Verlinde-Erik · Source catalogue

The paper proposes that gravity is not a fundamental force but an entropic force, like the elastic force of a polymer in a heat bath (Section 2). Space is assumed to emerge from microscopic information stored on holographic screens (closed surfaces holding the information about the enclosed region). Section 3 postulates that when a particle of mass m moves one reduced Compton wavelength (hbar/mc) toward a screen the screen's entropy changes by 2 pi k_B, generalised to dS = 2 pi k_B (mc/hbar) dx. With the force-entropy relation F dx = T dS and the Unruh temperature k_B T = hbar a/(2 pi c) this gives F = ma (inertia as entropic). Putting N = A c^3/(G hbar) bits on a spherical screen of area A, with equipartition E = (1/2) N k_B T and E = Mc^2, gives F = GMm/R^2. The author notes that N = Ac^3/(G hbar) can be read as the definition of G, that the 2 pi was 'picked precisely' to give F = ma, and that Newton's laws were ingredients in the black hole thermodynamics and holography arguments, which are here 'reversed'. Section 4 generalises to arbitrary mass distributions: screens are placed on equipotential surfaces of the Newton potential Phi, which is declared to be the coarse-graining variable (the measure of how much microscopic information has been averaged away). Equipartition over the screen reproduces Gauss's law M = (1/4 pi G) integral grad Phi . dA (4.20), and requiring this for arbitrary screens is said to force the Poisson equation (4.21). Section 4.2 obtains Newtonian forces on test particles from virtual displacements, using the first law and an entropy-change identification (4.22), with a Green's-identity proof outlined but not written out. Section 5 assumes Lorentz symmetry and a static curved spacetime (following Wald's textbook). With phi = (1/2) log(-xi.xi), temperature T = (hbar/2 pi) e^phi N.grad phi and the same entropy postulate it obtains F = T grad S = -m e^phi grad phi (5.30). The same equipartition step gives the Komar mass formula (5.34). A Jacobson-style sketch, with the stress-energy combination fixed 'presumably' by comparing both sides, is offered as the route to the full Einstein equations. Section 5.3 imposes constant entropy under virtual displacement, grad_i M + T grad_i S = 0 (5.39). Section 6 covers string theory and AdS/CFT (UV/IR connection, open/closed string duality as support for emergent gravity), and revisits Bekenstein's black-hole thought experiment as 'perhaps the clearest argument for the fact that gravity is an entropic force' (6.3, p.25). Section 6.4 (pp.26-27) admits the role of hbar is 'subtle and not yet fully understood', that the Bekenstein-Hawking entropy implied for the screens (6.41, p.27) appears to violate the Bekenstein bound, and that the screens may not be in equilibrium. It suggests gravitational force fluctuations as a possible prediction but says 'we need a better understanding of the theory to turn this in to a prediction' (p.27). The author calls the arguments 'rather heuristic' and states the conclusion that 'it is time we ... do away with gravity as a fundamental force'. Deep-read findings (the checking is by the deep reader, not the paper): each equation presented as a consequence follows correctly by algebra from its stated inputs; magnitudes are right, with loose sign conventions in (4.25) versus Newtonian work and in (5.27)/(5.30) and a misprinted extra k in (4.17). A numeric check for Earth gives T = 3.98e-20 K and N = 1.95e84 bits at the surface, with (1/2) N k_B T = Mc^2 as the algebra guarantees. The two field-equation steps (Poisson, Einstein) are one paragraph each; the Einstein one is self-described as 'a sketch'. The stress-energy side of (5.37) is posited, not derived. The new content is the interpretation (entropic force, coarse-graining variable), not field equations different from Newton's or Einstein's. QE-remediation deep read (Sections 2-3, pp.4-11) [read: p.N]. Section 2: the author's microcanonical polymer treatment already says 'the total entropy will remain constant' for an infinite heat bath (p.5) and, from d/dx S(E+Fx, x) = 0 (2.4), that 'the entropy in first approximation stays the same' (p.5); an