Table Of ContentEntropic corrected Newton’s law of gravitation and the Loop Quantum Black
Hole gravitational atom
R.G.L. Arag˜aoa, C.A.S.Silvaa
aInstituto Federal de Educac¸˜ao Ciˆencia e Tecnologia da Para´ıba (IFPB),
Campus Campina Grande - Rua Tranquilino Coelho Lemos, 671, Jardim Dinam´erica I.
Abstract
One proposal by Verlinde [1] is that gravity is not a fundamental, but an entropic force. In this way, Verlinde
6 has provide us with a way to derive the Newton’s law of gravitation from the Bekenstein-Hawking entropy-area
1 formula. On the other hand, since it has been demonstrated that this formula is susceptible to quantum gravity
0
corrections, one may hope that these corrections could be inherited by the Newton’s law. In this way, the entropic
2
interpretation of Newton’s law could be a prolific way in order to get verifiable or falsifiable quantum corrections
n
to ordinary gravity in an observationally accessible regimes. Loop quantum gravity is a theory that provide a way
a
to approach the quantum properties of spacetime. From this theory, emerges a quantum corrected semiclassical
J
black hole solution called loop quantum black holes or self-dual black holes. Among the interesting features of
6
1 loop quantum black holes is the fact that they give rise to a modified entropy-area relation where quantum gravity
correctionsarepresent. Inthiswork,weobtainthequantumcorrectedNewton’slawfromtheentropy-arearelation
] given by loop quantum black holes. In order to relate our results with the recent experimental activity, we consider
c
q the quantum mechanical properties of a huge gravitational atom consisting in a light neutral elementary particle in
- the presence of a loop quantum black hole.
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Email addresses: [email protected](R.G.L.Arag˜ao),[email protected](C.A.S.Silva)
Preprint submitted to Elsevier
1. Introduction formalism, have been used in several contexts including
cosmological ones [11].
Sincetherisingofblackholethermodynamics, inthe On the other hand, by using the measurement result
seventies, through the Hawking demonstration that all ofquantumstatesofultra-coldneutronundertheEarth’s
black holes emit blackbody radiation [2], investigations gravity, Kobakhidze presented an argument in opposi-
about these objects break up the limits of astrophysics. tion to Verlinde’s proposal [12]. The problem pointed
In fact, black holes have been put in the heart of the de- byKobakhidzecomesfromthefactthattheentropyfor-
bateofthemostfascinatingissuesintheoreticalphysics. mula defined by Verlinde formalism, in principle, leads
Among these issues, the search for a better understand- toaquantumneutronmixedstate. However,itdisagrees
ing of the quantum nature of gravity, since the quantum with the results from the ultra-cold neutron experiment.
behavior of spacetime must be revealed within the pres- Kobakhidze’s criticism have been questioned in [13] and
ence of a black hole strong gravitational field. one resolution was suggested by Abreu et al [14]. This
Among the most important lessons from black hole resolution can be found out by abandoning the implicit
thermodynamics,arisestheBekenstein-Hawkingformula assumption in [12] that the entropy on the holographic
which establishes that, in a different way from other screen is additive.
usual thermodynamical systems, the entropy of a black Ontheotherhand,itisalsoknownthat,inothercon-
holeisnotgivenasproportionaltoitsvolume, buttoits texts than Einstein’s gravity, the area formula of black
horizon area: S = kBc3A/4(cid:125)G. A deep intersection be- holeentropymaynotbeheld. Forexample,whenhigher
tweengravity,quantummechanics,andthermodynamics ordercurvaturetermappearsinsomegravitytheory,the
couldbecontainedinBekenstein-Hawkingformula,since area formula has to be modified [15]. Modifications to
it gives us one of the few situations in physics where the Bekenstein-Hawking formula also appear when quantum
Newton’sgravitationalconstantGandthespeedoflight gravityeffectsareincluded. Forexample,whenaGener-
c meet the Planck constant (cid:125) and the Boltzmann con- alizedUncertaintyPrinciple(GUP)istakenintoaccount
stant kB. In fact, it has been shown by String theory [16, 17]. In this way, it was investigated modifications
and Loop Quantum Gravity that the black-hole thermo- of the entropic force due to corrections imposed on the
dynamics must have its origin in the atomic structure area law by quantum effects and extra dimensions [18].
