Holographic bound in covariant loop quantum gravity
We investigate puncture statistics based on the covariant area spectrum in loop quantum gravity. First, we consider Maxwell-Boltzmann statistics with a Gibbs factor for punctures. We establish formulae which relate physical quantities such as horizon area to the parameter characterizing holographic degrees of freedom. We also perform numerical calculations and obtain consistency with these formulae. These results tell us that the holographic bound is satisfied in the large area limit and correction term of the entropy-area law can be proportional to the logarithm of the horizon area.
Second, we also consider Bose-Einstein statistics and show that the above formulae are also useful in this case. By applying the formulae, we can understand intrinsic features of Bose-Einstein condensate which corresponds to the case when the horizon area almost consists of punctures in the ground state. When this phenomena occurs, the area is approximately constant against the parameter characterizing the temperature. When this phenomena is broken, the area shows rapid increase which suggests the phase transition from quantum to classical area.
Canonical quantization of general relativity has a long history beginning in the 1960s Wheeler (). Basically, metric and its conjugate momentum had been used as canonical variables in these early days. In this case, the Hamiltonian constraint is nonpolynomial about these variables. So, it is almost impossible to solve its quantized counterpart. In complex (), it was shown that if we use the complex Ashtekar connection and its conjugate, the Hamiltonian constraint can be written as polynomial about these variables. Surprisingly, it was found that Wilson loop for this connection is a solution of quantized Hamiltonian constraint Jacobson (). Using Wilson loop, spin network, basic ingredients of the loop quantum gravity(LQG), has been constructed Smolin (). Discrete area spectrum is one of the main predictions in LQG Rovelli (); Ash1 ().
However, it has been recognized that the reality conditions for the physical quantities to be real are difficult to be solved. Then, the SU(2) real connection has been introduced where imaginary number in the complex Ashtekar connection was replaced by the real parameter called the Barbero-Immirzi(BI) parameter Immirzi (). Although the Hamiltonian constraint becomes nonpolynomial, this complication can be relieved if we rewrite it using the technique developed in Thiemann’s method (). If we apply it in the symmetry-reduced model, we can discuss singularity avoidance which has been paid much attention LQC ().
The microscopic origin of black hole entropy in LQG had also been discussed in Rovelli-entropy (), where the number of degrees of freedom of the edge configuration for a fixed SU(2) area spectrum was counted. Then, Ashtekar, Baez, Corichi, and Krasnov refined this idea based on the isolated horizon framework (so-called ABCK framework where the isolated horizon itself was described by U(1) connection ABCK (); isolated ()) and determined to satisfy the Bekenstein-Hawking entropy-area law where , and are black hole entropy, horizon area and the gravitational constant, respectively. Here, ambiguity of turned out to be the merit of using the real connection. Including the correction of error in original counting Domagala (), or ambiguity in counting Alekseev (); Mitra (); Tamaki (), relation with the quasinormal mode Schiappa (); Dreyer (); Hod (), various aspects have been discussed related to the ABCK framework projection (); Sahlmann (); Ansari ().
The situation slightly changed when it was found that the isolated horizon can be written using the SU(2) connection Engle (). This means that the horizon Hilbert space can be described by the SU(2) Chern-Simons state. Its dimension is written by the spin freedom and the level of the Chern-Simons state . Using a suitable analytic continuation of these variables to complex variables, it was obtained that the complex Ashtekar connection is desirable to reproduce BTZ (); analytic (). Furthermore, when we introduce the geometric temperature by demanding the horizon state be a Kubo-Martin-Schwinger state, we can also arrive at the complex connection temperature ().
Is there an essential reason why the complex Ashtekar connection is preferable? One of the reasons would be that the covariance is satisfied in this connection while it is violated in the real connection Samuel (). This should be taken seriously, and we should pay attention how to choose the Lorentz covariant connection which has been investigated in Choice (). The connection obtained in Choice () is called shifted connection which includes the BI parameter. Surprisingly, the Hamiltonian constraint can be written as a polynomial equation in this case again. Using a shifted connection, covariant LQG has been formulated, and covariant area spectrum has been obtained areaCLQG (); HilbertCLQG (). Making the consistent relation between covariant LQG and the spin foam models became the important realm recently reviewCLQG (). We should also notice that the covariant area spectrum does not include the BI parameter although the shifted connection itself does. Then, it is natural to ask whether or not we can obtain consistency with the entropy-area law if we consider counting microscopic freedom of black holes in covariant LQG. In countingCLQG (), by assuming the horizon area consists of the minimum area eigenvalue, it was argued that the answer is in the affirmative.
