Hawking temperature of black holes using the EUP and EGUP formalisms based on the Einstein-Bohr's photon box
N. Farahani, H. Hassanabadi, W.S. Chung, B.C. Lutfuoglu, S. Zarrinkamar
aa r X i v : . [ g r- q c ] F e b epl draft Hawking temperature of Black Holes using theEUP and EGUP formalisms based on the Einstein-Bohr’s Photon Box
N. Farahani , H. Hassanabadi , , W.S. Chung , B.C. L¨utf¨uo˘glu , and S. Zarrinkamar Faculty of Physics, Shahrood University of Technology, Shahrood, Iran. Department of Physics, University of Hradec Kr´alov´e, Rokitansk´eho 62, 500 03 HradecKr´alov´e, Czechia. Department of Physics and Research Institute of Natural Science, College of Natural Sci-ence, Gyeongsang National University, Jinju 660-701, Korea. Department of Physics, Akdeniz University, Campus 07058 Antalya, Turkey. Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran.
PACS – Quantum aspects of black holes, evaporation, thermodynamics
PACS – Physics of black holes
PACS – Classical black holes
Abstract – In this paper, by studying the Einstein-Bohr’s photon box for weighting a photon, wefind that the effective Newton constant can be proposed by the extended uncertainty principle andextended generalized uncertainty principle. We obtain the modified Hawking temperature, mass,specific heat, and entropy by using the modified Schwarzschild metric.
Introduction. –
According to the ordinary quantum mechanics, there is a fundamen-tal lower limit for the simultaneous measurement of certain pairs of physical quantities suchas momentum and length. This limit is proportional to the Planck constant and derived fromthe Heisenberg algebra, therefore it is called the Heisenberg uncertainty principle (HUP) [1].However, HUP does not predict a minimum uncertainty in position or momentum, solitary.On the other hand, the studies regarding the quantum gravity point to the existence ofsome additional terms not found in the HUP [2–4]. The predicted modification of Heisen-berg algebra is associated with a parameter in the order of Planck length [5] and it yieldsto derive a minimum uncertainty in position. This generalization of the HUP is named asthe generalized uncertainty principle (GUP) [6] and it is a well-known fact that its origin isbased on the string theory [7] and the Gedanken experiments [8]. It is shown that in anothergeneralization of the HUP, namely in the extended uncertainty principle (EUP), a minimumuncertainty in momentum can be obtained associated with an extension parameter that isproportional to the cosmological constant [9]. Meanwhile, in another work, it is shown thatif the (anti)-de Sitter ((A)dS) background is taken into account instead of the Minkowskispace-time, then HUP is needed to be extended with an additional term that is inverselyproportional to the square of the (A)dS radius [10, 11]. It is worth noting that the GUPand EUP are thought to play an important role in the early and later stages of the universe,p-1. Farahani et al. respectively [12]. Therefore, a combination of these two generalized forms, hereafter theextended generalized uncertainty principle (EGUP), is formulated to consider the minimaluncertainties together in position and momentum in various studies [13, 14].In 1930, Bohr and Einstein, two of the greatest physicists in history drew swords to eachother on a very fundamental property of quantum mechanics, HUP [15–18]. Einstein deviseda Gedanken experiment, namely the photon-box experiment, to show up an inconsistencyin the time-energy uncertainty relation. After one long and sleepless night, Bohr defendedthe HUP and saved his career [19]. In this letter, we revisit the photon-box experiment byconsidering the EUP and EGUP formalisms. We aim to realize how the gravitational fieldstrength and correspondingly, the Newton constant would be modified based on the EUP andEGUP. Next, we want to derive the Hawking temperature [20] and altered thermodynamicproperties of the black hole, taking into account the effective (modified) Newton constantin a Schwarzschild black hole [21–24].We construct this letter as follows: In the following section, we briefly discuss theEinstein-Bohr debate. Then, in the third and fourth sections, by introducing EUP andEGUP in AdS space-time, we derive the effective gravitational field strength, thus, themodified Schwarzschild black holes metric. After that, we discuss the modified Hawkingtemperature, the mass-temperature, specific heat-temperature and entropy-temperature re-lations for the EUP and EGUP black holes based on Einstein-Bohr’s photon box Gedankenexperiment, respectively. In the conclusion, we compare the modified Hawking temperaturefor EUP, GUP, and EGUP approaches.
