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Thursday, November 10, 2016 9:27

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Eric Verlinde has published article with title * Emergent Gravity and the Dark Universe* > (see this). The article represents his recent view about gravitational force as thermodynamical force described earlier and suggests an explanation for the constant velocity spectrum of distant stars around galaxies and for the recently reported correlation between the real acceleration of distant stars with corresponding acceleration caused by baryonic matter. In the following I discuss Verlinde's argument and compare the physical picture with that provided by TGD. I have already earlier discussed Verlinde's entropic gravity from TGD view point (see this). The basic observation is that Verlinde introduces long range quantum entanglement appearing even in cosmological scales: in TGD framework the hierarchy of Planck constants does this in much more explicit manner and has been part of TGD for more than decade. It is nice to see that the basic ideas of TGD are gradually popping up in literature.

Before continuing it is good to recall the basic argument against the identification of gravity as entropic force. As Kobakidzhe notices neutron diffraction experiments suggests that gravitational potential appears in the Schrödinger equation. This cannot be the case if gravitational potential has thermodynamic origin and therefore follows from statistical predictions of quantum theory: to my opinion Verlinde mixes apples with oranges.

Consider now Verlinde's recent argument.

- Verlinde wants to explain the recent empirical finding that the observed correlation between the acceleration of distant stars around galaxy with that of baryonic matter (see this) in terms of apparent dark energy assigned with entanglement entropy proportional to volume rather than horizon area as in Bekenstein-Hawking formula. This means giving up the standard holography and introducing entropy proportional to volume.

To achieve this he replaces anti-de-Sitter space (AdS) with de-Sitter space(dS) space with cosmic horizon expressible in terms of Hubble constant and assign it with long range entanglement since in AdS only short range entanglement is believed to be present (area law). This would give rise to an additional entropy proportional to volume rather than area. Dark energy or matter would corresponds to a thermal energy assignable to this long range entanglement.

The physical picture has analogies with my own approach (see this) to the explanation of the correlation between baryonic acceleration with observed acceleration of distant stars. In particular, long range entanglement has the ieentification of dark matter in terms of phases labelled by the hierarchy of Planck constants as TGD counterpart.

- Concerning the emergence of space and gravitation TGD leads to a different view. It is not 3-space but the experience about 3-space – proprioception -, which would emerge via tensor nets realized in TGD in terms of magnetic flux tubes emerging from 3-surfaces defining the nodes of the tensor net (see this) . This picture leads to a rather attractive view about quantum biology.
- Long range entanglement even in cosmic scales would be crucial and give the volume term in entropy breaking the holography in the usual sense. In TGD framework hierarchy of Planck constants h
_{eff}=n× h satisfying the additional condition h_{eff}=h_{gr}, where h_{gr}=GMm/v_{0}(M and m are masses and v_{0}is a parameter with dimensions of velocity) is the gravitational Planck constant introduced originally by Nottale , and assignable to magnetic flux tubes mediating gravitational interaction makes. This makes possible quantum entanglement even in astrophysical and cosmological long length scales since h_{gr}can be extremely large. In TGD however most of the the galactic dark matter and energy is associated with cosmic strings having galaxies along it (like pearls in necklace). Baryonic dark matter could correspond to the ordinary matter which has resulted in the decay of cosmic strings taking the role of inflaton field in very early cosmology. This gives automatically a logarithmic potential giving rise to constant spectrum velocity spectrum modified slightly by baryonic matter and a nice explanation for the correlation, which served as the motivation of Verlinde. - Also glass analogy has TGD counterpart. Kähler action has 4-D spin glass degeneracy giving rise to 4-D spin-glass degeneracy. In twistor lift of TGD cosmological term appears and reduces the degeneracy by allowing only minimal surfaces rather than all vacuum extremals. This removes the non-determinism. Cosmological constant is however extremely small implying non-perturbative behavior in the sense that the volume term for the action is extremely small and depends very weakly on preferred extremal. This suggests that spin glass in 3-D sense remains as Kähler action with varying sign is added.
- The mere Kähler action for the Minkowskian (at least) regions of the preferred extremals reduces to a Chern-Simons terms at light-like 3-surfaces at which the signature of the induced metric of the space-time surface changes from Minkowskian to Euclidian. The interpretation could be that TGD is almost topological quantum field theory. Also the interpretation in terms of holography can be considered.

Volume term proportional to cosmological constant given by the twistorial lift of TGD (see this) could mean a small breaking of holography in the sense that it cannot be reduced to a 3-D surface term. One must be however very cautious here because TGD strongly suggests strong form of holography meaning that data at string world sheets and partonic 2-surfaces (or possibly at their metrically 2-D light-like orbits for which only conformal equivalence class matters) fix the 4-D dynamics.

Could TGD allow to resolve the basic objection against gravitation as entropic force of generalize this notion?

- In Zero Energy Ontology quantum theory can be interpreted as “complex square root of thermodynamics”. Vacuum functional would be analogous to an exponent of E
_{c}/2, where E_{c}is complexified energy. I have also considered the possibility that vacuum functional is analogous to the exponent of free energy but following argument favors the interpretation as exponent of energy. - The variation of Kähler action would give rise to the analog of TdS term and the variation of cosmological constant term to the analog of -pdV term in dE= TdS- pdV. Both T and p would be complex and would receive contributions from both Minkowskian and Euclidian regions. The contributions of Minkowskian and Euclidian regions to the action would differ by a multiplication with imaginary unit and it is possible that Kähler coupling strength is complex as suggested in (see this).

If the inverse of the Kähler coupling is strength is proportional to the zero of Riemann zeta at critical line, it is complex and the coefficient of volume term must have the same phase: otherwise space-time surfaces are extremals of Kähler action and minimal surfaces simultaneously. In fact, the known non-vacuum extremals of Kähler action are surfaces of this kind, and one cannot exclude the possibility that preferred extremals have quite generally this property.

The variation of the volume term gives rise to a term proportional to the trace of the second fundamental form, which is 4-D generalization of ordinary force and vanishes for the vacuum extremals of Kähler action in which case one has analog of geodesic line. More generally, Kähler action gives rise to the generalization of U(1) force so that the field equations give a 4-D generalization of equations of motion for a point like particle in U(1) force having also interpretation as a generalization of entropic force.

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