In this letter, we use the Parikh-Wilczek tunneling model of Hawking
radiation to illustrate that a reformulation of Verlinde's entropic gravity is
needed to derive the Newton's law for a temperature-varying screen, demanded by
the conservation of energy. Furthermore, the entropy stored in the holographic
screen is shown to be additive and its temperature dependence can be obtained.Comment: 5 pages, 1 figure, version submitted to EPJ

Wen, Wen-Yu

English

arXiv

Wen-Yu Wen

Crossref

Eur. Phys. J. C (2014) 74:2912DOI 10.1140/epjc/s10052-014-2912-5LetterThoughts on entropic gravity in the Parikh–Wilczek tunnelingmodel of Hawking radiationWen-Yu Wen1,2,a1 Department of Physics and Center for High Energy Physics, Chung Yuan Christian University, Chung Li, Taiwan2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 106, TaiwanReceived: 20 December 2013 / Accepted: 17 May 2014 / Published online: 4 June 2014© The Author(s) 2014. This article is published with open access at Springerlink.comAbstract In this letter, we use the Parikh–Wilczek tunnel-ing model of Hawking radiation to illustrate that a refor-mulation of Verlinde’s entropic gravity is needed to deriveNewton’s law for a temperature-varying screen, required bythe conservation of energy. Furthermore, the entropy storedin the holographic screen is shown to be additive and its tem-perature dependence can be obtained.1 Verlinde’s entropic gravity and holographic screenTo explain why gravity is so different from the other threeforces in the nature, Verlinde proposed that gravity couldbe regarded as an entropic force caused by changes in theinformation associated with the positions of material bod-ies [1]. His idea can be illustrated by a particle with massm approaching some hypothesized holographic screen. Thescreen bounds the emerged part of space containing the par-ticle, and it stores data that describe the part of space that hasnot yet emerged, as well as some part of the emerged space.The change in entropy on the screen, denoted S, is assumedto be linear in the particle’s displacement x as follows:S = 2πkB mch¯ x . (1)If one further assumes that the screen is thermalized at theUnruh temperature set up by particle’s acceleration, thenNewton’s second law can be derived from the entropic forcerelation:Fx = T S. (2)One opposition to this proposal is given by [2] using themeasurement result of quantum states of ultra-cold neutronunder the Earth’s gravity. According to Kobakhidze’s argu-ment, a pure state neutron would have evolved into a mixeda e-mail: steve.wen@gmail.comstate thanks to S > 0 in (1). This criticism, however,is questioned in [3] and one resolution was suggested in[4] by abandoning the implicit assumption in [2] that theentropy on the holographic screen is additive. Instead, uni-tarity could still be restored even with S > 0 if bits onthe screen were entangled in some delicate way. Neverthe-less, it remains unclear how the entropy is entangled on theholographic screen.The other restriction of (2) is that a holographic screen atthermal equilibrium corresponds to the particle moves in uni-form acceleration. A generalization of the entropic force rela-tion is needed to describe a particle with generic motion. Sim-ilar to the generalization of Newton’s second law to includea varying mass, it is natural to generalize the entropic grav-ity to include a varying screen temperature. Having said that,we consider an adiabatic process such that the temperature ofthe holographic screen is slowly varying but still well definedwhile a massive particle moves relatively to the screen. Thegeneralized entropic force formula then becomesFr = (T S), (3)for varying screen temperature T and screen entropy S, whilea particle is displaced a distance r from the screen at fixedlocation.In the following, we will illustrate that Verlinde’s screencould be realized in the tunneling model of Hawking radia-tion if a reformulation of entropic gravity is adopted to com-pensate for the effect of varying temperature. Following asimilar argument as in [2] but for temperature-varying screen,we show that the entropy can still be additive on the screenand its temperature dependence can be derived.2 Entropic gravity with temperature-varying screenThe original treatment of Hawking radiation by Hawkingis to consider a perturbation in a fixed background of the1232912 