Black holes in effective loop quantum gravity:
Covariant holonomy modifications
Idrus Husin Belfaqih,1***e-mail address<EMAIL_ADDRESS>Martin
Bojowald,2†††e-mail address<EMAIL_ADDRESS>Suddhasattwa Brahma1‡‡‡e-mail
address<EMAIL_ADDRESS>and Erick I. Duque2§§§e-mail address:
<EMAIL_ADDRESS>
1Higgs Centre for Theoretical Physics, School of Physics & Astronomy,

University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK

2 Institute for Gravitation and the Cosmos,

The Pennsylvania State University,

104 Davey Lab, University Park, PA 16802, USA

###### Abstract

Emergent modified gravity provides a covariant, effective framework for
obtaining spherically symmetric black hole solutions in models of loop quantum
gravity with scale-dependent holonomy modifications. Exact solutions for
vacuum black holes in the presence of a cosmological constant are derived here
and analyzed in four different gauges, explicitly related to one another by
standard coordinate transformations. The global structure is obtained by
gluing space-time regions corresponding to the gauge choices, reconstructing a
non-singular wormhole space-time for an arbitrary scale-dependent holonomy
parameter. This outcome demonstrates the robustness of black-hole models with
covariant holonomy modifications under quantization ambiguities. Compared with
previous constructions, full covariance of the resulting space-time models as
derived here implies subtle new effects and leads to a novel understanding of
the parameters in holonomy modifications, distinguishing a constant holonomy
length from a possibly scale-dependent function that may change coefficients
of holonomy terms. New physical results are obtained for instance in the
context of a non-trivial zero-mass limit of holonomy-modified space-times. The
existence of a consistent effective space-time structure implies various novel
aspects of a net gravitational stress-energy and related thermodynamical
properties.

###### Contents

  1. 1 Introduction
  2. 2 Emergent modified gravity in spherical symmetry
    1. 2.1 Spherically symmetric canonical gravity
    2. 2.2 Modified canonical gravity
    3. 2.3 Covariance conditions
    4. 2.4 Covariant modified constraints
    5. 2.5 Effective loop quantum gravity
      1. 2.5.1 Traditional ingredients
      2. 2.5.2 Shortcomings
      3. 2.5.3 New features
      4. 2.5.4 Self-consistent treatment of holonomy effects
    6. 2.6 Gravitational observable
    7. 2.7 Reflection symmetry of the transition surface
  3. 3 Black-hole solution for general triangulation schemes
    1. 3.1 Non-periodic phase-space coordinates
      1. 3.1.1 Schwarzschild gauge: Static region
      2. 3.1.2 Schwarzschild gauge: Homogeneous regions
      3. 3.1.3 From static Schwarzschild to Gullstrand–Painlevé
      4. 3.1.4 From homogeneous Schwarzschild to Gullstrand-Painlevé
      5. 3.1.5 Gullstrand-Painlevé gauge
    2. 3.2 Internal-time gauge of homogeneous regions
    3. 3.3 Proof of extremality
  4. 4 Global structure
    1. 4.1 Asymptotic flatness and zero-mass limit
    2. 4.2 The maximal extension
    3. 4.3 Physical global structure
    4. 4.4 Presence of matter
  5. 5 Interior physics
    1. 5.1 Curvature around reflection-symmetry hypersurfaces
    2. 5.2 Energy conditions in an emergent space-time theory
    3. 5.3 Congruences near a minimum-radius hypersurface
      1. 5.3.1 Lightlike congruences
      2. 5.3.2 Timelike congruences and their implications
    4. 5.4 The net stress-energy tensor and gravitational energy
  6. 6 Exterior physics
    1. 6.1 Newtonian limit
    2. 6.2 Geodesic properties
      1. 6.2.1 Timelike geodesics and effective potential
      2. 6.2.2 Nearly-circular orbits of massive objects
      3. 6.2.3 Null rays: Deflection angle and redshift
    3. 6.3 Energy and thermodynamics
      1. 6.3.1 Curvature tensors
      2. 6.3.2 Geometric conditions, net stress-energy tensor, and gravitational energy
    4. 6.4 Black-hole thermodynamics
  7. 7 Constant holonomy function
    1. 7.1 Minimum radius and geometric conditions
    2. 7.2 Observational features
      1. 7.2.1 Newtonian and zero-mass limits
      2. 7.2.2 Nearly-circular orbits of massive objects
      3. 7.2.3 Deflection angle
      4. 7.2.4 Net stress-energy tensor
      5. 7.2.5 Black-hole thermodynamics
    3. 7.3 Maximal radius in the presence of a cosmological constant
  8. 8 Decreasing holonomy function
    1. 8.1 Minimum radius and geometric conditions
    2. 8.2 Observational features
      1. 8.2.1 Newtonian limit
      2. 8.2.2 Zero-mass limit
      3. 8.2.3 Nearly-circular orbits of massive objects
      4. 8.2.4 Deflection angle
      5. 8.2.5 Net stress-energy tensor
      6. 8.2.6 Black-hole thermodynamics
    3. 8.3 Presence of a cosmological constant
  9. 9 Conclusions
  10. A Equations of Motion
    1. A.1 Non-periodic variables
    2. A.2 Periodic variables

