a lower bound on the bekenstein hawking temperature of black holes

The quantum buoyancy effect and the lower bound on the black-hole temperature Thus far, we have analyzed the gedanken experiment at the classical level.Itisimportanttoemphasize,however,t

Contents lists available atScienceDirect

www.elsevier.com/locate/physletb

Shahar Hoda,b, ∗

aThe Ruppin Academic Center, Emeq Hefer 40250, Israel

bThe Hadassah Institute, Jerusalem 91010, Israel

Article history:

Received 14 April 2016

Received in revised form 9 June 2016

Accepted 9 June 2016

Available online 15 June 2016

Editor: M Cvetiˇc

We present evidence for the existence of a quantum lower bound on the Bekenstein–Hawking temperatureofblack holes.The suggestedboundis supportedbyagedankenexperiment inwhicha chargedparticleisdroppedintoaKerrblackhole.Itisprovedthatthetemperatureofthe finalKerr– Newmanblack-holeconfigurationisboundedfrombelowbytherelationTBH×rH> ( h¯ rH)2,whererHis thehorizonradiusoftheblackhole

©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3

1 Introduction

It is well known [1,2] that, for mundane physical systems of

spatialsize R,thethermodynamic(continuum)descriptionbreaks

downinthelow-temperatureregime T∼ ¯h/R (we shalluse

grav-itationalunits in which G=c=kB=1) In particular, these low

temperature systems are characterized by thermal fluctuations

whosewavelengthsλthermal∼ ¯h/T areoforder R,thespatial size

ofthesystem,inwhichcasetheunderlyingquantum(discrete)

na-tureofthesystemcan nolongerbeignored.Hence,formundane

physicalsystemsofspatial size R,thephysicalnotion of

temper-atureisrestrictedtothehigh-temperaturethermodynamicregime

[1,2]

Interestingly,blackholesareknowntohaveawell-defined

no-tion of temperature in the complementary regime of low

tem-peratures In particular, the Bekenstein–Hawking temperature of

genericKerr–Newmanblackholesisgivenby[3,4]

TBH= ¯h( +−r−)

where

aretheradiioftheblack-hole(outerandinner)horizons(here M,

J≡Ma, and Q are respectively the mass, angular momentum,

* Correspondence to: The Ruppin Academic Center, Emeq Hefer 40250, Israel.

E-mail address:shaharhod@gmail.com

and electric charge of the Kerr–Newman black hole) The rela-tion (2) implies that near-extremal black holes in the regime

(+−r−)/r+1 arecharacterizedbythestronginequality[5]

Itisquiteremarkablethatblackholeshavea welldefined no-tionoftemperatureintheregime(4)oflowtemperatures,where mundane physical systemsare governed by finite-size (quantum) effects and no longer have a self-consistent thermodynamic de-scription

One naturally wonderswhetherblackholes canhave a physi-cally well-definednotionoftemperatureallthewaydownto the extremal(zero-temperature)limit TBH×r+/h¯ →0?Inorderto ad-dress this intriguing question, we shall analyze in this paper a gedankenexperimentwhichisdesignedtobringaKerr–Newman blackholeascloseaspossibletoitsextremallimit.Weshallshow belowthat theresultsofthisgedankenexperimentprovide com-pellingevidencethat theBekenstein–Hawkingtemperatureofthe black holes is bounded from below by the quantum inequality

TBH×r+ (¯h/r+)2

2 The gedanken experiment

Weconsiderasphericalbodyofproperradius R,restmass μ, andelectricchargeq whichisslowlyloweredtowardsaKerrblack

sym-metry axis of the black hole (we shall assume q>0 and a>0 withoutloss ofgenerality) Theblack-holespacetime isdescribed

bythelineelement[6,7]

ds2= − 

ρ2(dt−a sin2θdφ)2+ ρ2

dr

2+ ρ2dθ2

http://dx.doi.org/10.1016/j.physletb.2016.06.021

0370-2693/©2016 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) Funded by 3

