Deformed General Relativity and Quantum Black Holes Interior

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance can not be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study homogeneous solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition.


Introduction
Black hole are iconic predictions of general relativity. Their classical description is very well established, and strong evidences for their existence is now regularly reported through the gravitational wave astronomy. Yet, their quantum description is still an open problem. The presence of a classical singularity in their core prevents from having a complete description of their interior geometry, and more generally of their formation. It is widely believed that, once a consistent UV completion of general relativity will be available, classical singularities will be replaced by a well defined "quantum geometry", allowing for a consistent unitary evolution of the degrees of freedom at the quantum level. Since a complete and consistent quantum theory of gravity is still missing, important efforts towards the understanding of the quantum description of black hole have been focused on developing instead effective approaches. The construction of effective regular black holes geometries have triggered an important activity over the last decades, initiated by the work of Bardeen (see [1] for a review). However, such ad hoc models suffer from several inconsistencies, and in particular, fail to provide a framework to discuss completely the end point of the evaporation 1 , as they predict generically an infinite time of evaporation [3].
Alternative approaches to quantum black holes rely on canonical quantizations of the phase space of spherically symmetric gravity in vacuum (see [4][5][6] for early works on this topic). The polymer quantization, inspired by loop quantum gravity, has led to several proposals for the effective description of the Schwarzschild interior. Early works in this direction, motivated by previous results on the polymer quantization of cosmological backgrounds, were presented in [7][8][9][10][11][12][13][14][15], suggesting new scenarii for the end of the collapse [16,17]. Since then, the polymer quantization of this model has been revisited by several authors [18][19][20][21][22], with several improvements along the years. Later on, effective approaches based on a "quantum" deformation of the classical phase space were used to provide effective and regular metrics for the Schwarzschild interior. Such effective models appear as a shortcut to investigate possible quantum corrections to the interior geometry without going through the whole quantization procedure 2 . Following this strategy, one found effective geometries where the singular vacuum Schwarzschild interior has been replaced by a black-to-white hole bounce, first in [25,26], and more recently in [27,28]. These new solutions have been discussed in [29][30][31] and an alternative effective polymer model of black-to-white hole transition, providing several improvements, has been introduced in [32][33][34] (see also [35][36][37][38][39][40][41][42][43][44] for additional investigations on polymer black hole geometries and [45][46][47][48] for yet another class of effective models of blackto-white hole transition). While these classical polymer constructions provide a straightforward and interesting platform to investigate quantum corrected effective metrics, the strategy employed in these hamiltonian constructions turns out to suffer from several shortcomings related to the covariance of the quantum corrections.
To explain this problem, we start giving the basic building bocks of these polymer effective models. First, let us recall that the hamiltonian formulation of vacuum spherically symmetric gravity, where there are no local degrees of freedom, leads to two first class constraints, the hamiltonian and (radial) vectorial constraints, which generate time and radial diffeomorphisms. Then, the common strategy employed in these effective polymer models is to introduce some 1 See however [2] for a new construction with potentially interesting features. 2 The common underlying assumption of these effective polymer models is that the modified hamiltonian constraint actually corresponds to the expectation value of the hamiltonian operator in some suitable coherent states at the quantum level. However, it is fair to say that the concrete realization of this assumption for polymer black holes is usually left aside, and one only deals with a modified classical system. Recently, efforts towards computing this expectation value of the hamiltonian operator for spherically symmetric quantum geometries were presented in [23], based on the framework of quantum reduced loop quantum gravity. Consequences for the black hole interior geometry were discussed in [24]. regularization at the phase space level through some modifications of the hamiltonian constraint known as point-wise holonomy corrections, while keeping the vectorial constraint unchanged. In general, this regularization breaks the covariance of the system, as the algebra of the first class constraints is no longer closed. Then the solutions to the modified equations of motion one can extract from this modified hamiltonian system do not have a classical space-time interpretation, as covariance is lost. In models such as [25][26][27][28] as well as in [32], where only the homogeneous interior region is considered, the issue of covariance is not visible. However, if one tries to extend these models to take into account inhomogeneities, the issue of the covariance breaking becomes important.
In this paper, we further investigate the issue of covariance in effective black holes models. As we have already said, there exist modifications of spherically symmetric general relativity, inspired from loop quantum gravity, which lead to a deformed but closed constraints algebra. This property makes the effective theories particularly interesting because they are invariant under a deformed covariance, thus there is no effective quantum anomalies, and then the classical solutions have a clear geometrical meaning. Now, we can ask the question whether one can extend such a construction beyond spherical symmetry. We explore this question in section 2. We consider a large class of modified theories of gravity whose deformation is not only inspired from (holonomy corrections induced by) loop quantum gravity but is also a natural generalization of the deformation introduced in the context of spherical symmetry. In particular, these theories are invariant under three-dimensional space-like diffeomorphisms. We prove that any theory of this class which has a closed constraints algebra reduces, after some canonical transformations, to general relativity with a cosmological constant. Hence, at the level of the full theory, we show that there is no non-trivial deformations of general relativity in a large class of modified gravity theories with the condition that they produce a closed constraints algebra. On one hand, this result clearly contrasts with what happens when space-times are reduced to spherical symmetry 4 . On the other hand, this finding parallels the well known Hojman-Kuchar-Teteilboim theorem which allows to reconstruct uniquely, from the Dirac's hypersurface algebra, the Einstein-Hilbert action [81].
In section 3, we revisit and study new aspects of loop (holonomy) deformations of spherically symmetric general relativity. We start with a quick Hamiltonian analysis of general relativity reduced to spherical symmetry using Ashtekar like variables, and we obtain the two expected constraints, the Hamiltonian constraint and the radial diffeomorphisms constraint. Then, we ask the question whether one can deform these constraints to keep their algebra closed. First, we show that there is no non-trivial deformations (up to canonical transformations) of the sub-algebra generated by the radial diffeomorphisms constraint. In other words, space-like diffeomorphisms cannot be consistently deformed in an effective theory of general relativity and they remain, in that sense, classical. This result is consistent with loop quantum gravity where kinematical quantum states are invariant under classical space-like diffeomorphisms. However, it is possible to deform the Hamiltonian constraint in a way that the constraints algebra is deformed but closed. We compute and recover the most general deformation which keeps the algebra closed.
In section 4, we compute the equations of motion for the effective metric components and we look for their solutions corresponding to "static" black holes interiors, which means that the metric components are time dependent only. Notice that, in that case, the closeness of the constraints algebra is not an issue and one can safely relax this condition. Hence, we consider in this section very general deformations of the Hamiltonian constraint with the only restriction that it still transforms as a scalar under radial diffeomorphisms. Remarkably, we completely solve the deformed Einstein equations in that very general case and we give the expressions of the components of the effective metric in terms of the functions that govern the deformations. For concreteness, we illustrate this general result with simple examples.
We conclude the article with a brief summary and a discussion of the results. We also present some interesting perspectives.

