For the ordinary value of Planck constant, TGD predicts the value of TG has upper bound in the range 10-7-10-6. The flat velocity spectrum for distant stars around galaxies determines the value of TG: one has v2=2TG from Kepler law so that the value of TG is determined from the measured value of the velocity v. The value of TG can be also deduced from the energy density of cosmic string-like objects predicted by TGD and is consistent with this estimate. If one takes the empirical evidence for a large value of TGseriously one must ask whether TGD can explain the claimed finding.
Could a large value of heff solve the discrepancy? String tension T as the linear energy density of the cosmic string is determined by the sum of Kähler action and volume term. The contribution of Kähler action to T is proportional to 1/αK = gK2/4πℏ. If cosmic string represents dark matter in TGD sense, one must make the replacement ℏ→ ℏeff so that the Kähhler contribution to T is proportional to ℏeff/ℏ. If the two contributions are of same order of magnitude or Kähler contribution dominates, ℏeff/ℏ=n≈ 105 would give the needed large value TG. The physical interpretation would be that cosmic string is an n-sheeted structure with each sheet giving the same contribution so that the value of T is scaled up by n≈ 105.
The physical interpretation would be that the cosmic string is an n-sheeted structure with each sheet giving the same contribution so that the value of T is scaled up by n≈ 105. There are two options. The n-sheetedness is with respect to M4 so that one has a n-fold covering of M4 or with respect to CP2 in which case one quantum coherent structure consisting of n parallel flux tubes.
It is intereting to consider in more detail the quantum model for the particles in the gravitational field of cosmic string.
- The gravitational field of a straight cosmic string behaves like 1/ρ as a function of the radial distance ρ from string, and Kepler's law predicts a constant velocity v2= 2TG for circular orbits irrespective of their radius. This explains the flat velocity spectrum of stars rotating around galaxies.
- Nottale proposed that planetary orbits obey Bohr quantization for the value of gravitational Planck constant ℏgr= GMm/β0 assignable to a pair of masses M and M associated with the gravitational flux tube mediating the gravitational interaction between M and m.
- If the mass M corresponds to a cosmic string idealized as straight string with an infinite length, the definition of ℏgr is problematic since M diverges. Therefore the application of Nottale's quantization to a distant star rotating cosmic string is problematic.
What is however clear that ℏgr should be proportional to m by Equivalence Principle and one should have ℏgr= GMeffm/β0 for the cosmic string. Meff= TLeff, where Leff is the effective length of the cosmic string is also a reasonable parametrization.
- Kepler law does not tell anything about the value of the radius r of the circular orbit. If the value of ℏgr is fixed somehow, one can apply the Bohr quantization condition ∮ pdq= nhgr of angular momentum to circular orbits to obtain vr= nGMeff giving
A reasonable guess is that β0 and the rotation velocity v/c=(2TG)1/2 have the same order of magnitude. v/c= xβ0< 1 would give β0= (2TG)<1/2/x. The minimal value of the orbital radius would be r1=rS,eff/[2xβ02].
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.