Entropic gravity in difficulties?

Eric Verlinde's Entropic Gravity is one of the fashions of recent day theoretical physics which come and go (who still remembers Lisi's "Exceptionally simple theory of everything", which raised Lisi for a moment a potential follower of Einstein?) That this would happen was rather clear to me from the beginning and I expressed my views in several postings: see this, this, and this. The idea that gravitons are there not all and gravitational force is purely thermodynamical force looks nonsensical to me on purely mathematical grounds. But what about physics? Kobakhidze wrote a paper in which he demonstrated that the neutron interferometry experiments disfavor the notion of entropic gravity. Neutron behaves like a quantal particle obeying Schrödinger equation in the gravitational field of Earth and it is difficult to understand this if gravitation is entropic force.

I wrote detailed comments about this in the second posting and proposed different interpretation of the basic formulas for gravitational temperature and entropy based on zero energy ontology predicting that even elementary particle are at least mathematically analogous to thermodynamical objects. The temperature and entropy would be associated with the ensemble of gravitons assigned with the flux tubes mediating gravitational interaction and temperature behaves naturally as 1/r² in absence of other graviton/heat sourcers and entropy is naturally proportional to the flux tube length and therefore to the radial distance r. This allows to understand the formulas deduced by Sabine Hossenfelder who has written one of the rather few clear expositions about entropic gravity (Somehow it reflects the attitudes towards women in physics that her excellent contribution was not mentiond in the reference list of the Wikipedia article. Disgusting.). Entropic gravitons are of course quite different thing than gravitation as entropic force.

The question about the proper interpretation of the formulas was extremely rewarding since it also led to ask what is the GRT limit of TGD could be. This led to beautiful answer and in turn forced to ask what black holes really are in TGD Universe. We have no empirical information about their interiors so that general relativistic answer can be taken only as one possibility which is even plagued by mathematical difficulties. Blackhole horizon is quite concretely the door to the new physics so that one should have be very openminded here- we really do not know what is behind the door!

The TGD based answer was surprising: black holes in TGD Universe correspond to the regions of space-time with Euclidian signature of the induced metric. In particular, the lines of generalized Feynman diagrams are blackholes in this sense. This view would unify elementary particles and blackholes. This proposal also leads to a concrete proposal for how to understand the extremely small value of the cosmological constant as the average value of cosmological constant which vanishes for Minkowskian regions but is large for Euclidian regions and determined by CP₂ size.

The first article of Kobakhidze appeared in arXiv already two years ago but was not noticed by bloggers (except me but as a dissident I am of course not counted;-). Here the fact that I was asked to act as a referee helped considerably. Unfortunately I did not have time for this!).) . The new article Once more: gravity is not an entropic force of Kobakhidze was however noticed by media and also by physics bloggers.

Lubos came first: Lubos however had read the article carelessly (even its abstract) and went to claim that M. Chaichian, M. Oksanen, and A. Tureanu state in their article that Kobakhidze's claim is wrong and that they support entropic gravity. This was of course not the case: the authors agreed with Kobakhidze about entropic gravity but argued that there was a mistake in his reasoning. In honor of Lubos one must say that he noticed the problems caused by the lack of quantum mechanical interferene effects already much earlier.

Also Johannes Koelman wrote about the topic with inspiration coming from the popular web article Experiments Show Gravity Is Not an Emergent Phenomenon inspired by Kobakhidze's article.

To my opinion Verlinde's view is wrong but it would be a pity if one would not try to explain the highly suggestive formulas for entropy and temperature like parameters nicely abstracted by Sabine Hossenfelder from Verlinde's work. I have already described briefly my own interpretation inspired by zero energy ontology. In TGD framework it seems impossible to avoid the conclusion that also the mediators of other interactions are in thermal equilibrium at corresponding space-time sheets and that the temperature is universally the Unruh temperature determined by acceleration. Also the expression for the entropy can be deduced as the following little argument shows.

What makes the situation so interesting is that the sign of both temperature and entropy are negative for repulsive interactions suggesting thermo-dynamical instability. This leads to the question whether matter antimatter separation could relate to are reversal of the arrow of geometric time at space-time sheets mediating repulsive long range interactions. This statement makes sense in zero energy ontology and the arrow of time has a concrete mathematical content as a property of zero energy states. In the following I will consider identification of the temperature and entropy assignable to the flux tubes mediating gravitational or other interactions. I was too lazy to deduce explicit formulas in the original version of the article about this topic and added the formulas also into it.