ρ2



adt− (r2+a2)dφ 2

where ≡r2−2Mr+a2 and ρ2≡r2+a2cos2θ [here(t r, θ, φ)

aretheBoyer–Lindquistcoordinates]

Thetest-particleapproximation impliesthat theparameters of

thebodyarecharacterizedbythestronginequalities

Theserelations implythat theparticle whichis loweredintothe

blackholehasnegligibleself-gravity(thatis, μ /R1)andthatit

ismuchsmallerthanthegeometriclength-scalesetbythe

black-holehorizonradius.Inaddition,theweak(positive)energy

condi-tionimpliesthattheradiusofthechargedbodyisboundedfrom

belowbyitsclassicalradius[8–10]

R≥Rc≡ q2

Thisinequalityensuresthattheenergydensityinsidethespherical

chargedbodyispositive[11]

Theenergy[12]ofthechargedbodyinthenear-horizon

black-holespacetimeisgivenby[11,13]

E ( ) = μ



r02−2Mr0+a2

r2

0+a2 + Mq2

2( 2

wherer=r0 istheradialcoordinateofthebody’scenterofmass

intheblack-holespacetime.Thefirsttermonther.h.s of(8)

rep-resentsthe energy associated withthe rest mass μ of the body

red-shiftedbytheblack-holegravitationalfield [3,14].Thesecond

termonther.h.s of(8)representstheself-energyofthecharged

bodyinthecurvedblack-holespacetime[11,13,15,16]

The proper height l of the body’s center of mass above the

black-holehorizonisrelatedbytheintegralrelation[3]

l( 0) =

r0



r+



r2+a2

totheBoyer–Lindquistradialcoordinater0.Inthenear-horizonl

r+regiononefindstherelation

r0(l) −r+= (r+−r−)l

2

4α [1+O(l2/r2+) ] , (10)

where α ≡r2++a2 Taking cognizance of Eqs (8) and(10), one

finds

E (l) = (+−r−) μl+Mq2

2α · [1+O(l2/r+2) ] (11)

fortheenergyofthebodyinthenear-horizonlr+region

Supposenowthatthechargedobjectisslowlyloweredtowards

theblackholeuntilitscenterofmassliesaproperheightl0(with

l0≥R)abovetheblack-holehorizon.Theobjectisthenreleasedto

fallintotheblackhole.Theassimilationofthechargedbodybythe

blackholeproducesafinalKerr–Newmanblack-holeconfiguration

whosephysicalparameters(mass,charge,andangularmomentum)

aregivenby

M→Mnew=M+ E (l0) ;

a→anew=a[1− E (l0)/M+O( E2/M2) ] ;

Thechangeintheblack-holetemperaturecausedby the

assimila-tionof thecharged body canbe quantified by the dimensionless

physicalfunction

(a¯ ) ≡ TBH

where a¯ ≡a M is the dimensionless angular momentum of the blackhole[17]

Ourgoalistobringtheblackholeascloseaspossibletoits ex-tremal (zero-temperature)limit.Thus, wewouldliketominimize thevalueofthedimensionlessphysicalparameter Inparticular,

wewouldliketoexaminewhether (a¯ ),thedimensionlesschange

in the black-holetemperature, can be madenegative all theway downtotheextremala¯ →1 (zero-temperature,TBH→0)limit

Weshallhenceforthconsiderblackholesintheregime

¯

a≥



2√

in which case a minimization of the energy delivered to the blackholealso correspondstoa minimizationofthe Bekenstein– Hawking temperature of the final black-hole configuration [18] ThefactthattheenergyE(l0)ofthechargedparticleinthe black-holespacetimeisan increasingfunctionofthedroppingheightl0

[see Eq.(11)]impliesthat, inorder tominimize thephysical pa-rameter (a¯ )in theregime (14),one shouldrelease thebody to fall into the black holefrom a point whoseproper height above the black-hole horizon is assmall as possible We therefore face theimportantquestion:Howsmallcanthedropping heightl0 be made?

