Eur. Phys. J. C (2018) 78:627 https://doi.org/10.1140/epjc/s10052-018-6107-3

Regular Article - Theoretical Physics

Comments on the entropic proposal

Sourav Bhattacharya1,a, Panagiotis Charalambous2,b, Theodore N. Tomaras3,c, Nicolaos Toumbas2,d 1 Department of Physics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140 001, India 2 Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus 3 ITCP and Department of Physics, University of Crete, 700 13 Heraklion, Greece

Received: 2 June 2018 / Accepted: 25 July 2018 / Published online: 6 August 2018 © The Author(s) 2018

Abstract Explicit tests are presented of the conjectured the horizon. Placing a particle at some distance away from the entropic origin of the gravitational force. The gravitational horizon perturbs the geometry and causes the horizon area force on a test particle in the vicinity of the horizon of a large to change. In the thermodynamical picture, thermal equilib- Schwarzschild in arbitrary dimensions rium is disturbed and the acquires dependence on is obtained as . The same conclusion can be the distance of the particle from the horizon. We investigate reached for the cases of a large electrically charged black whether the gravitational force on the particle (as seen by the hole and a large slowly rotating Kerr black hole. The gener- static observer) can be interpreted as an entropic force. Once alization along the same lines to a test mass in the field of the particle gets absorbed by the black hole, thermal equi- an arbitrary spherical star is also studied and found not to librium is restored and the entropy of the system becomes be possible. Our results thus reinforce the argument that the maximal. entropic gravity proposal cannot account for the gravitational In the limit of large black hole mass, the near horizon force in generic situations. region becomes sufficiently weakly curved so as to obtain the backreaction on the geometry due to the test particle. More specifically when the particle is slowly moving, we 1 Introduction obtain the shift of the horizon area δ A, the entropy change δS and the temperature T in terms of the particle’s distance Motivated by black hole physics [1Ð3], the holographic prin- ρ from the horizon. Using the formula ciple [4,5] and string theoretic developments on emergent dS space and the AdS/CFT correspondence [6], Verlinde argues F = T (1) that gravity should be understood as an entropic effect caused dρ by the tendency of the underlying microscopic theory to max- imize entropy [7,8]. Other pertinent work includes [9Ð17]. we compute the entropic force and find that it agrees with The purpose of this paper is to demonstrate some concrete the gravitational force on the particle irrespectively of the tests of the conjecture in various spacetime dimensions. number of spacetime dimensions. The same conclusion is In particular, we revisit the problem of a particle freely reached next for the case of a charged test particle in the falling in the vicinity of the horizon of a large D-dimensional vicinity of the horizon of a large charged black hole, as well Schwarzschild black hole. As is well known, a large black as for a test particle in the near horizon region of a slowly hole is in a state of (near) thermal equilibrium, with entropy rotating large Kerr black hole. To the best of our knowledge, given by (one quarter of) the horizon area in Planck units. The no explicit calculations exist in the literature, demonstrating static observer outside the black hole experiences a thermal that the gravitational force on a test particle in the field of environment with increasing temperature as we move toward a black hole of arbitrary mass (angular momentum and/or other charges) can be obtained as an entropic force. In this work we present such a computation in the limit of large black hole mass and test particle near the horizon. a e-mail: [email protected] We then proceed to investigate the case of a test mass b e-mail: [email protected] moving in the field of an arbitrary spherically symmetric c e-mail: [email protected] mass distribution, not necessarily a black hole. The static d e-mail: [email protected] Schwarzschild observer is locally equivalent to a Rindler 123 627 Page 2 of 13 Eur. Phys. J. C (2018) 78 :627 observer, uniformly accelerating with respect to an inertial 2 Test particle in the gravitational field of a frame. However, in this case our approximations cannot be Schwarzschild black hole in D dimensions used to obtain the backreaction of the test mass on the Rindler horizon. Instead, we use a holographic spherical screen suf- We are interested in the motion of a test particle of mass m ficiently close to the observer and associate to it an entropy in the near horizon region of a Schwarzschild black hole in and a temperature. We find that the entropy shift needed D = d + 1 spacetime dimensions (D ≥ 4). The metric is to interpret the gravitational force on a nearby test parti- given by cle as an entropic force is precisely given by one quarter 2 2 2 dr 2 2 of the change of the screen area in Planck units. The shift ds =−f (r) dt + + r d − , f (r) D 2 in the area arises due to the backreaction of the test particle   − R D 3 on the geometry. We argue that this result and the assump- f (r) = 1 − S (2) tion of thermal equilibrium imply that the entropy on the r screen already saturates the holographic bound. This in turn where requires the mass distribution to have collapsed to a black   / − hole and the screen to be the black hole horizon. It seems dif- 16πGM 1 D 3 RS = (3) ficult to modify some of the underlying assumptions in order (D − 2)D−2 to realize the entropic scenario in the more generic cases. The only explicit examples for which the gravitational force is the Schwarzschild radius; M is the mass of the black hole 2 − can be obtained as an entropic force involve a slowly mov- and d D−2 is the metric on the unit D 2-dimensional ing test particle in the near horizon region of a large black sphere. The area of the unit sphere is denoted by D−2.The hole. horizon of the black hole is at r = RS. We consider the case Our arguments reinforce other objections concerning the for which the mass of the black hole is much greater than the entropic gravity proposal, such as the irreversibility effects of mass of the particle, M  m, so that the backreaction on the entropic forces and possible inconsistencies with the inter- Schwarzschild geometry is small. ference patterns in ultracold neutron experiments [18Ð20]. Recall that in the Newtonian limit, the 00-component of For a possible inconsistency of the entropic gravity proposal Einstein’s field equation reduces to the Poisson equation for with that of MOND [21], we refer our reader to [22]. the Newtonian potential φ We also refer our reader to discussions on the so called 8πG (D − 3) emergent gravity paradigm pertaining to the thermodynamic ∇2φ = δ (4) aspects of gravity [23,24], where the thermodynamic char- D − 2 acteristics of the Einstein equations in various contexts have with δ being the mass density. Thus the Newtonian potential been discussed. See also [25Ð27] for recent applications of of a point particle of mass m at position r0 is normalized in this formalism to null hypersurfaces which are not necessar- terms of the D-dimensional G to be ily Killing horizons, to higher derivative alternative gravity theories and also in . π φ =− 8 Gm 1 The paper is organized as follows. In Sect. 2 we con- D−3 (5) (D − 2) D−2 | r − r0 | sider radial motion of a test particle in the gravitational field of a Schwarzschild black hole in D dimensions. We obtain The black hole entropy is given by one quarter of the the backreaction on the geometry in the limit of large black horizon area in Planck units hole mass and the corrected thermodynamic quantities at the D−2 A R D−2 moment the particle is instantaneously at rest. We proceed to S = = S (6) compute the entropic force and find agreement with the grav- 4G 4G itational force for all D ≥ 4. Some results on the metric per- and the Hawking temperature is turbation are reviewed in Appendix A. The D-dimensional mean value theorem Ð reviewed in Appendix B Ð plays a − = D 3 crucial role in the computation. We also consider the case TH (7) 4π RS of a large ReissnerÐNordstrom black hole and the effect of spacetime rotation by considering the Kerr metric. In Sect.3 Our goal is to determine how these thermodynamic quantities we analyze the more general spherically symmetric case and get modified due to the presence of a perturbing mass near argue that the entropic gravity proposal cannot account for the the horizon. gravitational force in generic situations. Finally our results, In order to probe the near horizon region, we introduce perspectives and open problems are summarized in the dis- a new coordinate ρ that measures proper distance from the cussion Sect. 4. horizon: 123 Eur. Phys. J. C (2018) 78 :627 Page 3 of 13 627    r  (D − 2) ρ = dr 2 =−( − )(ρ) − (ρ) +··· 2  (8) ds D 3 1 dt RS f (r ) 2 + ρ2 + 2 ( + (ρ) +···) 2 d RS 1 2 d D−2 (16) In terms of ρ, the Schwarzschild metric is written as  √ In the region where   1 ρ  2RS/ D − 3 ,the ds2 =−f (r (ρ)) dt2 + dρ2 + r 2(ρ)d2 (9) Schwarzschild metric can be approximated by 2 −( − )(ρ) 2 + ρ2 + 2 2 ds D 3 dt d RSd D−2 =¯ ρ(¯) =¯ρ   A static observer at r r is at distance r from the ( − ) 2 =−ρ2 D 3 + ρ2 + 2 2 horizon and measures a temperature dt d RSd D−2 (17) 2RS