entropic force 'can be conservative, at least when the temperature is kept constant', with an emergent potential (p.5). Section 3.3 (p.10): the author calls equipartition 'perhaps the least obvious assumption', and defends it: 'When the lost bits are randomly chosen among all bits, one expects the energy change associated with Delta S still to be proportional to the energy per unit area E/A. This fact could therefore be true even when equipartition is not strictly obeyed.' He states hbar 'can be chosen at will, and defined so that (3.5) is exactly valid'. Screen-moving remark (p.10): if the particle stays at R while the screen moves to R0 < R, the bit count is multiplied by (R0/R)^2 and T divided by the same factor, so 'effectively ... only hbar is multiplied by that factor, and since it drops out, the resulting force will stay the same'. Footnote 4 (p.10): in d dimensions (3.10) needs a factor (1/2)(d-2)/(d-3) to match Newton's constant. Novelty claim (p.9): 'That is the main statement, which is new and has not been made before.' Section 3.4 (p.11) introduces n bits for the particle, mc^2 = (1/2) n k_B T (3.14), giving Delta S/n = k_B a Delta x/(2c^2) (3.15) and Delta S/n = -k_B Delta Phi/(2c^2) (3.16), so Phi 'keeps track of the depletion of the entropy per bit' and is identified with a coarse-graining variable. Page images of pp.7-8 confirm (3.5)-(3.10) as transcribed from the text layer (including the factor 2 pi 'picked precisely' for F = ma, and the 'definition of G' reading of N = Ac^3/(G hbar)). QE-remediation deep read (Section 6, pp.22-27). 6.1 (p.22): 'it is time we not only notice the analogy ... but finally do away with gravity as a fundamental force'; 'If gravity is emergent, so is space time geometry'; the author says 'we uncovered what is the origin of force and inertia', 'a cause, a mechanism', and cites Jacobson [7] and 'for instance [14]' for earlier entropic/thermodynamic proposals. 6.2 (pp.23-24), the string-theory support in the author's words: gravity is a long-distance phenomenon that 'knows about short distance physics'; string theory invalidates the 'general wisdom' of Wilsonian effective field theory (that integrating out short-distance degrees of freedom produces only local, low-energy-irrelevant terms) because the number of high-energy open-string states grows fast enough that integrating them out has long-range effects; 'Their one loop amplitudes are equivalent to the tree level contributions due to the exchange of close string states, which among other are responsible for gravity'; so the open-string sum 'equivalently represent[s]' gravity (p.23). Holographic screens are identified with D-branes carrying open strings with a UV cut-off; the emerged space holds closed strings with a cut-off; pushing one cut-off to the UV pushes the other to the IR, set by the screen location (Figure 6, p.24). Condensed matter (p.23): AdS/CFT is now applied to condensed matter, where 'no one doubts that ... gravity emerges only as an effective description'; 'The entropic mechanism described in this paper should be applicable to these physical systems'. The author offers an analogy with elasticity theory by stress tensors predating knowledge of atoms (p.24). Section 6.3 (pp.25-26), Black hole horizons revisited, reconstructed step by step. (i) Bekenstein's thought experiment is likened to the polymer: the black hole is the heat bath and the particle the polymer end-point; in AdS/CFT a black hole is a thermal boundary state and the particle a delocalised operator being thermalised (p.25). (ii) Bekenstein places the particle about one Compton wavelength outside the horizon, where particle and hole are declared one system; this location is declared a holographic screen, and 'the precise location of this screen can not be important' since no natural preferred distance exists (p.25). (iii) Pulling the particle out a bit changes its energy by the work done by gravity; dropping it in raises M by that amount; 'Consistency of the laws of black hole thermodynamics implies that the additional change in the Bekenstein Hawking entropy, when multiplied with the Hawking temperature T_H, must be precisely equal to the work done by