Space-Time Structure of Loop Quantum Black Hole
In this paper we have improved the semiclassical analysis of loop quantum black hole (LQBH) in the conservative approach of constant polymeric parameter. In particular we have focused our attention on the space-time structure. We have introduced a very simple modification of the spherically symmetric Hamiltonian constraint in its holonomic version. The new quantum constraint reduces to the classical constraint when the polymeric parameter goes to zero. Using this modification we have obtained a large class of semiclassical solutions parametrized by a generic function . We have found that only a particular choice of this function reproduces the black hole solution with the correct asymptotic flat limit. In the semiclassical metric is regular and the Kretschmann invariant has a maximum peaked in . The radial position of the pick does not depend on the black hole mass and the polymeric parameter . The semiclassical solution is very similar to the Reissner-Nordström metric. We have constructed the Carter-Penrose diagrams explicitly, giving a causal description of the space-time and its maximal extension. The LQBH metric interpolates between two asymptotically regions, the region and the region. We have studied the thermodynamics of the semiclassical solution. The temperature, entropy and the evaporation process are regular and could be defined independently from the polymeric parameter . We have studied the particular metric when the polymeric parameter goes towards to zero. This metric is regular in and has only one event horizon in . The Kretschmann invariant maximum depends only on . The polymeric parameter does not play any role in the black hole singularity resolution. The thermodynamics is the same.
Quantum gravity is the theory attempting to reconcile general relativity and quantum mechanics. In general relativity the space-time is dynamical, then it is not possible to study other interactions on a fixed background because the background itself is a dynamical field. The theory called “loop quantum gravity” (LQG) (1) is the most widespread nowadays. This is one of the non perturbative and background independent approaches to quantum gravity. LQG is a quantum geometric fundamental theory that reconciles general relativity and quantum mechanics at the Planck scale and we expect that this theory could resolve the classical singularity problems of General Relativity. Much progress has been done in this direction in the last years. In particular, the application of LQG technology to early universe in the context of minisuperspace models have solved the initial singularity problem (2), (3).
Black holes are another interesting place for testing the validity of LQG. In the past years applications of LQG ideas to the Kantowski-Sachs space-time (4) lead to some interesting results in this field. In particular, it has been showed (5) (6) that it is possible to solve the black hole singularity problem by using tools and ideas developed in full LQG. Other remarkable results have been obtained in the non homogeneous case (7).
There are also works of semiclassical nature which try to solve the black hole singularity problem (8),(9), (9). In these papers the authors use an effective Hamiltonian constraint obtained replacing the Ashtekar connection with the holonomy and they solve the classical Hamilton equations of motion exactly or numerically. In this paper we try to improve the semiclassical analysis introducing a very simple modification to the holonomic version of the Hamiltonian constraint. The main result is that the minimum area (11) of full LQG is the fundamental ingredient to solve the black hole space-time singularity problem in . The sphere bounces on the minimum area of LQG and the singularity disappears. We show the Kretschmann invariant is regular in all space-time and the position of the maximum is independent on mass and on polymeric parameter introduced to define the holonomic version of the scalar constraint. The radial position of the curvature maximum depends only on and .
This paper is organized as follows. In the first section we recall the classical Schwarzschild solution in Ashtekar’s variables and we introduce a class of Hamiltonian constraints expressed in terms of holonomies that reduce to the classical one in the limit where the polymer parameter . We solve the Hamilton equations of motion obtaining the semiclassical black hole solution for a particular choice of the quantum constraint. In the third section we show the regularity of the solution by studying the Kretschmann operator and we write the solution in a very simple form similar to the Reissner-Nordström solution for a black hole with mass and charge. In section four we study the space-time structure and we construct the Carter-Penrose diagrams. In section five section we show the solution has a Schwarzschild core in . In section six we analyze the black hole thermodynamic calculating temperature, entropy and evaporation. In section seven we calculate the limit of the metric and we obtain a regular semiclassical solution with the same thermodynamic properties but with only one event horizon at the Schwarzschild radius. We analyze the causal space-time structure and construct the Carter-Penrose diagrams.
I Schwarzschild solution in Ashtekar variables
The components variables in the phase space have length dimension , , , . The Hamiltonian constraint is
Using the general relation ( is the metric on the spatial section) we obtain .