Einstein-Bohr’s photon box. –
Einstein’s purpose in designing this Gedanken ex-periment was to reject the HUP, explicitly. He considered an experiment box with a clockinside and a clock-related gap [25]. The gap associated with the clock is set against anothergap at a specified interval. At a time interval, the light pulse passes through these two gaps.If this light pulse is located at a farther distance, at a certain distance from the gap, it willbe detected and its energy will be measured with arbitrary precision. Measuring energy andtime with arbitrary precision would have meant the collapse of Heisenberg’s energy-timerelationship [26].Heisenberg believed in successive measurements of the position and momentum. How-ever, the uncertainty of the momentum or position of a particle is related to the position ormomentum that is measured at that moment. Hence, the HUP is not valid for retroactivemeasurements. If we want to turn this retroactive measurement into a predictive one, wehave to consider the energy exchange during the gap movement. Let us assume that the gapmoves at the speed of v . Then, the uncertainty of the momentum ∆ p during the radiationof the photon has to result in the uncertainty of ∆ E = v ∆ p . Since this energy exchange isexecuted in the constant gap width, d , then the momentum uncertainty has at least to beequal to ¯ hd . Therefore, we get ∆ E ≥ ¯ hvd . (1)To increase the accuracy of energy determination, vd ratio has to be reduced as much aspossible either by decreasing v or by increasing d . In both cases, the precision of the timingis disturbed, thereby, according to ∆ t ≈ dv we arrive at [25]∆ E ∆ t ≥ ¯ h. (2)Einstein developed another Gedanken experiment scheme that will be named as the ”Clockin the Box” experiment. He considered a box that contains electromagnetic radiation withfull reflective inner walls. He assumed one of the walls has a shutter on and it is openingand closing by a clock-related mechanism inside the box. The setting is such that the slotopens at time t = t for a short and arbitrary time interval, i.e. t − t , thereby a photoncan be emitted. By weighing the mass of the box just before and after the photon emission,p-2itleit is possible to determine the energy difference of the box with the least error ∆ E regardingthe mass-energy relation E = mc . According to the energy conservation law, the energydifference is related to the energy of the emitted photon. Therefore, the energy of the photonand the time that it needs to reach the plate from the device can be predicted by arbitraryuncertainty ∆ E and ∆ t . This is, of course, contrary to the Heisenberg uncertainty relation.However, after just one night, Bohr came up with a solution that demonstrated thatthe HUP remains valid. His defense was based on Einstein’s general theory of relativity.Einstein did not take the time-dilation effect into account. One can prove the validity ofHUP as follows: During the emission of the photon, the uncertainty in the momentum ofthe box is ∆ p = ∆( mv ) where v = gt . Here, t is the time for the observer outside the box.Then, we employ them in the HUP. We find∆ X ≥ ¯ h ∆ mgt . (3)Since there is a time dilation, the observer in the box (clock) experiences a time uncertainty,∆ t . In [27], it is given in terms of the vertical position uncertainty as follows:∆ t = g ∆ Xc t. (4)By substituting t from eq. (4) into eq. (3) we arrive at∆ t ≥ ¯ h ∆ mc . (5)Following the mass-energy relation we can express the conventional HUP as given in eq. (2).On the other hand, by substituting eq. (4) into eq. (2), we find [28]∆ E ≥ c ¯ hgt ∆ X . (6)
Einstein-Bohr’s photon box based on the EUP. –
