Coulomb force as an entropic force

Tower Wang*

Center for High-Energy Physics, Peking University, Beijing 100871, China(Received 5 February 2010; published 27 May 2010)

Motivated by Verlinde’s theory of entropic gravity, we give a tentative explanation to the Coulomb’s

law with an entropic force. When trying to do this, we find the equipartition rule should be extended to

charges and the concept of temperature should be reinterpreted. If one accepts the holographic principle as

well as our generalizations and reinterpretations, then Coulomb’s law, the Poisson equation, and the

Maxwell equations can be derived smoothly. Our attempt can be regarded as a new way to unify the

electromagnetic force with gravity, from the entropic origin. Possibly some of our postulates are related to

the D-brane picture of black hole thermodynamics.

DOI: 10.1103/PhysRevD.81.104045 PACS numbers: 04.70.Dy, 04.20.Cv

I. MOTIVATION

Starting from the holographic principle and an equipar-tition rule, E. Verlinde proposed an intriguing explanationfor Newton’s law of gravity as an entropic force [1]. Thisidea turns out to be very powerful.1 From it, one can deriveboth Newton’s laws and Einstein’s equations. In this paper,we try to understand its implication to the Reissner-Nordstrom (RN) spacetime. With a few generalizationsand reinterpretations, we find the same idea can produceCoulomb’s law and Maxwell’s equations.

To make our narration clear, for the most part we willconcentrate on the 4-dimensional spacetime, in which theRN solution is

ds2 ¼ �fðrÞdt2 þ 1

fðrÞdr2 þ r2d�2 (1)

with

fðrÞ ¼ 1� 2GNM

c2rþG2

NQ2

c4r2; M � jQj: (2)

For conciseness, we work in the ‘‘geometrized’’ unit ofcharge so that the Coulomb force between point chargesQ,q at large separation r in flat space takes the form

Fem ¼ GNQq

r2: (3)

If one feels uneasy with such a unit, the more familiar formcan be recovered by replacing Q ! Q=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�"0GN

p, q !

q=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�"0GN

pfor electric charges, where "0 is the electric

constant and GN is the Newton’s constant.Now let us show a puzzle regarding the equipartition

rule. On the one hand, in [1] it was proposed that thetemperature T is determined by

Mc2 ¼ 12NkBT (4)

as the average energy per ‘‘bit’’ (in certain unit of infor-mation and in terms of the Shannon entropy). Here

N ¼ Ac3

GN@(5)

is the number of bits on the boundary with area A, assuggested by the holographic principle. One would bepuzzled when checking the validity of (4) on the horizonof the RN black hole, where

AH ¼ 4�G2N

c4ðMþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

qÞ2;

TH ¼ 2GN@

kBcAH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

q:

(6)

Obviously the relation (4) is violated as long as Q � 0. Itwould be violated even if we go away from the eventhorizon of a RN black hole.Actually this puzzle has driven us to pursue the entropic

origin of the Coulomb force. The puzzle will be solved insubsequent sections. To do this, we will reinterpret thetemperature and generalize the equipartition relation inSec. II. In the same section, we will derive Coulomb’slaw in two different but related ways. As we will see, thereare promising relations between our postulates and theD-brane picture of black hole thermodynamics. InSec. III, taking the Coulomb force as an entropic force,we will derive the Poisson equation and the Maxwell’sequations. The holographic principle and the generalizedpartition rules are vital in our derivation. Finally, in Sec. IV,we will comment on our results and open questions.

II. GENERALIZED EQUIPARTITION RULE ANDCOULOMB’S LAW

Our analysis is very similar to Verlinde’s [1]. We willassume the readers are familiar with the ideas and skills inRef. [1]. In this way, we can pay attention to our newingredients and results, while the readers can better reflecton our ideas.

*wangtao218@pku.edu.cn1As a partial list, some previous and recent papers along this

direction are [2–5].