##  1 Introduction

Black holes are now everyday objects encountered in astronomical observations
indicating that much of their exterior is well described by General Relativity
(GR). Yet, the region beyond the event horizon remains inaccessible to
observations and we must rely on theoretical extrapolations to understand the
interior physics. In particular, GR robustly predicts a singularity at the
center of black holes [1], which has long been interpreted as the result of
the breakdown of the theory. Hence, it has been argued that a quantum theory
of gravity would be able to resolve such singularities in analogy to how the
quantum theory of light resolves the ultraviolet catastrophe. One candidate
background-independent and non-perturbative theory is loop quantum gravity
(LQG) [2, 3]. We shall explore the fate of stationary, spherically symmetric
(vacuum) solutions due to quantum corrections from LQG in an effective
framework, making sure that full covariance is realized in any effective line
element used to describe the physics of black holes.

The canonical quantization in LQG is based on the holonomy-flux algebra. This
is so because the regularization scheme followed in LQG does not allow for
quantum operators corresponding to the connection, but rather its parallel
transport along one-dimensional curves. In effective black-hole models, one
focuses on the spherically symmetric reduced theory and incorporates quantum
corrections corresponding to this feature by modifying the Hamiltonian
constraint from a simple polynomial dependence on the connection to a
trigonometric one resulting from the real-valued combinations of holonomies.
This procedure is often done by hand, rather than derived from first
principles, and covariance of the resulting models is seldom checked or even
acknowledged as a problem.

General covariance is harder to grasp in the canonical formulation than in its
more common geometric or Lagrangian cousins. A modified Hamiltonian constraint
will in general modify its Poisson brackets. One must therefore ensure that
the brackets remain ‘first class’ for the equations of motion to be gauge
covariant. Even if this condition is fulfilled, there often remains a modified
structure function in the Poisson bracket of two Hamiltonian constraints,
whose proper interpretation is a modified kinematical dependence of the space-
time metric on the phase-space variables [4, 5]. Contrary to what is commonly
believed, preserving the first-class nature of the constraint brackets does
not imply that the space-time metric tensor is covariant, constructed in the
usual way from phase-space variables, the structure function, as well as a
lapse function and shift vector field: In a canonically modified theory, this
object is not guaranteed to be subject to gauge transformations that are on-
shell equivalent to the tensor transformation law. Rather, additional
covariance conditions must be imposed on the structure function, which further
restrict compatible modifications of the Hamiltonian constraint [6]. In order
to implement these conditions, we will use Emergent Modified Gravity (EMG) [5,
6] as the underlying framework for effective LQG. In particular, the
spherically symmetric model of EMG can be solved for the most general,
covariant, modified constraint in (3+1)-dimensions with a dependence on the
spatial derivatives of the phase space up to the second order, which contains
terms that can be interpreted as holonomy modifications.