TH In the limit of large M, we may further approximate a suf- T = √ (10) f (r¯) ficiently small patch of the sphere, e.g. around the positive xd -axis, as flat [28]. Note that this patch becomes arbitrarily Asymptotically this coincides with the Hawking temperature large in the limit RS →∞. So we end up with the D- and increases as we move toward the horizon. Of course the dimensional flat Rindler metric static observer accelerates relative to a freely falling observer − and can explain the motion of a freely falling particle in terms d 1 2 −ρ2 ω2 + ρ2 + ( )2 of a gravitational force. ds d d dxi (18) = Next we set i 1 where ω = (D − 3) t/2R is a dimensionless time variable. r − RS S = , r = RS(1 + ) (11) The horizon at ρ = 0 becomes a planar Rindler horizon. RS However in computing the entropy shift, it will be necessary We take the black hole mass M to be large, scaling r as above, to take into account the compactness of the horizon√ and the 1 ρ¯  / − and focus in the small  region. In the large mass limit, the finiteness of its area. As long as 2RS D 3, the curvature invariants in this near horizon region become small, static observer lies in the flat region and records a temperature and we can approximate it as flat: inversely proportional to the distance from the horizon:

  TH D − 3 1 ( − )( − )2( − ) 2(D−1) T √ = √ (19) μνκλ D 1 D 2 D 3 RS ( − )¯ π ( − )¯ πρ¯ μνκλ = D 3 4 RS D 3 2 R R 4 RS r (12) We recognize this as the Unruh temperature [29] associated with a uniformly accelerating observer. The proper time asso- So at the horizon ciated with the clock of the Rindler observer at ρ¯ is