In this section we considerBohr’s argument throughout the EUP. As far as we know, Xiang emp et al. recently used theGUP instead of the HUP and derived an effective Newtonian constant [29]. We believe thatthe use of EUP within the Bohr’s argument is a very interesting open problem, as Mignemireported that the EUP correction can be derived from the geometry of the (A)dS spacetime[14]. The EUP is based on the α parameter, that is defined as 3 α = | Λ | = 10 − m − where α = L H . Here, L H is the radius of the (A)dS spacetime and Λ is the cosmological constant[30,31]. It is worth noting that in quantum field theory, dS spacetime is defined in positivelycurved spacetime with positive cosmological constant and radius, while AdS spacetime isdefined in negatively curved spacetime with negative cosmological constant and radius [32].We start with the following modified commutation relation[ X, P ] = i ¯ h (1 ± α X ) , (7)where α = α | Λ | while α has the order of unity and dimensionless. Then, the EUP appearsas [33, 34] ∆ X ∆ P ≥ ¯ h α | Λ | X ) . (8)However, we can employ a more general form of the modified commutation relation[ X, P ] = i ¯ hξ ( X ) , ξ ( X ) = (cid:18) α | Λ | X (cid:19) , (9)p-3. Farahani et al. hereafter, ξ ( X ) ≡ ξ .In the Gedanken experiment, after the photon is released, the box moves upward as itbecomes lighter. In order to lower the box to its previous level the lowest and quantum-mechanically meaningful mass is added to the system. It corresponds to the released photon’sweight, namely g ∆ m . This procedure is supposed to be executed in a period of time t , [29].Thus, we arrive at ∆ P min t = ξ ¯ ht ∆ X ≤ g ∆ m, (10)or we can write ξ ¯ h = ∆ X ∆ P min ≤ ∆ m ( g ∆ X ) t. (11)Then, by substituting eq. (4) into eq. (11), we get ξ ¯ h ≤ c ∆ m ∆ t = ∆ E ∆ t. (12)We employ ∆ t from eq. (4) in the above inequality. We obtain the uncertainty in the energyof the released photon as follows∆ E EUP ≥ ξ ¯ hc g ∆ Xt = ¯ hc g ′ ∆ Xt . (13)When we compare our result with equation eq. (6), we observe the only difference as thereplacement of g ′ at g . Thus, we conclude that in the EUP formalism the modified gravityof Earth becomes g ′ = gξ = g α | Λ | X . (14)Note that when α = 0, we find the HUP limit. Then, we express the effective gravitationalfield strength from the well-known definition g = GMR [35]. G ′ = G α | Λ | X . (15)Therefore, we conclude that we come up with a curved space from a modified effective New-ton constant.Next, as done in [29], we change G with G ′ and examine a new Schwarzschildmetric with the form [36] ds = − (cid:18) − GMξc r (cid:19) c dt + (cid:18) − GMξc r (cid:19) − dr + r d Ω . (16)We explore solutions for the modified Schwarzschild black hole in which the mass is takenas ˜ M = Mξ ( X ) . We obtain the Hawking temperature, [37], as follows: T H = ¯ hc πGk B ˜ M . (17)On the other hand, we consider X of order ∆ X . Close to the black hole the uncertainty isproportional to the black hole event horizon [38], thus, we write ∆ X = ηr s , where r s = GMc .We find T = ¯ hc πGk B M (cid:18) α | Λ | G M η c (cid:19) . (18)p-4itleWe solve the above equation while considering η = 2 π , we obtain M EUP = 3 k B T c α | Λ | Gπ ¯ h ± s − α | Λ | c ¯ h k B T , (19)which implies a minimum temperature as T ≥ T EUPmin = ¯ hck B r α | Λ | . (20)We employ eq. (20) in eq. (19). In order to obtain the minimum mass for EUP, we use thenegative quantity. We find M EUPmin = 3 k B T c α | Λ | Gπ ¯ h . (21)Considering eq. (19) for small values of α | Λ | with a Taylor expansion, we find M EUP = ¯ hc πGk B (cid:18) T + α | Λ | ¯ h c k B T + α Λ c ¯ h k B T + ... (cid:19) . (22)It is worth noting that, for α = 0, we recover the ordinary case result, M = ¯ hc πGk B T . Then,we demonstrate the mass function versus temperature in Fig. (1). Next, we examine thespecific heat of the EUP black hole which can be derived out of eq. (22) as done in [39, 40].We find C EUP = c dM EUP dT , (23)= − ¯ hc πGk B (cid:18) T + α | Λ | ¯ h c k T + 5 α Λ ¯ h c k B T + ... (cid:19) . We observe that the specific heat in the EUP black hole always is negative. We depict thespecific heat versus temperature in Fig. (2) to present its characteristic behavior. Next, wederive entropy for the EUP black hole by integrating eq. (23). We find S EUP = Z C EUP