An earlier, special case of EMG in spherical symmetry was studied in [7, 8]
where the holonomy modifications were shown to lead to a nonsingular black-
hole solution in vacuum. The global structure of such a solution is an
interuniversal wormhole joining a black hole to a white hole through their
interiors. The holonomy parameter $\mu$ in these works was assumed constant,
$\mu=\mu_{0}$, a framework referred to as the $\mu_{0}$ scheme in some of the
LQG literature. This holonomy parameter is associated with the coordinate
length of the links in the triangulation of space as required by LQG. More
precisely, in the spherically symmetric effective framework where the space-
time topology may be fixed to $\mathcal{M}=\mathbb{R}^{2}\times S^{2}$, one
only considers the angular discretization of the spheres (the submanifold
$S^{2}\subset\mathcal{M}$) and hence $\mu_{0}$ corresponds to the angular
coordinate length of these links. A constant $\mu_{0}$ is required if the
constraint operator is constructed from basic holonomy operators. However,
holonomies depend on a canonical variable such as a spatial connection
component or extrinsic curvature of the spheres, which unlike space-time
curvature scalars are not guaranteed to be small in low-curvature regimes.
Depending on specific properties of dynamical solutions, a constant $\mu_{0}$
could therefore imply large holonomy modifications in regions that should
remain close to the classical limit.

It is therefore necessary to introduce scale-dependent holonomy parameters
$\mu$ that become smaller in semiclassical regions in order to compensate for
potentially growing extrinsic curvature. Heuristically, such a behavior can be
motivated as including lattice-refinement effects of an underlying discrete
spatial state. As spherical areas in a semiclassical region grow larger, a
discrete lattice state would have to be refined more and more in order to
avoid discreteness being magnified to macroscopic sizes. The coordinate length
of lattice links correspondingly decreases, implying a scale-dependent $\mu$
as a function of the spherical area. The scale dependence is not unique and
determined largely by phenomenological reasoning, making sure that the
modified theory is consistent with classical behavior in the right regimes. In
homogeneous cosmological models, there is only one relevant semiclassical
regime given by large volumes. In spherical symmetry and the related black-
hole models, however, there are different semiclassical regimes, including
large radii as well as near-horizon geometries, which should also be
semiclassical for astrophysical black holes. In the latter case, however, the
conditions on a suitable scale dependence of $\mu$ are different from what is
required for large radii because they refer to different components of the
spatial metric. Moreover, these conditions are dependent on the space-time
gauge or slicing used to express the near-horizon geometry. This observation
re-inforces the condition that holonomy modifications must be made compatible
with general covariance of an effective line element, and that a large set of
scale-dependent $\mu$ must be compatible with a given class of modified
theories.

A simple ad-hoc version of scale-dependent holonomy parameters is often
referred to as a $\bar{\mu}$-scheme [9], but it is not uniquely defined. The
original motivation was to replace the coordinate length of the links of a
triangulation with a more tangible length such as the Planck length $\ell_{\rm
Pl}$, characteristic of quantum gravity. Since the holonomy around a closed
loop of links is related to the curvature of the gauge connection, one may use
the Planck area, or the smallest non-zero eigenvalue $\Delta$ in the area
spectrum of loop quantum gravity, as an ingredient of the holonomy parameter
$\mu$. If the exponent of $\Delta$ in the holonomy parameter is chosen, which
might be proportional to $\Delta$ itself as an area or $\sqrt{\Delta}$ as a
side length, or perhaps some other power law depending on one’s intuition,
dimensional arguments require that this quantity be divided by a function of
metric components with suitable length units. For this purpose, one can employ
the component $q_{\vartheta\vartheta}$ that determines the areas of spheres.
This procedure may then result in $\mu\propto q_{\vartheta\vartheta}^{-1}$ or
$\mu\propto q_{\vartheta\vartheta}^{-1/2}$ or with some other exponent. (The
square-root version is closest to the $\bar{\mu}$-scheme in spatially flat
cosmological models, where the relevant connection component is proportional
to $\dot{a}$ in terms of the scale factor. This component is then multiplied
by $\sqrt{\Delta}$ for dimensional reasons, and divided by $a$ in order make
the expression independent of spatial rescalings. Here, we have a simple
covariance argument, but, compared with spherical symmetry, only referring to
a small subset of possible coordinate transformations.) Such a construction
does not lead to a unique form of a scale-dependent $\mu$ because the final
result depends on the exponent chosen for $\Delta$, and the radial metric
component $q_{xx}$ could also be used in such an ad-hoc argument [10].