Here, we consider the generality of the holographic bound and argue the correction term of the entropy-area law discussed in first (); gas (). These are motivated by the quasilocal first law of black hole thermodynamics where the quasilocal energy is defined using the horizon area quasilocal (). Then, regarding the puncture, which is an intersection of the edge at the horizon, as a particle, we can argue its statistical mechanics. One of the important points in first (); gas () is that if we assume the degeneracy of matter fields close to the horizon as where is a dimensionless constant, must approach in the large area limit when punctures are indistinguishable. The correction term of the entropy-area law is basically proportional to unless we assume the special form for the fugacity. Then, our concerns are whether these properties hold or not in the covariant area spectrum. The answer is in the affirmative for the holographic bound while the correction term depends on the ambiguity of the covariant area spectrum as we discuss later.
This paper is organized as follows. In section II, we introduce tools necessary for constructing puncture statistics following first (); gas (). In Sec. III, we consider the case when Maxwell-Boltzmann statistics with a Gibbs factor for punctures is assumed. We establish formulae which relate physical quantities such as horizon area to the parameter characterizing holographic degrees of freedom. We also perform numerical calculations and obtain consistency with these formulae. These results show that the holographic bound is saturated in the large area limit and that the correction term of the entropy-area law can be proportional to . In section IV, we consider the case when Bose-Einstein statistics is assumed and argue that the above formulae are also useful in this case. By applying the formulae, we can understand the intrinsic features of Bose-Einstein condensate which corresponds to the case when black holes almost consist of punctures in the ground state. We show that when this phenomena occurs, the area is approximately constant against the parameter characterizing the temperature. When this phenomena is broken, the area shows rapid increase which suggests the phase transition from quantum to classical area spectrum. Concluding remarks follow in section V.
Ii Preparation for puncture statistics
Following first (); gas (), we introduce several notions necessary for arguing puncture statistics. First, we mention the quasilocal law of black hole thermodynamics which holds for the stationary observer at proper distance from the horizon quasilocal (),
where is quasilocal horizon energy of black hole. We rewrite (1) using the inverse Unruh temperature as
has been used. Here, is a half-integer associated with the puncture . Here, we use the covariant area spectrum written as
where is a half-integer with and is a real number areaCLQG (). Notice that there is no ambiguity related to . In HilbertCLQG (), it has been shown that it is enough for counting the degrees of freedom to consider the simple representation , which we assume here.
Thus, the simplest possibility is to choose , which we include considering below. The next simplest possibility would be to regard as a dependent variable of . In this case, does not necessarily correspond to the ground state, which is important when we discuss Bose-Einstein condensate as shown in gas (). Although it is an interesting possibility, it is reasonable to assume that is monotonic with as a first extension of the previous case in first (); gas (). Here, we choose as
which correspond to the cases,
in the limit , respectively. The reason why we choose a monomial or an exponential as (6) is supposed by the observation that only the qualitative behavior in the limit determines the holographic property and the correction term of the entropy-area law in first (); gas ().
Let us consider puncture statistics. In general, we do not require that the inverse temperature is equal to . We write the inverse temperature using as
where is a parameter. We only demand that vanishes in the semiclassical limit to satisfy the relation (2).
We define as the number of punctures carrying spin and as the total number of punctures. So, we have
We also define as the number of holographic degrees of freedom for a given configuration . Here, we assume
where and is a free parameter. We suppose that the freedom comes from the matter fields close to the horizon motivated by the entanglement entropy hypothesis Bombelli ().