Deformed General Relativity
In this section, we would like to recall some aspects and also give new results on modified gravity which will be useful in the following sections when we study modifications of gravity induced by loop quantum gravity. In particular, we start by introducing a large class of theories of modified gravity which share some features with effective theories of loop quantum gravity when it is reduced to spherical symmetry. However, contrary to effective loop quantum gravity, these modified theories of gravity are defined in full generality and not only in the case of symmetry reduced backgrounds. Then, we discuss these theories and show in particular the impossibility to have a closed deformed algebra of constraints in 3+1 dimensions in this class of theories. This result raises the question of the possibility to extend the construction of effective loop quantum gravity theories with a closed deformed algebra of constraints beyond spherical symmetry.

From (effective) loop quantum gravity to modified gravity
As any theory of quantum gravity, loop quantum gravity is expected to modify Einstein equations at the effective level. Such modifications have been computed and studied for homogeneous and spherically symmetric space-times to understand the effects of quantum gravity in early cosmology and black holes physics. In general, one distinguishes between two types of deformations, known as holonomy and inverse-triad corrections. Here we are concerned only with (point-) holonomy corrections which are local modifications in the sense that they do not produce non-local equations of motion.
These (point-) holonomy corrections are assumed to modify only the Hamiltonian constraint and do not affect the vectorial constraints. Indeed, in the construction of the kinematical Hilbert space of loop quantum gravity, the invariance under diffeomorphisms is imposed "classically" from the action of diffeomorphisms on the graphs of spin-network states. Hence, in general, one does not use holonomies to quantize the vectorial constraints. For this reason, it is natural to require that any effective theory of loop quantum gravity remains invariant under (undeformed) spatial diffeormorphims. Furthermore, we will show in the case of spherical symmetry that it is impossible to construct a non-trivial deformation of the vectorial constraint which leaves the deformed diffalgebra closed (see section 3.2).
To construct such effective theories, we assume that the space-time M if of the form Σ × R where Σ is a space-like slice, and we consider the ADM parametrization of the metric, where γ ij is the spatial metric on Σ, N is the lapse function and N i the shift vector. We also need to introduce the second fundamental form (extrinsic curvature tensor) whose components are given by where D i denotes the spatial covariant derivative associated with the spatial metric h ij . In this parametrization, the Einstein-Hilbert action takes the form where R and R respectively the 4-dimensional and the 3-dimensional Ricci scalars, g and γ are the determinant of the metrics g ij and γ ij respectively. In the following, we shall omit the boundary contribution to the action even though a complete analysis would require to include them.
To mimic loop quantum gravity type modifications, we now introduce a class of modified theories of gravity defined by an action of the form where F is an arbitrary 3-dimensional scalar constructed from the components of the extrinsic curvature K j i . Only the part of the action which depends on K j i is modified. This choice is motivated by the fact that, as we are going to recall later on (for spherically symmetric spacetimes), (point-) holonomy corrections affect mainly the components of the extrinsic curvature, whereas the components of the 3-dimensional Ricci tensor are left unchanged 5 . Interestingly, these modifications contrast with deformationsà la Horava gravity [82,83] where the spatial "Ricci" part of the action is strongly modified whereas the "K-part" is (almost) unchanged compared to general relativity (up to a detuning relative coefficient between K ij K ij and K 2 ).
Hence, we consider the actions (2.4) as models for effective loop quantum gravity theories in any background. As K i j can be viewed as a 3-dimensional matrix, the most general function F (K i j ) which transforms as a scalar under space-like diffeomorphisms can be written as where tr(M ) ≡ M ij γ ij = M i i for any matrix M . This is a direct consequence of the Cayley-Hamilton theorem. For simplicity, we are using the same notation for the two (different) functions F in the l.h.d. and r.h.s. of (2.5).
Before analyzing further this class of theories, let us make a couple of remarks. First, modified theories of gravity which are invariant under spatial diffeomorphisms only have been studied intensively these last years on many aspects. Their Lagrangians involve not only the extrinsic curvature tensor, but also the 3-dimensional Ricci tensor, the lapse and the shift, and their covariant derivatives. Horava gravity is an example of such theories with remarkable ultra-violet properties [82,83]. Scalar-tensor theories in the unitary gauge (where the scalar is a function of time only) are other interesting examples which have been applied to late time cosmology [84][85][86].
The second remark concerns the relation between the theories (2.4) and effective loop quantum gravity. Once again, let us emphasize that the choice of such theories to model the effects of loop quantum gravity at an effective level is motivated by results in symmetry reduced situations where modifications affect the components of the momenta of the metric (see for instance [7,9,11,18,25,28,62,68]). More rigorous constructions of effective theories (based on the full canonical or covariant quantum theory) with no symmetry reduction have been proposed recently and they lead to much more complete and more involved descriptions [23,24]. For the purposes of this paper where we study the possibility to extend the symmetry reduced effective theories to general (nonsymmetry reduced) effective theories, the action (2.4) is the most general starting point provided that we require that the modifications are local and affect only the extrinsic curvature.