A more systematic approach is therefore needed, which we construct here based
on the condition of covariance. As we will see, this condition, while still
not leading to a unique analog of $\mu$, implies additional restrictions on
possible phase-space dependencies of this function, which turn out to include
a dependence on $q_{\vartheta\vartheta}$ but not on $q_{xx}$. Such a behavior
is well-defined even when crossing the horizon because the physical length of
the links remains spacelike everywhere and discretizes the submanifold
$S^{2}\subset\mathcal{M}$. In Schwarzschild coordinates, the spatial slices
have the topology $\Sigma=\mathbb{R}\times S^{2}$, such that the exterior is
foliated by spheres with squared areal radius $q_{\vartheta\vartheta}$, while
the interior geometry is that of hypercylinders with squared radius
$q_{\vartheta\vartheta}$. In the heuristic lattice picture, both cases can be
interpreted as discrete submanifold $S^{2}$ with plaquettes of constant size
if $\mu\propto q_{\vartheta\vartheta}^{-1/2}$ is used.

The purpose of this paper is to generalize the results of [7, 8] to an
arbitrary holonomy parameter as a function of the radius,
$\mu(q_{\vartheta\vartheta})$, and hence obtaining a general triangulation
scheme in its heuristic interpretation. We will note crucial conceptual
differences in our new scheme as well, in particular its general covariance
and its role in light of a complete discussion of canonical transformations
that may be used to change a functional dependence such as
$\mu(q_{\vartheta\vartheta})K_{\varphi}$ where $K_{\varphi}$ is one of the
momenta, classically related to extrinsic curvature. While some properties of
$\mu_{0}$ and $\bar{\mu}$-type schemes can still be recognized in specific
choices of canonical variables, our covariant construction (both in terms of
space-time and phase-space transformations) provides a novel and unified
picture of holonomy modifications. Since the relevant equations are obtained
in the framework of emergent modified gravity, we follow the latter’s notation
and refer to the new holonomy parameter as $\lambda(q_{\vartheta\vartheta})$,
highlighting its new origin and features.

The new conceptual understanding of holonomy modifications leads to novel
physical insights. In particular, we extend previous constructions for black
holes in LQG to asymptotically de Sitter space-times with a well-defined
asymptotic infinity, rather than ending at a maximal radius. For this result,
our covariant treatment of scale-dependent holonomy parameters is crucial.
Covariance also allows us to apply standard space-time analysis in an
unambiguous way, including the derivation of timelike and likelight geodesics,
effective stress-energy tensors, and thermodynamics.

After a general discussion of holonomy modifications, we apply our results to
specific versions, including one that in its dynamical implications
corresponds to the traditional choice of $\mu(q_{\vartheta\vartheta})\propto
q_{\vartheta\vartheta}^{-1/2}$. To this end we begin Section 2 with a
discussion of emergent modified gravity and describe how some modification
functions can be interpreted as holonomy corrections and hence used as a
covariant effective model for LQG. The black-hole solution with a general
holonomy parameter is obtained in Section 3. The procedure follows the
solution of the equations of motion in four different gauges which we show are
all related by standard coordinate transformations covering distinct but
overlapping regions of the spacetime. These four patches are then sewed
together to recover a global structure with the properties of a wormhole and
present the maximal extension in null coordinates in Section 4. We further
show that the physical global structure is non-unique, owing to ambiguities in
the slicing and gluing of the maximal extension. Section 5 is devoted to an
analysis of the interior structure of space-time with the black-hole horizon.
We highlight the robust feature that the classical singularity is replaced by
a non-singular hypersurface of reflection symmetry, at which the initially
collapsing interior can be glued to an expanding version. This outcome is
largely independent of the specific scale dependence of holonomy parameters.
Geodesic analysis can be used in order to relate this non-singular behavior to
heuristic effects such as repulsive modified gravity. In Section 6, we study
several physical effects of holonomy modifications in the black-hole exterior
including corrections to the precession frequency of nearly circular orbits,
to the deflection angle of null rays, and to black hole thermodynamics. In
Section 7, we revisit asymptotic problems in holonomy modifications that
correspond to the traditional $\mu_{0}$-type scheme of scale-independent
$\mu$, including asymptotic corrections to the deflection angle, to the Brown-
York quasilocal energy, and to the Bekenstein-Hawking entropy, as well as the
existence of a maximum radius of the universe. In Section 8 we study the
effects of a specific version of scale-dependent $\mu$ and confirm that it
removes unexpected and potentially problematic features of the $\mu_{0}$-type
scheme in asymptotic regimes.

