Newtonian gravity as an entropic force: Towards a derivation of G
aa r X i v : . [ h e p - t h ] M a y Newtonian gravity as an entropic force:Towards a derivation of G F.R. Klinkhamer
Institute for the Physics and Mathematics of the Universe, University of Tokyo,Kashiwa 277–8583, JapanandInstitute for Theoretical Physics, University of Karlsruhe, Karlsruhe Institute ofTechnology, 76128 Karlsruhe, Germany ‡ E-mail: [email protected]
Abstract.
It has been suggested that the Newtonian gravitational force may emergeas an entropic force from a holographic microscopic theory. In this framework, thepossibility is reconsidered that Newton’s gravitational coupling constant G can bederived from the fundamental constants of the underlying microscopic theory.Journal-ref: Class. Quantum Grav. 28 (2011) 125003Preprint: IPMU10–0098, KA–TP–15–2010, arXiv:1006.2094PACS numbers: 04.50.-h, 05.70.-a, 06.20.Jr, 04.80.Cc ‡ Permanent address. ewtonian gravity as an entropic force: Towards a derivation of G
1. Introduction
Recently, Verlinde [1] has given a heuristic argument of how space, inertia, and gravitycould emerge from a microscopic theory in a holographic approach [2, 3]. Gravity wouldarise as a type of entropic force. (Related ideas have been presented in, e.g., [4, 5].)Verlinde’s discussion of Newton’s law of gravity is particularly elegant, as it directlygives an inverse-square law for the attractive force between two macroscopic point masses M and M . Specifically, the force on the point mass M at position X due to aneffective point mass f M at an effective position e X (the mass f M corresponding to aspherical holographic screen) is given by F , grav = M A , grav = G M f M ( e X − X )/ | e X − X | , (1)with A , grav the acceleration of the mass M .In this article, a previous suggestion [6] is reconsidered that Newton’s gravitationalconstant G can be derived from more fundamental constants of nature, including anew fundamental length l (see also [7, 8] for a classic paper and a recent review). Theentropic explanation of Newtonian gravity then gives a new interpretation of an earlierformula [6] for the Newtonian gravitational acceleration originating from a macroscopicpoint mass. Moreover, having a new fundamental constant l may help in resolvinga potential problem of Verlinde’s approach regarding the total entropy of a generalequipotential screen. Restricting the screen to a black-hole horizon, this entropy canbe used to perform a model calculation of G and to get a numerical estimate for l byconnecting to the Bekenstein–Hawking entropy [9, 10]. At the end of this article, a fewcomments are presented on possible experiments to determine this new fundamentalconstant l , if really existent.
2. Nonfundamental G Consider the possibility that the true fundamental constants of nature are ~ , c , and l , where the last constant has the dimensions of area. This suggests (as mentioned inSec. 2 of [6]) that the classical Newton constant G arises from the appropriate ratio ofthe two quantum constants l and ~ : G = f c l / ~ , (2) ewtonian gravity as an entropic force: Towards a derivation of G f ∈ R + to be calculated from the microscopic theory.Note that, for f = 1, the fundamental length l equals the standard Planck length l P ≡ ( ~ G ) / c − / .Expression (2) leads to the following structure of the Newtonian gravitationalacceleration A grav ≡ | A grav | from a point mass with a macroscopic value M at amacroscopic distance R : A grav = GM/R = c ( f M c / ~ ) ( l /R ) , (3)with all microscopic quantities indicated by lower-case symbols. The structure on theright-hand side of (3) is suggestive: the fundamental velocity c is multiplied by a mass-induced decay rate of space, f M c / ~ with coupling constant f , and a geometric dilutionfactor, l /R .Precisely this structure can be seen to result from the reasoning of Verlinde (see,in particular, Sec. 3 of [1]) for a spherical holographic screen Σ sph with area A = 4 πR (Fig. 1): A grav (cid:13) = 2 π c ( k B T / ~ ) (cid:13) = 4 π c ( f N k B T / ~ ) ( f − /N ) (cid:13) = 4 π c ( f E/ ~ ) ( l /A ) (cid:13) = c ( f M c / ~ ) ( l /R ) , (4)where step (cid:13) relies on the Unruh effect [11], step (cid:13) on trivial mathematics, and step (cid:13) on the following relation between the effective number N of degrees of freedom ofthe holographic screen and the area A of the screen: N = f − A/l . (5)Step (cid:13) of (4) also assumes that the screen corresponds to a physical system in a stateof equilibrium (or close to it), with a uniform distribution of the microscopic degreesof freedom over the surface and equipartition of the total energy E over these degreesof freedom (both properties being consistent with having a screen given by a constant-curvature manifold, i.e., a spherical surface). Somewhat surprisingly, Lorentz invarianceis seen to play a role in steps (cid:13) and (cid:13) of (4): implicitly as the Unruh temperature ewtonian gravity as an entropic force: Towards a derivation of G m S sph mM R Figure 1.