Canonical analysis and deformed Hamiltonian constraint
The aim of this subsection is to make a canonical analysis of (2.4) to see how the Hamiltonian constraint is modified compared to general relativity. For simplicity, we first assume that the function F in (2.5) depends on tr(K) and tr(K 2 ) only. We will discuss later the general case where F depends also on tr(K 3 ).
We start by introducing the pairs of canonical variables, As the theory is invariant under spatial diffeomorphisms, we do not need to introduce momenta associated to the shift vector N i . In the deformed theories, the lapse is still not dynamical and we get the primary constraint π N = 0. An immediate calculation shows that the momenta π ij are given by where F (a) is the partial derivative of F with respect to its first (a = 1) or second (a = 2) variable. From the expression of the momenta above (2.7), it is immediate to see that which implies, as expected, that tr(p) and tr(p 2 ) are scalars that can be expressed in terms of tr(K) and tr(K 2 ). Locally (if F satisfies regularity conditions which we assume to be true), one can reverse these relations as follows where the explicit form of the functions A and B is not needed here. Then, one makes use of these relations to invert the equations between the velocities and the momenta, and to solve K ij in terms of p ij , In these equations, F (1) and F (2) are viewed as functions of tr(p) and tr(p 2 ) using (2.9). Finally, the Hamiltonian can be obtained immediately, and after a short calculation, we show that where H i ≡ −2D j p ij is the usual vectorial constraint whereas H is a deformed Hamiltonian constraint given by Here again, the function F and its derivative are viewed as functions of tr(p) and tr(p 2 ) using (2.9).
As there are no restrictions on the function F (up to some regularity conditions which allow the inversion (2.9)), the function G is also arbitrary, and the action (2.4) leads to a generic loop-like deformation of the Hamiltonian constraint where only the momenta component of the Hamiltonian is affected. Hence, as we already said previously, it provides a very general model to test effective loop quantum gravity theories where only the Hamiltonian constraint is modified. The theory remains invariant under spatial diffeomorphisms, and the vectorial constraint is unchanged.
Notice that, if we had started with a function F which would have depended also on the variable tr(K 3 ), we would have obtained an expression for the Hamiltonian constraint similar to (2.13) with a function G depending also on tr(p 3 ). In the rest of the paper, we still restrict our study to the case (2.13).

Deformed Hamiltonian constraint vs. closed algebra of constraints
Once the Hamiltonian has been computed, one continues the Hamiltonian analysis to see whether the deformed Hamiltonian constraint is first or second class. As H is a scalar with respect to spatial diffeomorphisms, it commutes obviously with the vectorial constraint and studying the stability of the Hamiltonian constraint under time evolution reduces to computing the Poisson algebra for any functions u, v on Σ. If C[u, v] is weakly vanishing, H is first class and the canonical analysis stops here with the conclusion that the theory propagates 2 degrees of freedom as in general relativity. On the other hand, when C[u, v] is not weakly vanishing, the theory admits a new (secondary) constraint and one has to analyse further the properties of this constraint to know the number of degrees of freedom.
The calculation of C[u, v] can be done as follows. From the very definition of the Poisson bracket, we have where (u ↔ v) means that we add the same expression where the roles of the functions u and v are inverted, so that C[u, v] is skew-symmetric. Then, using the definition of the 3-dimensional Ricci scalar, we obtain where ∆ = D i D i is the Laplacian, and G (a) is the derivative of G with respect to its first (a = 1) or second (a = 2) variable. Later, we will also use the notation G (ab) for the second partial derivatives of G. The initial 3 integrals in (2.15) reduces to only one integral (2.16) after some integrations by part. The rest of the calculation is straightforward and one obtains finally reduces to a sum of two terms in the last parantheses (2.17). The first one is nothing but the vectorial constraint which is obviously weakly vanishing. In general, the second term (2.18) is not vanishing and leads to new constraints in the theory. Before discussing these new constraints, let us ask the question whether one can get a closed algebra of constraints with a deformed Hamiltonian constraint in a non-symmetry reduced case. This happens only when C j vanishes (at least weakly). An immediate calculation shows that C j can be written as a sum of four terms, which have independent tensorial structures. Hence, C j vanishes only if each term vanishes independently, which leads to the following four conditions of the function G, The general solution, which can be computed immediately, reads (up to a global constant factor) where α and λ are constant. Without loss of generality, one can always fix α to zero from a canonical transformation of the form p ij → p ij + βγ ij whereas γ ij is unchanged. In that case, we recover the Hamiltonian constraint of general relativity supplemented with a cosmological constant. As a consequence, none of the theories in the class (2.4), which is different from general relativity with a cosmological constant, admits a closed algebra of constraints generated by the usual vectorial constraints and a deformed Hamiltonian constraint. Furthermore, the stability under time evolution of the deformed Hamiltonian constraint leads to a differential equation for the lapse function, which has to be understood as a new constraint in the theory. We are using the standard notation ≈ for the weak equality (equality on the constraints surface). Not only S does not commute with π N , but also their Poisson bracket is non-local in the sense that it involves derivatives of delta distributions. Such a situation is pathological and makes the theory ill-defined with an undefined number of degrees of freedom.
The conclusion of this analysis is that one cannot extend the condition of having a closed algebra of constraints associated to an effective loop quantum gravity theory for an arbitrary (non-symmetry reduced) background which satisfies the following properties: first, it is invariant under spatial diffeomorphisms; second, the deformation does not involve the 3-dimensional Ricci tensor but only the extrinsic curvature tensor; and finally the effective theory is a local theory of the metric. One should relax one of these hypothesis to construct an effective theory of loop quantum gravity. As the fundamental variables in loop quantum gravity are holonomies of a connection, one may expect to get instead a non-local or a non-metric deformation.
Nonetheless, requiring a closed algebra of constraints for an effective theory of loop quantum gravity (satisfying the previous properties) for spherically symmetric space-times becomes possible for some reasons we are going to explain later. Their constructions lead to very interesting scenarii, as we are going to see in the next two sections.