##  2 Emergent modified gravity in spherical symmetry

The framework of emergent modified gravity plays an important role in
discussions of general covariance in a canonical setting and is therefore
reviewed in this section. Its main feature is that the space-time metric is
emergent rather than effective: It is not a corrected version of a pre-
existing classical metric tensor. Rather, it is the only metric tensor
admissible in a modified canonical theory because the classical expression as
a phase-space function fails to obey the tensor transformation law when gauge
transformations are modified. The entire emergent metric is derived from
covariance conditions, in contrast to what is usally referred to as an
effective metric if perturbative correction terms are derived for the
classical metric.

###  2.1 Spherically symmetric canonical gravity

Canonical gravity, originally derived by Dirac, is conveniently based on the
Arnowitt–Deser–Misner (ADM) decomposition where the line element in spherical
symmetry takes the form [13, 14]

${\rm d}s^{2}=-N^{2}{\rm d}t^{2}+q_{xx}({\rm d}x+N^{x}{\rm
d}t)^{2}+q_{\vartheta\vartheta}{\rm d}\Omega^{2}\ ,$ (1)

where $q_{ab}$ is the three-dimensional metric induced on the spatial
hypersurfaces foliating the manifold, while $N$ and $N^{x}$ are the lapse
function and shift vector field associated to the observer frame with time
coordinate $t$. The Hamiltonian is composed of the Hamiltonian constraint $H$
and the diffeomorphism constraint $H_{x}$. As phase-space functions, depending
on the spatial metric and its momenta, these expressions vanish on physical
solutions, $H=0$ and $H_{x}=0$, bringing the system on-shell. The Hamiltonian
that generates evolution with respect to $t$ is given by
$H[N,N^{x}]=H[N]+H_{x}[N^{x}]$ where the square brackets indicate that these
terms are smeared (or spatially integrated) by the functions $N$ and $N^{x}$.
The constraints then generate time evolution via Poisson brackets, $\dot{\cal
O}=\\{{\cal O},H[N,N^{x}]\\}$ for any phase-space function ${\cal O}$.

The same constraints also generate gauge transformations by use of different
smearing functions $\epsilon^{0}$ and $\epsilon^{x}$ instead of the fixed $N$
and $N^{x}$ for a given frame, and hence the generator of a general gauge
transformation is $H[\epsilon^{0},\epsilon^{x}]$. The vanishing of the
constraints must be consistent in different gauges, which means that the
Poisson brackets of the constraints with themselves must vanish on-shell. The
specific brackets found in canonical gravity are known as the hypersurface-
deformation brackets, given by

$\displaystyle\\{H_{x}[N^{x}],H_{x}[\epsilon^{x}]\\}$ $\displaystyle=$
$\displaystyle-
H_{x}[\epsilon^{x}(N^{x})^{\prime}-N^{x}(\epsilon^{x})^{\prime}]\,,$ (2a)
$\displaystyle\\{H[N],H_{x}[\epsilon^{x}]\\}$ $\displaystyle=$
$\displaystyle-H[\epsilon^{x}N^{\prime}]\,,$ (2b)
$\displaystyle\\{H[N],H[\epsilon^{0}]\\}$ $\displaystyle=$ $\displaystyle-
H_{x}\left[q^{xx}\left(N^{\prime}\epsilon^{0}-N(\epsilon^{0})^{\prime}\right)\right]\,,$
(2c)

where the structure-function $q^{xx}$ is the inverse of the radial metric
component, $q^{xx}=1/q_{xx}$. These brackets indeed vanish on-shell when the
constraints equal zero. Furthermore, if the system is fully on-shell, also
imposing equations of motion generated by the constraints, the gauge freedom
of canonical gravity is equivalent to the freedom of choosing a space-time
slicing or a coordinate system.

