GUP-Corrected Black Hole Thermodynamics and the Maximum Force Conjecture
GGUP-Corrected Black Hole Thermodynamics and the Maximum Force Conjecture
Yen Chin Ong
1, 2, 3, l Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Nordita, KTH Royal Institute of Technology & Stockholm University,Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
We show that thermodynamics for an asymptotically flat Schwarzschild black hole leads to a forceof magnitude c / (2 G ). This remains true if one considers the simplest form of correction due to thegeneralized uncertainty principle. We comment on the maximum force conjecture, the subtletiesinvolved, as well as the discrepancies with previous results in the literature. I. BLACK HOLE THERMODYNAMICS ANDMAXIMAL FORCE CONJECTURE
Consider an asymptotically flat Schwarzschild blackhole. The first law of black hole thermodynamics is d M = T d S . This allows us to define a force [1] (see also [2] inthe context of cosmology) F = T d S d r h = d M d r h = c G , (1)where r h denotes the horizon, and T and S are, respec-tively, the Hawking temperature and the Bekenstein-Hawking entropy, given explicitly by T = (cid:126) c πk B GM , S = k B c A (cid:126) G . (2)In this work we will refer to F as the “thermodynamicsforce”, since it has the correct dimension c /G for a force.This quantity is reminiscent of – and closely related to– the “entropic force”, proposed by Verlinde [3], but thereare some subtle differences. Firstly, the entropic forcedescribes the force on a test mass m on a “holographicscreen” at a distance ∆ r from the gravitating body, whichinduces a change on the entropy of the holographic screenby ∆ S = 2 πk B mc ∆ r/ (cid:126) . The entropic force, F ent , satisfies F ent ∆ r = T ∆ S, (3)or equivalently, F ent = 2 πk B mcT (cid:126) . (4)However the thermodynamics force defined above satisfiesd S d r h = 2 πr h = 4 πM, (5)which is independent of the mass m of any test body .We also note that one could calculate a force from Eq.(4) l Electronic address: [email protected] To put this in another perspective, consider the test mass to by substituting in the Hawking temperature. We obtain˜ F = 2 πk B mcT (cid:126) = c G mM . (6)The Hawking temperature is of course the temperaturemeasured by an observer at infinity. For a local observerat finite r , one should use the local (Tolman) temperature, T L , and arrive at [4]˜ F = 2 πk B mcT L (cid:126) = 1 (cid:112) − r h r c G mM . (7)The series expansion of Eq.(7) does not recover the Newto-nian gravitational force law F = GM m/r , hence Myung[4] argued that this expression (and so also Eq.(6)) hasnothing to do with the entropic force (which providesa “derivation” of Newtonian gravity [3]), thus we havedenoted it with ˜ F here, instead of F ent .We emphasize that while the idea of gravity beingan entropic force is still being debated (for numerousobjections, see among others, [5–8]) and no consensushas been reached, Eq.(1) is just an established propertyof black holes, equivalent to the first law of black holethermodynamics. In view of these subtleties, we do notrefer to Eq.(1) as the entropic force, although we notethat this quantity defined in the context of cosmologywas referred to as such [2].In general relativity, it has been conjectured that thereexists an upper bound for forces acting between two bod-ies. In 4-dimensions, F max = c / G ≈ . × N.Note that c /G gives the correct physical dimension for aforce, but the factor 1 / be a black hole of mass m , then S M = 4 πM , S m = 4 πm .Consider dropping the test mass into the black hole. Then S M + m = 4 π ( M + m ) . The change in the entropy can be obtainedfrom taking the limit:d S d r h = lim ∆ r → S M + m − S M m = 2 πr h , since taking ∆ r → m →
0. That is, the“thermodynamic force” already considers small m limit, whereasin what usually referred to as the “entropic form”, m is finite. a r X i v : . [ g r- q c ] S e p [9], Schiller [10], and Barrow and Gibbons [11]. This isknown as the maximum force conjecture , or maximumtension conjecture. In [9], it is further remarked that thefactor 1 / c . We thereforedistinguish between two different forms of the maximumforce conjecture:(1) Maximum Force Conjecture (Strong Form):In 4-dimensions, forces are bounded from above by F max = c / (4 G ).(2) Maximum Force Conjecture (Weak Form):In 4-dimensions, there exists a positive number K < ∞ , such that forces are bounded from above by F max = c K/G . It is also possible that K is asupremum instead of a maximum.That is to say, the weak form is simply the statementthat forces cannot be arbitrarily large in general relativity,but makes no claim to the value of the lowest upperbound. The force defined in Eq.(6), for example, satisfiesthe strong form of maximum force conjecture, since forany test mass m (cid:28) M , we have ˜ F (cid:28) c / (4 G ). On theother hand the force defined in Eq.(1) only satisfies theweak form of the conjecture, though its deviation fromthe strong form is only of order unity (in Planck units).Black hole physics is expected to receive quantum cor-rections as the black hole size reduces to near Plancklength, as the result of losing mass via Hawking radiation.While a full quantum gravity theory is still lacking, thereexist phenomenological models that allow us to study atleast some properties of quantum black holes. The gen-eralized uncertainty principle (GUP) is one such widelystudied model. Since the thermodynamic force is relatedto the first law of black hole mechanics, it is interestingto explore how it is affected by GUP correction, whichis known to modify both the Hawking temperature andthe Bekenstein-Hawking entropy. This is the focus of thiswork. II. GUP-CORRECTED BLACK HOLE ANDMAXIMAL FORCE CONJECTURE
