Loop quantization of the Schwarzschild interior revisited
The loop quantization of the Schwarzschild interior region, as described by a homogeneous anisotropic Kantowski-Sachs model, is re-examined. As several studies of different –inequivalent– loop quantizations have shown, to date there exists no fully satisfactory quantum theory for this model. This fact poses challenges to the validity of some scenarios to address the black hole information problem. Here we put forward a novel viewpoint to construct the quantum theory that builds from some of the models available in the literature. The final picture is a quantum theory that is both independent of any auxiliary structure and possesses a correct low curvature limit. It represents a subtle but non-trivial modification of the original prescription given by Ashtekar and Bojowald. It is shown that the quantum gravitational constraint is well defined past the singularity and that its effective dynamics possesses a bounce into an expanding regime. The classical singularity is avoided, and a semiclassical spacetime satisfying vacuum Einstein’s equations is recovered on the “other side” of the bounce. We argue that such metric represents the interior region of a white-hole spacetime, but for which the corresponding “white-hole mass” differs from the original black hole mass. Furthermore, we find that the value of the white-hole mass is proportional to the third power of the starting black hole mass.
The singularity theorems of general relativity tell us that, under generic conditions, spacetime singularities will form. From the study of solutions to Einstein’s equations we have learned that spacelike singularities arise in two physical situations: in cosmological scenarios where we associate them with either the Big Bang or the Big Crunch, and in the interior regions of event horizons, that is, inside black holes. Just as in the cosmological scenario, FLRW cosmological spacetimes are taken as paradigmatic examples of space-times possessing a singularity, the Schwarzschild spacetime is the standard example of those singularities of the second kind. The standard interpretation within general relativity, when such singularities appear, is that the description one is using of the physical situation breaks down; general relativity is no longer valid. In terms of physical quantities, such as geometric scalars, these singularities are manifest when some of such quantities grow un-boundedly. One can therefore expect that in these physical situations, quantum gravitational effects will take over and become dominant. The problem is that we do not yet possess a complete theory that describes such quantum phenomena. A very conservative approach is to apply ‘standard’ quantization techniques to highly symmetric classical configurations, such as homogeneous spacetimes in the cosmological context and for the interior of black holes. This is the strategy that we shall consider in this manuscript. Since the interior (black hole) Schwarzschild solution can be seen as a particular example of a contracting anisotropic Kantowski-Sachs model, one natural question is to consider its quantization with the aim of probing what the fate of the final classical singularity may be.
In recent years, new quantization techniques have been applied to these minisuperpace scenarios. These techniques, motivated by loop quantum gravity (LQG) lqg (), are based on non-regular representations of the canonical commutation relations that make these quantum systems inequivalent to the standard Schrödinger representation, already at the kinematical level. In the cosmological scenario the resulting formalism is known as loop quantum cosmology (LQC) lqc (). The main features that distinguishes these quantum minisuperspace models from the standard Wheeler-De Witt (WDW) models is that, for simple systems, the singularity is generically avoided and the quantum evolution continues past the ‘would be singularity’ in a unitary way aps2 (); slqc (). These contrasting features have been rigorously understood via various analytical lqc () and numerical investigations in LQC ps12 (). It is therefore natural to ask whether similar results emerge when these technique are applied to the Schwarzschild interior. The pioneer work that embarked on this task was Ref. aa:mb (), and several other works followed Modesto (); b-k (); Campiglia (); jp ().
Unlike the regular representations, where the Stone-Von Neumann theorem warranties uniqueness (up to unitary equivalence), within the realm of loop quantizations (sometimes referred also as ‘polymer quantizations’), there are infinite inequivalent representations of the quantum constraints. In simple cases, this freedom translates into the liberty of choosing the basic variables for the quantization. In the original treatment of the black hole interior aa:mb (), the strategy followed a particular quantization of isotropic FLRW models abl (); aps1 (). As in the early quantization of cosmological models in LQC, the quantizations of black hole spacetime given in Refs. aa:mb (); Modesto (); Campiglia (); jp () suffer from lack of independence of the the auxiliary structure, such as the ‘size’ of the fiducial cell, needed for the Hamiltonian description. This results in spurious quantum gravitational effect at ultra-violet and infra-red scales cs2 (). Further, in these quantizations singularity resolution can not be consistently identified with a curvature scale cs3 (); ksbound (). A new proposal motivated by the ‘improved quantization’ in LQC aps2 () was put forward by Böhmer and Vandersloot b-k () (see also Ref.mb_cartin_khanna ()). Though this quantization is free from the auxiliary structure and results in universally bounded expansion and shear scalars ksbound (), it has the following limitation. The quantization leads to ‘quantum gravitational effects’ at the horizon due to the coordinate singularity. Nevertheless, Böhmer and Vandersloot’s prescription turns out to be free from these effects when the quantization of Kantowski-Sachs spacetime is performed in the presence of matter. Interestingly, after the would-be singularity is avoided the spacetime corresponds to a ‘charged’ Nariai spacetime in Böhmer and Vandersloot’s prescription charged (); ksphen (), a conclusion which remains unchanged for the vacuum case and hence for the Schwarzschild interior.