Therefore, the constraints are an essential ingredient of encoding general
covariance. However, starting with gauge transformations of phase-space
functions, two additional steps are needed for a full realization of general
covariance, understood as gauge transformations being equivalent to coordinate
changes, or as the existence of a tensor-transformation law for the space-time
metric: First, we need gauge transformations not only for the spatial part
$q_{xx}$, which is a basic phase-space degree of freedom, but also for the
time-time and the mixed space-time components, depending on $N$ and $N^{x}$.
While the latter space-time functions do not have momenta and therefore are
not included in the phase space (unless an extended version is used), their
gauge transformations are uniquely determined by the constraints and their
Poisson brackets through consistency conditions between evolution and gauge
equations. Secondly, coordinate changes and the tensor transformation law
require explicit time derivatives, rather than momentum components as they
appear in the constraints and the gauge transformation they generate. Time
derivatives are related to momenta by some of the canonical equations of
motion generated by the constraints. No additional ingredient is therefore
required in order to obtain full expressions of coordinate transformations,
but this can be achieved only if the theory is taken fully on-shell by
imposing not only the constraint equations but also some of the equations of
motion.

We will include these ingredients in our present review in the more general
setting of modified canonical gravity.

###  2.2 Modified canonical gravity

If one considers a modified Hamiltonian constraint $\tilde{H}$, as implied for
instance by the regularization procedure of LQG, then we must first ensure
that the constraint brackets remain first class for the constraints to vanish
consistently in all gauges. More specifically, we require that the general
form of hypersurface-deformation brackets be preserved,

$\displaystyle\\{H_{x}[N^{x}],H_{x}[\epsilon^{x}]\\}$ $\displaystyle=$
$\displaystyle-
H_{x}[\epsilon^{x}(N^{x})^{\prime}-N^{x}(\epsilon^{x})^{\prime}]\,,$ (3a)
$\displaystyle\\{\tilde{H}[N],H_{x}[\epsilon^{x}]\\}$ $\displaystyle=$
$\displaystyle-\tilde{H}[\epsilon^{x}N^{\prime}]\,,$ (3b)
$\displaystyle\\{\tilde{H}[N],\tilde{H}[\epsilon^{0}]\\}$ $\displaystyle=$
$\displaystyle-
H_{x}\left[\tilde{q}^{xx}\left(N^{\prime}\epsilon^{0}-N(\epsilon^{0})^{\prime}\right)\right]\,,$
(3c)

where the structure-function $\tilde{q}^{xx}$ is determined by demanding that
no additional terms appear on the right-hand sides. However, the specific
structure function $\tilde{q}^{xx}$ need not equal the classical one, given by
a specific function of one of the basic phase-space variables, $q_{xx}$, on
which the modified $\tilde{H}$ depends (or an equivalent version in triad
variables).

This condition is stronger than anomaly freedom, which would be compatible
with, say, an additional $H_{x}$-term appearing in the Poisson bracket of two
$\tilde{H}$. Both conditions, anomaly-freedom and the specific form of
hypersurface-deformation brackets, restrict permissible modifications of the
constraint. A comparison with the classical geometrical structure of space-
time, together with an analysis of the transformation properties of lapse and
shift to be discussed in the next section, suggests that the inverse of the
modified structure function $\tilde{q}^{xx}$ plays the role of the radial
component of an effective line element (in the stricter sense of emergence)
replacing $q_{xx}$ in (1). However, anomaly freedom or brackets of
hypersurface-deformation form do not imply that this effective line element is
invariant. Rather, a further condition must be imposed on $\tilde{H}$, derived
from the tensor transformation law of $(1/\tilde{q}^{xx},N,N^{x})$, such that
the gauge transformations of the effective metric it generates correspond to
infinitesimal coordinate transformations [6]. (The angular component
$\tilde{q}_{\vartheta\vartheta}$ also plays a role in the covariance
condition, but it is less direct because this component does not appear in the
structure functions of spherically symmetric gravity.)

