Investigation of Unruh temperature of Black holes by using of EGUP formalism
Hassan Hassanabadi, Nasrin Farahani, Won Sang Chung, Bekir Can Lütfüoğlu
aa r X i v : . [ g r- q c ] S e p Investigation of Unruh temperature of Blackholes by using of EGUP formalism
Hassan Hassanabadi , , Nasrin Farahani , † , Won Sang Chung , andBekir Can Lütfüoğlu , Faculty of Physics, Shahrood University of Technology, Shahrood, Iran. Department of Physics and Research Institute of Natural Science,College of Natural Science,Gyeongsang National University, Jinju 660-701, Korea. Department of Physics, Akdeniz University, Campus 07058, Antalya, Turkey. Department of Physics, University of Hradec Králové, Rokitanského 62, 500 03 HradecKrálové, Czechia.
E-mail: † [email protected] Abstract
In this paper, we have used the extended generalized uncertainty principle to investi-gate the Unruh temperature and thermodynamic properties of a black hole. We startedwith a brief perusal of the Heisenberg uncertainty principle and continue with some phys-ical and mathematical discussion for obtaining the generalized and the extended general-ized uncertainty principle. Then, we obtained the Unruh temperature, mass-temperature,specific heat, and entropy functions of a black hole. We enriched the paper with graphicalanalysis as well as their comparisons.
Keywords: Extended uncertainty principle; Generalized uncertainty principle; Unruh tem-perature; Thermodynamic property.
According to the de Broglie relationship, every microscopic particle exhibits both wave andparticle characters. In 1927 W. Heisenberg stated that the exact position and exact momentumof microscopic particles as small as electrons could never be measured simultaneously. Later,this phenomenon was called the Heisenberg uncertainty principle (HUP) [1]. Recent researcheson the fields of the string theory [2], non-commutative geometries [3], black hole physics [4],and quantum gravity [5] proved the existence of a minimal length. However, the HUP doesnot speculate a minimal length, instead, generalized uncertainty principle (GUP) predicts avalue in the scale of the Planck length [6]. Alike the GUP, extended uncertainty principle(EUP) envisages a minimum measurable momentum value. Two authors of this manuscript1nvestigated the specific heat and entropy functions of a Schwarzschild black hole with thesimplest form of the EUP in [7]. In an (anti)-de Sitter ((A)dS) background, the EUP yields acorrection term that is proportional to the (A)dS radius [8, 9, 10]. Mignemi reported that thisterm could be derived from the geometry of the (A)dS spacetime as well [11, 12]. The extendedgeneralized uncertainty principle (EGUP) is the linear combination of the GUP and EUP,and it demonstrates the high energy and large length scale modifications together [13]. TheEGUP is employed to obtain the Hawking temperature in the (A)dS black holes [14]. Hawkingtemperature is a measure of black hole radiation for an observer who is in an inertial referenceframe outside the black hole [15, 16]. Unlike the Hawking temperature, Unruh temperature isa measure of black hole radiation observed by a uniformly accelerating detector in a vacuumfield [17, 18, 19, 20, 21, 22, 23].In this paper, we intend to obtain a modified expression of the Unruh temperature, mass,specific heat, and entropy functions of a black hole by employing the EGUP. We organized thepaper as follows: In section 2 we briefly introduce the HUP, GUP, EUP, and EGUP formalisms.In section 3 we derive an expression for the Unruh temperature in the EGUP and compare itwith the HUP, GUP, EUP limit ones. In section 4, we explore the mass-temperature, specificheat and entropy functions. We conclude the paper in section 5.
