An effective model for the quantum Schwarzschild black hole

We present an effective theory to describe the quantization of spherically symmetric vacuum in loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation of the Hamiltonian of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambigous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model.

The singularities predicted by general relativity (GR) are expected to disappear once a complete quantum description of gravity is achieved. Loop quantum gravity predicts a quantized spacetime presumably mending those defects. However, a complete quantum description of the regions close to a singularity is not at hand and one must consider effective descriptions that implement the expected corrections. In particular, the accuracy shown by effective techniques for homogeneous models [1][2][3][4], where the initial singularity is replaced by a quantum bounce, has been the motivation to extend the so-called holonomy corrections to spacetimes with less symmetry.
Concerning non-homogeneous models, the most simple scenario is that of a spherically symmetric black hole. The main approach in the literature has dealt just with its interior part by using the same techniques as for homogeneous models [5][6][7][8][9]. Nonetheless, the implementation of the isometry between the homogeneous interior and Kantowski-Sachs cosmology is only partially satisfactory and a comprehensive geodesic analysis is mandatory. In this respect, there are several proposals [10][11][12][13][14][15][16] which, however, present crucial problems that we address in our model. For instance, the extension to the exterior static region, the asymptotic flatness, the slicingindependence and the confinement of quantum effects to large-curvature regions are open issues present in most of the models in the literature. Moreover, none of the mentioned studies addresses explicitly the covariance of the theory [17][18][19][20][21][22]: quantum effects may thus depend on the particular gauge choice and not yield conclusive physical predictions.
Here we introduce holonomy corrections through a canonical transformation and implement a regularization of the deformed Hamiltonian constraint. We then construct the spacetime that solves this effective theory and obtain its global causal structure. In particular, a single chart covers a singularity-free (black-hole/white-hole) interior region plus two asymptotically flat exterior regions, as depicted in Fig. 1. The main features are listed at the end of the manuscript.
In the 3 + 1 setup of a manifold M based on the level hypersurfaces of some function t, the diffeomorphism invariance of GR is encoded in four constraints: the Hamiltonian constraint H, that generates deformations of the hypersurfaces (as a set), and the diffeomorphism constraint D, which has three components and generates deformations within the hypersurfaces. Spherical symmetry allows the introduction of another function x, constant on the symmetry orbits. In this case the two angular components of D trivially vanish and, in terms of the Ashtekar-Barbero variables, we have this simple polymerization may give rise to anomalies since the deformed constraint algebra does not generically close. Although a careful choice of the functions allows to define an anomaly-free polymerized Hamiltonian in vacuum, the presence of matter with local degrees of freedom rules out that possibility [17,20,21]. In view of the above, the idea introduced in [23,24] is to consider not just modifications of K ϕ but also of its conjugate variable E ϕ . For instance, if one performs the canonical transformation K ϕ = sin(λK ϕ )/λ, E ϕ = E ϕ / cos(λK ϕ ), K x = K x , and E x = E x , the theory remains free of anomalies even when adding matter fields. Note that this transformation leaves invariant the diffeomorphism constraint D = −E x K x + E ϕ K ϕ . As long as cos(λK ϕ ) does not vanish, the canonical transformation is bijective and, essentially, the dynamical content of the theory is the same as that given by GR. However, the surfaces (see below) cos(λK ϕ ) = 0 may contain novel physics. Since the Hamiltonian constraint diverges there, we regularize it, and define the linear combination along with H[f ] := f Hdx, so that the canonical algebra follows, with the non-negative structure function

Now, let us define
which is a constant of motion. It is important to note now that the condition cos(λK ϕ ) = 0 holds if and only if √ E x = 2mλ 2 /(1 + λ 2 ), which is a gauge-independent statement because E x is a scalar. Therefore, although K ϕ is not a scalar quantity, cos(λK ϕ ) = 0 covariantly defines surfaces on M . For convenience, we introduce r 0 := 2mλ 2 /(1 + λ 2 ), so that From now on we will assume m > 0 and λ = 0, and thus 0 < r 0 < 2m. The classical theory is recovered in the limit λ → 0, which implies r 0 → 0. Let us stress that the characteristic scale r 2 0 arises naturally from the constraint algebra and will show up in the model as a minimal area.
To construct a consistent geometric description, we use the functions t and x on M , plus the unit sphere metric dΩ 2 , to produce a chart {t, x} (we omit the angular part) in which a spherically symmetric metric g is given in the general form The lapse L, shift S, q xx and q ϕϕ depend on t and x.