μνκλ 1 τ =¯ρ ω μνκλ ∼ d R d (20) R R 4 (13) RS We define the coordinates scaling to zero in the large mass limit.  ( ) ρ Indeed when is small, f r and admit the following tM = ρ sinh ω, xd = ρ cosh ω (21) expansions   (D − 3)(D − 2) appropriate for a locally inertial (freely falling) observer. The f (r) = (D − 3) − 2 +··· metric becomes the D-dimensional Minkowski metric: 2 ds2 −(dt )2 + (dx )2 +···+(dx )2 2R √ M 1 d (22) ρ = √ S  +··· (14) D − 3 2 The trajectory ρ =¯ρ is equivalent with the hyperbola (xd ) − 3/2 2 2 where in the last expression the ellipses are of order  .So (tM ) =¯ρ . The Rindler observer accelerates relative to an inertial observer. When t = 0, the relative velocity is zero M √ 2 and the acceleration is 1/ρ¯. The Unruh temperature is 1/2π D − 3 ρ  = +··· (15) times this acceleration. 2RS 1 The entropy density of the horizon is always maximal, given by one The metric becomes quarter in Planck units. 123 627 Page 4 of 13 Eur. Phys. J. C (2018) 78 :627

2.1 Test particle motion in the near horizon geometry be an entropic force, reproducing the gravitational attraction to the particle. To this end, we study the perturbation of the Consider a probe particle in the Schwarzschild geometry, near horizon geometry due to the presence of a small mass moving along a radial geodesic. Assume that the particle is m, initially at rest on the positive xd -axis, at distance ρ0 from instantaneously at rest at r = r0 (along the positive xd axis). the black hole horizon. The proper velocity and acceleration satisfy Having approximated the near horizon region as flat, we   2 can easily obtain the leading effect due to the backreaction to dr − 1 1 = R D 3 − (23) the small mass.2 As we discuss in Appendix A, the leading dτ S r D−3 D−3 r0 order perturbed metric is 2 ( − ) D−3 d r D 3 RS   =− (24) d 2 D−2 2φ dτ 2 r  2 −( + φ) 2 + − ( )2   ds 1 2 dtM 1 dxi (29) 2 D−3 D−3 − d r (D − 3)R R D 3 = =− S 1 − S i 1 dt2 2 r D−2 r −   where 3(D − 3)R D 3 2 + S dr 1 2 r D−2 dt − ( / )D−3 8πGm 1 1 RS r φ =−   ( − )  (D−3)/2 (25) D 2 D−2 d−1( )2 + ( − ρ )2 i=1 xi xd 0   Assume that 0 1 but fixed; that is, the particle starts (30) its motion in the near horizon region. Eq. (25) becomes Transforming to Rindler coordinates, Eq. (21), we get for d2ρ =−ρ +··· (26) ω 0 dω2   2φ ds2 −ρ2(1 + 2φ)dω2 + 1 − dρ2 where the ellipses stand for velocity dependent terms, vanish- D − 3   ing at the initial moment. This last equation can be obtained φ d−1 2 2 by studying the motion directly in Rindler space. Indeed, + 1 − (dxi ) (31) D − 3 from the point of view of the freely falling (Minkowski) i=1 = ρ ρ = observer, the particle stays still at xd 0. Then The Rindler horizon is still at ρ = 0 but the proper distance ρ / ω 0 cosh , producing Eq. (26). to it gets shifted. In order to obtain the area shift and the Now suppose that the initial position of the particle coin- entropic force, we first compactify the Rindler horizon to a cides with the position of the observer at ρ¯ (ρ =¯ρ). Relative 0 sphere of (unperturbed) radius RS so as to regulate its area, to this observer, the particle’s initial acceleration is and take the large mass limit at the end. Letting ω 0, the perturbation on this spherical horizon is given by d2ρ 1 =− (27) τ 2 ρ¯ π d R 8 Gm 1 φh(ϑ) =−   (D − 2) − (D−3)/2 D 2 R2 + L2 − L R ϑ Thus, he concludes that an initial (ω = 0) attractive force S 0 2 0 S cos acts on the particle, given by (32) m F =− where ϑ is the angle of a point on the horizon with the positive ρ¯ (28) xd -axis; L0 = RS +ρ0 > RS, where ρ0 is the initial distance of the particle from the horizon (Fig. 1). where m is the mass of the particle. We would like to interpret Thus to leading order in m, the shift in the horizon area is this force as an entropic force.  − δ =−D 2 φ (ϑ) D−2  2.2 Perturbing the near-horizon geometry of a large mass A h RS d D−2 (33) D − 3 D−2 black hole S where we integrate the perturbation over the horizon sphere We treat the black hole-particle system as an entropic system. (of radius RS). As we review in Appendix B, the integral can The particle will move in a direction to maximize entropy. be evaluated exactly in terms of the potential at “the center Once it gets absorbed by the black hole, there is an increase in the horizon radius and area, and the entropy increases. When 2 For a study of perturbations in the full Schwarzschild metric see [30, the particle is at some distance from the horizon, there must 31]. 123 Eur. Phys. J. C (2018) 78 :627 Page 5 of 13 627

xd ticle should be explainable via entropic and electromagnetic m forces. We denote the charge and the mass of the test particle by q and m, which we take to be sufficiently small. ρ0 x1,...,xd−1 The black hole geometry is described by the ReissnerÐ Nordstrom metric, given by