Deformations of Spherically Symmetric Gravitation
From now on, we restrict our study to (non-static) spherically symmetric space-times. Even if the deformed gravity theory in this symmetry reduced sector can not be embedded into a fully covariant theory, it is still of interest to consider such specific models for at least two reasons. First, it provides a rare enough example of a gravitational system with deformed covariance, and leads to interesting peculiar effects, such as transition between lorentzian and euclidean regimes in the deep interior for which we have explicit solutions. Second, this deformed covariance is not tied to spherical symmetry, but is also realized in more general system, such as cylindrical symmetry reduced gravity (or Gowdy systems). Hence, while deformed covariance can only be realized in symmetry reduced systems, it provides an interesting testbed to understand the spacetime description beyond standard GR.
Focusing therefore on the spherically symmetric sector, one can choose a coordinate system such that the spatial line element d 2 takes the simple form where γ rr and γ θθ are functions of (t, r) only. We also assume that the lapse function N and the (radial component of the) shift vector N r are functions of (t, r) only. We start this section with a quick review of the Hamiltonian analysis of general relativity reduced to spherical symmetry. This will give us the opportunity to introduce some notations. Afterwards, we will study the possibility to deform the theoryà la loop quantum gravity. We will adopt a Hamiltonian point of view and ask the question of the possibility to deform the constraints keeping a closed Poisson algebra. First, we will prove the impossibility to deform the vectorial constraint without introducing anomalies (in the algebra of constraints). However, as it is well known, we will show on the other hand that it is possible to deform the Hamiltonian constraint keeping a closed algebra, and we will classify these deformations. This results contrasts with the results we have obtained in the previous section where we have shown the impossibility to deform (with some hypothesis) the Hamiltonian constraint for a generic background with no particular symmetries. We will discuss why reducing to spherical symmetry makes the deformation possible.

Reduction to spherical symmetry
We start with a review of the Hamiltonian analysis of general relativity for non-static spherically symmetric geometries. The ADM parametrization of the metric is given by (2.1) where N r is the only non-trivial component of the shift vector and the spatial metric is (3.1). The only non-vanishing components of the extrinsic curvature (2.2) are Therefore, the Einstein-Hilbert action (2.3) reduces to (up to an irrelevant global constant that we neglect but can be computed from the integration over the angles θ and ϕ) We recall that R is the 3-dimensional Ricci scalar for spherically symmetric background whose expression will be given later on.
To go further, it is convenient to introduce the "electric fields" E r and E ϕ associated to the metric (3.1) which are defined by the relations, with the condition that E ϕ > 0. Hence, (3.4) is now considered as an action for the dynamical variables E r and E ϕ . To perform its canonical analysis, one introduces the two pairs of conjugate momenta, whereas N and N r are considered as Lagrange multipliers. Notice that the notations x and y refer to the radial coordinate. The momenta are easily computed and read Then we compute the Hamiltonian of the theory, where the Hamiltonian and vectorial constraints are respectively given by, up to a rescaling of the lapse function, with the following expression of the Ricci scalar, We are using the notation f for the derivative of any function f with respect to r. It is well-known that H and H r define first class constraints, they generate the invariance under diffeomorphisms for non-static spherically symmetric backgrounds, and they satisfy the closed Poisson algebra, are the smeared constraints, u being a regular function of r. As a consequence, the Dirac analysis of the constraints stops here, and the theory propagates no degrees of freedom.
One expects loop quantum gravity to modify, at the effective level, the Einstein equations. In the Hamiltonian framework, this means that one expects modifications of the constraints of the theory. Even though we have shown in the previous section that it is not possible to obtain a non-trivial effective deformation of the full theory with local modifications of the Einstein-Hilbert action (of the form (2.4)) which keep the constraints algebra closed, we can ask the same question when we consider spherically symmetric space-times. It is well-known that this problem has a solution in that case. Now, we are going to derive the conditions for deformed vectorial and Hamiltonian constraints to still have a closed Poisson algebra.

Deformation of the vectorial constraint
From the construction of kinematical states in loop quantum gravity, we know that one has to keep the invariance under spatial diffeomorphisms, or at least to keep a closed algebra for an eventual deformation of the vectorial constraint (to avoid having anomalies). Indeed, spatial diffeomorphisms are still symmetries of the kinematical Hilbert space of loop quantum theory, and there is no reason to violate such a symmetry at the effective level. However, one can wonder if the (spatial) diffeomorphism algebra could be eventually deformed (compared to the classical case) but still closed. We are going to explore this problem and show that, under some general hypothesis, this is not possible.