In conventional quantum mechanics, the well-known HUP is written as [24]∆ X i ∆ P j ≥ ~ δ ij . (1)In an other representation where energy and time are considered instead of the momentum andposition quantites, the HUP becomes [25, 26, 27, 28]∆ E ∆ T ≥ ~ . (2)The GUP that was proposed in the context of the string theory [29] and black hole Gedankenexperiment [29], has the generic form of [31]∆ X i ∆ P j ≥ ~ δ ij (cid:20) β (∆ P i ) (cid:21) . (3)Here, β ≡ β ~ l p where β is the unit correction constant and l P is the Planck length. Unlikethe GUP which is assumed to have played an important role in the early days of our universe,EUP is considered to play a role in the latter times of the universe and is written in the form[32, 33, 34, 35]: ∆ X i ∆ P j ≥ ~ δ ij (cid:20) α (∆ X i ) (cid:21) . (4)2ere, α ≡ l H where l H denotes the (A)dS radius. In [13], Bolen et al. defined the EGUP∆ X i ∆ P j ≥ ~ δ ij (cid:20) α (∆ X i ) + β (∆ P i ) (cid:21) . (5)to predict the temperature of the event horizon in (A)dS spacetime. In the next section, weare going to employ the EGUP to explore the Unruh temperature. We use ∆ E = c ∆ p relation in eq. (5) and we find∆ X ∆ E ≥ ~ c | Λ | X ) + β l p ~ c (∆ E ) ! , (6)Note that α ≡ | Λ | , and | Λ | is the cosmological constant [11, 13, 35]. We solve the quadraticequation for ∆ E and take the solution with negative sign in front of the square root becausethe other one does not provide a physically meaningful result [34].∆ E = ~ cβ l P ∆ X − vuut − β l P (∆ X ) − β l P | Λ | . (7)We expand the square root term to its Taylor series up to second order terms and we neglectedfrom order O ( | Λ | ) and O ( β ). We obtain∆ E = ~ c X " | Λ | (∆ X ) β l P X ) + | Λ | ! . (8)We follow [36, 37, 38], and then, the distance along which each particle must be accelerated inorder to create N pairs is ∆ X that we use ∆ X = Nc a and ∆ E = k B T in eq. (8). Note that∆ E denotes the energy fluctuation during the N pair production while a is the acceleration ofthe frame. We obtain the temperature function in the EGUP as T EGUP = T U " | Λ | π c a + β l P a π c + | Λ | ! . (9)Here, T U is the Unruh temperature [18, 39], it is written in the form of T U = ~ a πk B c . (10)Alternatively, we express the derived temperature in terms of the Unruh temperature as follows: T EGUP = T U | Λ | ~ ck B T U ! + 9 β l P k B T U ~ c ! + β l P | Λ | . (11)3ote that, when the cosmological constant tends to zero, the Unruh temperature in the EGUPformalism reduces to the Unruh temperature in the GUP formalism. Therefore, we employ | Λ | → T GUP = T U β l P k B T U ~ c ! , (12)This result is in agreement with [7]. Alike, in the limit where the Planck length goes to zero, T EGUP → T EUP . In this case, we obtain T EUP = T U | Λ | ~ ck B T U ! , (13)as found in [36]. Finally, we would like to emphasize that in the HUP limit, where the cosmo-logical constant and the Planck length vanish, we get T HUP = T U . (14)To have a better understanding of the characteristic behavior of the modified Unruh tempera-tures in the different formalisms, we define ξ ≡ T EGUP T U , ξ α ≡ T EUP T U , ξ β ≡ T GUP T U . (15)Then, we plot ξ , ξ α , and ξ β versus the T U in fig. 1. We observe similar behaviors. In thelow Unruh temperature values ξ and ξ α , and in the high Unruh temperature values ξ and ξ β illustrate the same characteristic changes. However, ξ α and ξ β do not present a similar behavior.This result confirms the agreement on the fact that the GUP and EUP have modified the earlyand late time dynamic of the universe. At a certain temperature T CU = ~ c k B s α ~ β . (16) ξ α and ξ β have the same universal value