Lagrangian manifolds representing CP_{2} projection of the most general vacuum extremal, can fill densely D> 3-dimensional sub-manifold of CP_{2}. 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 CP_{2}.

The M^{4} projection of the imbedding corresponds to the region a^{2}>-R^{2} containing future and past lightcones. If u varies only in range [0,u_{0}] only hyperboloids with a^{2} in the range [-R^{2},-R^{2}+u_{0}^{2}] are present in the foliation. In zero energy ontology the space-like boundaries of this piece of De Sitter space, which must have u_{0}^{2}>R^{2}, would be carriers of positive and negative energy states. The boundary corresponding to u_{0}=0 is space-like and analogous to the orbit of partonic 2-surface. For u_{0}^{2}<R^{2} there are no space-like boundaries and the interpretation as zero energy state is not possible. Note that the restriction u_{0}^{2}>R^{2} 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 CP_{2} 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.

- Could the complete failure of the perturbation theory around canonically imbedded M
^{4}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? - 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.
- 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 M
^{4}× CP_{2}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 CP_{2}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?

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

## No comments:

Post a Comment