From now on we will work with the Planck units inwhich G = c = (cid:126) = 1. We also set k B = 1. As mentioned,in this work we would like to calculate the thermody-namic force as defined in Eq.(1), but in the context ofGUP-corrected Schwarzschild black hole. In GUP theHeisenberg’s uncertainty principle receives a correctionterm from gravitational effect:∆ x ∆ p (cid:62) (cid:34) (cid:126) + αL p ∆ p (cid:126) (cid:35) , (8)where α , the GUP parameter, is often taken to be O (1)in theoretical considerations. GUP is usually treated phenomenologically to model quantum gravitational ef-fects [12–15]. In the context of black hole physics, GUPcorrection yields [16] T [ α ] = M απ (cid:18) − (cid:114) − αM (cid:19) (9)for the Hawking temperature. The negative sign in frontof the square root was chosen so that the α → S [ α ] = (cid:90) T d M = 2 π (cid:104) M + M (cid:112) M − α − α ln( M + (cid:112) M − α ) (cid:105) + const. (10)Note the appearance of the logarithmic term, which seemsto be a common features in various quantum gravitationalmodels (see, e.g., [17–20]). Note that in deriving Eq.(9),it is assumed that the uncertainty in the photon positionis ∆ x ∼ r h , about the size of the black hole horizon. Thehorizon position is assumed to be still r h = 2 M . Thuswe should expect that the thermodynamics force Eq.(1)is unchanged since d M/ d r h = 1 / . Indeed,d S [ α ]d r h = 2 π ( M + M √ M − α − α ) √ M − α , (11)which upon multiplying with the GUP-corrected tem-perature Eq.(9), we obtain exactly /
2, independent ofthe value (and sign) of α , consistent with our previousobservation.In terms of series expansion (asymptotic series in M ),we have T [ α ] = 18 πM + 132 απM + 164 α πM + · · · = T (cid:18) α M + α M + · · · (cid:19) = T (cid:0) π αT + 512 π α T + · · · (cid:1) , (12)where T = T [ α = 0] is the original Hawking temperature T = 1 / (8 πM ). Similarly, the series expansion of the Thus S [ α ] = A/ not hold. Note our α is the α of [1]. GUP-corrected entropy is S [ α ] = S (cid:18) − α M ln M + α M + · · · (cid:19) = S (cid:18) − παS ln S + α π S + · · · (cid:19) , (13)where S = S [ α = 0] is the original Bekenstein-Hawkingentropy S = 4 πM . Consequently, F [ α ] = F (cid:16) − απS + 16 π αT + · · · (cid:17) , (14)where F = 1 / F [ α ] depends on S and T , thisseries cannot possibly diverge (we know that it is exactlyequal to 1 / S → T → ∞ , whichcorresponds to the end stage of Hawking evaporation.With GUP correction both T and S are not allowed totend to these limits. This can be appreciated from, e.g.,Eq.(12): RHS consists of series of T , which would divergeif T is allowed to diverge as in the usual scenario of Hawk-ing evaporation. However, we know that GUP providesan upper bound for Hawking temperature (how this isachieved depends on whether α is positive or negative,see [21]). Therefore we must be careful about the range M can take in the expansion, it should be the same asthat of the defining equation itself. In fact, the situationis better still: the terms − απS + 16 π αT (15)in Eq.(14) cancel exactly . The same holds for the higherorder terms in the series expansion, so that F ≡ F [ α ] , ∀ α .We note that there are also typos in the series expan-sions in [1], which were likely transferred from the lastreference therein, i.e., Tawfik et al. [22] (Eq.(34) in thearXiv version). The mistake seems to have been correctedin the next paper of partially the same authors of [1]:A. Alonso-Serrano et al. [23]. However, series expan-sion was again used in that work. It should again beemphasized that using the full expression for the Hawk-ing temperature while defining the thermal wavelengthwould yield a quantitatively different result, comparedto that obtained from just the first few terms of the se-ries expansion [Eq.(26) of [23]]: namely the latter led toFig.2 therein, in which the black hole mass can go to zero,whereas we should have a nonzero remnant mass at whichpoint evaporation stops. See [24] for more discussions ofthis technical difference.We thus conclude that the thermodynamic force of anasymptotically flat Schwarzschild black hole is indeed 1 / r = 2 M independent of GUP-correction, and the definition of thethermodynamic force.One might take a somewhat different perspective. If oneassumes that the GUP-corrected Hawking temperature can be derived via the usual Wick-rotation trick from themodified Schwarzschild geometry [25]d s = − f ( r )d t + f ( r ) − d r + r dΩ , (16)where f ( r ) = 1 − Mr + εM r , | ε | (cid:28) , (17)then the horizon is located at r h = 2 M (cid:18) √ − ε (cid:19) . (18)In such a scenario the entropic force receives a smallcorrection, but it remains finite. (The GUP parameter α is negative in this model [25], which has some virtuesincluding preventing white dwarfs from getting arbitrarilylarge [21].) III. DISCUSSION