Page 2 of 4 Eur. Phys. J. C (2014) 74:2912Schwarzschild black hole. The thermal spectrum broughtabout controversial debates over the Information Loss Para-dox. Parikh and Wilczek considered radiation as an outgo-ing tunneling particle where the conservation of energy isenforced [5]. It was later confirmed that in their model infor-mation is also conserved by computing the mutual infor-mation for two successive radiations [6]. If gravity has anentropic origin, it is desirable to describe the Hawking radi-ation (as a tunneling process), at least for Schwarzschildblack holes, as some form of entropy change in a holo-graphic screen. We recall that back reaction from radiatedparticle to the black hole makes it a system away from ther-mal equilibrium, namely with temperature varying or beingnot well defined. As a result it is proper to apply the gen-eral formula (3) rather than (2) to the tunneling model. Fora screen which stores the same amount of information as theblack hole entropy and possesses the same temperature asthe black hole, the change on the right-hand side before andafter radiation can be computed as(T S) = 18πG(M − ω) · 4πG(M − ω)2− 18πG M· 4πG M2 = −ω2, (4)which can be understood as a quantum of mass (energy) ωthat is discarded from the black hole up to a factor 1/2.1 Theleft-hand side computes the required work by pulling a point-like varying mass (energy) from the black hole. Accordingto Newton’s law of gravity, that is2G(M−ω)∫2G MF(r)dr = −G(M −r2G )(r2G )r∣∣∣∣2G(M−ω)2G M= −ω2,(5)which is consistent with the right-hand side. Therefore, thegravitational force can again be identified with an entropicforce, even for a nonequilibrium system.3 Holographic screen in the Parikh–Wilczek tunnelingmodel of Hawking radiationInspired by the reformulation of entropic gravity in the pre-vious discussion, we are motivated to study a holographicscreen at arbitrary position but admitting a varying temper-ature, in reaction to the displacement of a radiation particle.For symmetry reasons, a spherical screen enclosing the black1 This factor has been pointed out in the [7] as equipartition of energyin the horizon degrees of freedom. In brief, each quantum of area fluc-tuation contributes an energy T/2, that is, E = 12 nT for n = A = 4S.One can derive (4) for loss of energy (mass) ω.Before radiationAfter radiationBlack holeHolographic screenFig. 1 A schematic drawing of a particle radiated out of a black holeand their holographic screenhole seems to be a good choice. We denote the informationstored on a holographic screen associated to a black holewith mass M as SM (r). The screen is located at some dis-tance r from the center of the black hole. We may want tohave r > 2G M to avoid ambiguity of a screen inside theblack hole. Before proceeding with the general formula (3),we would like to first point out some difficulties appearing inthe original formulation of the entropic gravity (2) in the tun-neling model of black hole radiation. If we assume that theentropy stored on the screen follows a simple coarse grainedrelation as shown in Fig. 1, following the neutron–earth sys-tem in [2]:SM (r + r) = Sω(r + r) + SM−ω(r), (6)where r is the distance between radiated mass and thescreen. For an infinitesimal displacement, we have SM (r +r)  SM (r) + S. If the entropy on the screen were addi-tive, that is,SM−ω(r) = SM (r) − Sω(r), (7)one obtainsSM  Sω(r + r) − Sω(r). (8)Since SM ∝ r for its entropic origin, we come to thesame conclusion: that the translation operator is no longerHermitian and disobeying quantum mechanics. In [4], a gen-eralized entropy formula by Tsallis [8] was called upon torestore the unitarity of translation, where an additional entan-gled term was essential. In our radiation–black hole sys-tem, it was known that the Parikh–Wilczek model predicteda non-thermal radiation spectrum and correlation betweentwo emissions ω1, ω2 was computed as 8πω1ω2, known as123Eur. Phys. J. C (2014) 74:2912 Page 3 of 4 2912mutual information [6]. The entanglement is necessary forthe conservation of entropy. Therefore, it is possible to cor-rect the additivity condition (7) to include a term Sent > 0encoding entanglement between black hole and radiation:SM−ω(r) = SM (r) − Sω(r) − Sent(r), (9)such that entropy is conserved before and after radiation.2Notice that the relative sign for the last term