of the spacetime [3, 4, 5]. Moreover, in [6, 7, 8] it has QuantumgravitycorrectionstoBekenstein-Hawkingfor-
been argued that a topology change process due to the mulaalsoappearinthecontextofLoopQuantumGrav-
dynamics of the quantum spacetime could be the origin ity. The most popular form to these corrections ap-
ofblackholeentropyandtheGeneralizedSecondLawof pear as logarithmic corrections which arises due to ther-
black hole thermodynamics. mal equilibrium fluctuations and quantum fluctuations
In1995,asurprisingresultbyJacobsonhasdeepened [19, 20, 21, 22].
the significance of the Bekenstein-Hawking formula. As- AnotherwaytogetquantumcorrectionstoBekenstein-
sumingtheproportionalitybetweenentropyandhorizon Hawking formula, which we shall follow in this work,
area, Jacobson derived the Einstein’s field equations by arises in the context of loop quantum black holes [23,
using the fundamental Clausius relation [9]. The pro- 24, 25, 26, 27, 28]. A loop black hole, also called self-
cedure behind this result is to require that the Clau- dual black hole, consists in a quantum gravity corrected
sius relation, δQ = TdS, associating heat, temperature Schwarzschild black hole that appears from a simplified
and entropy, holds for all the local Rindler causal hori- modelofLoopQuantumGravity. Oneofthemostinter-
zon through each spacetime point, with δQ and T inter- estingresultsoftheloopblackholescenarioistheresolu-
preted, respectively, as the energy flux and Unruh tem- tionoftheblackholesingularitybytheself-dualityprop-
perature seen by an accelerated observer just inside the erty. This property guarantees that an asymptotic flat
horizon. In this way, the spacetime could be viewed as region corresponding to a Planck-sized wormhole arises
a kind of gas whose entropy is given by the Bekenstein- in the place of the black hole singularity. The wormhole
Hawkingformula,andtheEinstein’sfieldequationasan throatisdescribedbytheKantowski-Sachssolution. The
equation of state describing this gas. thermodynamicalpropertiesofloopblackholeshasbeen
FollowingJacobson’sresults,severalauthorshavead- addressedinthereferences[24,25,26,27,28]. Moreover,
dressed the issue of the relation of Einstein’s equations in the reference [29], the thermodynamical properties of
andthermodynamics(Forareviewandavoluminouslist loop quantum black holes were obtained by the use of a
of references see [10]). More recently, Verlinde [1] con- tunnelingmethodwiththeintroductionofback-reaction
jectured that gravity is a non fundamental interaction effects. On the other hand, in the reference [30], the
butwouldbeexplainedasanentropicforce. Inthisway, tunnelingformalismhasbeenappliedinordertoinclude
the second law of Newton is obtained when one tie up correctionsduetoaGeneralizedUncertaintyPrincipleto
the entropic force with the Unruh temperature. On the loop quantum black hole’s thermodynamics. Among the
otherhand,Newtonslawofgravitationisobtainedwhen results related with the thermodynamics of loop black
associatingtheseargumentswiththeholographicprinci- holes,wehaveaquantumcorrectedBekenstein-Hawking
ple and using the equipartition law of energy. Verlinde’s
2
formulafortheentropyofablackholeinwhichquantum Inthisway,twohorizonsappearsintheLQBH’sscenario
gravity ingredients have been included. - an event horizon at r and a Cauchy horizon at r .