In this work, we have re-examined a “thermodynamicforce” – not quite Verlinde’s “entropic force” thoughrelated to it – that can be defined via the first law ofblack hole thermodynamics. Such a force is identicallyequal to 1 / F max = 1 /
4. Let us emphasize thatthe original maximum force conjecture only considersforces that act between two bodies . It is not so clear whatthese “two bodies” refer to in the context of black holethermodynamics, so perhaps it is not too surprising that itdoes not satisfy the strong form of the conjecture, whereasthe force defined by considering a test mass in Eq.(6) does– when the temperature is measured at spatial infinity. Onthe other hand, in the context of asymptotically flat Kerrblack holes, we can define an effective spring constant [26]: k := M Ω , where M is the mass of the black hole, and Ω + the angular velocity of the event horizon. In the extremallimit, it turns out that Hooke’s law F = kx gives exactly1 /
4, which saturates the strong form of the maximum forceconjecture [26], although such a “force” does not seem tobe acting between any two physical bodies either; it is justa convenient effective description. This “harmonic force”vanishes for a Schwarzschild black hole (since k = 0),which is distinct from the thermodynamic force F = 1 / any force naturally defined in the context of generalrelativity satisfies the maximum force conjecture? (Onecan of course multiply a force by any factor, hence therequirement that it should be “naturally” defined fromthe context is crucial.)If one applies the generalized uncertainty principle to anasymptotically flat Schwarzschild black hole, we show thatits thermodynamic force is still finite. We explained theerror made in [1], which claimed that GUP-corrected blackhole could violate even the weak form of the maximumforce conjecture (i.e., forces can diverge). However, notethat [1] also discussed other gravity theories, and evenconsidered the possibilities that the values of c and G maynot be a constant. We do not entertain such possibilitiesof running c and G in this work.It is possible to consider the entropic force `a la Ver-linde, to derive the correction to the Newtonian force lawbetween two bodies with GUP correction. This yieldswith α = 1 [27], F N [ α = 1] = F N { β (2 − ln β )+ β [4 − β + (ln β ) ] + · · · (cid:9) , (19)where F N = GM m/R and β := G (cid:126) / ( c R ). Newtonianforces are of course not required to be bounded fromabove.Let us note that the GUP modifications of the entropyand temperature of black hole rely on taking the micro-canonical corrections, as opposed to the canonical one.This subtle point was recently raised in [23]. What thisstatement means is that we consider quantum correctionsto microstate counting, while keeping the horizon areafixed. This contrasts with the canonical correction, whichconsiders the thermal fluctuation of the horizon area, notrelated to the fundamental degrees of freedom. Follow-ing [23], since the object of study is correction to blackhole physics which becomes important near the Planckscale, it is arguably the more fundamental microcanonicalcorrection that should be considered. However, it wouldbe interested to see in future works how the canonicalcorrection affects the maximal force conjecture.Lastly, although the current work concerns asymptoti-cally flat Schwarzschild black holes, it is of some interestto comment on cosmology. An immediate observation isthat Eq.(14) can presumably be applied to the Hubble horizon r H = c/H , where H is the Hubble parameter.This gives a non-zero GUP correction to the thermody-namic force in the context of cosmology. Secondly, thereexist a vast literature of applying Verlinde’s entropic forceto cosmology. In such applications, the Hawking temper-ature of the associated cosmological horizon comes witha prefactor, usually denoted γ , which is related to theposition of the holographic screen, see [28, 29] and thereferences therein. The value of γ is often taken to beof order unity in theoretical considerations, however in[30], observational fitting suggested that γ is two to fourmagnitude smaller. This is expected to affect the asso-ciated force in Eq.(6). Despite a similar prefactor beingincluded in [1] while discussing GUP-corrected black hole,such a factor should not be included in the context of thermodynamical force , in which the event horizon is theobject of study, not the holographic screen as in the caseof entropic force. The distinction between the two forcesis subtle yet important. Acknowledgments [1] Mariusz P. D¸abrowski, Hussain Gohar, “Abolishing theMaximum Tension Principle”, Phys. Lett. B (2015)428, [arXiv:1504.01547 [gr-qc]].[2] Damien A. Easson, Paul H. 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