As pointed out by Bekenstein [3], the expression (11)for the energyofourchargedsphericalobjectintheblack-holespacetime

is onlyvalidin therestrictedregime l0≥R, whereevery partof the bodyis stilloutside thehorizon Thisfact implies, in particu-lar,thattheadiabatic(slow)descentofthechargedsphericalbody towards the blackhole muststop when its centerof massliesa properheightl0→R+abovethehorizon.Atthispointthebottom

of thebody isalmost swallowedby theblackhole andthebody [having aminimized(red-shifted) energyE(l0→R)] should then

be releasedtofallintotheblackhole[3].Inaddition, remember-ingthattheweak(positive)energyconditionsetsthelowerbound

(7)on theproperradius ofthechargedsphericalbody,one finds therelation[19]

lmin0 =Rmin= q2

for the optimaldropping point of the charged body [thatis, the dropping point forwhich theenergy deliveredto the blackhole, andthusalsothephysicalparameter (a¯ ),areminimized].

Substituting (15) into (11), one finds the remarkably simple (anduniversal[20])expression

Emin(a¯ ) = q2

fortheminimalenergydeliveredtotheblackholebythecharged body.TakingcognizanceofEqs.(2),(12),(13),and(16),onefinds theuniversalexpression[21,22]

min(a¯ ) = − q2

forthe smallestpossible(mostnegative)value ofthe dimension-less physicalparameter (a¯ ) whichquantifies thechange in the black-hole temperaturecausedbytheassimilation ofthecharged body[Notethattherelationq2=2μRr2

+forourcharged

spher-ical object [see Eqs (6) and (7)] implies |Tmin

BH | TBH Here

Tmin

BH denotes the most negative value which is physically al-lowedfor thechangeTBH intheblack-hole temperatureinour gedankenexperiment.].Interestingly,onefindsfrom(17)the char-acteristicinequality

min(a¯ ) <0, (18)

which is valid for all values a¯ ∈ [0,1) of the black-hole rotation

parameter The simple inequality (18) impliesthat, by absorbing

chargedparticles,theblackholecan approacharbitrarily closeto

theextremal(zero-temperature)TBH→0 limit

Itisimportanttoemphasizeagainthatthisconclusionisbased

on the assumption [3] that the charged body can be lowered

adiabatically (slowly) until its bottom almost touches the

black-hole horizon [23] In the next section we shall show, however,

thatThorne’s famous hoopconjecture[24] impliesthat,for

near-extremalblackholes,thechargedbodycannot belowered

adiabat-icallyallthewaydowntothehorizonoftheblackhole

3 The hoop conjecture and the lower bound on the black-hole

temperature

In the previous section we have seen that, by absorbing a

chargedparticle,ablackholecanapproacharbitrarilyclosetothe

extremal (zero-temperature) T B H →0 limit As we have

empha-sized above, this interesting conclusion rests on the assumption

that the charged body can be lowered slowly all the way down

to the horizonof the black hole [23] In the presentsection we

shall show, however, that Thorne’s famous hoop conjecture [24]

sets a lower bound on the minimal proper height lmin0 that the

chargedbodycan approachthe black-holehorizonwithoutbeing

absorbed,aboundwhichmaybestrongerthanthepreviously

as-sumedbound(15)

TheThornehoopconjecture[24]assertsthataphysicalsystem

oftotal mass (energy) M formsa blackhole ifits circumference

radius rc is equal to (or smaller than) the corresponding radius

rSch=2M oftheSchwarzschildblackhole.Itisworthemphasizing

thatthevalidityofthisversionofthehoopconjectureissupported

byseveralstudies[25].However,itisalsoimportanttoemphasize

thefactthat thereare knownspacetimesolutions oftheEinstein

fieldequationswhichprovideexplicitcounterexamplestothis

ver-sionofthehoopconjecture[26,27]