gravity. Hence F_gravity = T_H dS_BH/dx' (6.40). The author: 'This fact is surely known, and probably just regarded as a consistency check.' He says gravity 'becomes an entropic force near the horizon, and ... the equations presented in section 3 are exactly valid' (p.25) and that near the horizon hbar is the actual Planck constant (p.26). (iv) The extension: take the screen further out, ignore that it moved, and lower the particle to the old screen location; that process happens behind the new screen and 'should have a holographic representation on the screen'; the perpendicular coordinate x is 'just some scale variable' and has no microscopic meaning, nor acceleration; the mass is the energy of the particle's holographic image, 'presumably ... a near thermal state', slightly out of equilibrium 'just like the polymer'; the change in energy 'can be nothing else then an entropic effect ... due to the thermalization process'; 'the correspondence rules ... must be such that F = T grad S translates in to the gravitational force. It is straightforward to see that this indeed works' (p.26). (v) Rindler version (p.26): the same experiment for a Rindler horizon gives F = T grad S translating to F = ma; also, the horizon force can be arbitrarily weak for an arbitrarily large black hole, so the argument 'is not just teaching us about black holes'. Classification of the steps (analysis by the deep reader; no verdict): step (iii) is standard black-hole thermodynamics (first law dM = T_H dS_BH, Hawking temperature, Bekenstein-Hawking entropy) [memory: high]. Steps (ii) and (iv) are assumed or asserted: the screen's location is arbitrary by a symmetry/naturalness argument, and the claim that the outward-moved screen and the holographic image behave as a near-thermal state is stated, not derived; 'It is straightforward to see' is not shown on the page. Step (v) is asserted by parallel. Check (calculated by the deep reader): at a horizon, R = R_s = 2GM/c^2, the Bekenstein-Hawking entropy S = k_B c^3 A/(4 G hbar) with A = 4 pi R_s^2 equals k_B pi R_s^2 c^3/(G hbar), and the Bekenstein bound S <= 2 pi k_B E R/(hbar c) with E = M c^2 equals k_B 2 pi M c R_s/hbar = k_B pi R_s^2 c^3/(G hbar); the ratio is exactly 1, so the excess over the bound seen at R > R_s (Earth surface about 7.2e8, from the earlier numeric check) falls to 1 at the horizon [calc]. Hence the near-horizon form is the one regime where T, S and hbar are the standard ones. The paper itself notes the bound only in natural units ('an entropy larger than ER', p.27). The step from horizon to non-horizon screens is the point contested by Wang and Braunstein's objection that the first-law relation fails off horizons (as reported by the Investigator in research/; not re-read here). Section 6.4 (pp.26-27): 'we have no direct confirmation or proof that the same is true when we go away from the horizon, or eespecially when there is no horizon present at all'; (6.41) S = (c^3/(4 G hbar)) integral dA is 'very high' and 'violates the Bekenstein bound [15]'; possible reconciliations are that Bekenstein's argument does not apply to holographic screen degrees of freedom, or that the screens are far from equilibrium, or rescaling hbar by f(Phi_0) so that T is multiplied and S divided by the same factor, or weighted averages over screens; 'Even then the essence of our conclusions would not change'. Prediction (p.27): fluctuations of the gravitational force 'may turn out to be more pronounced for weak gravitational fields between small bodies of matter', but 'we need a better understanding of the theory to turn this in to a prediction'. The Hooke-Newton closing remark (p.27) says holography allows 'contact' interactions between degrees of freedom of different bodies. Page correction: the digest and earlier files placed the 6.4 caveats at p.28; they are on pp.26-27 (p.28 is Acknowledgements and references).

Key claims

• Gravity is an entropic force: F dx = T dS with dS = 2 pi k_B (mc/hbar) dx for a particle near a holographic screen (3.5-3.7).
• Newton's second law F = ma follows from the entropy postulate combined with the Unruh temperature, so inertia is also entropic (Section 3.1; abstract).