is differ-ent from the usual entanglement between two subsystems.This difference in fact makes the situation worse since nowSM  Sω(r +r)−Sω(r)−Sent(r). For a translation to beunitary, say Sω(r +r) = Sω(r), one has SM < 0, imply-ing a violation of the second law of thermodynamics, whichreflects the fact that the black hole cannot evaporate clas-sically. If the Hawking radiation were thermal, one mightexpect the increased entropy during thermal process to belarge enough to compensate for the reduced entropy due tosplitting of the radiated particle from the black hole. How-ever, in Parikh–Wilczek’s tunneling model, one has no suchadditional source for the entropy since the process is non-thermal.Now we turn to an application of the general entropicformula (3). Because the temperature may vary for screens atdifferent locations, we assume that the coarse grained relation(6) is modified asT (r + r)SM (r + r) = T (r + r)Sω(r + r)+ T (r)SM−ω(r). (10)If we assume a linear relation between screen temperatureand small displacement, that is, T (r + r)  T (r) + T ,one obtainsSM  [Sω(r +r)− Sω(r)]+[TTSω(r + r) − Sent].(11)For the translation of the radiated particle being unitary,Sω(r) = Sω is a constant. Vanishing of SM for r = 0suggests Sent(r) = 0. According to (9), the entropy on thescreen is additive. The temperature dependence can also beobtained by integration:SM (r) − SM (r0) = Sω ln T (r)T (r0) , (12)for some reference position r0. The movement of the radi-ated particle with respect to the screen does not change corre-2 Classically, a black hole is forbidden to split because SM (r) >SM−ω(r)+ Sω(r). In the Parikh–Wilczek model, the entanglement pro-vides an additional source of entropy to conserve the entropy during theevaporation process.sponding to the entropy Sω. Instead, it feels different tempera-tures from screens at different locations. If one recalls the sce-nario of a massive particle tunneling out of black hole hori-zon: the radiated particle experiences less gravitational pullwhile it moves away from the black hole, as if it feels colderfrom holographic screen. The screen temperature reflects theacceleration or deceleration as what happens in the Unruheffect.4 DiscussionIn this letter, with the explicit example of Parikh–Wilczek’stunneling model of Hawking radiation, we illustrate thatVerlinde’s holographic screen could have violated unitarityof quantum mechanics if it were additive in entropy. Thissickness cannot be simply cured by the assumption that theentropy on screen is entangled. Motivated by a general for-mula of entropic gravity, one shows that the entropy on thescreen can still be additive if the relation between screenentropy and temperature is given by (12). Some commentsare in order.Firstly, the Parikh–Wilczek is a semiclassical model,which could receive a further quantum correction. Forinstance, a one-loop correction to the surface gravity wasconsidered in [9]. As a result, the Bekenstein–Hawking arealaw receives a logarithmic correction, denoted Sα(M) =4π M2 − 4πα ln (1 + M2α), and the Hawking temperature ismodified as TH = M2+α8π M3 for a large black hole, where thecoefficient α is related to the trace anomaly. One can showthat the relation of the generalized entropic force (3) is nolonger satisfied, since(TH Sα) = −ω2 +α[32M2− 3α + M22M4ln(1 + M2α)]ω.(13)Nevertheless, this might be fixed from two different view-points. One way is to modify Newton’s gravitational forceby a quantum correction up to O(α2), such that ∫ Fα(r)dragrees with the above result. The other way is to rederive theHawking temperature, instead of that obtained in the smallω limit. The proper definition of the Hawking temperatureTα(M) is given byTα(M − ω)Sα(M − ω) − Tα(M)Sα(M) = −ω2 . (14)It seems unlikely to have analytic forms for loop correctedFα(r) and Tα(M), but at least in principle one can get theorder expansion for small α. Once the relation (3) can befixed, our conclusion for the temperature-varying screen willstill be valid, at least perturbatively in terms of α.1232912 Page 4 of 4 Eur. Phys. J. C (2014) 74:2912Secondly, although Verlinde’s adoption of entropic forcewas meant to tame the wild gravity beast, it is natural to askhow to incorporate the other forces in his entropic formalism.This generalization has been considered for the electromag-netic force, by inclusion of the chemical