A weaker (and therefore a more robust) version of the hoop

conjecture for spacetimes with no angular momentum was

sug-gestedin [28,29] Here we would like to generalize this weaker

version ofthe hoopconjecture to thegeneric caseof spacetimes

whichpossessangularmomentumandelectriccharge.In

particu-lar,weconjecturethat:AphysicalsystemofmassM,angular

mo-mentum J ,andelectriccharge Q formsablackholeifits

circum-ference radius rc isequal to (or smaller than)the corresponding

Kerr–Newmanblack-hole radius rKN=M+ M2− (J/M)2−Q2

Thatis,weconjecturethat

rc≤M+ M2− (J/M)2−Q2 =⇒ Black-hole horizon exists.

(19)

Inthecontextofourgedankenexperiment,thisweakerversion

of the hoop conjecture implies that a new (and larger) horizon

isformed ifthe chargedbody reachesthe radial coordinater0=

rhoop,whererhoop( μ ,q)isdefinedbytheKerr–Newmanfunctional

relation[seeEq.(3)]

rhoop=M+ E (hoop)

+  [M+ E ( hoop) ]2− {J/ [M+ E ( hoop) ]}2− (Q+q)2.

(20)

Substituting(8)into(20),andassumingrhoop−r+r+−r−r+

[30],onefinds

rhoop−r+= 2β2μ2

fortheradiusofthenewhorizon(herewehaveusedthe approx-imatedrelationsa¯ 1 and α 2r2

+ fornear-extremalblackholes

witha M r+),where

β ≡1+



1− q2

8μ2· τ with τ ≡r+−r−

Substitutingtheradialcoordinate(21)intoEq.(10),onefinds

l( hoop) =4βμ

Taking cognizance of Eqs (15)and(23) one realizes that, in the regime

l( hoop) >Rmin= q2

a new (and larger) horizon is formed [31] before the spherical chargedbody [32] touchesthehorizonoftheoriginal blackhole Thus,intheregime(24),oneshouldtake[33]

inEq.(11)inordertominimizetheenergydeliveredtotheblack holeby the chargedbody[It is worth emphasizingagainthat, in theregime(14),aminimizationoftheenergy(11)whichis deliv-eredto theblack holealsocorresponds to aminimization of the dimensionlessphysical parameter which quantifies the change

in the black-hole temperature (see [18] and [22]).] This implies (here we use the approximated relations a¯ 1 and α 2r2

+ for

near-extremalblackholeswitha M r+)

Emin(a¯ ) =4βμ2+q2

forthesmallestpossibleenergydeliveredby thechargedparticle

totheblackholeintheregime(24).TakingcognizanceofEqs.(2),

(12),(13),and(26),onefindstherelation

min(a¯ ) =

8 μ2

2r2+

√

intheregime(24) Interestingly,one finds from(27) thatthe black-hole-charged-bodysystemischaracterizedbytheinequality

inthe regime (24).Note,in particular,that theinequality (24) is satisfied by near-extremalblack holes whosedimensionless tem-perature τ ischaracterizedbytherelation[seeEqs.(22)and(23)]

τ <8μ2

Taking cognizance ofEqs (28) and(29)one realizesthat, in our gedankenexperiment,theBekenstein–Hawkingtemperatureofthe blackholescannot beloweredbelowthecriticalvalue

TBHc ×r+= ¯h

where μandq arethepropermassandelectricchargeofthe ab-sorbedparticle,respectively(here wehaveusedtheapproximated relation TBH×r+ τh¯ /8 π for the near-extremal Kerr–Newman blackholeswitha¯ 1)