RS dr2 R ds2 =−f (r) dt2 + + r 2d2, S f (r) 2 L0 =RS +ρ0 ϑ 2GM Q2G f (r) = 1 − + (38) O r r 2 where Q is the charge of the black hole and M its mass. The electric field is radial given by E = Q/r 2. To avoid a naked Fig. 1 The initial position of the test mass m. O is the center of the r singularity we take M2G ≥ Q2, and so spherical horizon of radius RS

(r − r+)(r − r−) f (r) = (39) of the sphere”, using the mean value theorem in d = D − 1 r 2 spatial dimensions. The shift in the horizon area is    − = ± − 2/ 2 D 2 D−2 with r± MG 1 1 Q M G . Thus the black hole δ A =− D−2 R φC D − 3 S has two horizons with radiir±. The radius of the outer horizon − 2 8πGmRD 2 is r+. The associated entropy is given by S = πr+/G and = S (34) 2 D−3 the Hawking temperature by TH = /4πr+, where  = (D − 3) (RS + ρ0) r+ − r−. This temperature vanishes for an extremal black R → Differentiating the expression above, and letting S hole, for which M2G = Q2 and r+ = r− Ð see e.g. [28]for ∞ ,gives more details. It is a rather challenging problem to obtain the backreac- d(δ A) =−8πGm (35) tion on the black hole geometry (and calculate the shift in the dρ 0 horizon area) for generic motions of the particle, requiring irrespectively of the number of spacetime dimensions D. one to incorporate relativistic effects, including the radiation The shift in the horizon area amounts to a change in emitted by the particle. In this work, we restrict to the case entropy. The corresponding entropy gradient in the large of a particle of sufficiently small mass and charge, instan- mass limit is taneously at rest in the near horizon region of a large, non- extremal black hole for which M2G > Q2. More specif- (δ ) dS = 1 d A =− π ically, we take the mass M and the charge Q of the black 2 m (36) 2 2 dρ0 4G dρ0 hole to be arbitrarily large, keeping the ratio M G/Q fixed (and greater than unity). In this limit, the near horizon region, yielding an entropic force r − r+  r+, becomes sufficiently weakly curved so as to 4 dS m obtain the backreaction. F = T =− (37) Indeed in the near horizon region, the proper distance from entropic dρ¯ ρ¯ the horizon is given by This is in agreement with Eq. (28). ( − )1/2 ρ 2r+ r r+ / (40) 2.3 Considering charged black holes and spacetime rotation 1 2 and the metric can be approximated by Next we consider the case of an electrically charged test par- = ticle interacting with a large charged black hole in D 4. 2ρ2 We would like to check if the gravitational force on the par- 2 − 2 + ρ2 + 2 2 ds 4 dt d r+d 2 (41) ticle can be interpreted as an entropic force. In addition, the 4r+ particle interacts with the electric field produced by the black Furthermore, since we take r+ to be very large, we approx- hole.3 So if Verlinde’s theory is correct, the motion of the par- imate a sufficiently small patch of the sphere, around the 3 Notice that only the gravitational force on the charged particle should be obtained as an entropic force and not the net or the electric component 4 The case of an extremal black hole, M2G = Q2, for which the near of the force. horizon geometry is AdS2 × S2 [32] will be considered in future work. 123 627 Page 6 of 13 Eur. Phys. J. C (2018) 78 :627 positive z-axis, as flat. So we end with the four-dimensional where flat Rindler metric: 2 2 2 2 2 2 r = (r + a ) − 2MGr,ρ= r + a cos θ ds2 −ρ2dω2 + dρ2 + dx2 + dy2 (42) The parameter a is called the rotation parameter. The black hole event horizon radius, rH , is given by the largest positive 2 where ω = t/2r+. A static observer at small distance ρ¯  root of r = 0. r+ from the horizon (along the positive z-axis) measures a We could not find an appropriate generalization of the temperature mean value theorem and Eq. (90)forEq.(45) for generic values of the rotation parameter. This is chiefly due to the 2 T 2r+TH 1 lack of SO(3) invariance of the spacetime. Instead, we wish T¯ = √ H = (43) f (ρ)¯ ρ¯ 2πρ¯ to present a discussion on a weakly rotating version of the Kerr spacetime, keeping in terms only linear in a: Notice also that the electric field in this region becomes arbi- 2 2 −1 2 2 2 2 2 2 ds ≈−f (r) dt + f (r) dr + r d trarily small in the limit: Er Q/r+ Q/M G .   2MG Let the charged particle be instantaneously at rest at a −2a sin2 θ dt dφ (46) distance ρ¯ from the horizon along the positive z-axis. It is r easy to see that the initial acceleration of the particle with where f (r) = 1 − 2MG/r. The surface gravity of the black respect to the static observer at ρ¯ is given by hole event horizon is given by ( = ) d2ρ m κ = f r rH m =− + qE (44) H τ 2 ρ¯ r 2 d R where the prime denotes differentiation with respect to the =− φ/ φφ where τR =¯ρω is the proper time associated with the clock radial coordinate. The angular speed gt g at the hori- of the static observer at ρ¯. We would like to check if the first zon is given by term on the RHS can be interpreted as an entropic force. The 2MGa H = second term is the electric force on the particle due to the r 3 interaction with the electric field of the black hole. H Since the near horizon region is approximately flat in the Near the black hole event horizon, we make further coordi- ¯ large M, Q limit, we can obtain the backreaction on the nate transformation, dφ = dφ + H dt,e.g.[34], to cast the geometry due to the presence of the charged particle. Keep- metric in the diagonal form ing terms only linear in m and q, the perturbed metric is given 2 2 −1 2 ds |r→r ≈−f (r) dt + f (r) dr by Eq. (29) in (locally) inertial coordinates and by Eq. (31) H   2 2 2 ¯2 2 in Rindler coordinates, as can be verified by examining the +r dθ + sin θ dφ + O(a ) (47) ReissnerÐNordstrom metric in isotropic coordinates.5 So fol- which is analogous to Eq. (2) with D = 4. We next follow lowing similar steps as in the previous section, we can verify similar steps described in the previous subsections to reach that the gradient of the shift in the horizon area is given by Eq. the same conclusions. (35). Therefore, the corresponding entropic force reproduces Despite these agreements in the limit of large black hole the first term of Eq. (44). mass, it is not clear to us that entropic forces can account Finally it seems to be an interesting task to include the for the gravitational forces and motion in generic systems effect of spacetime rotation into the above discussions. We with charges and/or angular momentum. In fact by examining take the line element of the Kerr spacetime in four spacetime simpler, spherically symmetric neutral systems in the next dimensions [33] section, we argue that this is not the case. 2 2 2   r − a sin θ 2a sin θ ds2 =− dt2 − r2 + a2 −  dtdφ ρ2 ρ2 r   sin2 θ 3 General spherically symmetric distribution + (r2 + a2)2 −  a2 sin2 θ dφ2 ρ2 r ρ2 In this section we generalize the computation to the case of 2 2 2 + dr + ρ dθ (45) a test mass moving in the field of an arbitrary spherically r symmetric mass distribution of finite radius (not necessarily comprising a black hole). Away from the distribution, the 5 Even if q2 > m2G, at a Compton wavelength away from the particle, the correction to the metric due to the electric field of the particle is geometry is described by the Schwarzschild metric, Eq. (2), suppressed (by a factor of q2) compared to the correction −2Gm/r. where M is the total mass of the distribution. 