potential in the firstlaw of thermodynamics [10]. In particular, for the tunnelingmodel of Reissner–Nordström black hole radiation [11], oneexpects that the relation (3) is generalized asFr = (T S − AQ), (15)where the gauge potential and total black hole charges areidentified as the chemical potential and number density onthe screen, respectively. On the left-hand side, one has toinclude the electromagnetic force for a consistent result. Ifthere exists a holographic screen for charged black holes, itmust also carry degrees of freedom for charges. The blackhole censorship constraint Q ≤ M would translate into anupper bound for the charge density on the screen. The breakdown of the screen for excess charge would correspond toa possible naked singularity or closed time-like curve. Therecently found charge–mass ratio bound for emission fromthe RN black hole [12] seems to support this picture by stat-ing that a part of screen, which corresponds to the radiatedparticle, also has an upper bound for the charge density.Acknowledgments WYW is supported in part by the Taiwan’sNational Science Council under Grant NO. 102-2112-M-033-003-MY4and the National Center for Theoretical Science.Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.Funded by SCOAP3 / License Version CC BY 4.0.References1. E.P. Verlinde, On the origin of gravity and the laws of Newton.JHEP 1104, 029 (2011)2. A. Kobakhidze, Gravity is not an entropic force. Phys. Rev. D 83,021502 (2011)3. M. Chaichian, M. Oksanen, A. Tureanu, On gravity as an entropicforce. Phys. Lett. B 702, 419 (2011)4. E.M.C. Abreu, J.A. Neto, Considerations on gravity as an entropicforce and entangled states. Phys. Lett. B 727, 524 (2013)5. M.K. Parikh, F. Wilczek, Hawking radiation as tunneling. Phys.Rev. Lett. 85, 5042 (2000)6. B. Zhang, Q.-Y. Cai, L. You, M.-S. Zhan, Hidden messengerrevealed in Hawking radiation: a resolution to the paradox of blackhole information loss. Phys. Lett. B 675, 98 (2009)7. T. Padmanabhan, Equipartition of energy in the horizon degreesof freedom and the emergence of gravity. Mod. Phys. Lett. A 25,1129 (2010)8. C. Tsallis, Possible generalization of Boltzmann–Gibbs statistics.J. Stat. Phys. 52(1–2), 479–487 (1988)9. R. Banerjee, B.R. Majhi, Quantum tunneling and back reaction.Phys. Lett. B 662, 62 (2008)10. Y.-X. Chen, J.-L. Li, First law of thermodynamics on holographicscreens in entropic force frame. Phys. Lett. B 700, 380 (2011)11. J.-Y. Zhang, Z. Zhao, Hawking radiation of charged particles viatunneling from the Reissner–Nordström black hole. JHEP 0510,055 (2005)12. K.K. Kim, W.-Y. Wen, Charge–mass ratio bound and optimizationin the Parikh–Wilczek tunneling model of Hawking radiation. Phys.Lett. B 731, 307 (2014)123

The European Physical Journal C

Thoughts on entropic gravity in the Parikh–Wilczek tunneling model of Hawking radiation

Springer - Publisher Connector

Eur. Phys. J. C (2014) 74:2912
DOI 10.1140/epjc/s10052-014-2912-5
Letter
Thoughts on entropic gravity in the Parikh–Wilczek tunneling
model of Hawking radiation
Wen-Yu Wen1,2,a
1 Department of Physics and Center for High Energy Physics, Chung Yuan Christian University, Chung Li, Taiwan
2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 106, Taiwan
Received: 20 December 2013 / Accepted: 17 May 2014 / Published online: 4 June 2014
© The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract In this letter, we use the Parikh–Wilczek tunnel-
ing model of Hawking radiation to illustrate that a refor-
mulation of Verlinde’s entropic gravity is needed to derive
Newton’s law for a temperature-varying screen, required by
the conservation of energy. Furthermore, the entropy stored
in the holographic screen is shown to be additive and its tem-
perature dependence can be obtained.
1 Verlinde’s entropic gravity and holographic screen
To explain why gravity is so different from the other three
forces in the nature, Verlinde proposed that gravity could
be regarded as an entropic force caused by changes in the
information associated with the positions of material bod-
ies [1]. His idea can be illustrated by a particle with mass
m approaching some hypothesized holographic screen. The
screen bounds the emerged part of space containing the par-
ticle, and it stores data that describe the part of space that has
not yet emerged, as well as some part of the emerged space.