First necessary condition
For that purpose, we start with the simple remark that π ϕ and E r transform as scalars under the action of the vectorial constraint H r whereas E ϕ and π r are densities. Hence, for convenience (in this subsection only), we introduce the notations, so that From the point of view of the vectorial constraint, the two degrees of freedom (q 1 , p 1 ) and (q 2 , p 2 ) are decoupled (there are no cross terms involved), and we look for modifications which conserve this decoupling. Hence, we ask the question whether there exists a function D(q, q , p, p ) such that the deformed vectorial constraint, H r,def ≡ D(q 1 , q 1 , p 1 , p 1 ) + D(q 2 , q 2 , p 2 , p 2 ) ≈ 0 , where we omit, for simplicity, the labels (1 or 2) for the position and momentum variables. Variations of D[u] induced by variations δq and δp of p and q respectively are easily computed and read δD[u] = dr δq (uD q − (uD q ) ) + δp (uD p − (uD p ) ) , (3.18) where D z denotes the derivative of D with respect to z ∈ {q, q , p, p }. Hence, after an immediate calculation, we show that For the algebra to be closed, the terms proportional to p and q in (3.19) must vanish, which implies the two necessary conditions These two equations are obviously similar and then are solved in the same way. we concentrate on the first one which is solved as follows: where the function A is arbitrary. Similarly, the second condition leads to where the function B is also arbitrary. As a consequence, A(q, p, p ) = B(q, p, q ) = C(q, p) and the general solution of the two previous conditions is D(q, q , p, p ) = D q + C(q, p)p , (3.23) where D is an arbitrary function of one variable. Hence, any deformation of the diffeomorphisms algebra is necessarily of the form (3.23). Before going further and solving the remaining conditions, we make a canonical transformation that will drastically simplifies the analysis.

Canonical transformation: simplification of the problem
Indeed, we introduce a new pair of canonically conjugate variables (Q, P ) related to (q, p) via the generating function Φ(p, Q) by the relations Hence, the combination q + C(q, p)p that appears in (3.23) transforms under this canonical transformation according to and then we can always choose a generating function Φ such that its partial derivative α satisfies the condition so that q + C(q, p)p = α Q Q does not depend on P , and the deformed constraint takes the simple form D(q, q , p, p ) = D(α Q (Q, P )Q ) , (3.27) where we used the shorthand α Q (Q, P ) for α Q (p(Q, P ), Q). When the function C is regular enough, the first order partial differential equation (3.26) admits always a solution at least locally. As a consequence, up to canonical transformations, the deformed constraint can be reduced, without loss of generality, to the simple form D(q, q , p, p ) = D(X) , X ≡ J(q, p)q , (3.28) where J is an arbitrary function.

No-go: no closed algebra for deformed diffeomorphisms constraints
At this stage, we return to the expression of the Poisson bracket (3.19) between the deformed constraints and we immediately see that it simplifies considerably when D is of the form (3.28), Hence, the full constraint (3.16) has a closed Poisson algebra if and only if where λ is necessarily a constant (independent of q 1 , p 1 , q 2 , p 2 and their derivatives). Substituting (3.28) into this equation leads to which can be solved easily for the function D that is given by As D is a function of X only by definition, then J p is necessarily a constant. Finally, we arrive at the conclusion that the only constraints (3.16) which admits a closed Poisson algebra are such that D(q, q , p, p ) = λ(p + µ(q))q , where we recall that λ is a constant, and the arbitrary function µ(q) can be set to zero without loss of generality by a simple canonical transformation. Hence, there is no deformation of the diffeomorphisms Poisson algebra which preserves the decoupling (3.16).

Deformation of the scalar constraint
Now, we review the possibility to deform the Hamiltonian constraint keeping a closed (but deformed) constraints Poisson algebra. Hence, we look for deformed Hamiltonian constraints H def , which are functions on the phase space, such that (3.12) remains unchanged and (3.13) is deformed but closed.

General point-holonomy deformation of the scalar constraint
First of all, the fact that the Poisson bracket (3.12) is unchanged implies that the deformed Hamiltonian constraint H def is a scalar density of weight of +1. Furthermore, as we are considering point-holonomy corrections only, the Ricci part of the classical Hamiltonian constraint (3.9) is unchanged and then H def can be written in the form where S is an arbitrary function of the three independent scalars, which generate the algebra of scalar functions in the phase space. Notice that the factor 4 that comes with the Ricci scalar R has been introduced to make the comparisons with the classical constraint (3.9) easier. Now, we compute the Poisson bracket between two deformed Hamiltonian constraints (3.34). For simplicity, we introduce the notation which, after a long but straightforward calculation, reduces to If one substitutes the expression of G in terms of the function S (3.36), this Poisson bracket becomes where S (a) ≡ ∂S/∂X a and S (ab) ≡ ∂ 2 S/(∂X a ∂X b ) denote the partial derivatives of the function S with respect to the variables X a (3.35).

Closeness of the deformed algebra
The constraints algebra is closed if and only if where γ rr def could be an arbitrary function on the phase space which represents the deformation of the (inverse) metric component γ rr . As a consequence, from (3.39), we immediately see that a necessary condition for the algebra to be closed is that S (22) = 0 which implies that S is an affine function of X 2 and then, as where A and B are arbitrary functions. In that case, it is easy to see that (3.39) reduces to with the deformed (inverse) metric coefficient given by As a consequence, the algebra is closed if and only if the following condition is satisfied otherwise there is an anomaly and the Hamiltonian constraint is no more first class, which means that it does not generate any symmetries. This conclusion is consistent with what has already been found in the literature [62,66,68,87]. Before going further and discuss this condition, notice that in general relativity the Hamiltonian constraint is of the form (3.34) with (3.41) where the functions A and B are explicitly given by which obviously satisfy the closeness condition (3.44). Furthermore, the deformed inverse metric (3.43) reduces, in that case, to the classical inverse metric γ rr (3.5).

