What about the relationship of gravitational Planck constant to ordinary Planck constant?
Gravitational Planck constant is given by the expression hbargr= GMm/v0, where v0<1 has interpretation as velocity parameter in the units c=1. Can one interpret also hbargr as effective value of Planck constant so that its values would correspond to multifurcation with a gigantic number of sheets. This does not look reasonable.
Could one imagine any other interpretation for hbargr? Could the two Planck constants correspond to inertial and gravitational dichotomy for four-momenta making sense also for angular momentum identified as a four-vector? Could gravitational angular momentum and the momentum associated with the flux tubes mediating gravitational interaction be quantized in units of hbargr naturally?
- Gravitational four-momentum can be defined as a projection of the M4-four-momentum to space-time surface. It's length can be naturally defined by the effective metric gαβeff defined by the anticommutators of the modified gamma matrices. Gravitational four-momentum appears as a measurement interaction term in the modified Dirac action and can be restricted to the space-like boundaries of the space-time surface at the ends of CD and to the light-like orbits of the wormhole throats and which induced 4- metric is effectively 3-dimensional.
- At the string world sheets and partonic 2-surfaces the effective metric degenerates to 2-D one. At the ends of braid strands representing their intersection, the metric is effectively 4-D. Just for definiteness assume that the effective metric is proportional to the M4 metric or rather - to its M2 projection: geffkl= K2mkl.
One can express the length squared for momentum at the flux tubes mediating the gravitational interaction between massive objects with masses M and m as
gαβeff pαpβ= gαβeff ∂αhk∂βhl pkpl == geffkl pkpl = n2hbar2/L2 .
Here L would correspond to the length of the flux tube mediating gravitational interaction and pk would be the momentum flowing in that flux tube. geffkl= K2mkl would give
p2= n2hbar2/K2L2 .
hbargr could be identifed in this simplified situation as hbargr= hbar/K.
- Nottale's proposal requires K= GMm/v0 for the space-time sheets mediating gravitational interacting between massive objects with masses M and m. This gives the estimate
pgr =[GMm/v0] 1/L .
For v0=1 this is of the same order of magnitude as the exchanged momentum if gravitational potential gives estimate for its magnitude. v0 is of same order of magnitude as the rotation velocity of planet around Sun so that the reduction of v0 to v0≈ 2-11 in the case of inner planets does not mean that the propagation velocity of gravitons is reduced.
- Nottale's formula requires that the order of magnitude for the components of the energy momentum tensor at the ends of braid strands at partonic 2-surface should have value GMm/v0. Einstein's equations T= κ G+Λ g give a further constraint. For the vacuum solutions of Einstein's equations with a vanishing cosmological constant the value of hgr approaches infinity. At the flux tubes mediating gravitational interaction one expects T to be proportional to the factor GMm simply because they mediate the gravitational interaction.
- One can consider similar equation for gravitational angular momentum:
gαβeff LαLβ= geffkl LkLl = l(l+1)hbar2 .
This would give under the same simplifying assumptions
This would justify the Bohr quantization rule for the angular momentum used in the Bohr quantization of planetary orbits.
Maybe the proposed connection might make sense in some more refined formulation. In particular the proportionality between mkleff= Kmkl could make sense as a quantum average. Also the fact, that the constant v0 varies, could be understood from the dynamical character of mkleff.