Modified dispersion relations and (A)dS Schwarzschild Black holes

In this paper we investigate the impact of modified dispersion relations (MDR) on (Anti)de Sitter-Schwarzschild black holes. In this context we find the temperature of black holes can be derived with important corrections. In particular given a specific MDR the temperature has a maximal value such that it can prevent black holes from total evaporation. The entropy of the (A)dS black holes is also obtained with a logarithmic correction.


I. INTRODUCTION
It is well known that Planck mass M p or Planck length l p plays an important role in quantum gravity. General belief is that the Planck length may be the minimal observable length [1,2]. So it is natural to take the Planck length as a universal constant. But this seems leading to a puzzle, saying that the Lorentz symmetry at Planck scale is not preserved since the length is obviously not an invariant under linear Lorentz boost. One of approaches to solve this paradox is non-linear special relativity or doubly special relativity (DSR) [3], which may preserve the relativity principle and at the same time treat Planck energy as an invariant. In this framework, Lorentz symmetry is deformed such that the usual energymomentum relation or dispersion relation may be modified at Planck scale. As pointed out in [4], a general modified dispersion relation may be written as where f 1 and f 2 are two functions of energy from which a specific formulation of boost generator can be defined. η is a dimensionless parameter(We set = c = 1 through the whole paper) characterizing the strength of the correction. It is also interesting to see that this formula can be incorporated into a general relativity framework, see ref. [5]. Besides this, modified dispersion relations and its implications have been greatly investigated in recent years, references can be found in [6,7,8,9,10,11,12,13,14]. In particular, it has been known that MDR may change the thermodynamical properties of black holes greatly so as to provide novel mechanism for understanding the late fate of the black hole evaporation [15,16]. In particular, in [15]  In this paper, we intend to extend above analysis to (A)dS Schwarzschild black holes. We firstly review that the usual Hawking temperature of (A)dS Schwarzschild black holes can be heuristically derived by employing the standard dispersion relation as well as extended uncertainty principle (EUP) to radiation particles in the vicinity of horizon. Then we investigate how a general form of MDR may affect the temperature as well as the entropy of black holes in section three. The combination of both effects due to MDR and the generalized EUP (GEUP) is also presented.

II. UNCERTAINTY PRINCIPLES AND HAWKING TEMPERATURE
It has been argued in [17,18] that there is an intrinsic uncertainty about the Schwarzschild radius R for those photons in the vicinity of horizon. With the use of this fact one can heuristically derive the the Hawking temperature of Schwarzschild black holes. However, as pointed out in Ref. [19,20], the usual uncertainty principle can not be naively applied to the space with large length scales like as in (A)dS space. As a result, to properly obtain the temperature of (A)dS Schwarzschild black holes in this manner the usual uncertainty relations should be extended to include the effects of the cosmological constant, which may be named as the extended uncertainty principle (EUP) [21,22]. It can be written as where β is a dimensionless real constant of order one, and L is the characteristic large length scale related to the cosmological constant as Λ ∼ 1/L 2 . In contrast to the ordinary uncertainty relation, in EUP there exists an absolute minimum for the momentum uncertainty However, it is easy to see that for a very large L, Eq.(2) goes back to the usual Heisenbergs uncertainty relation.

Now, we consider a 4-dimensional AdS black hole with the metric
where The event horizon r + can be obtained by setting N 2 = 0. Now, using the standard results in statistical mechanics we expect that the energy of photons emitted from the horizon can be identified as the characteristic temperature of the Hawking radiation, namely [17,23] T where p is the momentum of photons emitted from the horizon. Next, we propose that photons emitted from the black hole satisfy the extended uncertainty principle(EUP). By modelling a black hole as a black box with linear size r + , the position uncertainty ∆x of photons emitted from the black hole is just the horizon r + , i. e.
while the momentum of photons in a quantum mechanical region approximately satisfy the relation p ∼ ∆p. Then with (3), (6)and (7), we immediately obtain the Hawking temperature where a "calibration factor" 4π is introduced [17] and the parameter β 2 is set to 3 for four dimensional black holes .
It is straightforward to obtain the temperature for dS black holes by setting L 2 → −L 2 since the cosmological constant Λ ∼ 1/L 2 [24], As pointed out in [22], it is easy to see that in the AdS case, the temperature has an absolute minimum due to EUP, while in dS case, there exists a maximal radius for black hole horizon but no minimal one. Nevertheless, in both cases the temperature will suffer from the divergency as the size of the horizon approaches to zero. This is a unsatisfactory point implying that the conventional picture of Hawking radiation may not be applicable to the late stage of black hole evaporation. To provide a more reasonable picture for this, in next section we propose to modify the usual dispersion relation for photons and discuss its possible impact on the thermodynamics of (A)dS black holes.

III. THE IMPACT OF MDR ON BLACK HOLE PHYSICS
It has been studied that the existence of a minimum length can prevent black holes from total evaporation [17,25], where the generalized uncertainty principle(GUP) plays an essential role. In this section, we show that MDR may provide a similar mechanism to describe the late stage of (A)dS black hole radiation in a reasonable manner. As far as the uncertainty relations is concerned, we will first consider the EUP case and then turn to the GEUP one.