We restrict integration over to a finite interval and the Hamiltonian takes the form (6)
The rescaled variables are: , , , . The length dimensions of the new phase space variables are: , , , . From the symmetric reduced connection and density triad we can read the components variables in the phase space: , , with Poisson algebra , . We choose the gauge and the Hamiltonian constraint reduce to
The Hamilton equations of motion are
This is exactly the Schwarzschild solution inside and also outside the event horizon as we can verify passing to the metric form defined by ( contains the gravitational constant parameter ). The line element is
Introducing the solution (6) in (7) we obtain the Schwarzschild solution in all space-time except in where the classical curvature singularity is localized and except in where there is a coordinate singularity
where . To obtain the Schwarzschild metric we choose . In this way we fix the radial cell to have length and disappears from the metric. In the semiclassical LQBH metric does not disappears fixing . At this level we have not fixed but only the dimension of the radial cell. This is the correct choice to reproduce the Schwarzschild solution. We have defined the dimension of the cell in the direction to be obtaining the correct Schwarzschild metric in all space time, we will do the same choice for the semiclassical metric. With this choice will not disappears from the semiclassical metric and in particular from the solution. We will use the minimum area of the full theory to fix . For the semiclassical solution at the end of section (V) we will give also a possible physical interpretation of .
Ii A general class of Hamiltonian constrains
The correct dynamics of loop quantum gravity is the main problem of the theory. LQG is well defined at kinematical level but it is not clear what is the correct version of the Hamiltonian constraint, or more generically, in the covariant approach, what is the correct spin-foam model (13). An empirical principle to construct the correct Hamiltonian constraint is to recall the correct semiclassical limit (14). When we impose spherical symmetry and homogeneity, the connection and density triad assume the particular form given in (1). We can choose a large class of Hamiltonian constraints, expressed in terms of holonomies , which reduce to the same classical one (4) when the polymeric parameter goes towards to zero. We introduce a parametric function that labels the elements in the class of Hamiltonian constraints compatible with spherical symmetry and homogeneity. We call the constrain for the full theory and the constraint for the homogeneous spherical minisuperspace model. The reduction from the full theory to the minisuperspace model is
where the arrow represents the spherical symmetric reduction of the full loop quantum gravity hamiltonian constraint. To obtain the classical Hamiltonian constraint (4) in the limit we recall that the function satisfies the following condition
We are going to show that just one particular choice of gives the correct asymptotic flat limit for the Schwarzschild black hole. In fact the asymptotic selects the particular form of the function .
The classical Hamiltonian constraint can be written in the following form
where and ( is the extrinsic curvature, and ). The holonomies in the directions for a generic path are defined by
We define the field straight in terms of holonomies in the following way
it’s a simple exercise to verify that when (13) we obtain the classical field straight. The Hamiltonian constraint in terms of holonomies is
is the spatial section volume. We have introduced modifications depending on the function only in the field straight but this is sufficient to have a large class of semiclassical hamiltonian constraints compatible with spherical simmetry. The Hamiltonian constraint in (14) can be substantially simplified in the gauge
From (15) we obtain two independent sets of equations of motion on the phase space
Solving the first three equations and using the Hamiltonian constraint , with the time parametrization and imposing to have the Schwarzschild event horizon in , we obtain
where we have defined the quantities
Now we focus our attention on the term . The choice of this term and in particular the choice of the exponent will be crucial to have the correct flat asymptotic limit. The exponent is in the form and expanding in powers of the small parameter we obtain at large distance () (we remember that outside the event horizon the coordinate t plays the rule of spatial radial coordinate). It is straightforward to see that exists only one possible way to obtain the correct asymptotic limit and it is given by the choice . In other words we can say that any function diverges logarithmically for small and large distance ().
Let as take . In force of the correct large distance limit and in force also of the regularity of the curvature invariant in all space time, we will extend the solution outside the event horizons with the redefinition . I will come back to this extension in the next section.
A crucial difference with the classical Schwarzschild solution is that has a minimum in , and . The solution has a spacetime structure very similar to the Reissner-Nordström metric and presents an inner horizon in
For , . We observe that the inside horizon position (we recall is the Barbero-Immirzi parameter).
Now we study the trajectory in the plane and we compare the result with the Schwarzschild solution. In Fig.1 we have a parametric plot of ; we can follow the trajectory from where the classical (dashed trajectory) and the semiclassical (continuum trajectory) solution are very close. For , and (this point corresponds to the Schwarzschild radius). From this point decreasing we reach a minimum value for . From , starts to grow again until , this point corresponds to a new horizon in localized. In the time interval , grows together with and as it is very clear from the picture the solution approach the second specular black hole for . In particular we have a second asymptotic region for .
Metric form of the solution.
In this section we write the solution in the metric form and we extend that to the all space-time. We recall the Kantowski-Sachs metric is . The metric components are related to the connection variables by
We have introduced by a coordinate transformation ,
This coordinate transformation is useful to obtain the Minkowski metric in the limit . The explicit form of the lapse function in terms of the coordinate is
Using the second relation in (20) we can obtain the metric component,
The function corresponds to given in (17). The metric obtained has the correct asymptotic limit for and in fact , , . The semiclassical metric goes to a flat limit also for . We can say that LQBH interpolates between two asymptotic flat region of the space-time. The metric obtained in this paper has the correct flat asymptotic limit for and reproduce the Minkowski metric for . Both those limit are not satisfied in the work (8). The small modification introduced in the holonomy form of the Hamiltonian is necessary for those two fundamental consistency limit.