Discussion
We found deformations of spherically symmetric reduced general relativity that lead to a closed constraints algebra where the vectorial constraint is unchanged but the Hamiltonian constraint is deformed by (point-) holonomy like corrections. These theories admit two different symmetries which are the classical invariance under diffeomorphisms and an invariance under a deformed time reparametrisation. Hence, they have no local degrees of freedom, exactly as general relativity when it is reduced to spherical symmetry. The deformed symmetry has been discussed recently in [73]. The possibility to have a closed (deformed) constraints algebra relies on the fact that we are considering spherically symmetric backgrounds only. The reason is simply that there is only one spatial diffeomorphisms constraint in the symmetry reduced theory whereas there are obviously three of them in the full theory. As we showed in section (2.3), requiring the invariance under 3 dimensional spatial diffeormorphisms leads to no-go results and to the impossibility of having a non-trivial Hamiltonian (local and point-holonomy like) deformation of general relativity.
To illustrate better the specificity of spherically symmetric models compared to fully (3 dimensional) covariant theories, let us study the spherically symmetric reduced version of (2.4) when F is of the form (2.5) with no tr(K 3 ) dependency. For that purpose, we use the same notations as in section (3.1) and we compute the momenta canonically conjugated to E r and E ϕ which are given by the relations where F (1) and F (2) are the derivatives of F with respect to tr(K) and tr(K 2 ) respectively. From these expressions, we immediately show the two following relations which imply that tr(K) and tr(K 2 ) can be (at least implicitly) expressed in terms of the combinations E r π r / √ γ and E ϕ π ϕ / √ γ. As a consequence, one shows after a long but straightforward calculation that the Hamiltonian is of the usual form (3.8) with a classical vectorial constraint and a deformed Hamiltonian constraint (3.34) with where, in the last equation, tr(K) and tr(K 2 ) are expressed in terms of As a consequence, these theories have a deformed Hamiltonian constraint of the type (3.34). Hence, they fall in the class of theories with a closed deformed constraints algebra if the conditions (3.41) and (3.44) are fulfilled even though its fully covariant constraints algebra has an anomaly. Thus, this analysis illustrates clearly that a fully covariant theory with a non-closed constraints algebra can lead, in the spherical symmetric sector, to a reduced theory with a closed constraints algebra. This remark raises the possibility to extend the closeness property beyond spherical symmetry in a fully covariant effective (local and metric) action for loop quantum gravity.

The case of a non-closed constraints algebra
When the conditions (3.41) and (3.44) for having a closed constraints algebra are not fulfilled, then the conservation under time evolution of the deformation Hamiltonian constraint leads to a new constraint given by, which reduces to the constraint when only the condition (3.41) is satisfied and (E r ) = 0. Generically, the new constraint C has a non-vanishing Poisson bracket with the deformed Hamiltonian constraint, for any non-vanishing function u. As a consequence, in that situation, the deformed Hamiltonian constraint and the secondary constraint (3.51) form a pair of second class constraints. There is no more invariance under deformed time reparametrization and the lapse function is a Lagrange multiplier which is fixed by the requirement that the secondary constraint has to be stable under time evolution. Hence, the equation satisfied by the lapse is simply given bẏ which is, in general, a partial differential equation. Clearly N = 0 is a solution, but it is physically unacceptable. A necessary condition for the theory to be physically relevant is the existence of a non-vanishing solution for the lapse function.
As in the previous case, these theories do not have local degrees of freedom because one is replacing the first class constraint H def by a pair of second class constraints (H def , C). Let us emphasize again that, in this case, the lapse function is not free and can be, in principle, expressed in terms of the phase space variables. Contrary to the fully covariant case where the non-closeness of the constraints algebra leads to inconsistencies, spherically symmetric models with a non-closed constraints algebra is well-defined, even though it does not exhibit anymore any deformed reparametrization invariance. Furthermore, it is not clear whether these theories have, in general, physically relevant solutions.

Effective Black Holes Interior Solutions
The purpose of this section is to study (interior) black hole solutions of the theory defined by a deformed Hamiltonian constraint of the form (3.34) while the vectorial constraint is unchanged. First, we will compute the equations of motion. Then, we will show how to resolve them in full generality. Finally, we will give some concrete examples, and we will compare with solutions already found in the literature.

Equations of motion
Here, we look for static spherically symmetric black hole solutions where the metric components (3.1), the lapse and the shift vector depend on the "radial" coordinate only. Furthermore, as the deformations of general relativity are induced by quantum gravity, we are interested in the geometry inside the black hole where quantum gravity effects are supposed to become important (at least close enough to the classical singularity). In the black hole interior (behind the horizon), the radial coordinate plays the role of time and we will denote it t instead of r. In other words, the role of these variables changes when one crosses the horizon, this corresponds to considering time-dependent degrees of freedom only.
Thus, the equations of motion and the constraints reduce to ordinary differential equations involving time derivatives only. Hence, the vectorial constraint is trivially satisfied and one can formally forget about the closedness property of the constraints algebra. Furthermore, the secondary constraint C itself, introduced in (3.51), is also trivially satisfied when radial derivatives vanish, and then the lapse function is free. Now, let us compute the equations of motion. For that, we recall that the time evolution of any function O on the phase space is defined from the Poisson bracket bẏ (4.1) Hence, the equations of motion for the phase space variables (3.6) are easily obtained, and they are partial differential equations that we cannot solve in full generality. As we said while introducing this section, we restrict our study to time dependent solutions only. Therefore, one can drop all radial dependency, and the Hamiltonian constraint together with the effective Einstein equations dramatically simplify. Indeed, the Hamiltonian constraint becomes and the Einstein equations arė 3) where the weak equality ≈ means an equality up to a term proportional to the constraint H def . As the dynamical system reduces to a classical mechanical system (and not a classical field theory because the radial dependency has disappeared), the time evolution of the deformed Hamiltonian constraint is trivially satisfied, which can be verified explicitly. As a consequence, one can forget about the secondary constraint (3.51) and still have a consistent dynamical system. Indeed, one immediately sees that (3.51) is trivially satisfied when radial derivatives are vanishing.

General solution of the equations of motion
Let us show how to solve these equations in full generality. What makes the system "solvable" is that it admits a "triangular" structure in the sense that one can first decouple the variable E r from the other variables, then solve the equation for E r , and then successively decouple the equations for π ϕ , E ϕ and π r that can be solved in principle (at least numerically). This method works for any function S provided that S (2) does not vanish and generalizes the results of [72].