4 The quantum buoyancy effect and the lower bound on the

black-hole temperature

Thus far, we have analyzed the gedanken experiment at the

classical level.Itisimportanttoemphasize,however,thatthewell

knownquantum buoyancy effect[34] intheblack-holespacetime

shouldalsobetakenintoaccount inthepresentgedanken

exper-iment.Thisquantumbuoyancyeffectstemsfromthefactthatthe

slowly loweredobject interactswith the quantumthermal

atmo-sphereoftheblack-holespacetime[34,35]

Inparticular,asshownbyBekenstein[35],thequantum

buoy-ancy effectshifts the optimaldropping point of the object (that

is,thedroppingpointforwhichtheenergydeliveredtotheblack

holeisminimized)fromlmin0 =R [seeEq.(15)]toaslightlyhigher

pointwhoseproperradialdistancefromtheblack-holehorizonis

givenby[35]

wherethedimensionlessfactor isgivenby[35,36]



N

720π · ¯h

andN isthe effectivenumberofquantumradiationspecies[35]

Thequantumshift(increase) R [seeEq.(31)]intheradialproper

distance ofthe optimaldropping point results in a quantum

in-crease · (r+−r−) μR/ α [35]intheenergydeliveredtotheblack

hole.Taking into account thisquantumbuoyancy increase inthe

energy delivered to the black hole, one finds that the classical

expression (17) for the dimensionless function (a¯ ) acquires a

positive quantum correction term In particular,for near-extremal

black holes the quantum-mechanically corrected expression for

(a¯ ) is given by (here we use the approximated relations a¯ 1

and α 2r2

+fornear-extremalblackholeswitha M r+)[32]

min(a¯ →1)= − q2

4M2· 1−  · 8r+

r+−r−

Interestingly,one finds from(33)that the

black-hole-charged-bodysystemischaracterizedbytheinequality

intheregime

Therelations(34)and(35)implythat,duetothequantum

buoy-ancy effect [34,35], the Bekenstein–Hawking temperature of the

black holes cannot be loweredbelow the critical value (here we

use the approximated relation TBH×r+ τh¯ /8 π for the

near-extremalKerr–Newmanblackholeswitha¯ 1)

TBHc ×r+=  · ¯h

Interestingly, the quantum lower bound (36) becomes stronger

than the classical lower bound (30) in the regime > μ2/ 2,

which corresponds to charged objects [32] in the regime q>

(360 π /N¯h)12μ2

5 Summary and discussion

We have analyzed a gedanken experimentin which a

spher-ical charged particle is lowered into a Kerr black hole It was

shownthat if the chargedparticlecan be loweredslowly all the

waydownto thehorizonoftheblackhole,then theBekenstein– Hawkingtemperatureofthefinalblack-holeconfigurationcan ap-proacharbitrarilyclosetotheextremal(zero-temperature)TBH→

0 limit

However, we have shown that Thorne’s famous hoop conjec-ture [24] [and also its weaker (and more robust) generalization

(19)]impliesthat,fornear-extremalblackholesintheregime(29),

a new (and larger) horizon isalready formed before the charged particle touchesthe horizonof theoriginal black hole.The hoop conjecturethereforeimpliesthat,inourgedankenexperiment,the temperature ofthe final [37] black-hole configurationcannot

ap-proacharbitrarilyclosetozero[38].Inparticular,wehaveproved thattheBekenstein–Hawkingtemperatureoftheblackholesisan

irreducible quantity inthenear-extremalregime TBH<TBHc deter-minedbythecriticaltemperature(30)

It is worth emphasizing that we have provided in this paper onlyonespecificexample,not ageneralproof,tothefactthatthe black-hole temperature cannot approach arbitrarily closeto zero Nevertheless, this intriguing conclusion of our gedanken experi-ment [39] makes it tempting to conjecture that the Bekenstein– Hawkingtemperatureofblackholesisboundedfrombelowbythe simpleuniversalrelation[seeEq.(30)][40–42]

TBH×r+  ¯h

r+

2

We believe that itwouldbe highly importantto test thegeneral validity of the conjectured lower bound (37) on the Bekenstein– Hawkingtemperatureoftheblackholes

Acknowledgements

This research is supported by theCarmel Science Foundation

I wouldliketothankYaelOren,ArbelM.Ongo,AyeletB.Lata,and AlonaB.Teaforhelpfuldiscussions

References

[1] J.D Bekenstein, Phys Rev D 23 (1981) 287.