123 Eur. Phys. J. C (2018) 78 :627 Page 7 of 13 627   2(D−3) It will be more convenient to work in isotropic coordinates. μνκλ 1 RS Rμνκλ R | ∼ (54) To this end we set R0 4 R0 R0    /( − ) D−3 2 D 3 1 R so that a sufficiently small region of proper size L  R0 r = R 1 + S (48) 4 R around the static observer can be approximated to be flat. Next consider the Rindler observer at ρ =¯ρ, with uniform /ρ¯ ρ =¯ρ + x and the metric becomes [35] acceleration 1 . Setting , we may expand the Rindler metric for small x. We get ⎡   ⎤ D−3 2   − 1 RS d−1 ⎢1 4 R ⎥ 2 ds2 =−⎣   ⎦ 2 ds2 −¯ρ2 1 + x dω2 + dx2 + (dx ) D−3 dt i R ρ¯ + 1 S i=1 1 4 R   −    /( − ) d 1 D−3 4 D 3   =− + 2 τ 2 + 2 + ( )2 1 RS 2 2 2 1 x d R dx dxi (55) + 1 + dR + R d − ρ¯ 4 R D 2 i=1 (49) So locally we may identify the static observer with the Rindler observer if we set ρ¯ = 1/g(R0). The static observer  When R RS, the metric can be approximated by at R records a temperature proportional to the acceleration  0   − due to gravity R D 3 ds2 − 1 − S dt2 R 1 g(R )  T (R ) = = 0 (56)   −   0 πρ¯ π 1 R D 3 2 2 + + S dR2 + R2d2 1 − D−2 D 3 R Construction of local thermal field theory using such local (50) Unruh temperature can be seen in e.g., [36].  The horizon corresponding to the Rindler observer is at Introducing Cartesian coordinates such that d x2 = R2, i=1 i distance this metric takes the form derived in Appendix A:   D−3 1 R0   d ρ¯ = R (57) 2 g(R ) 0 R ds2 −( + ) dt2 + − (dx )2 0 S 1 2 1 − i (51) D 3 = i 1 much outside the region which we can approximate as flat.   − So, we cannot apply the arguments of the previous section where  =−8πGM/ (D − 2) − R D 3 is the Newto- D 2 on this horizon to compute the entropic force. nian potential associated with the mass distribution. Instead, we use a spherical holographic screen of coordi- Next consider a static observer sufficiently far from the nate radius R r Ð the Schwarzschild coordinate radius distribution, at fixed radial coordinate R so that R  R . 0 0 0 0 S r is given by Eq. (48) Ð that intersects the x axis at the Without loss of generality, we take the observer to lie on the 0 d location of the observer. This radius of the screen is suf- positive x axis. Define x ≡ x − R and focus on a suffi- d d 0 ficiently large to enclose the mass distribution. According ciently small region |x|, |x |R around the location of this i 0 to the [4,5], the interior region, and observer. There, the metric in Eq. (50) can be approximated everything that fits inside it, can be described in terms of by a boundary theory on the screen. Furthermore, the number of microscopic degrees of freedom of the boundary theory d−1 2 2 2 2 should scale with the area in Planck units. Indeed, the maxi- ds −(1 + 2g(R0) x) dt + dx + (dxi ) (52) mal entropy allowed in the region is equal to the entropy of i=1 a Schwarzschild black hole that just fills the region [1]. This where black hole has radius r0, entropy equal to A/4G, and mass D−3 (D − 2) − r 8πGM(D − 3) 1 = D 2 0 g(R ) = Mmax 0 D−2 (53) 16πG (D − 2)D−2 R 0 ( − ) ( )(D−3)/(D−2) = D 2 D−2 A π (58) is the acceleration due to gravity at radial distance R0. Notice 16 G that the metric Eq. (52) is flat. Indeed the curvature invariant where A is the proper area of the holographic spherical at R0 is of order screen. The maximal entropy of any system is proportional 123 627 Page 8 of 13 Eur. Phys. J. C (2018) 78 :627 to the number of fundamental degrees of freedom needed to Differentiating with respect to x0 and taking the x0 → 0 describe the system. limit, gives We suppose, as in [7], that in the underlying holographic d(δ A) description, the energy of the mass distribution is suitably =−8πGm (63) partitioned among the microscopic degrees of freedom, so dx0 that the system becomes entropic with temperature given by Therefore the entropy gradient Eq. (59) is precisely given by Eq. (56). Wewill demonstrate in the following subsection that one quarter of the area gradient in Planck units: the equipartition principle and thermal equilibrium require the mass of the distribution to be an appreciable fraction of 1 d(δ A) dS the maximal mass allowed in the region. Moreover, we argue = (64) 4G dx0 dx0 below that the interpretation of gravity as an entropic force requires the entropy on the screen to be maximal. If this is However, as we argue in the next subsection, the princi- the case, the original mass distribution must be a black hole ple of equipartition and thermal equilibrium are compatible and the holographic screen the black hole horizon. with Eq. (64) only if the entropy on the holographic screen is We consider a freely falling particle of mass m  M,ini- maximal. If this is the case, the original distribution must be tially at rest at position xd = R0 + x0, x0 > 0onthexd axis. a large black hole that fills the interior region and the screen The particle lies in the region outside the screen. According the black hole horizon. The valid computation of the entropic to Verlinde, in the holographic description, the particle per- force in the black hole case has been presented in Sect. 2.Itis turbs the system and the entropy changes as a function of not clear to us how to interpret the gravitational force in more the location of the particle. He postulates that when the par- general cases as an entropic force. Of course it could be that ticle is sufficiently close to the screen, the entropy gradient one or more of our assumptions break down, with the entropy is proportional to the mass of the particle [7]: on the screen being small and Eq. (64) holding, but we would like to understand how. Another possibility is to relate the dS entropy shift necessary to reproduce the gravitational force =−2πm (59) dx0 with a change in the entanglement entropy between the inte- rior and the exterior regions. The entanglement