+ −
Experimentalissuesrelatedwithloopquantumblack Furthermore, we have that
holes have also been addressed in the literature. In this
way, gravitational lenses effects due to this kind of black
√
holes have been investigated in [31]. On the other hand, r∗ = r+r− =2mP . (6)
loopquantumblackhole’squasinormalmodeshavebeen
where P is the polymeric function given by
calculated in [32], [33] and [34]. In the last, axial gravi-
tational perturbations have been taken into account. √
1+(cid:15)2−1
In the present work, we shall address how the New- P = √ , (7)
1+(cid:15)2+1
ton’slawofgravitationwouldbemodifiedinthepresence
ofloopquantumblackholes,whenquantumpropertiesof
and
spacetime are taken into account. Moreover, motivated
by the results of recent experiments, we shall consider A
a = min , (8)
the quantum mechanical system of a gravitational atom 0 8π
consisting in a light neutral elementary particle in the
where A is the minimal value of area in LQG.
min
presenceofaloopquantumblackhole. Inparticular, we In the metric (1), since g is not just r2, r is only
θθ
apply the Bohr Somerfeld formalism to this system, by
asymptotically the usual radial coordinate. From the
the use of the modified Newtons potential, in order to
form of the function H(r), one obtains a more physical
obtain its energy levels.
radial coordinate given by
(cid:114)
2. Loop quantum black holes R= r2+ a20 . (9)
r2
Loop quantum black holes (LQBHs) appeared at the
In this way, the proper circumferential distance is mea-
firsttimefromasimplifiedmodelofLoopQuantumGrav-
sured by R.
ity(LQG) [23]. The LQBH’s scenario is described by a
Moreover, the parameter m in the solution is related
quantumgravitationallycorrectedSchwarzschildmetric,
to the ADM mass M by
and can be written in the form
M =m(1+P)2 . (10)
The equation (9) reveals important aspects of the
ds2 =−G(r)dt2+F−1(r)dr2+H(r)dΩ2 (1) LQBH’s internal structure. From this expression, we
have that, as r decreases from ∞ to 0, R first decreases
√ √
with from ∞ to 2a at r = a and then increases again to
0 0
∞. The value of R associated with the event horizon is
given by
dΩ2 =dθ2+sin2θdφ2 , (2)
(cid:114)
where,intheequation(1),themetricfunctionsaregiven R =(cid:112)H(r )= (2m)2+(cid:16) a0 (cid:17)2 . (11)
EH + 2m
by
A peculiar feature in LQBH’s scenario is the prop-
erty of self-duality. This property says that if one in-
G(r)= (r−r+)(r−r−)(r+r∗)2 , (3) troduces the new coordinates r˜= a0/r and t˜= tr∗2/a0,
r4+a20 withr˜± =a0/r∓ themetricpres√ervesitsform. Thedual
radius is given by r = r˜ = a and corresponds to
dual 0
(r−r )(r−r )r4 the minimal possible surface element. Moreover, since
F(r)= + − , (4) √
(r+r )2(r4+a2) the equation (9) can be written as R = r2+r˜2, it
∗ 0
is clear that, in the LQBH’s scenario, we have another
and
asymptotically flat Schwazschild region in the place of
a2 the singularity in the limit r → 0. This new region cor-
H(r)=r2+ 0 , (5)
responds to a Planck-sized wormhole. Figure (1) shows
r2
the Carter-Penrose diagram for the LQBH.
where
r =2m ; r =2mP2 .
+ −
3
Figure 1: Carter - Penrose diagram for the LQBH metric. The
diagramshowstwoasymptoticflatregions,onelocalizedatinfinity
and the other near the origin, which can not be reached by an
observerinafinitetime. Figure 2: The LQBH temperature solid line in contrast with the
Schwarzschildblackholetemperaturedashedline.
