It is well known that one way to study canonical quantum cosmology is through the Wheeler DeWitt (WDW) equation where the quantization is performed on the minisuperspace variables. The original ideas of a deformed minisuperspace were done in connection with noncommutative cosmology, by introducing a deformation into the minisuperspace in order to incorporate an effective noncommutativity. Therefore, studying solutions to Cosmological models through the WDW equation with deformed phase space could be interpreted as studying quantum effects to Cosmology. In this chapter, we make an analysis of scalar field cosmology and conclude that under a phase space transformation and imposed restriction, the effective cosmological constant is positive. On the other hand, obtaining the wave equation for the noncommutativity Kantowski-Sachs model, we are able to derive a modified noncommutative version of the entropy. To that purpose, the Feynman-Hibbs procedure is considered in order to calculate the partition function of the system.
- quantum cosmology
- thermodynamics of black holes
Since the initial use of the Hamiltonian formulation to cosmology, different issues have been studied. In particular, thermodynamic properties of black holes, classical and quantum cosmology, dynamics of cosmological scalar fields, and the problem of cosmological constant among others. In this chapter, we present some results in deforming the phase space variables, discussing recent advances on this special topic by presenting three models. In the first model (Section 2), we analyze the effects of the phase space deformations over different scenarios, we start with the noncommutative on Λ cosmological and comment on the possibility that the origin of the cosmological constant in the (4 + 1) Kaluza-Klein universe is related to the deformation parameter associated to the four-dimensional scale factor and the compact extra dimensions. In Section 3, we study the effects of phase space deformations in late time cosmology. To introduce the deformation, we use the approach given in Refs. . We conclude that for this model an effective cosmological constant Λ
In Section 4, the thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state are explicitly given, and the effect of noncommutativity and quantum correction is discussed; in this sense, the goal of this section is to explore how these considerations introduced in Bekenstein-Hawking (BH) entropy change the thermodynamic information contained in this new fundamental relation. Under these considerations, Section 4 examines the different thermodynamic equations of state and their behavior when considering the aforementioned modifications to entropy.
In this chapter, we mainly pretend to indulge in recollections of different studies on the noncommutative proposal that has been put forward in the literature by the authors of this chapter [2–4]; in this sense, our guideline has been to concentrate on resent results that still seem likely to be of general interest to those researchers that are interested in this noncommutative subject.
2. Model 1: Kaluza-Klein cosmology with Λ
Let us begin by introducing the model in a classical scenario which is an empty (4+1) theory of gravity with cosmological constant Λ as shown in Eq. (1). In this setup, the action takes the form:
where are the coordinates of the 4-dimensional spacetime and
For the purposes of simplicity and calculations, we can rewrite this Lagrangian in a more convenient way:
where the new variables were defined as
and . The Hamiltonian for the model is calculated as usual and reads
which describes an isotropic oscillator-ghost-oscillator system. A full analysis of the quantum behavior of this model is presented in Ref. .
2.1. Noncommutative model
As is well known, there are different approaches to introduce noncommutativity in gravity . In particular, to study noncommutative cosmology [6, 7], there exist a well-explored path to introduce noncommutativity into a cosmological setting . In this setup, the noncommutativity is realized in the minisuperspace variables. The deformation of the phase space structure is achieved through the Moyal brackets, which are based on the Moyal product. However, a more appropriate way to introduce the deformation is by means of the Poisson brackets rather than the Moyal ones.
The most conventional way to understand the noncommutativity between the phase space variables (minisuperspace variables) is by replacing the usual product of two arbitrary functions with the Moyal product (or star product) as
The noncommutative deformation has been applied to the minisuperspace variables as well as to the corresponding canonical momenta; this type of noncommutativity can be motivated by string theory correction to gravity [6, 8]. In the rest of this model, we use for the noncommutative parameters and .
If we consider the following change of variables in the classical phase space
it can be verified that if obeys the usual Poisson algebra, then
Now that we have defined the deformed phase space, we can see the effects on the proposed cosmological model. From the action Eq. (4), we can obtain the Hamiltonian constraint Eq. (6); inserting relations Eq. (11), a Wheeler DeWitt (WDW) equation can be constructed as:
By a closer inspection of the equation, it is convenient to make the following definitions:
With these definitions, we can rewrite Eq. (12) in a much simpler and suggestive form:
which is a two-dimensional anisotropic ghost-oscillator . From Eq. (14), we can see that the terms can be associated to a minimal coupling term as is done in electromagnetic theory. From this vector potential, we find that and the vector potential
We found that
This was obtained by a definition of the effective cosmological constant as was done in Section 2, to finally get a redefinition of the effective cosmological constant due to noncommutative parameters:
Now if we choose the case
We can see from Eq. (17) that the noncommutative parameter
3. Model 2: Scalar field cosmology
Let us start with a homogeneous and isotropic universe with a flat Friedmann-Robertson-Walker (FRW) metric:
Now, we make the following change of variables:
where . Then the Hamiltonian is
with . To find the dynamics, we solve the equations of motion; for this model, it can easily be integrated .