area contains Then, indeed, the entropic force a divergent piece that scales with the area of the boundary (in units of the cutoff). However it is not clear how to regularize dS the entanglement entropy, including the fluctuations of the Fentropic = T (R0) =−mg(R0) (60) dx0 background geometry, and get the required coefficient. Let us also make some comments about the choice of holo- coincides with the gravitational force on the particle at R0. graphic screens. In this work we consider (quasi) static space- α But, let us take a closer look at the x0 → 0 limit. Let us, in times with a time-like Killing vector ξ (that would become particular, compute the shift in the area of the screen due to null at the horizon of a black hole), and which can be foliated the backreaction of the particle on the background geometry. with space-like surfaces that satisfy the holographic bound α Since the screen radius R0  RS and the particle mass is on the entropy [37]. The norm of the Killing vector, −ξ ξα, taken to be very small, the perturbed metric is of the form represents the redshift factor that relates the proper time of Eq. (51) a static clock at some point in the interior of spacetime to that of a clock at infinity. As in [7], we take the screens to be    d surfaces of constant redshift. These would naturally include 2  2 2 2 ds −(1 + 2 ) dt + 1 − (dxi ) the horizon of a black hole and coincide with equipotential perturbed D − 3 i=1 surfaces in regions of space with weak gravitational fields. (61) Notice that the deformation of the equipotential surface due to the introduction of a test particle induces a higher order  with  =  + φm being the total Newtonian potential, effect to the induced metric (and hence the area of the screen) including the contribution of the particle. Using the metric with respect to the mass of the particle. In the cases of spher- above and the mean value theorem, we obtain the shift in the ically symmetric we are mostly interested in, the area of the holographic screen choice of screens is dictated by symmetry.  Equipotential surfaces play an important role in the orig- − D 2 D−2 δ A =− φm(ϑ) R dD−2 inal proposal of [7]: the direction of the entropic force on − − 0 D 3 S D 2 a test particle, initially at rest at a small distance from the − 8πGm RD 2 screen, is precisely given by the unit vector perpendicular to = 0 (62) D−3 the screen. So this choice has a priori a chance to reproduce (D − 3) (R0 + x0) 123 Eur. Phys. J. C (2018) 78 :627 Page 9 of 13 627 the gravitational force. Any other choice for screens (includ- Assuming thermal equilibrium, the first law gives (for ing dynamical ones) should account for both the magnitude fixed N) and direction of the force and also be compatible with black hole horizons. 2M dM = TdS = dS (67) It would be interesting to generalize the above results for N non-spherically symmetric cases, to see how the notion of the entropic force would compare between surfaces with differ- Integrating this equation, we obtain   ent topologies but otherwise with the same area (and hence N M S = S − ln max the same maximal entropy). Clearly, in such cases one needs max 2 M to develop a covariant and coordinate independent formalism    Mmax to address the issue. = S 1 − k ln max M Finally for more general asymptotically flat spacetimes,    A M one would need to foliate them with null hypersurfaces [38] = 1 − k ln max (68) and make use of the covariant entropy bound [37] and other 4G M thermodynamics aspects of gravity. It is not clear to us how- where Mmax is the mass for which the entropy becomes max- ever how to implement Verlinde’s proposal in such situations. imal Ð see Eq. (58) Ð and k = (D − 2)/(D − 3).Inthelast expression, we expressed the maximal entropy in terms of 3.1 Statistical the area of the holographic screen. When the energy in the interior region becomes equal to Mmax, the system collapses Here we shall try to build a toy model in which Eq. (64) does to a black hole of radius r0. Systems with greater energies do not necessarily imply that the entropy on the holographic not fit in the interior region (and a screen of bigger area must screen is maximal and the case studied above does not actu- be used). ally correspond to the black hole case studied in Sect. 2. For Eq. (68) to be valid, the mass M cannot be arbitrarily According to the holographic principle, all configurations small. In fact, thermal equilibrium and equipartition can be that fit in the interior region can be described in terms of a established when the mass of the distribution is an apprecia- boundary theory living on the screen. The exterior region is ble fraction of the maximal mass, namely empty space. Let us denote by N the number of fundamental degrees of freedom in the boundary theory. The total energy M M ≥ max (69) in the system is given by the mass M of the distribution, e1/k which cannot exceed Mmax given by Eq. (58). The energy M will be distributed among the microscopic We expect that for smaller masses or temperatures, equipar- degrees of freedom N according to some non-trivial distri- tition breaks down, with the temperature/energy relation bution function. When M is sufficiently large, we assume depending strongly on the details of the interactions of the as in [7] that the principle of equipartition holds to a good boundary theory. approximation. The resulting temperature T is proportional Next consider the variation of the entropy, while keeping to the mean energy: the mass M < Mmax fixed. Such a variation can arise from a shift in the area of the screen A, due to the shift of the 1 location of the test particle. When M = M , S = A/4G M = NT (65) max 2 and S = A/4G. However when M < Mmax, we get    Requiring that the temperature is given by Eq. (56), the num- A Mmax A Mmax ber of degrees of freedom N scales with the area of the screen S = 1 − k ln − k (70) 4G M 4G M in Planck units, in accordance with the holographic principle: max Using Eq. (58), it is easy to see that A(R ) N ∼ 0 (66) G M A k max = (71) Notice that the precise way the energy is divided among the Mmax A microscopic degrees of freedom depends on the interactions and the details of the holographic mapping, which we do in any spacetime dimension. So not know. The interactions may modify the precise relation     A M A S A between the energy and the temperature, but we expect the S =−k ln max =− 1 − = basic conclusions regarding the realization of the entropic 4G M 4G Smax 4G scenario to continue to hold. (72) 123 627 Page 10 of 13 Eur. Phys. J. C (2018) 78 :627