Graviton temperature

Consider first the gravitonic temperature. The natural guess for the temperature parameter would be as Unruh temperature

T_gr= (hbar/2π) a ,

where a is the projection of the gravitational acceleration along the normal of the gravitational potential = constant surface. In the Newtonian limit it would be acceleration associated with the relative coordinates and correspond to the reduced mass and equal to a=G(m₁+m₂)/r².

One could identify T_gr also as the magnitude of gravitational acceleration. In this case the definition would involved only be purely local. This is in accordance with the character of temperature as intensive property.

The general relativistic objection against the generalization is that gravitation is not a genuine force: only a genuine acceleration due to other interactions than gravity should contribute to the Unruh temperature so that gravitonic Unruh temperature should vanish. On the other hand, any genuine force should give rise to an acceleration. The sign of the temperature parameter would be different for attractive and repulsive forces so that negative temperatures would become possible. Also the lack of general coordinate invariance is a heavy objection against the formula.

In TGD Universe the situation is different. In this case the definition of temperature as magnitude of local acceleration is more natural.

1. Space-time surface is sub-manifold of the imbedding space and one can talk about acceleration of a point like particle in imbedding space M⁴× CP₂. This acceleration corresponds to the trace of the second fundamental form for the imbedding and is completely well-defined and general coordinate invariant quantity and vanishes for the geodesics of the imbedding space. Since acceleration is a purely geometric quantity this temperature would be same for flux sheets irrespective of whether they mediate gravitational or some other interactions so that all kinds of virtual particles would be characterized by this same temperature.

2. One could even generalize T_gr to a purely local position dependent parameter by identifying it as the magnitude of second fundamental form at given point of space-time surface. This would mean that the temperature in question would have purely geometric correlate. This temperature would be alwas non-negative. This purely local definition would also save from possible inconsistencies in the definition of temperature resulting from the assumption that its sign depends on whether the interaction is repulsive or attractive.

3. The trace of the second fundamental form -call it H- and thus T_gr vanishes for minimal surfaces. Examples of minimal surfaces are cosmic strings, massless extremals and CP₂ vacuum extremals with M⁴ projection which is light-like geodesic. Vacuum extremals with at most 2-D Lagrangian CP₂ projection has a non-vanishing H and this is true also for their deformations defining the counterpart of GRT space-time. Also the deformations of cosmic strings with 2-D M⁴ projection to magnetic flux tubes with 4-D M⁴ projection are expected to be non-minimal surfaces. Same applies to the deformations of CP₂ vacuum extremals near the region where the signature of the induced metric changes. The predicted cosmic string dominated phase of primordial cosmology would correspond to the vanishing gravitonic temperature. Also generic CP₂ type vacuum extremals have non-vanishing H.

4. Massless extremals are excellent macroscopic space-time correlate for gravitons. The massivation of gravitons is however strongly suggested by simple considerations encouraged by twistorial picture and wormhole throats connecting parallel MEs definine the basic building bricks of gravitons and would bring in non-vanishing geometric temperature, (extremely small but non-vanishing) graviton mass, and gravitonic entropy.
   1. The M⁴ projection of CP₂ type vacuum extremal is random light-like curve rather than geodesic of M⁴ (this gives rise to Virasoro conditions). The mass scale defined by the second fundamental form describing acceleration is non-vanishing. I have indeed assigned this scale as well as the mixing of M⁴ and CP₂ gamma matrices inducing mixing of M⁴ chiralities giving rise to massivation. The original proposal was that the trace of second fundamental form could be identifiable as classical counterpart of Higgs field. One can speak of light-like randomness above a given length scale defined by the inverse of the length of the acceleration vector.

   2. This suggests a connection with p-adic mass calculations: the p-adic mass scale m_p is proportional to the acceleration and thus could be given by the geometric temperature: m_p=n R^-1p^-1/2∼ hbar H=hbar a, where R∼ 10⁴L_Pl is CP₂ radius, and n some numerical constant of order unity. This would determine the mass scale of the particle and relate it to the momentum exchange along corresponding CP₂ type vacuum extremal. Local graviton mass scale at the flux tubes mediating gravitational interaction would be essentially the geometric temperature.

   3. Interestingly, for photons at the flux tubes mediating Coulomb interactions in hydrogen atom this mass scale would be of order

      hbar a ∼ e²hbar/[m_pn⁴a₀²]∼ 10^-5 /n⁴ eV,

      which is of same order of magnitude as Lamb shift, which corresponds to 10^-6 eV energy scale for n=2 level of hydrogen atom. Hence it might be possible to kill the hypothesis rather easily.