The derivation of the black hole’s thermodynamical
propertiesfromthemetric(1)proceedsintheusualway. situations that arise from LQBH structure [27]. In the
The Bekenstein-Hawking temperature T can be ob- firstofthesepossibilities,theeventhorizonstaysoutside
BH
tained by the calculation of the surface gravity κ by the wormhole throat. In order to have this situation,
√
T =κ/2π, with the condition r > a is necessary, which implies that
BH √ + 0
m > a /2. In this case, the bounce takes place after
0
theblackholeformsforasuper-PlanckianLQBHandthe
1
κ2 =−gµνg ∇ χρ∇ χσ =− gµνg Γρ Γσ , (12) exterior, is similar, in a qualitative way, to that would
ρσ µ ν 2 ρσ µ0 ν0
beproducedbyaSchwarzschildblackholewiththesame
where χµ = (1,0,0,0) is a timelike Killing vector and mass. In this way, outside the event horizon, the LQBH
Γµ are the connections coefficients. scenario is different from the Schwarzschild’s one only
σρ
By connecting with the metric, one obtains that the by Planck-scale corrections. On the other hand, in the
LQBH temperature is given by sub-Planckian regime, we have a more instigating situa-
tion. In this case, the event horizon becomes the other
(2m)3(1−P2) side of the wormhole throat. Moreover, the deviations
T = . (13)
H 4π[(2m)4+a2] from the Schwarzschild metric are very expressive and
0
the bounce takes place before the event horizon forms.
It is easy to see that one can recover the usual Hawk-
Consequently,evenlargeeventhorizons(whichitwillbe
ing temperature in the limit of large masses. However, √
form(cid:28)m )itwillbeinvisibletoobserversatr > a .
differently from the Hawking case, the temperature (13) P 0
The thermodynamics properties of LQBHs has been
goestozerofor m→0, ashavebeenshowninthefigure
alsoobtainedthroughtheHamilton-Jacobiversionofthe
(2). In this point, we remind that the black holes ADM
mass M =m(1+P)2 ≈m, since P (cid:28)1. tunneling formalism [29]. By the use of this formalism,
back-reaction effects could be included. Moreover, ex-
Theblackhole’sentropycanbefoundoutbymaking
(cid:82) tensionsoftheLQBHsolutiontoscenarioswherecharge
useofthethermodynamicalrelationS = dm/T(m).
BH
and angular momentum are preset can be found in [35].
4π(1+P)2(cid:104)16m4−a2(cid:105) The issue of information loss has been also addressed in
S = 0 . (14)
(1−P2) 16m2 the context of loop black holes. In this case, it has been
pointed that the problem of information loss by black
Moreover, the black hole entropy canbe expressedin
holes could be relieved in this framework [26, 29, 36].
terms of its area [28]
This result may be related with the absence of a singu-
(cid:112) larity in the loop black hole interior, and consists in a
A2−A2 (1+P)
S =± min , (15) forceful result in benefit of this approach.
4 (1−P)
Another interesting result in the realm of LQBH is
where we have set the possible additional constant to the fact that, as have been demonstrated in [37], it can
√
zero. Sispositiveform> a0/2andnegativeotherwise. been sees as the building blocks of Loop Quantum Cos-
The double possibility in the signal of the loop black mology(LQC),inthesensethat,startingfromtheLQBH
hole entropy is related with the two possible physical
4
entropy expression, LQC equations can been obtained of bits on the screen is proportional to the horizon en-
throughtheuseofJacobsonformalismtoobtaintheEin- tropy, from the equation (15), we shall assume that this
stein’s gravitational equations. relation is given by
In the next sections, following the formalism devel-
(cid:112)
oped by Verlinde [1], we will derive the quantum cor- (1+P) A2−A2
N = min (20)
rected Newton’s from the modified entropy-area relation (1−P) L2
P
given by the equation (15).
In this way, from the equation above together with
(16), (19) and E =Mc2, we shall have
3. Quantum corrected Newton’s law from loop
quantum black holes
GMm(1+P) 1
F =− × . (21)
Recently,Verlindeconjecturedthatgravityisnotfun- R2 (1−P) (cid:112)1−A2 /16π2R4
min
damental but can be explained as an entropic force. In
(cid:82)
this section, following the Verlind’s entropic force ap- Moreover,forthegravitationalpotentialV(r)=− F(R)dR,
proach to gravity, we shall derive quantum corrected we shall have
Newton’s law of gravitation from LQBHs entropy-area
relation (15). GMm(1+P)(cid:18) A2 A4 (cid:19)
We have that in thermodynamics, if the number of V(r)=− 1− min − min +...