Therefore if M = Mmax, it is not possible for the change of the screen must be a small fraction of the maximal entropy entropy to equal to A/4G. Within our set of assumptions, allowed in the interior region. We found that the entropy gra- this relation holds only for M = Mmax or S = Smax.Ifthis dient needed to interpret the gravitational force on a nearby is the case, the mass distribution already comprises a large slowly moving test particle as an entropic force is equal to one black hole and was studied in Sect. 2. quarter of the area gradient in Planck units. We argued that this result, the principle of equipartition and thermal equilib- rium imply that the entropy on the screen must be maximal, 4 Discussion given by one quarter of the screen area in Planck units. So the original mass distribution must be a large black hole that In this work we attempted to carry out some tests of the fills the region inside the screen. Within this set of assump- entropic gravity proposal in arbitrary spacetime dimensions tions, as usually stated, it is not clear that the entropic gravity (D ≥ 4). First we considered a large Schwarzschild black proposal can account for the gravitational interactions in the hole interacting with a test particle, which is initially at rest most general cases. This work, as we mentioned earlier, hope- at a small distance from the horizon. The particle perturbs fully thus adds to the existing debates on the entropic gravity the near horizon geometry and induces a shift in the area proposal, e.g. [18,22]. of the horizon. We obtained the perturbation in the metric in the limit of large black hole mass and at the moment the Acknowledgements NT wishes to thank the ITCP and the Depart- particle is instantaneously at rest. As the mass of the black ment of Physics of the University of Crete where parts of this work were done for its hospitality. SB wishes to acknowledge S. Chakraborty hole increases, the near horizon region becomes sufficiently for discussions. SB’s work is partially supported by the ISIRD grant weakly curved, and can be well approximated with Rindler 9-298/2017/IITRPR/704. space. We then computed the shift in the horizon area, and Open Access This article is distributed under the terms of the Creative hence the shift in the black hole entropy, as a function of Commons Attribution 4.0 International License (http://creativecomm the particle’s distance from the horizon. We found that the ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, entropic force agrees with the gravitational force on the par- and reproduction in any medium, provided you give appropriate credit ticle (as seen by a static observer outside the black hole hori- to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. zon), irrespectively of the number of spacetime dimensions. Funded by SCOAP3. The cases of a large charged black hole and a slowly rotating Kerr black hole were discussed next and the force on a test Appendix A: Metric perturbation particle near the horizon was verified to satisfy the entropic force conjecture. Consider a system of non-relativistic particles moving in D- It would be interesting at this point to obtain the per- dimensional flat space. Assume that the particle masses are turbed metric for more general particle motions and to incor- small and the particle separations large, so that the gravita- porate velocity dependent terms and relativistic effects, at tional forces between them are weak. As a first step, we ignore least perturbatively. Then we could investigate if it is possi- the motions of the particles and treat the system as a static ble to reconstruct (perturbatively) the geodesic equations in mass distribution. Then we may obtain the corrections to the terms of entropic forces and other thermodynamic quantities. flat spacetime metric perturbatively. We may also incorpo- Since entropic forces eventually cause irreversible changes rate particle motion and obtain the relativistic corrections, in a system, it is not obvious if these can account for the employing the post-Newtonian approximation. For the pur- most general particle motion in a gravitational background, poses of this work, it will be sufficient to obtain the leading including the case of a black hole background. It would also correction to the metric, which is linear in the masses. We be interesting to extend the study of the perturbations in the write full Schwarzschild geometry, letting the mass of the black hole be large but finite and the distance of the particle from gμν = ημν + hμν (73) the horizon arbitrary. We then proceeded to generalize the computation to the where hμν is the metric perturbation, linear in the masses. case of an arbitrary spherically symmetric mass distribution, Einstein’s field equations can be written in terms of the not necessarily comprising a black hole. We chose a spheri- energy-momentum tensor of the matter system as follows cal holographic screen of sufficiently large radius to enclose the mass distribution, and associated to it an entropy and Rμν =−8 π GSμν (74) the Unruh temperature measured locally by a static observer. This temperature is dictated by the principle of equipartition where of energy and the holographic scaling of the number of fun- 1 λ Sμν = Tμν − Tλ gμν (75) damental degrees of freedom on the screen. The entropy on D − 2 123 Eur. Phys. J. C (2018) 78 :627 Page 11 of 13 627