A. EUP case
After setting m 0 = 0 in (1) for photons we may rewrite the general form of modified dispersion relations as As discussed in previous section, we expect that the energy of photons emitted from black holes can be identified as the Hawking temperature, but with MDR this identification will lead to a recursion relation for the black hole temperature Similarly, through the EUP in Eq.(3) the Hawking temperature of (A)dS black holes can be written as where T 0 is given by It is easy to see that for the low energy case both functions f 1 and f 2 approach to one, the temperature in Eq.(12) goes back to the original one. However, at high energy level the temperature will receive important corrections and the expression depends on the specific form of f 1 and f 2 . For explicitness, we take the ansatz f 2 1 = 1 − (l p E) 2 and f 2 2 = 1. Then Eq.(12) becomes From this equation we obtain the Hawking temperature as where M p = l −1 p (The other solution with plus sign ahead of the square root is ruled out as it does not provide reasonable physical meanings). From this equation it is easy to see that for large black holes where T 0 ≪ M p , the modified temperature goes back to the usual one T ∼ T 0 . However, when the temperature increases with the evaporation, we find that the modified temperature has a upper limit T ≤ M p / √ 2 and the inequality saturates when The corresponding radius of the black hole horizon reaches a minimal value r + = l p /2π. This situation is similar to the case presented in [15] while the difference here is that due to EUP, T ∼ T 0 is also bounded from below with a minimum value for large black holes. The existence of the minimal size of horizon implies that black hole maybe have a final stable state and this can be testified by calculating the heat capacity of the (A)dS black holes, which turns out to be where the upper sign and lower sign correspond to AdS and dS black holes respectively.
Obviously the heat capacity vanishes as T 0 = M p /2.
As a result, we argue that MDR may provide a mechanism to prevent (A)dS black holes from total evaporation such that an explosive disaster of black holes can be avoided.
In the end of this section we briefly discuss the correction to the entropy of (A)dS black holes due to MDR. It is expected that the first thermodynamical law still holds for large black holes (i.e. 8G ≪ A). Plugging Eq.(15) into dM = T dS, we have where G = 1/(8πM 2 p ) and A = 4πr 2 + is the area of the horizon. Eq.(17) can be written in a simpler way, Now using the condition for large black holes 8G ≪ A ≪ L 2 , we can obtain the entropy of the black hole, The first term is the conventional Bekenstein-Hawking entropy while the other terms are corrections due to MDR. It is interesting to notice that the entropy has a logarithmic term, agreement with the results obtained in string theory and loop quantum gravity [26,27,28,29,30].

B. GEUP case
In previous section we have demonstrated that an appropriate form of MDR will provide a cutoff for the temperature of (A)dS black holes. Definitely we may consider this effect in the context of generalized EUP(GEUP), which is given by [22] ∆x∆p ≥ 1 + α 2 l 2 where α is a new dimensionless parameter with a "± ′′ sign corresponding to AdS and dS black holes respectively. As shown in [22], for GEUP one has the inequality where Then, one can find that ∆x has the absolute minimum Now closely following the procedure in previous section, one can obtain a modified Hawking temperature as where T GEU P = 1 4π with β 2 = 3. We notice that Eq.(24) is just an extension of Eq.(12) with T 0 → T GEU P .
In hence, if one fixes both functions f 1 and f 2 as given in the previous section, the final modified temperature of black holes becomes Comparing Eqs. (25) and (26), it is interesting to notice that this specific form of MDR leads to a similar modification to the temperature as GEUP . Thus, both MDR and GEUP may independently provide a upper limit for the temperature of black holes. In the context of GEUP this maximal value is given by where α is supposed to be non-zero and order one (α → 0 leading to T GEU P → T 0 as seen from Eq.(25)). Our above discussion indicates that the combination of MDR and GEUP does not change the whole picture obtained in the subsection of EUP case greatly, but provides further modifications to the final temperature as well as the mass of black holes.

IV. SUMMARY AND DISCUSSIONS
In this paper we have investigated the impact of modified dispersion relations on the thermodynamics of (A)dS black holes. We have shown that MDR contributes corrections to the usual Hawking temperature as well as the entropy of black holes. Such corrections may play an important role in the understanding of the final fate of black holes. In particular, it provides a vanishing heat capacity at the late stage of black hole evaporation such that it can prevent black holes from total evaporation, but arriving a stable state with a minimum horizon radius and the remnant can be treated as a candidate for cold dark matter.
It is worthwhile to point out that through the paper we identify the expectation value of the energy as the temperature of black holes, which is a standard relation in statistics for radiation particles. However, as pointed out in [31] and [32], MDR may also change the statistical properties of the ensemble such that E ∼ T is not strictly satisfied. It may be modified as E ∼ T (1 + δlp 2 T 2 ). Therefore, more exact results of the Hawking temperature maybe have to take these modifications into account. However, a delicate calculation shows that this modification will not change the main picture about the final fate of black holes as we present here.
Through the paper, we choose a specific form of MDR in which both energy and momentum of particles are bounded. It is completely possible to extend our strategy to other MDR forms and a parallel analysis should be straightforward if such forms could provide a upper limit for the energy of particles as expected from the side of doubly special relativity. Furthermore, our discussions are applicable to (A)dS black holes in higher dimensional spacetime once the parameter β in EUP is properly fixed.