R (1−P) 160π2R5 18432π4R9
statesdependsonposition∆x,entropicforceF arisesas
thermodynamicalconjugateof∆x. Inthiscase, thefirst
Inthisway,correctionstoNewton’sgravitationallaw
law of thermodynamics can be written as
can be obtained from LQBH entropy-area relation. As
we can see, the deviations on the Newton’s law depend
F∆x=T∆S (16)
onthevalueofminimalareaA inLQG,aswellason
min
Based on the Bekenstein’s entropy bound, Verlinde the polymeric parameter P. In this way, the corrections
postulatedthatwhenatestparticlemovesapproachinga found out are important in the case of submilimeter dis-
holographic screen, the change of entropy on this screen tances,eventhoughitcouldberealizedinthecontextof
is proportional to the mass m of the particle, and the largedistancesthroughthedependenceontheparameter
distance ∆x between the test particle and the screen P.
mc
∆S =2πk ∆x (17)
B (cid:125) 4. The loop quantum black hole atom
to derive the entropic force hypothesis, (17) should hold In the seventies, Hawking introduced the possibil-
atleastwhen∆xissmallerthanorcomparablewiththe ity that a free charged particle could be capture by a
Compton wave-length of the particle. primordial charged black hole forming neutral and non-
The temperature that appears in (16) can be under- relativistic ultra-heavy black hole atoms [38]. After, the
stood in two ways: one can relate temperature and ac- term gravitational atom was coined by V. V. Flambaum
celeration using Unruh’s rule andJ.C.Berengutin2001[39]forgravitationallybound
neutral black hole and a charged particle.
1 (cid:125)a
k T = , (18) An interesting fact about gravitational atoms is that
B 2π c
they have been pointed as an important constituent of
or relate temperature, energy and the number of used
darkmatter. Infact,primordialblackholeremnantsleft
degrees of freedom using equipartition rule
after the Hawking evaporation have been considered as
asourceofdarkmatterbyseveralauthorsformorethan
1
E = 2NkBT. (19) two decades [40, 41, 42, 43, 44, 45, 46, 47, 48, 49] (for
a review see [50, 51, 52]). However, a central question
It is necessary to point that the temperature T in equa-
is whether some remnants could leave after the Hawking
tions (18) and (19) have different meaning. In the first
evaporation, forming a stable nucleus for the gravita-
equation, the temperature is defined in the bulk. How-
tional atom. In other words, in order to have a gravi-
ever, in the second, the temperature is defined on the
tational atom system as a suitable candidate to describe
holographic screen. To admit these two temperatures to
darkmatter,itwouldbenecessarythat,atsomepointof
be equal is an further supposition in Verlinde’s paper.
its evolution, the black hole nucleus establish a thermal
From the equations (16), (17) and (18) one obtains
stable equilibrium with its neighborhood.
the second Newtons’s law F = ma. In order to obtain
In the Schwarzschild scenario, this kind of situation
the Newton’s law of gravitation, one must have a way to
is possible for a black hole to be in equilibrium with the
relate the number of bits on the holographic screen with CMB radiation, for a black hole mass of 4.50×1022kg.
the black hole horizon area. Assuming that the number
However, this equilibrium is not a stable one because
5
for a Schwarzschild black hole the temperature always
increases as its mass decreases and vice versa (see the (cid:112)(P +1)(1−P)(cid:125)2j2
dashed line in the fig. (2)). R =
j m2GM(P +1)
On the other hand, new phenomenon emerges in the
(cid:112)
LQBH scenario. From equation (13), all light enough (P +1)(1−P)m6G3M3(P +1)
− A2
LQBHswouldradiateuntiltheirtemperaturecoolsuntil 2(cid:125)6j6(P −1)2 min
the point it would be in thermal equilibrium with the + ··· (25)
CMB. In fact a stable thermal equilibrium occurs for a
black hole mass given by m ≈ 10−19kg. Based on where the first term corresponds to the usual gravita-
stable
this feature of LQBH, Modesto et al have yet pointed to tional atom radius, unless the P parameter factors.