The change in entropy on the screen, denoted S, is assumed
to be linear in the particle’s displacement x as follows:
S = 2πkB mch¯ x . (1)
If one further assumes that the screen is thermalized at the
Unruh temperature set up by particle’s acceleration, then
Newton’s second law can be derived from the entropic force
relation:
Fx = T S. (2)
One opposition to this proposal is given by [2] using the
measurement result of quantum states of ultra-cold neutron
under the Earth’s gravity. According to Kobakhidze’s argu-
ment, a pure state neutron would have evolved into a mixed
a e-mail: steve.wen@gmail.com
state thanks to S > 0 in (1). This criticism, however,
is questioned in [3] and one resolution was suggested in
[4] by abandoning the implicit assumption in [2] that the
entropy on the holographic screen is additive. Instead, uni-
tarity could still be restored even with S > 0 if bits on
the screen were entangled in some delicate way. Neverthe-
less, it remains unclear how the entropy is entangled on the
holographic screen.
The other restriction of (2) is that a holographic screen at
thermal equilibrium corresponds to the particle moves in uni-
form acceleration. A generalization of the entropic force rela-
tion is needed to describe a particle with generic motion. Sim-
ilar to the generalization of Newton’s second law to include
a varying mass, it is natural to generalize the entropic grav-
ity to include a varying screen temperature. Having said that,
we consider an adiabatic process such that the temperature of
the holographic screen is slowly varying but still well defined
while a massive particle moves relatively to the screen. The
generalized entropic force formula then becomes
Fr = (T S), (3)
for varying screen temperature T and screen entropy S, while
a particle is displaced a distance r from the screen at fixed
location.
In the following, we will illustrate that Verlinde’s screen
could be realized in the tunneling model of Hawking radia-
tion if a reformulation of entropic gravity is adopted to com-
pensate for the effect of varying temperature. Following a
similar argument as in [2] but for temperature-varying screen,
we show that the entropy can still be additive on the screen
and its temperature dependence can be derived.
2 Entropic gravity with temperature-varying screen
The original treatment of Hawking radiation by Hawking
is to consider a perturbation in a fixed background of the
123
2912 Page 2 of 4 Eur. Phys. J. C (2014) 74:2912
Schwarzschild black hole. The thermal spectrum brought
about controversial debates over the Information Loss Para-
dox. Parikh and Wilczek considered radiation as an outgo-
ing tunneling particle where the conservation of energy is
enforced [5]. It was later confirmed that in their model infor-
mation is also conserved by computing the mutual infor-
mation for two successive radiations [6]. If gravity has an
entropic origin, it is desirable to describe the Hawking radi-
ation (as a tunneling process), at least for Schwarzschild
black holes, as some form of entropy change in a holo-
graphic screen. We recall that back reaction from radiated
particle to the black hole makes it a system away from ther-
mal equilibrium, namely with temperature varying or being
not well defined. As a result it is proper to apply the gen-
eral formula (3) rather than (2) to the tunneling model. For
a screen which stores the same amount of information as the
black hole entropy and possesses the same temperature as
the black hole, the change on the right-hand side before and
after radiation can be computed as
(T S) = 1
8πG(M − ω) · 4πG(M − ω)
2
− 1
8πG M
· 4πG M2 = −ω
2
, (4)
which can be understood as a quantum of mass (energy) ω
that is discarded from the black hole up to a factor 1/2.1 The
left-hand side computes the required work by pulling a point-
like varying mass (energy) from the black hole. According
to Newton’s law of gravity, that is
2G(M−ω)∫
2G M
F(r)dr = −G(M −
r
2G )(
r
2G )
r
∣∣∣∣
2G(M−ω)
2G M
= −ω
2
,
(5)
which is consistent with the right-hand side. Therefore, the
gravitational force can again be identified with an entropic
force, even for a nonequilibrium system.
3 Holographic screen in the Parikh–Wilczek tunneling
model of Hawking radiation
Inspired by the reformulation of entropic gravity in the pre-
vious discussion, we are motivated to study a holographic
screen at arbitrary position but admitting a varying temper-
ature, in reaction to the displacement of a radiation particle.