Iii Maxwell-Boltzmann statistics
We include the Gibbs factor in the Maxwell-Boltzmann statistics. The case without the Gibbs factor is discussed later in this section. Then the canonical partition function is given by
Here, we abbreviate the puncture index and write the spin index in the quasilocal energy as
Using (10), we can express the partition function as
Here, we defined
We introduce the fugacity where is a chemical potential. In this case, we can express the grand canonical partition function by
The total number and the mean energy are
We can express the entropy as
As we said above, we required in the limit . So, if , the correction term of the entropy-area law proportional to disappears as pointed out in gas ().
Since one of the purposes using the covariant area spectrum is to investigate the correction term, we consider the case . In other treatments, it is often argued that the correction term proportional to appears Carlip (); Das (); Sen (). From (17) and (18), we have
Thus, in discussing the ratio between and the correction term, it is enough if we investigate the ratio between and . Since plays a minor role for this reason, we set below, for simplicity.
How can we estimate the relation between and ? We should first notice that convergence of the sequence (14) highly depends on . So, our strategy is to analyze the dependencies of and as a function of for obtaining the relation between and .
Since it would be difficult in calculating (14) exactly, we suppose using numerical calculation. In this case, it is important to know we should sum up, which is a key to understand above property. In concrete, we assume that we need to sum up from to in obtaining the value for enough precision toward the true value , e.g., relative error . To accomplish the above task, we need to estimate the dependence of on as a first step, which is also a difficult task, in general. However, we can expect that in the limit , and we can estimate (14) using the asymptotic form in the limit . For this reason, we assume .
Let us consider the cases (7). If we have in , we can write as
where . If we define and , we can rewrite as
where and .
Using these notations, we comment on following important properties.
If we reduce ,
(i) we should change in preserving the same precision.
(ii) we obtain approximately.
To understand these properties, we should first notice that interval in the sum (22) becomes while does not change by (i). This means that there are terms we should sum up in the former case while terms in the latter case in (22). Thus, we obtain approximately. Since , we also have approximately. Therefore, we have same relative error and the precision is preserved.
For this approximation to be valid, following conditions should hold.
Changing is negligible.
does not have the property,
, or .
The former assumption is implicitly used when we use the asymptotic form in the limit . The latter assumption holds when is small enough in the above case.
From these consideration, we obtain
Since , we also have
Next, we consider the case in . In this case, we can write as
As in the previous case, if we want to obtain , we need to change the number of terms we should sum up from to for preserving the precision. This means that should change to satisfy
So we have . This means . As a result, we have
where is a constant. So we have
The case of (28) is included in the limit . It is surprising that fluctuations of both energy and horizon area are summarized in this simple manner.
In the above estimate, we used the asymptotic form in the limit . Thus, it is desirable to check consistency using a numerical calculation. For this purpose, we choose
which correspond to the cases,
in the limit , respectively. However, we stress that we use the exact expression (12) by substituting (30). We show -, relations in Figs. 1 which have complete consistency with (23), (24), (27), and (28). Especially, in all cases, for . So we confirmed that the holographic bound is saturated, i.e., , in the semiclassical limit where the temperature should approach Unruh temperature . This is a generalization of the result in gas ().
Then, we should also notice the results for derived by (27) and (28). To check its accuracy, we also show that is almost constant for in Fig. 2. The deviation from constant for large would be due to it from the asymptotic form. So, we obtain the correction if we use the freedom . This is also our new results obtained by considering the covariant area spectrum.
Iv Bose-Einstein statistics
Here, we consider Bose-Einstein statistics as a candidate of the puncture statistics. First, we discuss the case as an extension of the case in Maxwell-Boltzmann statistics. In this case, the grand canonical partition function can be written as
So, we have
We can perform an analogous discussion in the previous section. For example, if we have in , we replace by in (22). Then, the discussion below (22) holds, and we obtain (23) and (24). Similarly, for , we obtain (27) and (28).
The conclusions are that we have a holographic bound in the large area limit, and the correction term of the entropy-area law behaves same as the case in Maxwell-Boltzmann statistics qualitatively. The result for is consistent with gas () where the correction term is shown to be proportional to both in Maxwell-Boltzmann statistics and in Bose-Einstein statistics. We have shown that these can be understood in an unified way including the cases in covariant area spectrum.