Left panel: Spherical holographic screen Σ sph with area A = 4 πR andtest mass m in the emerged space (shaded) outside the screen [1]. The screenΣ sph has N microscopic degrees of freedom at an equilibrium temperature T withtotal equipartition energy E = N k B T . Right panel: The gravitational effects ofΣ sph for the emergent space correspond, in leading order, to those of a point mass M = E/c located at the center of a sphere with radius R . (The Schwarzschild radius R Schw ≡ GM/c is considered to be negligible compared to R and cannot be shownin the right panel, but the corresponding sphere would be a maximally-coarse-grainedscreen with smallest possible area, according to [1].) ultimately traces back to the Lorentz invariance of the Minkowski vacuum [11] andexplicitly through the energy-mass equivalence E ≡ M c from special relativity.The several steps of (4) constitute, if confirmed by the definitive microscopic theory,a derivation of Newton’s gravitational coupling constant G in the form (2). The pointof view of this article is not to consider (4) as mere dimensional analysis but to take allnumerical factors seriously. In that spirit, there is the new insight from (5) that, giventhe “quantum of area” l , the inverse of the constant f entering Newton’s constant (2)is related to the nature of the microscopic degrees of freedom on the holographic screen.For example, an “atom of space” with “spin” s atom gives f − = 2 s atom + 1, but this“spin” need not be half-integer. Still, the number of “atoms” needed to build-up thearea A is taken to be an integer, given by the ratio of the area A and the quantum l . ewtonian gravity as an entropic force: Towards a derivation of G
3. Two types of entropy
The introduction of two quantum constants, ~ and l , may also help to resolve a potentialproblem noted by Verlinde in Sec. 6.4 of [1]. There, he considers an equipotential screenΣ which is not a maximally-coarse-grained surface but is nevertheless assumed to bein thermal equilibrium. He, then, remarks that the required entropy S Σ appears tocontradict Bekenstein’s upper bound [12] on the entropy S Ξ of a material system Ξwith energy E Ξ and effective radius R Ξ , S Ξ /k B < π ~ − E Ξ R Ξ /c . (6)With the new fundamental constant l , Verlinde’s expression (6.41) for S Σ isreplaced by S Σ /k B = 14 f − l − Z Σ dA , (7)which generalizes (5). Expressions (6) and (7) involve essentially different physicscharacterized by, respectively, ~ and l (see also the discussion of Sec. 2 in [6] for ageneralized dimensionless action with ~ = 0 and l >
4. Model calculation of G For a maximally-coarse-grained spherical surface (horizon) with area A , the entropy (7)reproduces the Bekenstein–Hawking black-hole entropy [9, 10] S BH /k B = 14 A/ ( f l ) = (1 / N , (8)where the number N has already been defined by (5).Now, consider the “atoms of space” mentioned in the last paragraph of Sec. 2. Thecrucial new equation from (5) is then given by N = d atom N atom , (9)with the physical interpretation of l as the quantum of area giving N atom ≡ A/l ∈ N ≡ { , , , . . . } (10 a ) ewtonian gravity as an entropic force: Towards a derivation of G d atom ≡ f − ∈ R + . (10 b )The physical picture, suggested by the derivation (4), is that the “atoms of space” haveno translational degrees of freedom but only internal degrees of freedom.The number of configurations [2, 3] of these distinguishable “atoms of space” isreadily calculated: N config = N atom Y n =1 d atom = ( d atom ) N atom . (11)Equating this number of configurations with the exponential of the Bekenstein–Hawkingentropy (8) while using (9) gives the following set of conditions:( d atom ) N atom = exp [(1 / d atom N atom ] , (12)for all positive integer values of N atom . Remarkably, this infinite set of conditions reducesto a single transcendental equation for the effective dimension d atom ,ln d atom = (1 / d atom , (13)which has two solutions: d (+) atom ≈ .
613 169 456 , d ( − ) atom ≈ .
429 611 825 , (14)where a 1 ppb numerical precision suffices for the present purpose. § Given l , there are then two possible values for the gravitational coupling constant(2): G ± = ( d ( ± ) atom ) − c l / ~ . (15)The detailed microscopic theory must tell which of the two values from (14) enters (15).It could, for example, be that the microscopic theory demands d atom ≥
2, selecting thelarger value d (+) atom in (14) and (15).The experimental value G N of Newton’s gravitational coupling constant is, ofcourse, already known [13], albeit with a rather large relative uncertainty of 100 § Condition (13) would not be satisfied for any value of d atom if the factor 1 / g > /e , with e ≈ . d atom = 1, corresponding to f = 1 in the originalexpression (2) for Newton’s constant. ewtonian gravity as an entropic force: Towards a derivation of G l ± ) = d ( ± ) atom ( l P ) ≈ . × − m , . × − m , (16)with l P ≡ ( ~ G N ) / /c / ≈ . × − m for G N = 6 . − m kg − s − [14]. The microscopic theory would, again, have to choose between these alternativevalues. For either choice, the implication would be that l and l P are of the same orderof magnitude.Needless to say, the numerical estimates of (16) are only indicative because of theextreme simplification of the model calculation (for example, merely “tiles” of a singlesize l and design d atom have been used to cover the area A ). But, perhaps, the simplicityof the model is also its strength, as long as the effective quantum of area l is consideredand not the individual eigenvalues of the area operator.The real question is if this l can be measured directly. This question will beaddressed in the next section. Anticipating a positive outcome of that discussion andlooking far into the future, note that the accurate measurement of one of the values of l in (16) would allow for an equally accurate calculation of G from (15). For example,measuring for l the larger value in (16) with a relative uncertainty of 100 ppb wouldgive G also with an uncertainty of approximately 100 ppb from (15) by use of the d (+) atom value from (14), since ~ is already known with an uncertainty of 50 ppb [14].