Given that the quantizations of black hole spacetimes based on the original LQC suffer from auxiliary structures dependence and the lack of a consistent singularity resolution scale, and the attempt based on ‘improved quantization’ in LQC results in large ‘quantum gravity effects’ at low curvatures near the horizon, none of the available quantizations of the Schwarzschild interior in LQG can be deemed satisfactory. The purpose of this manuscript is to advance a new viewpoint to construct a quantization that is free from these limitations. The main new ingredient in the construction is the realization that, in order to define the Hamiltonian formulation, it is necessary to introduce a physical length scale into the problem. This is a true scale that, in the classical limit, selects a unique classical solution and becomes nothing but the Schwarzschild radius. This scale is to be contrasted with the auxiliary length that fixes the fiducial cell needed for defining the symplectic structure. The latter being an auxiliary structure has no physical significance and should be absent in the final quantum theory.
The proposal that we shall describe in detail is motivated by recent considerations regarding the loop quantization of symmetry reduced models cs2 (); cs3 (). There, it was proposed that rather natural physical criteria be used to select various choices in the quantization procedure. In particular one should require that the final quantum theory be independent of any auxiliary structure, have a well defined high curvature regime and recover the classical theory at low curvatures. In the case of the black hole interiors, no such construction exists to date. Here we are able to construct a quantum theory that satisfies all these criteria and, even when it resembles the original treatment of aa:mb (), it possesses subtle and important differences. These are manifested in the inputs necessary for regulating the curvature in the Hamiltonian constraint using holonomies which capture the fundamental discrete nature of the underlying quantum geometry. We notice that different prescriptions need to be specified for the two anisotropic directions, with distinct dependence on the two scales at hand, namely the (new) physical scale and the auxiliary . This subtle difference makes the resulting quantum theory singularity free and independent of , thus removing dependence on any auxiliary structure. A study of the semiclassical ‘effective dynamics’ also shows that there are no spurious quantum gravity effects appearing at low curvatures. From this viewpoint, this represents the first fully consistent quantization for black hole interiors using loop quantization methods which has a well defined and consistent ultra-violet regime where the singularity resolution occurs, and which is in agreement with general relativity in the infra-red limit. It should be noted that we arrive at this description without invoking ideas from improved dynamics of LQC which involved a subtle change of the original discretization variable. The classical singularity is avoided, and replaced by a ‘quantum bounce’ to a new branch. The spacetime does not end at the singularity. An interesting feature of the effective spacetime that arises after the bounce is that it represents the (expanding) interior region of a white hole, but of a different mass than the original black hole. Furthermore, there appears to be a cubic powerlaw relation between the two sets of masses. It should be noted that the difference in the white hole mass and the starting black hole mass is also present in the Ashtekar-Bojowald prescription as was first shown in Ref. b-k (). But, there is a striking difference in physics in comparison to our analysis. Unlike the quantization introduced in this manuscript, the white hole mass in the Ashtekar-Bojowald prescription depends on the fiducial length , and not on the starting black hole mass. Thus, while the white hole mass in our prescription is proportional to the cubic power of the initial black hole mass and independent of any fiducial structure, in the Ashtekar-Bojowald prescription it can take any arbitrary value through its dependence.
The structure of the paper is as follows. In Sec. II, we recall the classical minisuperspace in terms of connections and triads that we are considering and introduce the main assumption underlying the construction of the quantum theory. Sec. III is devoted to the study of the classical solution for the equations and their space-time interpretation. In Sec. IV we consider the loop quantization of these models and arrive at a quantum difference equation that selects the physical states of the theory. In Sec. V we consider the effective dynamics one expects to recover from the quantum theory and study its phenomenological implications. Here we discuss the boundedness of expansion and shear scalars in the effective spacetime description when the would-be central singularity is approached. We show the independence of the physics on the underlying fiducial structures and discuss the relationship between the starting black hole mass and the white hole mass. We conclude the article with a summary of results and a discussion in Sec. VI.