T dT, (24)= ¯ hc πGk B (cid:18) T + α | Λ | ¯ h c k B T + 5 α Λ ¯ h c k B T + ... (cid:19) . In Fig. (3), we plot the modified entropy-temperature function versus temperature. Fi-nally, by taking the negative sign and expanding eq. (19), we obtain the modified Hawkingtemperature based on EUP in terms of temperature. T EUPH = T H (cid:18) α | Λ | ¯ h c k B T (cid:19) . (25)In the following section, by following the same strategy we are going to obtain the EGUPformalism equivalents of the functions we have obtained in the EUP formalism. Einstein-Bohr’s photon box based on the EGUP. –
We start by introducingEGUP as a linear combination of the EUP and GUP [41, 42]∆ X ∆ P ≥ ¯ h (cid:18) α | Λ | X + β ℓ P ¯ h ∆ P (cid:19) , (26)p-5. Farahani et al. which can be obtained from the following modified commutation relation[ X, P ] = i ¯ h (cid:18) α | Λ | X + β ℓ P ¯ h P (cid:19) . (27)Here, ℓ P is the Planck length and β is the parameter of the order of unity. Then, werewrite the modified commutation relation in the most general form, [ X, P ] = i ¯ hζ , where ζ = ζ ( X, P ). We follow the same arguments of the previous sections and express∆ P min t = ζ ¯ ht ∆ X ≤ g ∆ m, (28)instead of eq. (10) [24, 29]. Alternatively, we can write ζ ¯ h = ∆ X ∆ P min ≤ ∆ m ( g ∆ X ) t. (29)By replacing eq. (4) into eq. (29), we find ζ ¯ h ≤ c ∆ m ∆ t = ∆ E ∆ t. (30)We observe that the uncertainty in the energy of the released photon is restricted by∆ E EGUP ≥ ζ ¯ hc g ∆ Xt = ¯ hc g ′ ∆ Xt . (31)Here, we use g ′ = gζ . Then, as we have done in the previous section, we find the effectivegravitational field strength in the form of G ′ = G (cid:16) α | Λ | X + β ℓ P ¯ h P (cid:17) . (32)Next, we use the EGUP effective gravitational field strength to examine the modifiedSchwarzschild black hole while M ′ = Mζ . Then, eq. (17) changes its form to T = ¯ hc πGk B M (cid:18) α | Λ | X + β ℓ P ¯ h P (cid:19) . (33)After that according to [43], we assume P = k B Tc and X = Gmηc . We find M = 3 k B cT α | Λ | ¯ hGπ ± s − α | Λ | β ℓ P − α | Λ | c ¯ h k B T . (34)In Fig. (4), we plot the mass-temperature function versus temperature. Then, we explorethe temperature expression that corresponds to the minimum mass value of the black hole.We get T ≥ T EGUP = ¯ hck B vuut α | Λ | (cid:16) − α | Λ | β ℓ P (cid:17) . (35)We substitute eq. (35) into eq. (34) and we use the positive quantity. We obtain the minimummass value as M ≥ M EGUPmin = 3 k B T c πα | Λ | ¯ hG . (36)p-6itleWe compare minimal mass values of the EUP and EGUP black holes by matching eqs. (21)and (36). Notably, we observe that M EGUPmin = M EUPmin . For the small values of α | Λ | and β ℓ P parameters we Taylor expand eq. (34) by considering the negative sign. We find [12] M EGUP = ¯ hc πGk B (cid:18) T + α | Λ | ¯ h c k B T + β ℓ P k B Tc + α | Λ | β ℓ P ¯ h T (cid:19) . (37)Then, we derive the specific heat function of the EGUP black hole case out of eq. (37) [12] C EGUP = c dM EGUP dT = ¯ hc πGk B (cid:18) − T + β ℓ P k B c − α | Λ | ¯ h c k B T − α | Λ | β ℓ P ¯ h T (cid:19) . (38)In Fig. (5), we plot the modified specific heat function versus temperature. We observe thatthe specific heat function is equal to zero at a particular temperature value. We find thistemperature from the following expression T = ¯ hck B β ℓ P s (6 + ( α | Λ | β ℓ P ¯ h ) )12¯ h × " s α | Λ | β ℓ P ¯ h ) (6 + ( α | Λ | β ℓ P ¯ h ) ) . (39)Next, we derive the entropy function for the EGUP black holes out of eq. (38). We find S EGUP = Z C EGUP