Resolution of the system
As we announced, we start by solving E r . For that, we fix the lapse function by the condition which is equivalent to change the time variable t into τ such that 2dτ = N S (2) dt. In that case, the equation for E r (4.3) simplifies and can solved explicitly according to where a is an integration constant that we fix to a = 0 in order to recover the Schwarzschild solution at the classical limit. Then, we concentrate on the equation (4.6) for π ϕ which becomes, where, to simplify notations, we replaced the weak equality ≈ by a standard equality. To go further and solve this equation, we use the Hamiltonian constraint (4.2) which enables us to write the relation 2t 2 S(π ϕ , π r E ϕ , t 2 ) = 1 . (4.10) As S (2) is supposed not to vanish, we can formally (and locally) invert this equation and solve π r /E ϕ in terms of π ϕ and t (by virtue of the implicit function theorem) according to, where P is the (implicit) function defined by the relation 2t 2 S(π ϕ , P (π ϕ , t 2 ), t 2 ) = 1 . Interestingly, when the necessary condition (3.41) to have a closed constraints algebra is satisfied, the function P can be explicitly computed and it is given by the expression In any case, the variable π ϕ (4.9) decouples and satisfies the equatioṅ π ϕ = 2tP (π ϕ , t 2 ) , (4.14) which can be solved, at least numerically. Later, we will propose examples where it can be solved analytically.
We continue with the equation for E ϕ (4.4) which can now be solved according to where S (a) (t) = S (a) (π ϕ (t), P (π ϕ (t), t 2 ), t 2 ) and π ϕ (t) is given by the solution of (4.14). Of course, E ϕ is defined up to a constant which can be fixed by physical conditions. Finally, the remaining variable π r is immediately obtained from (4.11).

Summary of the results: metric in the black hole interior
To conclude this section, we summarize the results. The resolution relies on the choice of the lapse function (4.7) (which is equivalent to a change of coordinate) and on the (implicit) inversion of the Hamiltonian constraint which state the existence of a function P such that where 2E r S(π ϕ , P (π ϕ , E r ), E r ) = 1 . In general, P is implicitly defined only and can be computed locally (at the vicinity of any points in the phase space when S (2) = 0). In some cases, the function P can be found explicitly. Then, the general solution is E r = t 2 ,π ϕ = 2tP (π ϕ , t 2 ) , E ϕ = exp 2 t du u S (1) (u) S (2) (u) , π r = E ϕ P (π ϕ , t 2 ) , (4.17) which depends on two integration constants, the first one coming from the integration of π ϕ , and the second one coming in the integral defining E ϕ . Notice that the constant a which appears in the integration of E r (4.8) has already been fixed to a = 0. Finally, the metric inside the spherical black hole is given by where the two functions F and G are defined by Such a metric corresponds to a black hole interior if it admits at least one event horizon, which imposes conditions on the functions F and G.

Conditions for describing a trapped interior region
At this stage, we have obtained an exact expression of the most general homogeneous solution of the very general effective theory introduced before. The components of the effective metric are expressed in terms of the functions that govern the deformation of the theory. Nonetheless, a priori, this solution could describe a cosmological background or a homogeneous black hole interior.
Here, we are interested in the case of black holes interiors.
In order for this solution to describe a well defined black hole interior, and therefore a trapped region, one has to impose some conditions.
• The geometry is bounded by an outer horizon located at t h .
• This outer horizon is null, and therefore it can be interpreted as a black hole horizon.
• The geometry can be consistently extended through this horizon, in the outer-communication region corresponding to t > t h .
In order to impose these three conditions, it is useful to introduce the Kodama vector defined for any spherically symmetric geometry as follows. Following [88], a general spherically symmetric spacetime can always be written as, where σ ab is the 2-dimensional metric on the base space with standard coordinates (t, r), the scalar function R(t, r) is the physical radius, and dΩ 2 is the metric on the normalized 2-sphere. With these notations, the Kodama vector is defined by where αβ ⊥ = αβ / |σ| is the densitized 2-dimensional Levi-Civita tensor (see [88,89] for details as well as [90][91][92]). In time-dependent and inhomogeneous geometries, where there are no time-like Killing vector, this vector turns out to be especially useful to locate the horizons. For asymptotically flat spacetime, it coincides with the time-like Killing vector as spatial infinity. The crucial property of the Kodama vector is that it encodes the causal structure of the spacetime. In particular, assuming that there is a single horizon at t h , surrounding a trapped region for t < t h , the Kodama vector turns out to be time-like, null and space-like for t > t h , t = t h and 0 < t < t h respectively. As such the Kodama vector becomes space-like in a trapped region.
Hence, we compute the Kodama vector in the homogeneous interior metric we found, and we obtain that Imposing the first two conditions above imply that there exists t h such that which signals the presence of an outer horizon. Now imposing that there is a coordinate system in which one can extend the spacetime beyond the horizon is equivalent to demand that the determinant of the metric remains regular on the horizon and keep the same sign both inside and outside the horizon, which ensures that the metric remains Lorentzian in the whole spacetime. As the determinant is trivially given by 24) then, at the horizon where F (t h ) = 0, one has to impose the condition As a consequence, one obtains in addition that g rr (t h ) = G(t h ) = 0, which ensures that the outer horizon is indeed a black hole horizon. Moreover, one has to impose that the deformation are such that the curvature invariants remain regular at the horizon. Once we have identified the generic conditions for the solution (4.18) to describe a black hole interior, bounded by at least an outer horizon, it would be interesting to derive the conditions for this geometry to be regular. However the general expression of the metric (4.18) prevents us from providing sharp and useful conditions on this issue. We leave therefore this interesting direction for future works.