The portion of the Schwarzschild spacetime corresponding to the interior region of the black hole can described by a homogeneous cosmological model given by a Kantowski-Sachs geometry with symmetry group . This corresponds to a foliation in which the 3-manifolds with topology approach a null-surface, the horizon, on the past and a space-like singularity on the future. As in the previous cases studied within minisuperspace Hamiltonian formulation, it is a standard procedure to define a fiducial metric on the 3-manifold and define the corresponding compatible triad and co-triad . However, unlike the isotropic and homogeneous FLRW and Bianchi-I models, there is only one non-compact direction and thus one has to introduce a line interval as an auxiliary structure to define the phase space. The choice of fiducial metric that we will here make, even when rather similar to that of aa:mb () (and used in all subsequent articles Modesto (); b-k (); mb_cartin_khanna (); Campiglia ()) is, however, motivated by different considerations. In aa:mb (), the fiducial metric on the sphere was chosen to be a unit sphere. Even when this choice is mathematically consistent, one needs to specify a scale, in this case with dimensions of area that, together with the dimension-full coordinate , yield a fiducial metric with consistent dimensions. The viewpoint that we shall adopt here is that, in order to perform a Hamiltonian analysis in the absence of a natural scale in GR, one has to specify it externally as a boundary condition. This situation is not new in black hole physics. For instance, as part of the definition of an isolated horizon IH (), one has to specify the precise values of all the multipole moments of the horizon. For the simplest case of spherical horizons (of Type I in the standard terminology), this means fixing the value of the area of the horizon. The classical Hamiltonian theory is thus dependent on this external parameter, and this is carried over to the quantum regime.
Our concrete proposal is to extend this viewpoint to the formalism that we are here considering, and fix a classical scale by asking that the area of be equal to . The fiducial metric takes the form:
with determinant . This externally prescribed length scale will turn out to be the Schwarzschild radius, thus providing a physical parameter that defines the configurations under consideration. Namely, just as in the analysis of entropy for isolated horizons where one chooses a given fixed area from the start, the system under consideration here will be the dynamics of a Kantowski-Sachs cosmology with the interpretation of being the interior region of a black hole of a given area . In practical terms, this does not represent a limitation, since at the end of the day one can let become a free parameter, so we have a description of all possible black hole interiors. We have been careful in explaining the role of since it represents the main conceptual difference from other treatments, and the one that will allow for a consistent formulation.
In order to define the symplectic structure, one has to specify a finite interval for the coordinate in . The usual procedure is to restrict the range of to the interval . With this choice the fiducial volume of the fiducial cylinder is equal to . With our choices, this quantity has the ‘right dimensions’ of volume.
Utilizing the symmetries of the spacetime (and after imposing the Gauss constraint), the connection and the triad can be written as follows aa:mb ():
with fiducial triad and cotriad:
The metric for the particular case of the Schwarzschild interior spacetime is
Thus and are related to components of the standard Schwarzschild metric as
where and is the ADM mass of the spacetime. The triads and are dimensionless, the latter being equal to unity at the horizon by the choice of parameter (with dimensions of mass parameter): . The symplectic structure expressed in terms of the conjugate pairs and is not invariant under the change of fiducial metric. We can introduce
There are two underlying freedoms associated with auxiliary structures of which the theory should be invariant. The redefinition (8) takes care of the freedom to rescale coordinates (for eg. ) keeping the metric invariant. However, freedom to rescale size of the interval of integration exists, and as in the cosmological model one of the primary tasks will be to identify suitable variables for the phase space. Under the rescaling of :
Note that in our model, there is no underlying freedom to change . As stated before it amounts to considering a different physical situation. This is quite different from the quantity that is an auxiliary (and arbitrary) construct with no physical meaning.