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A. Scheme one

There are two schemes to solve the puzzle we broughtout. The first choice is to naively replace the equipartitionrule (4) with

c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

q¼ 1

2NkBT: (7)

This relation is well satisfied on the horizon of the RNblack hole. In the past, we are familiar with the equiparti-tion rule for energy. When writing down (7), we general-

ized the rule toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

p. On the event horizon, the

temperature T is identical to the Bekenstein-Hawking tem-perature. Outside the horizon, T can be considered as ageneralized Bekenstein-Hawking temperature on the holo-graphic screen. But such a generalization makes sense onlyif M � jQj. This is a shortcoming of rule (7).

Now suppose there is a test particle with mass m andcharge q near a holographic screen which encloses avolume with total mass M and total charge Q. InRef. [1], without a charge, it was shown that theNewton’s law emerges as an entropic force2

F ¼ T@xS; (8)

where x is the emergent coordinate, i.e., the directionperpendicular to the holographic screen. Generalizing theassumption of [1], we write down

@xS ¼ 2�kBc

@

Mm�QqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

p (9)

WhenQ ¼ 0, it reduces to the assumption made in [1], alsoin the third equation of (12) in the next subsection. In [1]there is an elegant argument for (12), but for (9) we havenot found such a nice argument. This is another short-coming of scheme one.

Nevertheless, by substituting (5), (7), and (9) into (8), wecan unify gravity and the Coulomb force as one entropicforce

F ¼ GN

r2ðMm�QqÞ: (10)

It would be interesting to notice that in the limitm � Mand q � Q, Eq. (9) can be reformed as

@xS ’ 2�kBc

@�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 �Q2

q¼ �k2B

@c�ðNTÞ (11)

if we identify �M ¼ m, �Q ¼ q. This might give us a hint

to incorporate the angular momentum, but we will not gothat far in this paper.In the present scheme, masses and charges appear in a

mixed form, or, namely, in a better-unified form. In thecoming subsection, we will propose a scheme where themass and the charge are ‘‘disassociated.’’

B. Scheme two

The second choice is by postulating equipartition rulesand entropy changes in the emergent direction as follow-ing:

Mc2 ¼ 1

2NkBTg; @xSg ¼ 2�kBmc

@;

Qc2 ¼ 1

2NkBTem; @xSem ¼ � 2�kBqc

@:

(12)

Remember that, as mentioned in the very beginning ofSec. I, we are working in the geometrized unit of charge,in which Coulomb’s law takes almost the same form as theNewton’s law except for the difference in signature. In ourassumptions, the temperature and entropy should be rein-terpreted. In this scheme, the gravitational and the electro-magnetic parts have their own ‘‘temperatures’’ and‘‘entropies,’’ as indicated by the subscripts.The first line of (12) corresponds to the gravitational

part, which has been investigated by Verlinde in [1]. Thenew ingredients are these given in the second line, whichare essentially independent of the gravitational part. Thisline corresponds to the electromagnetic part. Even theholographic screen and the emergent direction for electro-magnetic force can be different from those for gravity.Here again we generalized the equipartition rule, to theelectric or magnetic charge Q. The electric or magnetic‘‘temperature’’ Tem is dictated by the average charge perbit. It is important to notice that Tem can be positive ornegative, depending on the signature of Q. This soundsbizarre. However, as will be explained in the next subsec-tion, we can never observe Tem directly. Amusingly, thedefinition of entropy change in the second line implies anew interpretation for the electric or magnetic charge,parallel to Verlinde’s interpretation for mass [1].With these assumptions at hand, it is straightforward to

derive Coulomb’s law from the entropic force Fem ¼Tem@xSem. When the emergent directions of gravity andelectromagnetic force coincide, we can reobtain Eq. (10) as

Fg þ Fem ¼ Tg@xSg þ Tem@xSem ¼ GN

r2ðMm�QqÞ:

(13)

Now return to our puzzle. In view of this scheme, thetemperature T in (4) should be interpreted as gravitationaltemperature Tg, and it does not equal to the Bekenstein-

Hawking temperature (6) on the horizon. Then the puzzleis solved.

2We would like to regard (8) as the definition of entropic forcerather than the first law of thermodynamics. Some people take itas the first law of thermodynamics, then the second law ofNewton should be implicit, because F ¼ @xE ¼ @xðmv2=2Þ ¼mva=v ¼ ma. If we took their point of view, then Verlinde’sderivation for the second law of Newton [1] would look likecircular reasoning.