For symmetry reasons, a spherical screen enclosing the black
1 This factor has been pointed out in the [7] as equipartition of energy
in the horizon degrees of freedom. In brief, each quantum of area fluc-
tuation contributes an energy T/2, that is, E = 12 nT for n = A = 4S.
One can derive (4) for loss of energy (mass) ω.
Before radiation
After radiation
Black holeHolographic screen
Fig. 1 A schematic drawing of a particle radiated out of a black hole
and their holographic screen
hole seems to be a good choice. We denote the information
stored on a holographic screen associated to a black hole
with mass M as SM (r). The screen is located at some dis-
tance r from the center of the black hole. We may want to
have r > 2G M to avoid ambiguity of a screen inside the
black hole. Before proceeding with the general formula (3),
we would like to first point out some difficulties appearing in
the original formulation of the entropic gravity (2) in the tun-
neling model of black hole radiation. If we assume that the
entropy stored on the screen follows a simple coarse grained
relation as shown in Fig. 1, following the neutron–earth sys-
tem in [2]:
SM (r + r) = Sω(r + r) + SM−ω(r), (6)
where r is the distance between radiated mass and the
screen. For an infinitesimal displacement, we have SM (r +
r)  SM (r) + S. If the entropy on the screen were addi-
tive, that is,
SM−ω(r) = SM (r) − Sω(r), (7)
one obtains
SM  Sω(r + r) − Sω(r). (8)
Since SM ∝ r for its entropic origin, we come to the
same conclusion: that the translation operator is no longer
Hermitian and disobeying quantum mechanics. In [4], a gen-
eralized entropy formula by Tsallis [8] was called upon to
restore the unitarity of translation, where an additional entan-
gled term was essential. In our radiation–black hole sys-
tem, it was known that the Parikh–Wilczek model predicted
a non-thermal radiation spectrum and correlation between
two emissions ω1, ω2 was computed as 8πω1ω2, known as
123
Eur. Phys. J. C (2014) 74:2912 Page 3 of 4 2912
mutual information [6]. The entanglement is necessary for
the conservation of entropy. Therefore, it is possible to cor-
rect the additivity condition (7) to include a term Sent > 0
encoding entanglement between black hole and radiation:
SM−ω(r) = SM (r) − Sω(r) − Sent(r), (9)
such that entropy is conserved before and after radiation.2
Notice that the relative sign for the last term is differ-
ent from the usual entanglement between two subsystems.
This difference in fact makes the situation worse since now
SM  Sω(r +r)−Sω(r)−Sent(r). For a translation to be
unitary, say Sω(r +r) = Sω(r), one has SM < 0, imply-
ing a violation of the second law of thermodynamics, which
reflects the fact that the black hole cannot evaporate clas-
sically. If the Hawking radiation were thermal, one might
expect the increased entropy during thermal process to be
large enough to compensate for the reduced entropy due to
splitting of the radiated particle from the black hole. How-
ever, in Parikh–Wilczek’s tunneling model, one has no such
additional source for the entropy since the process is non-
thermal.
Now we turn to an application of the general entropic
formula (3). Because the temperature may vary for screens at
different locations, we assume that the coarse grained relation
(6) is modified as
T (r + r)SM (r + r) = T (r + r)Sω(r + r)
+ T (r)SM−ω(r). (10)
If we assume a linear relation between screen temperature
and small displacement, that is, T (r + r)  T (r) + T ,
one obtains
SM  [Sω(r +r)− Sω(r)]+
[
T
T
Sω(r + r) − Sent
]
.
(11)
For the translation of the radiated particle being unitary,
Sω(r) = Sω is a constant. Vanishing of SM for r = 0
suggests Sent(r) = 0. According to (9), the entropy on the
screen is additive. The temperature dependence can also be
obtained by integration:
SM (r) − SM (r0) = Sω ln T (r)T (r0) , (12)
for some reference position r0. The movement of the radi-
ated particle with respect to the screen does not change corre-
2 Classically, a black hole is forbidden to split because SM (r) >
SM−ω(r)+ Sω(r). In the Parikh–Wilczek model, the entanglement pro-
vides an additional source of entropy to conserve the entropy during the
evaporation process.
sponding to the entropy Sω. Instead, it feels different tempera-
tures from screens at different locations. If one recalls the sce-
nario of a massive particle tunneling out of black hole hori-
zon: the radiated particle experiences less gravitational pull
while it moves away from the black hole, as if it feels colder
from holographic screen. The screen temperature reflects the
acceleration or deceleration as what happens in the Unruh
effect.