Ignoring the motion of the particles, the only non-vanishing integral over a d-dimensional ball Bd of radius R, centered component of the energy-momentum tensor is T 00, which at the origin: δ equals the density of rest mass . Then    ψ∇2φ − φ∇2ψ ddr D − 3 1 S = T 00, S = δ T 00 Bd  00 − ij − ij (76) D 2 D 2 d−1 = R (ψ∇φ − φ∇ψ) ·ˆrdd−1 (85) Sd−1 Imposing further the harmonic coordinate conditions We choose ψ to be μν λ g μν = 0 (77)   1 1 1 ψ (r) =− − and dropping corrections non-linear in the masses, the Ricci d−2 d−2 (86) (d − 2) d−1 r R tensor simplifies to the following

which satisfies ψ (R) = 0 and ∇2ψ = δd (r), and φ to 1 2 1 2 R = ∇ h , Rij = ∇ hij, R i = 0 (78) 00 2 00 2 0 satisfy Laplace’s equation

Therefore, the 00-component of Einstein’s equation becomes ∇2φ = 0 (87) to this order

πG(D − ) 2 16 3 00 inside Bd . Then Eq. (85) reduces to the mean value theorem ∇ h00 =− T (79) D − 2 in d spatial dimensions, namely with solution  φ dd−1 = d−1 φC (88) Sd−1 h00 =−2 (80)

d−1 where  is the Newtonian potential in D-dimensions satis- where φC is the value of φ at the center of the sphere S . 2 fying If ∇ φ = 0insideBd , then the integral gets an extra term   8πG (D − 3) ∇2 = δ (81) φ  =  φ −  ψ∇2φ d − d d−1 d−1 c d−1 d r (89) D 2 d−1 S Bd The spatial components give As an example, we apply the mean value theorem for the π Newtonian potential of a point particle of mass m, located 2 16 G 00 ∇ hij =− δijT (82) D − 2 at distance L0 from the origin on the positive xd axis. In spherical coordinates, the potential is given by with solution 1 8πGm 2 φ =−  h =− δ (83) m (d−2)/2 ij ij (d − 1) d−1 2 + 2 − ϑ D − 3 r L0 2L0r cos (90) The corrected metric is   ϑ  d where is the angle between the position vector r and the 2 2 2 2 ds −(1 + 2)dt + 1 − (dxi ) (84) positive xd axis. The mean value theorem gives M D − 3 i=1  π φ  =− 1 8 Gm m d d−1 − (91) d−1 d − 1 d 2 Appendix B: Mean value theorem S L0

The mean value theorem can be extended in d spatial dimen- The center of the sphere is chosen to be at the origin and the sions. We start by applying Green’s identities to the following radius R is taken to be R < L0. 123 627 Page 12 of 13 Eur. Phys. J. C (2018) 78 :627

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