Examples
To illustrate the previous results, we consider theories where S is of the form (3.41), i.e. it is an affine function of π r /E ϕ . In that case, the function P entering in (4.17) exists globally and can be computed explicitly. Furthermore, we assume that where f 1 and f 2 are functions of π ϕ only. If one imposes the closeness of the constraints algebra, then A and B satisfy the condition (3.52) which translates into 2f 2 = df 1 /dπ ϕ , and one recovers obviously the well-known anomaly-free condition [62,72,87]. To be general, we assume for the moment that f 1 and f 2 are independent. Let us compute the building blocks to find the explicit form of the line element (4.18). A direct calculation shows that where f a (a = 1, 2) is the derivative of f a with respect to π ϕ . Using E r = t 2 and, after a short calculation, one shows that the equation for π ϕ (t) can be reformulated as On integrates easily this equation and shows that π ϕ is obtained by inverting the relation where r s is a constant of integration. Notice that the constant that comes from the integral in the r.h.s. of (4.29) can be reabsorbed into r s . The expression of E ϕ follows immediately and, after direct calculations, one obtains where π ϕ (t) is given by (4.29) and b is a constant of integration. Without loss of generality, we can fix 2b = r s (as in [72]) by a redefinition of the constant in the integral defining f (4.29). As a conclusion, the line element is given by (4.18) with (4.31) As f = 2f 2 /(f 1 + 1), we can simplify further the expression of G(t) which can be written in the form G(t) = C(t)F (t) as in (4.25) with (4.32) Interestingly, as f 1 and f 2 are generically independent, F and G are themselves independent and one obtains the most general "static" and spherically symmetry solution for the metric. Such a geometry describes effectively a black hole interior if it admits an event horizon which imposes the conditions we have described in the previous subsection. In particular, if we assume that the horizon is located at t = r s (as for the Schwarzchild solution), these conditions simplify and read (f 2 • f −1 )(0) = 0 , (f 1 • f −1 )(0) = −1 . In the case where the condition 2f 2 = f 1 to have a closed constraints algebra is satisfied, then f + 1 = λ(f 1 + 1) where λ is an integration constant. A quick calculation shows that the expression (4.32) reduces to C(t) = λ and one recovers the result of [72] where λ has been fixed to λ = 1.

Discussion
In this article, we have revisited and studied further the issue of deformed covariance in the socalled polymer models of quantum spherically symmetric general relativity in vacuum. We have obtained several new results.
First, in section 2, we introduced a large class of modified gravity theories which mimic, at the level of the full theory (hence beyond spherical symmetry), loop quantum gravity (holonomy like) deformations that one uses in standard symmetry reduced polymer models. In these theories, the invariance under space-time diffeomorphisms is broken whereas 3 dimensional diffeomorphisms remain symmetries, as it is expected from loop quantum gravity, and only the extrinsic curvature part of the Lagrangian is deformed compared to general relativity. Obviously, these theories have a deformed Hamiltonian constraint. Hence, we ask the question whether it is possible that the constraints algebra remains closed even though it is deformed, and we show that this is impossible. In the full class of these theories of modified gravity, only general relativity leads to a closed constraints algebra, and such a no-go result is very similar to the uniqueness Hojmann-Kuchar-Teitelboim theorem [81]. As a consequence, it seems that the notion of deformed covariance found in spherically symmetric model and discussed in detail in [62], could be a peculiar consequence of the spherical symmetry. Nevertheless, we must emphasize that the effective models we introduced are based on the implicit assumption that quantum gravity effects can be described in terms of a local action whose dynamical variable is still a metric. Hence, it is in principle possible to evade the no-go result considering instead non-local actions or non-metric and more exotic variables.
Second, we focus on spherically symmetric models. We consider the Hamiltonian theory whose phase space consists in two pairs of conjugate variables (schematically the two free functions in the most general spherically symmetric metric together with their conjugate momenta) which satisfy two constraints, the Hamiltonian and the vectorial constraint. We ask the question to which extend one can deform these constraints, compared to general relativity, with the condition that their algebra remains closed. We show that there is no non-trivial deformation of the vectorial constraint, which provides a very strong argument for keeping the vectorial constraint undeformed as it is usually done in the context of polymer models. Then, we compute the most general deformation of the Hamiltonian constraint and we recover known results in the literature. Our construction provides a large class of effective theories with a deformed covariance which encompass most of polymer models introduced in the literature. Notice that the model introduced recently in [27,32] does not fall in this class mainly because the corresponding Hamiltonian constraint is, a priori, not a scalar with respect to the vectorial constraint.
Finally, we compute the equations of motion and look for spherically symmetric static solutions. Remarkably, we found an explicit, exact and very general solution. In particular, we give the expressions of the components of the effective metric in terms of the functions that govern the deformations of the theory. This exact solution, given by (4.18) and (4.19), can describe both a cosmological or a black hole interior solution. Assuming that the solution corresponds to the interior of a black hole, we gave the general conditions (4.23) and (4.25) in order for the effective geometry to come with a well-defined trapped region.
This work opens many directions. First of all, one can extend our analysis to include different models as the ones introduced in [27,32]. One can try to find how to characterize more generally such models and to compute their general static solutions as we have done in the present work. Moreover, we have now a framework to address the question of the stability of the solutions we found with respect to linear (and spherical) perturbations. To our knowledge, stability properties of the perturbed polymer black holes have been investigated only in [13] and in [41]. Let us finally stress that the present modified gravity model could also be used to derive polymer-like deformations of the spherically symmetric exterior geometry, and could be extended to describe deformations of the axi-symmetric vacuum gravity phase space. This would allow us to discuss the effective dynamics of polymer deformations of the Kerr black hole (see [93] for current efforts in this direction). We leave these open directions for future works.