T dT, = ¯ hc πGk B (cid:18) T + k B β ℓ P c ln T + α | Λ | ¯ h c k B T + α | Λ | β ℓ P ¯ h T (cid:19) . (40)In Fig. (6), we depict the modified entropy-temperature function versus temperature. Weobserve a minimal value that corresponds to T temperature that is derived in eq. (39). Notethat at this temperature modified specific heat vanishes as well. Finally, we expand eq. (34)by considering the negative sign. We obtain T EGUPH T H = (cid:18) α | Λ | c ¯ h k B T + β ℓ P k B T ¯ h c + α | Λ | β ℓ P (cid:19) . (41)Finally, in Fig. (7), we plot T EGUPH , T EUPH and T GUPH versus temperature. Here, we takethe form of T GUPH function from the reference [29]. We see that T EGUPH and T EUPH havesimilar behaviors at high temperatures, i.e. T EGUPH and T EUPH are greater than T H . Wealso note that T EGUPH and T GUPH have similar behavior at low temperatures, i.e. T EGUPH and T GUPH are greater than T H . However, T EUPH and T GUPH don’t present a similar behaviorand they coincide at the temperature [12, 13]. T H = ¯ hck √ (cid:18) α β (cid:19) / . (42)p-7. Farahani et al. Conclusion. –
In this paper, we studied the Hawking temperature and the modi-fied thermodynamic properties for the Schwarzschild black hole in the EUP and EGUPapproaches. As shown in eqs. (14), (15) and (32), the effective gravitational field and thecorresponding Newton constant are modified in the presence of EUP and EGUP. This resultallows us to investigate the modified Hawking temperature and the corresponding thermody-namic properties for the black hole. Then, by obtaining the modified Hawking temperaturein the presence of EGUP, we obtained the remarkable consequence that its value was thesum of the modified Hawking temperature in the presence of EUP and GUP, and an addi-tional term which had both of generalization parameters ( β ℓ P α | Λ | ). This is the result weobtained after studying the mass, specific heat, and entropy of the Schwarzschild black hole.This result indicates that the modified Hawking temperature obtained for the Schwarzschildblack hole has a higher generality , which is clearly shown in Fig. (7). As can be seen, themodified Hawking temperatures of EGUP and GUP behave similarly at low temperatures.As the modified Hawking temperatures of EGUP and EUP behave similarly at high temper-atures. On the other hand, it can be pointed out that the modified Hawking temperaturesdo not resemble a similar behavior in GUP and EUP cases, they only coincide at the tem-perature given in eq. (42). While the modified Hawking temperature in the presence ofEGUP gives us a more accurate answer. Also, after examining and plotting the specificheat and entropy of the Schwarzschild black hole in the presence of EGUP, we concludethat the entropy reaches its lowest value at the temperature T in eq. (39), and this is thetemperature at which the specific heat is zero. Notably, we found that M EGUPmin = M EUPmin for small values α | Λ | and β ℓ P . ∗ ∗ ∗ The authors thank the referee for a thorough reading of our manuscript and for con-structive suggestion. B. C. L¨utf¨uo˘glu, was partially supported by the Turkish Science andResearch Council (T ¨UB˙ITAK).
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Fig. 2: Plot of the modified specific heat-temperature relation for the C EUP with α | Λ | = 0 . h = c = k B = G = 8 π = 1. S EUP
Fig. 3: Plot of the modified entropy-temperature relation for the S EUP with α | Λ | = 0 . h = c = k B = G = 8 π = 1 M EGUP
Fig. 4: Plot of the modified mass-temperature relation for the M EGUP with α | Λ | = 0 . β ℓ P = 0 . h = c = k B = G = 8 π = 1. p-10itle - - - - C EGUP
Fig. 5: Plot of the modified specific heat-temperature relation for the C EGUP with α | Λ | = 0 . β ℓ P = 0 . h = c = k B = G = 8 π = 1. S EGUP
Fig. 6: Plot of the modified entropy-temperature relation for the S EGUP with α | Λ | = 0 . β ℓ P = 0 . h = c = k B = G = 8 π = 1. T EGUP H T H T HEGUP T HGUP T HEUP
Fig. 7: Plot of the modified Hawking temperature-temperature relation for the T EGUPH , T EUPH and T GUPH with β ℓ P = 0 . α | Λ | = 0 . h = c = k B = G = 8 π = 1.= 1.