[2] L.D Landau, E.M Lifshitz, Statistical Physics, Addison–Wesley, Reading, Mass, 1969.

[3] J.D Bekenstein, Phys Rev D 7 (1973) 2333.

[4] S.W Hawking, Commun Math Phys 43 (1975) 199.

[5] It is worth emphasizing that there are several ways to define the effective size

of a black-hole system In particular, the black-hole spacetime is characterized

by an infinite throat in the extremal limit This infinite throat may suggest that the effective size of the black-hole system diverges in the extremal (zero-temperature) limit.

[6] S Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, New York, 1983.

[7] R.P Kerr, Phys Rev Lett 11 (1963) 237.

[8] K Gottfried, V.F Weisskopf, Concepts of Particle Physics, Volume 2, Oxford Uni-versity Press, 1986.

[9] H Levine, E.J Moniz, D.H Sharp, Am J Phys 45 (1977) 75;

T Erber, Fortschr Phys 9 (1961) 343.

[10] Note that the limiting caseR=Rc=q2/2μcorresponds to a charged spherical object whose rest mass μis attributed exclusively to the electrostatic self-energy it produces [8,9] The factor of 1/2 in (7) comes from the assumption that the electric charge is uniformly spread on a thin spherical shell [11] C Eling, J.D Bekenstein, Phys Rev D 79 (2009) 024019.

[12] We refer here to the energy as measured by asymptotic observers.

[13] S Hod, Phys Rev D 60 (1999) 104031.

[14] B Carter, Phys Rev 174 (1968) 1559.

[15] D Lohiya, J Phys A 15 (1982) 1815;

B Léauté, B Linet, J Phys A 15 (1982) 1821.

[16] The physical origin of this self-interactionO ( q2

/ M )term is attributed to the distortion of the body’s long-range Coulomb field by the curved black-hole spacetime [11,13,15]

[17] We recall that the parameters M , a}are the mass and angular momentum per unit mass of the original Kerr black hole.

[18] That is, Kerr black holes in the regime (14) are characterized by the relation

T M 0 [see Eqs (2) and (3) ].

[19] It is important to emphasize that our assumptionlmin

0 r+, [see Eq (10) ] cor-responds to charged particles in the regimeq2μ r+.

[20] The expression (16) for the minimal energy delivered to the black hole by the

charged body is universal in the sense that it is independent of the black-hole

rotation parametera.

[21] The expression (17) for the dimensionless physical quantity min(¯ a )is

univer-sal in the sense that it is independent of the black-hole rotation parametera.¯

[22] It is worth noting that, for generic values of the dropping heightl0 and in

the regimeq2r+(+−r−), one finds from Eqs (2) , (11) , (12) , and (13) ,

the relation ( l0;a )= 1+¯a2−2



1 −¯a2

· 2μ l0− 2 −1 −¯a2

·q2

2α

1 −¯a2 for the dimensionless physical parameter which quantifies the change in the black-hole

temper-ature caused by the absorbed particle This expression implies that, in the

regime (14) , the dimensionless physical parameter ( l0;a ) is an increasing

function of the dropping heightl0 (see [18] ).

[23] That is, the simple inequality (18) is based on the assumption [3] that the

proper distance of the body’s center of mass from the black-hole horizon at

the dropping point can approach arbitrarily close to the limiting valuel0→

Rmin=q2/2μ[see Eq (15) ].