ξ α (cid:16) T CU (cid:17) = ξ β (cid:16) T CU (cid:17) = 1 + ~ αβ. (17)that depends only on the cosmological constant and the Planck length. In this section, at first, we examine the mass-temperature relation for a Schwarzschild blackhole with a mass M in the EGUP formalism. Then, we investigate the specific heat and entropy4unctions of the black hole. We assume a pair of photons near the surface. We suppose one ofthe photons has a negative effective energy − E and it is absorbed by the black hole. Whereas,the other one has + E and is emitted to an asymptotic distance from the black hole. Thecharacteristic energy E of the emitted photons may be estimated from the HUP. Nearby thehorizon of the black hole, the position uncertainty of the photon is assumed to be proportionalto the Schwarzschild radius, r s ≡ GMc , [40].∆ X ≡ ηr s . (18)Here, G is the Newton’s universal gravitational constant and η is a scale factor. Since, photonis a massless quantum particle, its momentum uncertainty is defined with its temperature [7]∆ P ≡ k B Tc . (19)We substitute the position and momentum uncertainty expression into eq. (5). We obtain aquadratic equation of mass and temperature in the form of α (cid:18) Gηc (cid:19) M − (cid:18) Gηc (cid:19) k B T ~ c ! M + ~ β ) k B T ~ c ! = 0 , (20)We exclude the solution with plus sign since it has no evident physical meaning [7]. Then, weexpress the mass-temperature function in the EGUP with M EGUP = 1 α c Gη ! k B T ~ c ! − vuut − ( α ~ β ) − α ~ ck B T ! . (21)By way of same manipulation done in sec. 3, we Taylor expand the square root term and presentthe mass-temperature equation of the Schwarzschild black hole under the EGUP formalism as M EGUP = 12 c Gη ! ~ ck B T ! α ~ ck B T ! + ( ~ β ) k B T ~ c ! + 2 α ~ β ! . (22)We take α = β = 0 to deduce the HUP limit. We find M HUP = 12 c Gη ! ~ ck B T ! . (23)We determine the value of the scale factor be equal to 2 π by matching eq. (23) with theHawking temperature [7, 40]. Then, we derive the EUP limit of the mass-temperature functionby employing β = 0 in eq. (22) M EUP = c πG ! ~ ck B T ! α ~ ck B T ! . (24)5like, we deduce the GUP limit of the mass-temperature function by employing α = 0 ineq. (22) M GUP = c πG ! ~ ck B T ! ~ β ) k B T ~ c ! . (25)We find a minimum value of the temperature out of eq. (21) T ≥ T EGUPmin = ~ ck B ! α q − ( α ~ β ) , (26)As beta tends to zero, we obtain the minimum value of the temperature in the EUP [7] T ≥ T EUPmin = α ~ ck B ! , (27)In the EGUP limit for ( α ~ β ) <
1, we investigate the lowest value of the temperature, T EGUP ,that minimizes the black hole mass function. We find T EGUP = ~ ck B ! vuut ( α ~ β ) + 2(2 ~ β ) vuuut α ~ β ) (cid:16) ( α ~ β ) + 2 (cid:17) / . (28)Next, we follow [7] and examine the difference of mass-temperature functions in the differentlimits.∆ M EGUP ≡ M EGUP − M HUP = M HUP α ~ ck B T ! + ( ~ β ) k B T ~ c ! + 2 α ~ β ! (29)∆ M EUP ≡ M EUP − M HUP = M HUP α ~ ck B T ! , (30)∆ M GUP ≡ M GUP − M HUP = M HUP ( ~ β ) k B T ~ c ! . (31)Therefore, we obtain ∆ M EGUP = ∆ M EUP + ∆ M GUP + M HUP ( α ~ β ) . (32)Before we proceed to discuss the thermal properties of the black hole, we introduce threefunctions, η ≡ M EGUP M HUP , η α ≡ M EGUP M GUP , η β ≡ M EGUP M EUP . (33)to illustrate the characteristic behavior of the mass-temperature in three limits. We plot η , η α ,and η β versus temperature in Fig. 2. We observe that η has a similar behavior with η α and η β at low and high temperatures, respectively. At a certain temperature value, T C = ~ ck B s α ~ β . (34)6 α and η β become equal: η α (cid:16) T C (cid:17) = η β (cid:16) T C (cid:17) = 1 + ( α ~ β ) − ( α ~ β )( α ~ β ) + 2 . (35)Next, we investigate the specific heat function of the black hole. We employ the definition C ≡ c dMdT . First, we obtain the specific heat in the HUP limit as C HUP = − c T M