The unit normal to the hypersurfaces of constant t is given by n = L −1 (−∂ t + S∂ x ). Our purpose is to define these functions in terms of phase-space variables in such a way that infinitesimal coordinate transformations coincide with gauge variations. We start by imposing that the Hamiltonian construction is based indeed on t and x, that is, the Lagrange multipliers correspond to the lapse and shift, hence L = N and S = N x as functions on M . Now, on the one hand, an infinitesimal change of coordinates (t + ξ t , x + ξ x ) is given by the Lie derivative of the metric along the vector ξ = ξ t ∂ t + ξ x ∂ x . On the other hand, a gauge transformation of a function G on the phase space is given by with gauge parameters 0 and x . Since H and D satisfy the canonical algebra (2), these two deformations should coincide if the gauge parameters correspond to the components of the normal decomposition of the vector ξ [25], that is, In particular, the modification of the Lagrange multiplier N x under a gauge transformation is given by [18,26] δ whereas, under infinitesimal coordinate transformations, the shift changes as the dot being the time derivative. The equivalence of the two variations needs q xx = 1/F , which can be consistently imposed since δ ξ q xx = δ (1/F ). Also, we have for the lapse δ ξ N = δ N . Finally, we demand q ϕϕ to retain its classical form, q ϕϕ = E x , which has the correct transformation properties. The explicit details of the equivalence of gauge variations in phase space and coordinate transformations of this construction are shown in [27]. We thus end up with the metric, c.f. (3), Compared to its classical form, it contains the term Also, the precise form of E x , E ϕ , N and N x as functions of the coordinates will not be generically the same as in GR, since they must solve the de- ϕ, along with H = 0 and D = 0 [27]. Since our construction is consistent, different gauge choices will simply lead to different coordinate charts (with different domains of M in general) and corresponding expressions for the same metric. Next we find the solution to that system and obtain the corresponding unique geometry for four different gauges. (a) A static region: Using the labels {t, x} = {t,r} for this chart, and setting E x =r 2 and K ϕ = 0 we get withr ∈ (2m, ∞). This region is asymptotically flat, and will describe one exterior domain.
with T ∈ (r 0 , 2m), that will describe half of a homogeneous Kantowski-Sachs type interior.
None of these two coordinate systems crosses the horizon at r = 2m, nor the instant T = r 0 , and their domains on M do not intersect. The next gauge (c) produces a chart on a domain that will cover two regions (b), providing the full interior homogeneous region including the hypersurface T = r 0 ; whereas the gauge (d) yields a chart on a domain U ⊂ M that covers all the above.
(c) The whole homogeneous region: We set {t, x} = {T , Y } and demand E x = E ϕ = 0 as in (b), but now we take K ϕ = T /λ. Naming √ E x =:r, we obtain wherer(T ) = 2mr 0 /(2m sin 2 T + r 0 cos 2 T ), so that 2m and the range of coordinates is restricted to T ∈ (0, π). This region will describe the full homogeneous Kantowski-Sachs type interior, and contains the spacelike hypersurfacer = r 0 , located at T = π/2.
with (τ, z) ∈ R 2 . The function r in this chart, r(z), is even r(−z) = r(z) and it is implicitly given by Observe that r(0) = r 0 > 0 is its only minimum and r(z) is analytic on R, with image on [r 0 , ∞). The chart {τ, z} thus maps some domain U ⊂ M to the whole plane R 2 . In the search for the global structure of (U, g) we will produce appropriate coordinate transformations so that (9) takes the explicit conformally flat form on the (τ, z)-plane, see (10) and (13), that will coincide with (5), (6), and (7) on their corresponding domains. This will show that (U, g) covers any such static region (a) and homogeneous regions (b) and (c). The procedure will end by proving that (U, g) contains exactly one globally hyperbolic interior domain composed of one homogeneous region (c), which covers two regions (b), and two exterior regions (a). This whole process, along with the resulting spacetime diagram, is sketched in Fig. 1 (for further details see [27]).
We first define the sets: E := {r > 2m} ∩ U, I := {r < 2m} ∩ U, Z := {r = 2m} ∩ U, and T := {r = r 0 } ∩ U ⊂ I. Then we use the chart {τ, z} to decompose these sets (except T ) by taking their restrictions under the sign function sgn(z), and use the notation D σ := D| sgn(z)=σ (with σ = ±1) for any domain D. In particular, E = E − ∪E + is disconnected and I = I − ∪T ∪I + is a connected set. To ease the notation we will use the same letter for the domain in U and its image on R 2 under the chart {τ, z}. For instance, E + also stands for the half plane z ∈ (z s , ∞) in R 2 , where z s is the positive root of r(z s ) = 2m, and I is also the stripe z ∈ (−z s , z s ).