• Newton's law of gravitation F = GMm/R^2 follows from N = Ac^3/(G hbar) bits on a spherical screen with equipartition E = (1/2) N k_B T and E = Mc^2 (3.10-3.12).
• For a static mass distribution, screens on equipotential surfaces with local T = hbar|grad Phi|/(2 pi k_B c) and uniform bit density reproduce Gauss's law (4.20), hence the Poisson equation (4.21).
• Forces on test particles at arbitrary positions follow from virtual displacement, F . dr = integral of T ds with ds = k_B (dPhi/2c^2) dN, and are independent of which equipotential screen is used (4.22-4.25).
• In a static relativistic setting, F = T grad S = -m e^phi grad phi (5.30), and the same equipartition construction gives the Komar mass (5.34).
• A Jacobson-style local argument applied to timelike screens yields the full Einstein equations (5.37 and following); the author calls this a sketch.
• Redshift must be seen as a consequence of the entropy gradient, not the other way round (p.19).
• Space, spacetime geometry and gravity are emergent macroscopic concepts; gravity should be abandoned as a fundamental force (Section 6.1).
• The mechanism is the same as for Hooke's elasticity of a polymer in a heat bath, with the black hole (or holographic screen) acting as the heat bath (Sections 2, 6.3).
• Near a black-hole horizon, consistency of black-hole thermodynamics gives F_gravity = T_H dS_BH/dx (6.40); the author says gravity becomes an entropic force there with the Section 3 equations 'exactly valid' and hbar the actual Planck constant, and argues (without showing the steps) that moving the screen outward leaves the force entropic; the same thought experiment for a Rindler horizon is said to give F = ma (Section 6.3, pp.25-26).
• Gravitational force fluctuations, with size set by an effective temperature, may be larger for weak fields between small bodies (Section 6.4); the author says this is not yet a prediction.
• Rescaling hbar by a function f(Phi_0) of the potential on the screen would multiply T and divide S by the same factor and leave the force unchanged; this is offered as a possible way to reconcile the screen entropy (6.41) with the Bekenstein bound (Section 6.4, p.27).
• In the microcanonical picture the total entropy stays constant (constant-entropy condition (2.4) for the polymer, p.5; S(M(x+dx), x+dx) = S(M(x), x) for virtual displacement in Section 5.3, (5.38), p.21), so the 2010 paper already contains a constant-entropy form of the entropic force.

Relevance: The only document in the project and the whole of its subject: Verlinde's founding statement of entropic gravity. It sets out both the claim and the author's own list of assumptions, so later steps can separate what is derived from what is assumed (the circularity ledger is in the deep read). Section 6.3 (eq. 6.40) is the author's own 'clearest argument' and the one regime where the paper's identifications coincide with standard black-hole thermodynamics.

Mainstream conflict: Partial. The paper reproduces Newton's law, the Poisson equation and (as a sketch) the Einstein equations, so it does not predict different results in the regimes tested; its departure is interpretive: gravity as emergent and entropic rather than fundamental. The deep read lists inputs that resemble the outputs: Newton's laws, the Unruh temperature form, G defined by the bit count, the Newton potential and GR's static-spacetime machinery. The author discloses the Bekenstein-bound tension in Section 6.4. [memory: medium] Later literature by others tested entropic-gravity ideas against, for example, ultra-cold neutron bound states (the GRANIT experiment, with the Colella-Overhauser-Werner interferometry argument also raised), planetary and galactic data; the Investigator and reviewers should look this up and not take it from here. Precedence lead (to verify): [memory: medium] T. Padmanabhan, 'Equipartition of energy in the horizon degrees of freedom and the emergence of gravity', arXiv:0912.3165 (Dec 2009), derived Newtonian gravity from equipartition of horizon degrees of freedom shortly before this paper; the paper cites Padmanabhan only via arXiv:0911.5004 (ref. [14]) and states its own main statement 'is new and has not been made before' (p.9). Not verified on the web; for the Investigator. The author also says AdS/CFT condensed-matter systems are a possible application domain (p.23).