Iii Classical Solutions
The classical Hamiltonian constraint in terms and can be written as aa:mb ()
It is related to the classical Hamiltonian as: . Choosing in and using (9), we can find the equations of motion for the phase space variables:
where the derivative is with respect to ‘time’ . Integrating the equations for , and and using the constraint equation
It is convenient to make a change of variables from dimensionless to dimensionfull , with as a length scale to be determined. We also identify the free parameter in and , denoted as . With this change of variables, the solutions take the form:
So far, we have four free parameters, two of which have to be fixed to end up with only two (corresponding to the true degrees of freedom in the canonical formulation). The first simplification pertains to the freedom we have in choosing the initial time (given that the constraint generates constant translations in , and one has to reduce this freedom). Thus one can chose without losing generality which implies . Since the Schwarzschild radius, is a natural physical length scale in the model we further identify which implies , the mass parameter of the black hole. Note that has the interpretation, in terms of the spacetime metric of the geometric radius of the homogeneous spheres. Thus, we would like to associate the parameter with this quantity. This implies the choice . With these identifications the solutions can be rewritten as
We are thus left with only two free quantities, namely . We further note that the spacetime metric only knows of the one independent parameter (). This can be seen by considering the component that is given by,
From which one can fix to be , in order to have the standard form of the line element. Note that for any other choice of , say one could still bring the metric to the standard form by a constant rescaling of the coordinate by defining 111However such a rescaling, even when valid from the spacetime perspective, is not a canonical transformation from the Hamiltonian perspective, given that is also rescaled by when changing coordinates. This illustrates the fundamental difference between the Hamiltonian and the spacetime approaches to this system. This discussion corrects some statements of aa:mb () and b-k () that were based on a slightly different parametrizations..
There is another way of solving the equations of motion that is illustrative to understand the behavior. First note that there is a natural constant of the motion given by (in the standard terminology it is a Dirac observable). With this identification we see that the Hamiltonian constraint can be rewritten:
Thus, we can interpret as a reduced Hamiltonian that dictates the motion for the part of the phase space, that is decoupled from the sector. It is also a first integral that gives us the relational dynamics of as function of :
There is only one free parameter in this dynamics, namely the value of . In the sector since , we only have to fix one constant (for instance .). Thus we recover the two degrees of freedom. It is now immediate to identify and interpret the dynamical evolution, with parameter (or ), in the phase space. For , that in the spacetime pictures represents the horizon, we have that and . As approaches zero, and grow unboundedly, and increases from to where it reaches its maximum value , and then becomes a monotonically decreasing function that approaches zero as .
Iv Loop Quantization
The elementary variables for the quantization are the holonomies of the connections (considered over edges labelled by in direction) and (considered over edges labelled by in and directions)
The holonomies generate an algebra of the almost periodic functions with elements of the form and the resulting kinematical Hilbert space is a space of square integrable functions on the Bohr compactification of :. The eigenstates of and are:
which satisfy .
In order to quantize the gravitational constraint we treat extrinsic curvature as the connection and consider its curvature , using which the classical constraint can be written as
Here is the curvature corresponding to the spin connection . At the equator of , the spin connection vanishes and . Holonomies of extrinsic curvature along , and directions then turn out to be equal to the holonomies of connection (30,31,32) when computed from equator. To be precise, we consider loops in , an planes. The edge along direction in has length with and the edges along longitudes and equator of each having length with .
The term proportional to inverse triad can be casted in terms of holonomies by using an identity on the classical phase space:
where (and ) correspond to or (and similarly or ) depending on the edge over which a holonomy is computed. Here is the physical volume of the fiducial cell
The eigenvalues of can be found using (33) which yield .
Classically, the field strength can be written in terms of holonomies using
In the quantum theory, due to underlying discreteness of quantum geometry, the loop can only be shrunk to the minimum value of area as given in LQG: with of the order unity abl (). Hence the area of the loop in and planes is constrained as:
Unlike the loop in or , the loop in is an open loop. However due to homogenity one can still associate an effective area with this loop aa:mb () and constrain it with :
Both and depend on length scales in the model. However, the difference in their dependence is important. The label of holonomies along equator and longitudes , ‘knows’ only about the radius of where as which labels holonomies along ‘knows’ about – length of the fiducial interval. This dependence is consistent with the expectation that holonomies along the equator and longitude should not feel the auxiliary length scale . As will turn out, the relationship between and the auxiliary line interval plays an important role to obtain a consistent physics from this quantization.
Remark: Instead of constraining the area of the closed loops in and directions to be equal to , we could require them to be equal to the electric flux along the transverse edge, as in the case for a Bianchi-I model cs2 (); cs3 (). For both of the loops it corresponds to equating with times the minimum eigenvalue of : which using (40) implies . Though the numerical factors change, the crucial dependence of on does not. In comparison to (39), in this case also feels the radius of . In the present analysis, we will however work with Eqs.(39) and (40).