4 Discussion
In this letter, with the explicit example of Parikh–Wilczek’s
tunneling model of Hawking radiation, we illustrate that
Verlinde’s holographic screen could have violated unitarity
of quantum mechanics if it were additive in entropy. This
sickness cannot be simply cured by the assumption that the
entropy on screen is entangled. Motivated by a general for-
mula of entropic gravity, one shows that the entropy on the
screen can still be additive if the relation between screen
entropy and temperature is given by (12). Some comments
are in order.
Firstly, the Parikh–Wilczek is a semiclassical model,
which could receive a further quantum correction. For
instance, a one-loop correction to the surface gravity was
considered in [9]. As a result, the Bekenstein–Hawking area
law receives a logarithmic correction, denoted Sα(M) =
4π M2 − 4πα ln (1 + M2
α
), and the Hawking temperature is
modified as TH = M2+α8π M3 for a large black hole, where the
coefficient α is related to the trace anomaly. One can show
that the relation of the generalized entropic force (3) is no
longer satisfied, since
(TH Sα) = −ω2 +α
[
3
2M2
− 3α + M
2
2M4
ln
(
1 + M
2
α
)]
ω.
(13)
Nevertheless, this might be fixed from two different view-
points. One way is to modify Newton’s gravitational force
by a quantum correction up to O(α2), such that ∫ Fα(r)dr
agrees with the above result. The other way is to rederive the
Hawking temperature, instead of that obtained in the small
ω limit. The proper definition of the Hawking temperature
Tα(M) is given by
Tα(M − ω)Sα(M − ω) − Tα(M)Sα(M) = −ω2 . (14)
It seems unlikely to have analytic forms for loop corrected
Fα(r) and Tα(M), but at least in principle one can get the
order expansion for small α. Once the relation (3) can be
fixed, our conclusion for the temperature-varying screen will
still be valid, at least perturbatively in terms of α.
123
2912 Page 4 of 4 Eur. Phys. J. C (2014) 74:2912
Secondly, although Verlinde’s adoption of entropic force
was meant to tame the wild gravity beast, it is natural to ask
how to incorporate the other forces in his entropic formalism.
This generalization has been considered for the electromag-
netic force, by inclusion of the chemical potential in the first
law of thermodynamics [10]. In particular, for the tunneling
model of Reissner–Nordström black hole radiation [11], one
expects that the relation (3) is generalized as
Fr = (T S − AQ), (15)
where the gauge potential and total black hole charges are
identified as the chemical potential and number density on
the screen, respectively. On the left-hand side, one has to
include the electromagnetic force for a consistent result. If
there exists a holographic screen for charged black holes, it
must also carry degrees of freedom for charges. The black
hole censorship constraint Q ≤ M would translate into an
upper bound for the charge density on the screen. The break
down of the screen for excess charge would correspond to
a possible naked singularity or closed time-like curve. The
recently found charge–mass ratio bound for emission from
the RN black hole [12] seems to support this picture by stat-
ing that a part of screen, which corresponds to the radiated
particle, also has an upper bound for the charge density.
Acknowledgments WYW is supported in part by the Taiwan’s
National Science Council under Grant NO. 102-2112-M-033-003-MY4
and the National Center for Theoretical Science.
Open Access This article is distributed under the terms of the Creative
Commons Attribution License which permits any use, distribution, and
reproduction in any medium, provided the original author(s) and the
source are credited.
Funded by SCOAP3 / License Version CC BY 4.0.
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In this letter, we use the Parikh–Wilczek tunneling model of Hawking radiation to illustrate that a reformulation of Verlinde’s entropic gravity is needed to derive Newton’s law for a temperature-varying screen, required by the conservation of energy. Furthermore, the entropy stored in the holographic screen is shown to be additive and its temperature dependence can be obtained

Thoughts on entropic gravity in the Parikh-Wilczek tunneling model of Hawking radiation

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