To construct the deformed model, we usually follow the canonical quantum cosmology approach, where after canonical quantization , one formally obtains the WDW equation. In the deformed phase space approach, the deformation is introduced by the Moyal brackets to get a deformed Poisson algebra. To construct a deformed Poisson algebra, we use the approach given in Refs. [1, 9]. We start with the same transformation on the classical phase space variables that satisfy the usual Poisson algebra as shown in Section 2.1, Eqs. (10) and (11). With this deformed theory in mind, we first calculate the Hamiltonian which is formally analogous to Eq. (21) but constructed with the variables that obey the modified algebra Eq. (11)
where we have used the change of variables Eq. (10) and the following definitions:
Written in terms of the original variables, the Hamiltonian explicitly has the effects of the phase space deformation. These effects are encoded by the parameters
To obtain the dynamics for the model, we derive the equations of motion from the Hamiltonian Eq. (22). The solutions for the variables
where . For , the hyperbolic functions are replaced by harmonic functions. There is a different solution for , the solutions in the “
where we have taken . For the case , the hyperbolic function is replaced by a harmonic function. For the case , the volume is given by
where is the initial volume in the “
As already discussed, phase space deformation gives two physical descriptions. If we say that both descriptions should be equal, then comparing the late time behavior for the two frames with the scale factor of de Sitter cosmology, an effective positive cosmological constant exists and is given by
This result is the same as the one obtained from the WDW formalism of Kaluza-Klein cosmology. Therefore, one can start taking seriously the possibility that noncommutativity can shed light on the cosmological constant problem.
4. Model 3: Thermodynamics of noncommutative quantum Kerr black hole
Thermodynamics of black holes has a long history, focusing mainly on the problem of thermodynamic stability. It is known for a long time that this problem can be extended beyond the asymptotically flat spacetimes . For example, in de Sitter spacetimes, thermodynamic information of black holes exhibit important differences with the previous case [14, 15]. Gibbons and Hawking found that, in analogy with the asymptotically flat space case, such black holes emit radiation with a perfect blackbody spectrum and its temperature is determined by their surface gravity. However, a feature of de Sitter space is that exists a cosmological event horizon, emitting particles with a temperature which is proportional to its surface gravity. The only way to achieve thermal equilibrium is when both surface gravities are equal, which corresponds to a degenerate case [16, 17].
Regarding AdS manifolds, it was shown that thermodynamic stability of black holes in this spacetime can be achieved . In this manifold, gravitational potential produces a confinement for particles with nonzero mass, which acts as an effective cavity of finite volume, containing the black hole. An important feature of black holes in AdS manifolds is that their heat capacity is positive, opposite to the asymptotically flat case; additionally, this positiveness allows a canonical description of the system.
It is also known that thermodynamic stability of black holes is related with dynamical stability of those systems, which brings an additional motivation to study it. For example, in the asymptotically flat spacetime case, it is well known that Schwarzschild black holes are thermodynamically unstable, although they are dynamically stable . For AdS spacetimes, however, it is known that both thermodynamic and dynamical stability are closely related [20, 21].
In this study, we study black holes in asymptotically flat spacetime, whereby it seems very legitimate to ask whether corrections like the above discussed noncommutativity or even semiclassical ones can modify thermodynamic properties of black holes in order to have thermodynamic stable systems.
In a number of studies [22–24], black hole entropy proposed by Bekenstein and Hawking is postulated to be the fundamental thermodynamic relation for black holes, which contains all thermodynamic information of the system. Under this assumption, corresponding classical thermodynamic formalism is constructed, finding that its thermodynamic structure resembles ordinary magnetic systems instead of fluids.