   4. Note that momentum exchange is space-like for Coulomb interaction and the trace H^k of the second fundamental form would be space-like vector. It seems that one define mass scale as H=(-H^kH_k)^1/2 to get a real quantity.

   5. This picture is in line with the view that also the bosons usually regarded as massless possess a small mass serving as an IR cufoff. This vision is inspired by zero energy ontology and twistorial considerations kenociteallb/twistor. The prediction that Higgs is completely eaten by gauge bosons in massivation is a prediction perhaps testable at LHC already during year 2011.

Remark: In MOND theory of dark matter a critical value of acceleration is introduced. I do not believe personally to MOND and TGD explains galactic rotation curves without any modification of Newtonian dynamics in terms of dark matter assignable to cosmic strings containing galaxies like around it like pearls in necklace. In TGD framework the critical acceleration would be the acceleration above which the gravitational acceleration caused by the dark matter associated with the cosmic strings traversing along galactic plane orthogonally and behaving as 1/ρ overcomes the acceleration caused by the galactic matter and behaving as 1/ρ². Could this critical acceleration correspond to a critical temperature T_gr and could critical value of H perhaps characterize also a critical magnitude for the deformation from minimal surface extremal? The critical acceleration in Milgrom's model is about 1.2*10^-10 m/s² and corresponds to a time scale of 10¹² years, which is of the order of the age of the Universe.

The formula contains Planck constant and the obvious question of the inhabitant of TGD Universe is whether the Planck constant can be identified with the ordinary Planck constant or with the effective Planck constant coming as integer multiple of it (see this).

1. For the ordinary value of hbar the gravitational Unruh temperature is extremely small. To make things more concrete one can express the Unruh temperature in gravitational case in terms of Schwartschild radius r_S=2GMm at Newtonian limit. This gives

   T_gr= (hbar/4π r_S) [ (M+m)/M] (r_S/r)² .

   Even at Schwartschild radius the temperature corresponds to Compton length of order 4π r_S for m<<M.

2. Suppose that Planck constant is gravitational Planck constant hbar_gr= GMm/v₀, where v₀≈ 2^-11 holds true for inner planets in solar system (see this). This would give

   T_gr= (m/8π v₀) [(M+m)/M] (r_S/r)² .

   The value is gigantic so that one must assume that the temperature parameter corresponds to the minimum value of Planck constant.

Gravitonic entropy

A good guess for the value of gravitational entropy (gravitonic entropy associated with the flux tube mediating gravitational interaction) comes from the observation that it should be proportional to the flux tube length. The relationship dE= dS/T suggests S∝ φ_gr/T_gr as the first guess in Newtonian limit. A better guess would be

S_gr= -V_gr/T_gr= [(M+m)/M] (r/hbar m) ,

The replacement M→ M+m appearing in the Newtonian equations of motion for the reduced mass has been performed to obtain symmetry with respect to the exchange of the masses.

The entropy would depend on the interaction mediated by the space-time sheet in question which suggests that the generalization is

S=-V(r)/T_gr .

Here V(r) is the potential energy of the interaction. The sign of S depends on whether the interaction is attractive or repulsive and also on the sign of the temperature. For a repulsive interaction the entropy would be negative so that the state would be thermodynamically unstable in ordinary thermodynamics.

The integration of dE= TdS in the case of Coulomb potential gives E= V(r)-V(0) for both options. If the charge density near origin is constant, one has V(r) proportional to r² in this region implying V(0)=0 so that one obtains Coulombic interaction energy E=V(r). Hence thermodynamical interpretation makes sense formally.

The challenge is to generalize the formula of entropy in Lorentz invariant and general coordinate invariant manner. Basically the challenge is to express the interaction energy in this manner. Entropy characterizes the entire flux tube and is therefore a non-local quantity. This justifies the use of interaction energy in the formula. In principle the dynamics defined by the extremals of Kähler action predicts the dependence of the interaction energy on Minkowskian length of the flux tube, which is well-defined in TGD Universe. Entropy should be also a scalar. This is achieved since the rest frame is fixed uniquely by the time direction defined by the time-like line connecting the tips of CD: the interaction energy in rest frame of CD defines a scalar. Note that the sign of entropy correlates with the sign of interaction energy so that the repulsive situation would be thermodynamically unstable and this suggests that matter antimatter asymmetry could relate to thermal instability.

See the article TGD inspired vision about entropic gravity. For background see the chapter TGD and GRT of "Physics in Many-Sheeted Space-Time".