With the auxiliary α : Since R U (r 0 ) = 0, it is easy to check that U (τ, z) := τ + sgn(z)R U (r(z)) is analytic on the whole plane.
Finally, we use that Υ(r) strictly decreases on [r 0 , ∞) with maximum Υ(r 0 ) = 1, ensuring (12) has a solution for r everywhere on tan u σ tan v σ ≤ 1. Therefore each set A σ can be extended to the Kruskal-Szekeres-type regions Q σ (see Fig. 1). The purpose of these extensions is twofold. Firstly, the sets C σ can be mapped respectively to B + := {u + , v + ∈ (0, π/2); u + +v + < π/2} ⊂ Q + and in order to define the extended charts {u σ , v σ } Qσ so that they map all p ∈ I σ by Λ σ • Φ I (p) to the respective point on B σ . This ends the contruction of the full Penrose diagram for (U, g). Secondly, we extend (U, g) to two Kruskal-Szekeres-type analytic regions, Q + and Q − , by adding their remaining halfs. These can be used to build up the maximal analytic extension of M in the usual periodic fashion, and show that M is geodesically complete [27]. All test particles that cross the horizon at r = 2m, arrive to r 0 , continue towards negative values of z with increasing values of r, and cross again r = 2m after a finite proper time. In particular, radial infalling particles at rest at infinity take a time 8 3 (m+r 0 )(1− r0 2m ) 1/2 to cross the interior region. The singularity in Schwarzschild, at r = r 0 = 0, is not present here and the curvature is bounded. In particular, the curvature scalars take their maximum value at r = r 0 . For instance, the Ricci scalar R = 3mr 0 /r 4 is everywhere positive. Note that even if quantum-gravity effects (parametrized by r 0 ) are present outside the horizon, they decay as one moves to lowcurvature regions.
The computation of the expansions of ingoing and outgoing radial null congruences shows, as expected, that the spheres of constant t and x are non-trapped in the exterior region r > 2m, and that r = 2m is indeed a horizon. Moreover, in the interior region r 0 < r < 2m both expansions have the same sign, given by − sgn(z), and vanish at r = r 0 . Therefore, in I + (I − ) those spheres are trapped (anti-trapped) while in T , they have zero mean curvature. In fact, the hypersurface r = r 0 itself is minimal, reflecting the mirror symmetry z → −z.
Therefore, as one expects for a singularity resolution, some of the eigenvalues of the Einstein tensor G a b must attain negative values on I. Indeed, if one interprets G a b as an effective energy-momentum tensor, the eigenvalues on the angular part would define an angular pressure (r − m)r 0 /(2r 4 ). On E one would get a positive energy density 2mr 0 /r 4 and a negative radial pressure −r 0 /r 3 , while on I the energy density would be r 0 /r 3 and the radial pressure −2mr 0 /r 4 . With these values it is easy to check that none of the geometric energy conditions are satisfied at any point except at the horizon. However, let us recall that (M, g) solves the vacuum equations, thus satisfying trivially all the physical energy conditions. Let us finally summarize the main features of this effective quantum black-hole model: (i) The brackets between deformed constraints vanish on-shell and thus form an anomaly-free algebra. (ii) We provide a consistent, hence covariant, geometric setup so we can talk of a metric that solves the system. Different gauge choices on the phase space simply provide different charts (and domains) of some spacetime (M, g), with corresponding expressions for the same metric tensor. (iii) A convenient choice of gauge provides a single chart that covers a domain (U, g) with global structure shown in Fig. 1, which represents a globally hyperbolic interior (black-hole/white-hole) region and two asymptotically flat exteriors of equal mass. (iv) We have produced the maximal analytical extension (M, g). (v) Quantum-gravity effects introduce a length scale r 0 > 0, that defines a minimum of the area of the orbits of the spherical symmetry, and removes the classical singularity. More precisely, the surface r = r 0 is just a minimal hypersurface between a trapped and antitrapped region, and all causal geodesics cross it in finite time. (vi) All curvature scalars are bounded everywhere. (vii) Quantum-gravity effects die off as we move to lowcurvature regions. (viii) Schwarzschild is recovered for r 0 = 0 and Minkowki for m = 0.