We are interested in evaluating the corresponding symmetric operator: , where obtained from (42) is
We can now find the action of on the states :
The states are further required to satisfy the invariance under parity operation: . The quantum difference equation reduces to the one in of Ref. aa:mb () if we put (except for the factor of multiplying term). Thus various properties remain similar. These include its non-singular nature. If one considers as a clock, then the ‘evolution’ occurs in the steps of . By specifying the wavefunction at initial time steps and , it is possible to backward evolve the equation through the classical singularity at . For that let us consider the case when where is positive integer greater than four222The character of the quantum difference equation changes for , however it still remains non-singular.:
Since the coefficient of the term never vanishes, we can use above equation to determine the wavefunction at negative values of starting the backward evoluution from the positive values. Thus we can ‘evolve’ across the singularity.
It is to be emphasized that non-singular nature of above difference equation should be only seen as a indication of the resolution of singularity in this quantization. To have a detailed knowledge of the latter, it is important to construct a physical Hilbert space and finds expectation values of the Dirac observables (as accomplished for the isotropic LQC aps2 ()). This analysis will be performed elsewhere.
V Effective Dynamics
Let us analyse the dynamics resulting from an effective Hamiltonian corresponding to the quantum constraint. This will be on the lines of similar analysis in the isotropic LQC, where the effective Hamiltonian, derived using geometric methods in quantum mechanics, captures the underlying quantum dynamics extremely well even for states which may not be sharply peaked nlqc2 (). In the following, we assume that the effective spacetime description remains valid.
It remains to be seen if the same holds for the following effective Hamiltonian, which requires a detailed numerical analysis once we have the knowledge of physical states and observables in the model. The effective Hamiltonian is given by333In principle the effective Hamiltonian can have modifications coming from the inverse powers of similar to in the case of inverse triads in the isotropic LQC. There its effects turn out to be negligible when compared with the corrections coming from field strength encoded in terms such as . For this reason, we ignore these modifications in the following analysis.
For and , approximates (11).
To solve for the dynamics we choose and using (9) we obtain 444For analogous sets of effective dynamical equations arising in other prescription for Schwarzschild interior, see Refs. b-k (); chiou (); ksbound ().
where the derivatives are with respect to time . Integrating the equations for , and , using the relations between , and and determining using we obtain
The effective dynamics predicts a non-singular bounce of and as the classical singularity is approached. Of particular interest is the behavior of . It takes a minimum value at time :
(where we have used Eq.(41)). Since , with is of order one, we see that as , 555In the LQC literature is sometimes taken as , but one can also see it as a free parameter to be determined by physical considerations slqc ().. Thus the resolution of Schwarzschild singularity has a pure quantum gravitational origin.
The variation of and obtained by solving the above effective equations is shown in figures 1 and 2. The evolution starts from at the horizon. Classically the evolution breaks down at the singularity where both and . This does not happen in the effective dynamics. In the high curvature regime, both and bounce and the singularity is avoided leading to a new white hole spacetime. Fig. 1, shows the effects of changing the mass parameter on the evolution. Fig. 2, displays the curves if one changes the auxiliary length . As can be seen, it only translates the curve vertically without changing the value of the mass of the white hole. At the low curvature scales, the evolution approximates the classical theory.
To understand the details of the singularity resolution, let us consider the expansion () and shear () scalars. These are given as follows:
In the classical theory, the expansion and shear scalars diverge at the central singularity where and vanish simultaneously. Since is bounded below, the dynamical evolution in the effective spacetime description does not allow and to diverge when the central singularity is approached in the loop quantized model. Thus, the expansion and shear scalars are dynamically bounded and the central singularity is resolved. The boundedness of expansion and shear scalars is an indication that geodesics are complete in this effective spacetime, past the would-be singularity, as was the case for the isotropic and anisotropic models in LQC ps09 (); ps11 (). Finally, we note that at the horizon the expansion and shear scalars diverge both in the classical theory and the effective theory because of the coordinate singularity encountered there.
It is important to note that the expansion and the shear scalars are free from the underlying freedom of the rescaling of the fiducial length . This is in contrast to the Ashtekar-Bojowald prescription, where these scalars turn out to be dependent on the fiducial length ksbound (). As has been stressed earlier by the authors, a consistency check of any quantization prescription is that the geometric scalars such as and must be independent of any fiducial structure cs2 (); cs3 (). Otherwise no consistent physical predictions on the details of the singularity resolution can be drawn from the quantization. So far, in the context of the black hole spacetimes the Böhmer-Vandersloot prescription was the only available quantization which satisfies this criteria ksbound (). Our quantization prescription is the only other choice which passes this criteria, and which leads to a mathematically and physically consistent dynamics.