the possibility that these objects could be an important TheenergylevelsEj oftheLQBHgravitationalatom
component of dark matter [24]. In this way, one could are obtained from the expressions (22), (23) and (25),
think about the possibility of gravitational atoms where
a LQBH could appear as the atomic nucleus. 1
E = mv2+V =
In order to give a first glance on these kind of sys- j 2
tem, let us use the expression for the gravitational force m3G2M2(P +1)3/2 m11G6M6(P +1)7/2
− + A
between a LQBH and a neutral particle orbiting it given 2(cid:125)2j2(1−P)3/2 64(cid:125)10j10π2(P −1)7/2 min
by the equation (21):
m11G6M6(P +1)7/2
+ A2 ×
20480(cid:125)18j18π4(P −1)11/2 min
GMm(1+P) 1
F = × [15m8G4M4(P +1)2−(5120π2−128)(cid:125)8j8π2(P −1)2]
R2 (1−P) (cid:112)1−A2 /16π2R4
min + ···
mv2
= , (22) (26)
R
where the first therm corresponds to the usual expres-
where v is the particle velocity in the orbit.
sion to the gravitational atom energy levels (unless the
In this way, we shall have:
dependence on the polymeric parameter), which can be
obtained in the limit where the quantum gravity correc-
(cid:20) (cid:21)1 tions goes to zero.
v = (1+P)GM 2 ×(cid:0)1−A2 /16π2R4(cid:1)−14 (23)
(1−P) R min
5. Conclusions and Remarks
Using the Bohr-Somerfeld quantization method,
mvR=j(cid:125), we shall get the following equation We have derived quantum corrected Newton’s gravi-
tation law from the LQBH’s entropy-area relation using
theVerlindeentropicforceinterpretationtogravity. Our
(1−P) (j(cid:125))4 (1−P)(j(cid:125))4A2 results points to some quantum deviation from classical
R6− R4+ min =0,
(1+P)(GMm2)2 (1+P)(GMm2)2 Newton’s law that must have a important rule in sub-
(24) millimeter distances where Newton’s gravitation theory
whose only real solution is has not been tested yet.
Due to its self-duality property, LQBHs can have a
masslowerthanPlanckmass. Particularly,form ≈
(cid:104) (cid:125)4j4(P −1) (cid:16) stable
R = + (cid:125)4j4A (P −1)× 10−19kg, a LQBH would assume a stable thermal equi-
j 3m4G2M2(P +1) min librium with the CMB, which makes possible that this
(cid:112)
27m8A2 G4M4(P +1)2−4h8j8(P −1)2 kind of black holes can be seen as a good candidate for
min +
2(33/2)m8G4M4(P +1)2 dark matter. In this way, impelled by the current exper-
imental activity, we investigate the energy spectrum of
(cid:125)4j4(1−P)[2(cid:125)8j8(1−P)2−27m8A2 G4M4(P +1)2](cid:17)1/3
min a huge gravitational atom composed by a neutral parti-
54m12G6M6(P +1)3
cle orbiting a LQBH. As have been demonstrated, this
(cid:125)12j12(P −1)3 frequency depends on the quantum gravitational correc-
+ ×
9m8G4M4(P +1)2 tions inherited from the LQBH metric.
(cid:112)
(cid:16) 27m8A2 G4M4(P +1)2−4h8j8(P −1)2
min
2(33/2)m8G4M4(P +1)2 6. Acknowledgements
−2(cid:125)8j8(P −1)2−27m8A2minG4M4(P +1)2(cid:17)1/3(cid:105)1/2 TheauthorswouldliketothanktoConselhoNacional
54m12G6M6(P +1)3 deDesenvolvimentoCientficoeTecnolgico-CNPQ/Brazil
for the financial support.
and can be expanded as
.
6
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