5. Experiments
As promised in the previous section, let us briefly discuss the prospects of theexperimental determination of the factor f in (2), which may or may not be foundto agree with the inverse of one of the calculated values in (14). Given the numericalvalues for c and ~ from nongravitational experiments, at least two gravity/spacetimemeasurements would be needed to disentangle f and l .The first measurement is, of course, provided by the Cavendish experiment [13, 14],which determines the particular combination f l .A second measurement (without definite results, for the moment) can come from ewtonian gravity as an entropic force: Towards a derivation of G k from anontrivial small-scale structure of spacetime [15, 16]. Such a measurement may, in fact,determine f ≡ ( l P /l ) , if the average size of spacetime defects is set by l P and theiraverage separation by l (with l > l P ); see the discussion of the paragraph starting a fewlines below Eq. (10) in [6]. The value f & − suggested by (14) would, however, behard to reconcile with the data (cf. [15, 16] and references therein).A third type of measurement (entirely in the domain of Gedankenexperiments )could try to isolate pure-quantum-gravity effects of (primordial) gravitational waves.Such a measurement would only depend on l , if the generalized dimensionless actionof [6] is relevant.A fourth type of measurement (also in the domain of Gedankenexperiments )would look for quantum modifications of Newton’s gravitational acceleration (3) bya multiplicative factor [1 − e a l /R ], where the dimensionless number e a would trace backto a logarithmic correction of the entropy (7), as pointed out in [17]. More generally,an entropy modification S ( A ) = (1 / k B l − P [ A + l e s ( A/l )], for some dimensionlessfunction e s of A/l , would give a correction factor [1+ l d e s/dA ] for Newton’s gravitationalforce. A measurement of such a modification of the force could, in principle, be used todetermine l , if the function e s ( A/l ) is nontrivial and known from theory.Each of the last three possible experiments relies on a crucial assumption (indicatedby occurrence of the word ‘if’) and is, therefore, not yet conclusive in determining thevalue of l .
6. Conclusion
The two most interesting results of this article are the following. The first is thatthe interpretation of the Newtonian acceleration (3) as a mass-induced decay rate ofspace (together with a geometric dilution factor) may be explained by a Verlinde-type derivation (4) relying on the Unruh temperature and holography. The second k According to the discussion in Sec. 2, it may be that the fundamental theory is essentially Lorentzinvariant. Still, there may be effects from some type of spontaneous symmetry breaking of Lorentzinvariance (meaning that a particular ground-state solution breaks the symmetry), which show up asmodifications of the standard particle-propagation properties. ewtonian gravity as an entropic force: Towards a derivation of G f ≡ ( d atom ) − entering expression (2) for Newton’s gravitationalconstant, where the microscopic theory is still needed to choose between the two possiblevalues (14).Having a calculated value for f in the G formula (2) is, of course, only of interest if l can be determined directly (the numerical values of G , ~ , and c are already known). Theexperiments discussed in Sec. 5 are suggestive but, for the moment, still inconclusive,because each experiment involves one or more assumptions. The main outstanding task,therefore, is to design an experiment, real or imaginary, which allows for an unambiguousdetermination of the quantum-gravity length scale l , independent of the value of thePlanck length l P [even though, in the end, both may turn out to have approximatelythe same numerical value, as suggested by the calculated numbers (16)].The first version of the present article was released on June 10, 2010. Since then,it has been shown [18] that a more sophisticated tiling than the one used in Sec. 4can produce a single transcendental equation which gives a unique physical value for l . The numerical values for l from two such tiling models are both approximatelyequal to 2 . × − m , which is only 20 % above the maximal value found here. Moreimportantly, the quantity l of these models [18] would correspond to the true minimalquantum of area. Acknowledgements
The author thanks D. Easson for pointing out [1] and S. Hellerman, H. Sahlmann, L.Smolin, W. Unruh, and E. Verlinde for discussions. He also gratefully acknowledges thehospitality of the IPMU during the month of May 2010. This work was supported inpart by the World Premier International Research Center Initiative (WPI Initiative),MEXT, Japan. ewtonian gravity as an entropic force: Towards a derivation of G References [1] E.P. Verlinde, “On the origin of gravity and the laws of Newton,” arXiv:1001.0785v1.[2] G. ’t Hooft, “Dimensional reduction in quantum gravity,” in: A. Ali, J. Ellis, and S. Randjbar-Daemi (eds.),
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