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C. Relations between schemes and to D-brane

Scheme two has more general applications than schemeone, because scheme one can be applied only if M � jQjand the distribution of Newtonian potential coincides withthe distribution of Coulomb potential. It is quite revealingto derive scheme one from scheme two when these con-ditions are satisfied. This can be done quickly in virtue ofthe relations below:

T2 ¼ T2g � T2

em; (14)

T@xS ¼ Tg@xSg � Tem@xSem: (15)

In subsection II A, we mentioned that the temperature in(7) is a natural generalization of the Bekenstein-Hawkingentropy outside the event horizon of the RN black hole.From this point of view, only the temperature T on the left-hand side of (14) is possibly observable, while Tem on theright hand side is never. Tg is not observable if Tem � 0.

But the value of observable T is determined by Tg and Tem

together.Up to some numerical coefficients, expression (6) holds

well in a higher dimensional spacetime. As a result, inappropriate units, expressions (14) and (15) will hold inhigher dimensional spacetimes.

Relation (14) reminds us of the temperatures defined inRef. [6] in the D-brane picture, where the thermodynamicproperty of a 5-dimensional near-extremal RN black holehas been studied. Here we copy some relations from [6] intheir convention of notations and normalization. In thatreference, they defined the left-moving temperature andthe right-moving temperature as

TL ¼ 2

�R

ffiffiffiffiffiffiffiffiffiffiffiffiNL

Q1Q5

s; TR ¼ 2

�R

ffiffiffiffiffiffiffiffiffiffiffiffiNR

Q1Q5

s; (16)

where NL is the quantized number of left-moving momen-tum and NR is the right-moving one. In the case NL �NR � 1, they found the Bekenstein-Hawking temperatureis identical to the right-moving temperature TH ¼ TR.

Stimulated by the D-brane picture, we conjecture thatthe temperatures in (14) can be identified as Tg ¼ TL and

T ¼ TR. Then making use of some relations in [6], we find

T2em ¼ T2

L � T2R ¼

�2

�R

�2 NL � NR

Q1Q5

¼�

2

�re

�2; (17)

in which re is horizon radius of the near-extremal blackhole. The same result can be obtained by generalizing (12)to five dimensions.

III. FROM ENTROPIC FORCE TO MAXWELL’SEQUATIONS

A. To Poisson’s equation

In scheme two of the previous section, we treat theelectromagnetic force and gravity independently. This

scheme can be extended to general matter and chargedistributions. As argued in [1], the holographic screenscorrespond equipotential surfaces, so it seems natural todefine the gravitational temperature generally by the gra-dient of Newtonian potential

kBTg ¼ @

2�cr�g: (18)

If one generalizes relation (5) to the form

dN ¼ c3

GN@dA; (19)

the Poisson equation for gravity can be perfectly derived[1].Likewise, we define the electromagnetic temperature

generally by the gradient of Coulomb potential

kBTem ¼ � @

2�cr�em: (20)

Once again we choose the geometrized unit. To recover thefamiliar unit, one has to replace �em ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4�"0GN

p�em.

Then making use of the integral expression of the equipar-tition rule for charge

Qc2 ¼ 1

2kB

I@V

TemdN (21)

and the Gauss’ theorem, we straightforwardly work out

Q ¼ � 1

4�GN

ZVr2�emdV (22)

and hence the Poisson’s equation

r2�em ¼ �4�GN�em: (23)

Transformed from the geometrized unit to the ordinaryunit, it takes the familiar form

r2�em ¼ � 1

"0�em: (24)

B. To Maxwell’s equations

With the Poisson’s equation at hand, the derivation ofMaxwell’s equations is quite similar to or even simplerthan the derivation of Einstein’s equations in [1], whereJacobson’s method [2] was followed. In this subsection, wewill sketch the procedure but not go into details.In order to derive the Maxwell’s equations, we need a

timelike Killing vector �a, just like in [1]. Then �em andr2�em can be recast into the covariant form

�em ¼ �aja; r2�em ¼ 1ffiffiffiffiffiffiffi�g

p �a@bð ffiffiffiffiffiffiffi�gp

FabÞ: (25)

Subsequently (24) is generalized to the covariant form

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1ffiffiffiffiffiffiffi�gp �a@bð ffiffiffiffiffiffiffi�g

pFabÞ ¼ � 1

"0�aj

a: (26)

Following the trick in [1] and applying Jacobson’s reason-ing [2] to timelike screens, one can obtain the Maxwell’sequations with a source current ja.