4.1. Schwarzschild and Kerr black holes
As previously discussed, it is well known that for an asymptotically flat spacetime, temperature of black holes is proportional to its surface gravity
where, , , and . The area of the event horizon of a black hole is given by . Applying for the elements of the metric tensor given in Eq. (31), the resulting area is:
Assumed thermodynamic fundamental relation for Kerr black holes is found substituting the above result in Eq. (30); where
where the following constants appear:
where the role of temporal
Next, the Wheeler DeWitt (WDW) equation for Kantowski-Sachs metric with the above parametrization of the Schwarzschild interior solution is found, along with the corresponding Hamiltonian of the system
However, we are not interested in the usual case, rather our point of interest is the solution that can be found when the symplectic structure of minisuperspace is modified by the inclusion of a noncommutativity parameter between the coordinates
These modifications allow us to redefine the coordinates of minisuperspace in order to obtain a noncommutative version of the WDW equation:
With the wave function presented in Eq. (37) for the noncommutative Kantowski-Sachs cosmological model, a modified noncommutative version of the entropy can be obtained. In order to calculate the partition function of the system, the Feynman-Hibbs procedure is considered . Starting with the separated differential equation for
In this equation, the exponential in the potential term is expanded up to second order in
where is the mean value of
The above potential allows us to calculate the canonical partition function of the system:
This partition function allows us to calculate any desired thermodynamic observable by means of the thermodynamic connection of the Helmholtz free energy , with the internal energy and the Legendre transformation:
With this equation for 〈
With the aid of this relation and the Legendre transformation for Helmholtz free energy presented above, an expression for the noncommutative quantum-corrected black hole entropy can be found:
Functional form of
Likewise, natural units, , will be considered through the rest of this chapter. In this section, the previous result found in Eq. (43) for the Schwarzschild noncommutative black hole is extended to the rotating case, that is, the Kerr black hole. This is not straightforward as an analog expression for the noncommutative entropy of the rotating black hole is required, implying the application of a similar procedure to the one presented above: A diffeomorphism between the Kerr metric and some appropriated cosmological model and the procedure is presented in Ref. . To our knowledge, the implementation of this procedure has not been yet reported. However, we are interested to have an expression to study not only the static case but also the effect of angular momentum over the physical properties of the system. Our proposal to have an
A clarification must be made that Eq. (44) is not a unique valid generalization for the quantum-corrected noncommutative entropy of a rotating black hole in the neighborhood of small
Through the rest of this section, all thermodynamic expressions with superindex ⋆ will stand for noncommutative quantum-corrected quantities derived from Eq. (44), meanwhile, all thermodynamic functions without subindexes or superindexes will represent the corresponding noncommutative Bekenstein-Hawking counterparts. It is known that noncommutativity parameter
In Figure 1, plots for both Bekenstein-Hawking entropy and its quantum-corrected counterpart are presented for . Figure 1a shows plots for and ; Bekenstein-Hawking entropy is above the quantum-corrected one, in all its dominion, even in the region of low masses, where entropy is thermodynamically stable [22, 24]. Figure 1b presents the same curves as function of angular momentum instead, for ; a similar behavior can be noticed in this case. If this analysis is performed over the noncommutative relation, it is found that for small values of
4.2. Equations of state
Working in entropic representation, fundamental Bekenstein-Hawking thermodynamic relation for a Kerr black hole has the form . For these systems, partial derivatives of and play the role of thermodynamic equations of state; here,
For the entropy of the quantum-corrected entropy
The same relations for noncommutative Bekenstein-Hawking entropy are calculated as:
When the overall effect over
As previously mentioned, when values in the vicinity of Γ = 1 are considered, temperature is minimally affected by noncommutativity. We also tested smaller values of noncommutativity parameter, it was found that the maximum values that
An interesting result is obtained for angular velocity Ω, this property seems to be independent of both quantum and noncommutative corrections to entropy, namely:
In Figure 3, plots for angular velocity are presented. As this equation of state is not modified by any of the considered corrections, only one curve per graphic appears; first, in Figure 3a, Ω as a function of the black hole internal energy is presented, as can be noticed, angular velocity steadily decreases as black hole mass is increased, asymptotically going to zero. Figure 3b considers instead the case where the black hole mass is fixed at , for which Ω grows until it reaches a maximum value determined by the square root that appears in the denominator of Eq. (48), beyond this value angular velocity becomes complex.
In section 2, if we turn our attention to the case where there is no deformation on the coordinates. Taking the noncommutative parameter
From the last equation, we get the most interesting result of this section. We can see that noncommutative parameter
Then, in Section 4, an analysis on the thermodynamic properties of noncommutative quantum-corrected Kerr black holes using an approximate relation was presented. Although the resulting expressions are mathematically more complicated, the thermodynamic properties still retain the same functional behavior with respect to those calculated through Bekenstein-Hawking entropy. It can be proved that Kerr black holes do not pass through a first-order phase transition ; since the local criteria to find the critical point are not fulfilled for any value in the domain, corresponding isotherms do not exhibit van der Waals loops, and the Maxwell construction cannot be obtained; all of which are characteristic of this kind of transition. Regarding the effective noncommutativity incorporated in the coordinates of minisuperspace, outside the vicinity where , changes introduced by this parameter over the thermodynamic information of the system are relevant. For a complete analysis using this phase deformations, for example, thermodynamic response functions, thermodynamic stability, and phase transitions for Kerr black holes, see Ref. .
Eri Atahualpa Mena Barboza thanks the financial support from C.U.CI., U. de G. project Desarrollo de la investigación y fortalecimiento del posgrado 235506.