[24] K.S Thorne, in: J Klauder (Ed.), Magic Without Magic: John Archibald Wheeler,

Freeman, San Francisco, 1972.

[25] See A.M Abrahams, K.R Heiderich, S.L Shapiro, S.A Teukolsky, Phys Rev D 46

(1992) 2452, and references therein.

[26] See J.P de León, Gen Relativ Gravit 19 (1987) 289, and references therein.

[27] H Andreasson, Commun Math Phys 288 (2009) 715.

[28] S Hod, Phys Lett B 751 (2015) 241, arXiv:1511.03665.

[29] Interestingly, it was shown in [28] that the (weaker version [28] of the) hoop

conjecture must be invoked in order to guarantee the validity of the

general-ized second law of thermodynamics [3] in a gedanken experiment in which an

entropy bearing object is lowered slowly into a near-extremal black hole.

[30] We shall henceforth assume that the original Kerr black hole is a near-extremal

one.

[31] As discussed above, this new (and larger) horizon is expected to be formed

according to the original hoop conjecture [24] and its generalized version (19)

[32] We recall that the proper radius of the charged spherical body is given by

R=Rmin=q2

/2μ[see Eq (7) ].

[33] It is important to emphasize that our assumptionlmin

0 r+ corresponds to particles in the regimeμr+−r−[see Eq (23) ].

[34] W.G Unruh, R.M Wald, Phys Rev D 25 (1982) 942.

[35] J.D Bekenstein, Phys Rev D 49 (1994) 1912;

J.D Bekenstein, Phys Rev D 60 (1999) 124010.

[36] As shown by Bekenstein [35] , the relations (31) and (32) are valid for macro-scopic and mesoscopic bodies in the regimeR ¯h μ This strong inequality corresponds to the regime1.

[37] That is, after the assimilation of the charged particle by the black hole [38] It is worth noting that, had we used the original hoop conjecture [24] instead

of its weaker version (19) , we would have found that the Bekenstein–Hawking temperature of the final Kerr–Newman black-hole configuration (after the as-similation of the charged particle) is higher than the temperature of the origi-nal Kerr black hole in the entire regimea∈ [0,1].

[39] That is, the fact that, in our gedanken experiment, the Bekenstein–Hawking temperature of the black holes is an irreducible quantity in the near-extremal regimeTBH< TcBH, determined by the critical temperature (30)

[40] Here we have used the relationμ2/2=μ / 2R [seeEq (7) ] for our charged massive particle In addition, we have used the inequalitiesh μ≤Rr+[see

Eq (6) ] which characterize the physical parameters of the captured particle (As emphasized by Bekenstein [3] , the inequalityR≥ ¯h μreflects the fact that the proper radius of the particle is bounded from below by its Compton length [3] ).

[41] It is interesting to note that the suggested lower bound (37) on the Bekenstein–Hawking temperature of the black holes is universal in the sense that it is independent of the physical parameters (proper mass and electric charge) of the captured particle which was used in our gedanken experiment

in order to infer the bound.

[42] It is worth noting that, taking cognizance of Eq (36) and using the strong inequalities μr+ and Rr+ [see Eq (6) ], one can obtain the stronger lower boundTc

BH×r+ ¯h3 / r+ on the Bekenstein–Hawking temperature of the black holes Note, however, that this bound, which is a direct consequence

of the quantum buoyancy effect, is probably of no relevance if, instead of being lowered slowly towards the black hole, the charged particle splits off from a larger body which falls freely (and thus experiences no buoyant force) towards the black hole (note that, in order to deliver as small as possible energy to the black hole, the splitting of the larger body into two particles should take place

in the near-horizon region and, in addition, the second particle should escape the black hole) We therefore believe that the relation (37) should be regarded

as the more fundamental bound on the Bekenstein–Hawking temperature of the black holes (that is, a generic bound which is independent of the manner

in which the charged object arrives at the near-horizon region).