Tuesday, March 30, 2021

Does the quantal gravitational force vanish below critical radius in average sense?

Nottale's gravitational constant hbargr= GMDm/v0 contains dark mass MD as a parameter. At the surface of Earth MD much smaller than MD and for the planets  one has MD=MSun. It turns out that in the  average sense  MD must grow to M.   This is  required by  the condition that  Bohr radii correspond to the classical radii in the average sense. The actual dependence of MD on r  is expected to  be a staircase like function.

At the quantum level, this   effectively eliminates  the average  gravitational force in the scales below the critical radius rcr above  which MD=M is true.   Indeed, due to the average MD∝ r dependence,  gravitational potential would be constant on the average. 

 Could one regard this   effective elimination of  the gravitational force as a kind of    Quantum Equivalence Principle or   as an analog of asymptotic freedom?

See the article Two alternative generalizations of Nottale's hypothesis or the chapter About the Nottale's formula for hgr and the relation between Planck length and CP2 length.

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

Articles and other material related to TGD.


Crow- said...

this does not seem realistic to me. i think after i derive thr cardinality of the continuum in R3 it will make more sense. Grothendieck sites

Matti Pitkänen said...

It does not seem realistic to me either;-). It was just a question. The correct answer came later. I was pondering a problem relaed to cyclotron frequency scale. It was too large if h_gr= GMm/v_0 for M = Earth's mass M_E. and v_0/c= 2^{-11} suggested by by Nottale's model for planetary orbits as Bohr orbits.

I had three proposals for a solution of the problem. G could be replaced with G_D: TGD indeed predicts the value of G and it could have a spectrum and there is evidence for small variations of G considerably larger than the measurement uncertainty. M_D discussed here was second solution: M is replaced with dark mass M_D considerably smaller than M_E. M_D could also be length scale dependent in many-sheeted space-time.

It however turned out that the correct solution is to increase v_0/c: v_0/c=1/2 near the surface of Earth. By the way, it gives for gravitational Compton length of particle with mass m, Lambda_gr= 2GMm = r_S, Schwarschild radius of Earth about 1 cm. It was stupid that I did not realize this at first. Also in Nottales model v_0 is smaller for outer planets.