Next, we discuss the case . The grand canonical partition function can be written as
So, we should require
Since we assumed that is monotonic function with , we obtain
as an extension of gas (). So, the high temperature region with should be described by a Maxwell-Boltzmann statistics. We concentrate on the case with .
We consider whether or not above discussion can be extendible for the case . We define
The rhs of this equation includes two independent parameters and . To avoid complication, we set below. Then, if we have in , we replace by to perform an analogous discussion.
However, in this case, does not necessarily satisfy Conditions in the previous section. This depends on the ratio between and . In concrete, if is small enough, can be much smaller than . Of course, if we take , we can obtain same conclusion as above. Below, we consider the case where Conditions are violated.
We can understand physical meaning of Conditions by using the number of punctures for general . Here, is represented by
If , the mean number of the grand state can be approximated as
which is quite large. Thus, is possible, which corresponds to the case .
Moreover, the total sum of the mean number , can be much smaller than which corresponds to the Bose-Einstein condensate state defined as a state where the horizon is almost dominated by spin puncture. Since () can satisfy Conditions, can be estimated by following the analogous discussion as above. That is, if we have or in , we can estimate that or , respectively. So the criteria for the Bose-Einstein condensate are
We show the relation between and its number density corresponding to or for and in Fig. 3. Although both are the cases of the Bose-Einstein condensate, decays of make a contrast in these cases.
We are interested in changes of physical quantities caused by the Bose-Einstein condensate. We show as a function of for the case in Fig. 4. Surprisingly, plateau appears for large while increases as following (24) or (28) for small . If we use the criteria (41) and (42), the Bose-Einstein condensate occurs for all in this diagram. Then, how can we understand this plateau?
We can discuss that in the plateau corresponds to the case where almost consists of the area spectrum , . The reason is as follows. To estimate the area consisting of the area spectrum , we use
If in , we have
Similarly, we can consider the case in and this case is included in the limit in (45). So the condition for can be estimated as
We can find that this is consistent with the results in Fig. 4. This result is also newly revealed in this paper.
If we consider what observables in black hole physics are, we may adopt the criterion (46) as a condition for the Bose-Einstein condensate. When this condition is broken, shows rapid grow as . If we can discuss this phenomena as a phase transition from the quantum black hole to the classical black hole, it is very interesting.
V Conclusion and discussion
We have investigated the puncture statistics based on the covariant area spectrum. First, we have considered Maxwell-Boltzmann statistics with a Gibbs factor for punctures. If we assume the fugacity , we have reconfirmed the results in gas () that the correction term of the entropy-area law disappears for . When we assume the fugacity , we have established formulae which relate physical quantities such as horizon area to the parameter characterizing holographic degrees of freedom using asymptotic form of the area spectrum in the large spin limit. We have also performed numerical calculations and obtained consistency with these formulae. From these results, we have obtained that the holographic bound is satisfied in the large area limit which is the extension of the previous research. We have found that the correction term of the entropy-area law can be proportional to the logarithm of the horizon area as it has been pointed out in other researches.
Second, we have also considered Bose-Einstein statistics and shown that above formulae are also useful in this case. By applying the formulae, we have understood intrinsic features of the Bose-Einstein condensate which correspond to the case when the horizon area almost consists of punctures in the ground state. We have shown that when this phenomena occurs, the area is approximately constant against the parameter characterizing the temperature. When this phenomena is broken, the area shows a rapid increase as , which suggests the phase transition from quantum to classical area.
What should we consider as a next step? Although we have assumed that is a dependent function of , the validity should be checked by other method. For example, to reveal the property of in the covariant area spectrum and the puncture statistics, it is important to investigate the Hawking radiation as in spectroscopy () which is one of our future work. It is also interesting to discuss possibility of the phase transition using covariant area spectrum as in phase (). In a long span, we should also investigate a covariant volume spectrum, which would lead us to the covariant loop quantum cosmology. This must be the interesting arena in the next decade.
We would like to thank Kei-ichi Maeda for continuous encouragement. We are thankful for financial support from the Nihon University.
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