HUP . (36)Then, we use eq. (22) to derive the specific heat function in the EGUP limit. C EGUP = C HUP α ~ ck B T ! − ( ~ β ) k B T ~ c ! + 2 α ~ β ! . (37)We employ β → α → C EUP = C HUP α ~ ck B T ! , (38) C GUP = C HUP − ( ~ β ) k B T ~ c ! . (39)Next, we define∆ C EGUP ≡ C EGUP − C HUP = C HUP α ~ ck B T ! − ( ~ β ) k B T ~ c ! + 2 α ~ β ! , (40)∆ C EUP ≡ C EUP − C HUP = C HUP α ~ ck B T ! , (41)∆ C GUP ≡ C GUP − C HUP = C HUP − ( ~ β ) k B T ~ c ! . (42)Similar to the analysis that is carried on the mass temperature function, we observe that thecontributions to the specific function can be expressed separately from each uncertainty limit.∆ C EGUP = ∆ C EUP + ∆ C GUP + C HUP ( α ~ β ) . (43)Before we proceed to examine the entropy function of the black hole, we introduce the followingfunctions ϑ ≡ C EGUP − C HUP , ϑ α ≡ C EGUP − C GUP , ϑ β ≡ C EGUP − C EUP . (44)In fig. 3 we present their behaviors versus the temperature. We observe that at low tempera-tures ϑ β and ϑ , at high temperatures ϑ α and ϑ have similar shapes. We point out that ϑ α is7lways in the negative value region.Finally we derive the entropy functions via the definitions given in [7, 41, 42]. We find S HUP = c T M
HUP , (45) S EGUP = S HUP α ~ ck B T ! + ( ~ β ) k B T ~ c ! ln T + 2 α ~ β ! , (46) S EUP = S HUP α ~ ck B T ! , (47) S GUP = S HUP ~ β ) k B T ~ c ! ln T . (48)Next, we define variation functions to illustrate the contributions of each uncertainty limit tothe total entropy function.∆ S EGUP ≡ S EGUP − S HUP = S HUP α ~ ck B T ! + ( ~ β ) k B T ~ c ! ln T + 2 α ~ β ! , (49)∆ S EUP ≡ S EUP − S HUP = S HUP α ~ ck B T ! (50)∆ S GUP ≡ S GUP − S HUP = S HUP ( ~ β ) k B T ~ c ! ln T . (51)Alike the mass-temperature and specific heat functions, these variation functions also satisfy∆ S EGUP = ∆ S EUP + ∆ S GUP + S HUP ( α ~ β ) . (52)We consider the following functions µ ≡ S EGUP − S HUP , µ α ≡ S EGUP − S GUP , µ β ≡ S EGUP − S EUP . (53)Then, we present the entropy function behaviors in fig. (4). We observe that µ mimics the µ α and µ β functions at low and high temperature values, respectively. µ α and µ β has the samevalue at a critic temperature which can be evaluated from the root of the following function. k B T ~ c ! ln T = 38 α ~ β ! . (54) In this letter, we examined the Unruh temperature and the thermodynamic properties of ablack hole by employing the extended generalized uncertainty principle which is a summationof generalized and extended uncertainty principles. We showed that the modified Unruh tem-perature of the EGUP limit has similar behavior with the GUP and EUP limits in high and8ow-temperature values, respectively. Our investigation on a black hole’s mass-temperaturefunction in the EGUP limit ended with an expression that describes the minimum temperaturevalue. Moreover, the relatively defined mass functions in the different limits yielded similarbehaviors at low and high temperatures. Furthermore, we investigated the specific heat andthe entropy functions of the black hole in the EGUP formalism. Alike to the mass-temperaturefunctions, we found that the contributions of the EGUP formalism can separately be expressedin the specific heat and entropy functions. The relatively defined functions of specific heatand entropy showed that at low-temperature and high-temperature EGUP functions behave asEUP and GUP functions.
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