IV. DISCUSSION

The results in this paper are tentative rather than con-clusive. We will make some comments to help the readersbetter understand our results and ponder on them. After apreliminary version of this paper was posted in the e-printarchive, some issues mentioned above were later discussedfrom different viewpoints in the literature [7–10]. Thecomments below are made after considering the literature.

We are motivated by the RN black hole whereM � jQj,but most of the analysis does not rely on the RN solution orthe conditionM � jQj. Actually, in most parts we treat thegravity and electromagnetic force independently.Therefore, as long as the gravitational background is neg-ligible, we can study the pure electrodynamics with theholographic principle and the generalized ‘‘equipartitionrules.’’ As clarified in [8], the key point is a straightforwardgeneralization from gravitational ‘‘charge’’ (mass) to elec-tric charge. The ‘‘puzzle’’ proposed in the beginning of thispaper was solved in a better way in [9], but it does notaffect the entropic derivation of electromagnetic forcehere.

In our formulation, the electromagnetic temperaturecould be negative. This looks bizarre. Especially, thisimpairs the physical significance of the formally thermo-dynamical interpretation. At first sight, one may avoid it byfocusing on the systems with the same signature of charge.If there is no signature difference in charges, there will beno signature difference in temperatures, and then one canalways choose a positive signature. However, in that situ-ation we can only study the repulsive Coulomb force.When physically interpreting our mathematical formula-tion, the problem of negative temperature is confronted as a

serious obstacle. The authors in [10] met the same problemwhen they try to accommodate inflation in the entropicforce scenario. At the moment, we do not have a goodsolution to this problem. We hope there will be progress inthe future.In the above, we investigated the electrostatics, which is

trivial to generalize to incorporate the static magnetic field.Although Maxwell’s equations have been derived based oncharge integral, the time-dependent case has not beenstudied and would be more complicated. Of course, thetime-dependent gravitational background is even harder totame.In subsection II C, we conjectured that our postulates

could have a close relation to the D-brane picture of thethermodynamics of RN black holes. We conjectured thecorrespondence Tg ¼ TL, T ¼ TR. It will be necessary to

do more serious and extensive comparison with the quan-tities given in [6].No matter how we formulate the theories and change our

logic, the Nature goes in its own way. It is possible that theNewton’s law originates from entropic force butCoulomb’s law does not, or neither does. In the past, wehave collected a lot of evidence that the gravitational forcehas a holographic nature. Little for the electromagneticforce. On the other hand, we have a quantum theory ofelectrodynamics to explain its microscopic degrees of free-dom, but this is not the case for gravity. Here the motiva-tion is to reconcile Verlinde’s equipartition rule on thehorizon of RN black hole. The other support is given bythe apparent similarity between Newton’s law andCoulomb’s law. If they really go in the same way, wehope the formulation in this paper can be regarded as aform of unification of gravity and electromagnetic force.

ACKNOWLEDGMENTS

This work is supported by the China PostdoctoralScience Foundation. We are grateful to Miao Li for intro-ducing Verlinde’s idea on his blog.

[1] E. P. Verlinde, arXiv:1001.0785.[2] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).[3] T. Padmanabhan, Mod. Phys. Lett. A 25, 1129 (2010).[4] F. Caravelli and L. Modesto, arXiv:1001.4364.[5] M. Li and Y. Wang, Phys. Lett. B 687, 243 (2010).[6] C. G. Callan and J.M. Maldacena, Nucl. Phys. B472, 591

(1996).

[7] X. Kuang, Y. Ling, and H. Zhang, arXiv:1003.0195.[8] S. Hossenfelder, arXiv:1003.1015.[9] E. Chang-Young, M. Eune, K. Kimm, and D. Lee,

arXiv:1003.2049.[10] M. Li and Y. Pang, arXiv:1004.0877.

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