Iii LQBH in all space-time
In this section we extend the (metric) semiclassical solution obtained obtained in the previous section to all space-time. As explained in the previous subsection the metric solution has the correct flat limit for and goes to Minkowski for . Now we shaw that the Kretschmann scalar is regular in all space-time. In terms of , and the Kretschmann scalar is
In Fig.2 is plotted a graph of , it is regular in all space-time and the large behavior is the classical singular scalar .
What about ? Now we fix the parameter using the full theory (LQG). In particular we choose in such way the position of the Kretschmann invariant maximum is independent of the black hole mass. This means the sphere bounces on a minimum radius that is independent from the mass of the black hole and from and depends only on . We consider the solution and we impose the minimum area of the sphere to be equal to the minimum gap area of loop quantum gravity . With the choice we obtain a significative physical result. We have not impose to have a minimum in but we have just impose that the minimum of is the minimum area of the full theory. The minimum area of the two sphere is a result and not a request. We observe that this choice of fixes the absolute maximum and relative minimum of to be independent of the mass as this is manifest from the plot in Fig.3.
We want to provide an argument to support the choice . In the paper (15) it is shown the phase space is parametrized by and the conjugated momentum and it is shown that are both constants of motion (in our notation ). As usual in elementary quantum mechanics to derive the Heisenberg uncertainty relation, we can introduce the state , where and are the mass and momentum operators and . From the positive norm we have the discriminant, of second order in , is negative or zero. The condition on the discriminant gives . Introducing the commutator we obtain . We can calculate on semiclassical gaussian states,
and the result is (for ). Using the Heisenberg uncertainty relation we determine . If we identify we obtain , which is exactly for , and . We have introduced explicitly all the coefficients but the main result is . However the presented here is just an argument and not a proof.
At the end of section (V) we will give a physical interpretation of .
We now want underline the similarity between the equation of motion for and the Friedmann equation of loop quantum cosmology. We can write the differential equation for in the following form
From this equation is manifest that bounces on the value . This is quite similar to the loop quantum cosmology bounce (16).
As it is evident from Fig.4 the maximum of the Kretschmann invariant is independent of the mass and it is in localized. At this point we redefine the variables (with the subsequent identification ) and the metric components to bring the solution in the standard Schwarzschild form
Schematically the properties of the metric are the following,
We consider the property (28) sufficient to extend the solution in all space-time.
The solution is summarized in the following table (in the table we have not fixed the parameter ).
We have said in the previous section the metric solution has two event horizons. An event horizon is defined by a null surface . The surface is a null surface if the normal is a null vector or satisfied the condition . The last identity says that the vector is on the surface itself, in fact and . The norm of the vector is given by
In our case (29) reduces to
and this equation is satisfied where and if the surface is independent from , . The points where are and .
We can write the metric in another form which is more similar to the Reissner-Nordström space-time. The metric can be written in the following form
If we develop the metric (LABEL:metricabella) by the parameter and the minimum area at the zero order we obtain the Schwarzschild solution: , and . We have correction to the metric from the polymer parameter and also from the minimum area .
To check the semiclassical limit we calculate the perturbative expansion of the curvature invariant for small and and we obtain a divergent quantity in at any order of the development. The regularity of is a non perturbative result, in fact for small values of the radial coordinate , diverges for . (For the semiclassical solution the trace of the Ricci tensor () is not identically zero as for the Schwarzschild solution. We have calculated also this operator and we have obtained a regular quantity in ).
We conclude this section showing the independence of the pick position of Kretschmann invariant from the polymeric parameter . We have plotted the invariant and we have obtained the result in Fig.(7).
From the picture is evident the position of the Kretschmann invariant maximum is independent from .
Corrections to the Newtonian potential.
In this paper we are interested to to singularity problem in black hole physics and not to the Post-Newtonian approximation, however we want give the fist correction to the gravitational potential. The gravitational potential is related to the metric by . Developing the component of the metric in power of to the order , for fixed values of the parameter and the minimal gap area , we obtain the potential
where is defined in (18).
Iv Causal structure and Carter-Penrose diagram
In this section we construct the Carter-Penrose diagrams (17) for the semiclassical metric (LABEL:metricabella). To obtain the diagrams we will do many coordinate changing and we enumerate them from one to eight.
1) We can put the metric (LABEL:metricabella) in the form introducing the tortoise coordinate defined by :