According to the classic analysis of [15, 16], if one considers the classical
spatial metric as a configuration variable of the phase space, then the
Hamiltonian constraint is uniquely determined by the hypersurface-deformation
brackets up to the choice of Newton’s and the cosmological constant. These
results therefore imply that $H$ must be the classical constraint of GR, and
generally covariant modifications are ruled out unless one includes higher-
derivative terms that in a canonical formulation enlarge the classical phase
space. However, in the recent formulation of emergent modified gravity [5, 6]
the assumption that the spatial metric $q_{ab}$ is a configuration variable of
the phase-space is not necessary in order to obtain a consistent field-theory
description of space-time. Instead, emergent modified gravity assumes that the
phase space is composed of certain fundamental fields different from the
metric, and the metric is an emergent object to be regained by the imposition
of anomaly freedom and covariance. In fact, the most general modified
Hamiltonian constraint depending on spatial derivatives of the phase space up
to second order and satisfying anomaly freedom and covariance in vacuum
spherical symmetry has been obtained exactly in [6]. If LQG is to be treated
in a covariant framework of effective line elements, without, as usually
assumed in this context, an extension of the classical phase space in order to
account for perturbative higher-curvature corrections or higher time
derivatives, its effective Hamiltonian constraint must be included in the
family of constraints allowed by emergent modified gravity, and its effective
line element must be given by the corresponding emergent space-time.

###  2.3 Covariance conditions

The hypersurface-deformation brackets (3) together with general properties of
a Poisson bracket, such as the Jacobi identity, imply that off-shell gauge
transformations for the lapse function and shift vector are given by [17, 18,
4]

$\displaystyle\delta_{\epsilon}N$ $\displaystyle=$
$\displaystyle\dot{\epsilon}^{0}+\epsilon^{x}N^{\prime}-N^{x}(\epsilon^{0})^{\prime},$
(4a) $\displaystyle\delta_{\epsilon}N^{x}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}^{x}+\epsilon^{x}(N^{x})^{\prime}-N^{x}(\epsilon^{x})^{\prime}+\tilde{q}^{xx}\left(\epsilon^{0}N^{\prime}-N(\epsilon^{0})^{\prime}\right)\,.$
(4b)

These equations play a central role in general covariance as explained below.

Building on [4], the brackets (3) and the transformations (4) suggest the
existence of the space-time metric

${\rm d}s^{2}=-N^{2}{\rm d}t^{2}+\tilde{q}_{xx}({\rm d}x+N^{x}{\rm
d}t)^{2}+\tilde{q}_{\vartheta\vartheta}{\rm d}\Omega^{2}\ ,$ (5)

where $\tilde{q}_{xx}$ is the inverse of the (possibly non-classical)
structure-function, assuming for now that $\tilde{q}_{xx}>0$, while
$\tilde{q}_{\vartheta\vartheta}$ is not given by structure functions and must
hence be chosen on different grounds. (This feature is a consequence of the
symmetry-reduced theory under consideration and its space-time structure.
Since the component is subject to canonical transformations, it is possible to
prescribe a specific form of $\tilde{q}_{\vartheta\vartheta}$ as a phase-space
function in order to reduce the freedom of performing canonical
transformations.) Because the spatial metric is not directly determined by
kinematical properties of the phase space but rather depends, at least in
part, on the inverse of the structure-function, we refer to it as an emergent
metric. This nomenclature will make more sense when dealing with non-classical
structure functions which arise from the requirement of anomaly-freedom of the
modified constraints incorporating potential non-perturbative quantum
corrections.

Anomaly-freedom of the brackets (3), even if they are of hypersurface-
deformation form, does not ensure covariance of the resulting space-time (5),
contrary to what is commonly stated. Instead, we have to impose the stronger
covariance condition

$\displaystyle\delta_{\epsilon}\tilde{g}_{\mu\nu}\big{|}_{\text{O.S.}}=$
$\displaystyle\mathcal{L}_{\xi}\tilde{g}_{\mu\nu}\big{|}_{\text{O.S.}}\ ,$ (6)

which ensures that canonical gauge transformations generate infinitesimal
diffeomorphisms. This requirement contains (4) as a necessary condition, but
it is also non-trivial when applied to the spatial part of the metric and
imposes yet another set of restrictions on allowed modifications of the
classical constraints.

###  2.4 Covariant modified constraints

Following [19, 20] we consider the vacuum spherically symmetric theory with
canonical gravitational variables
$\left\\{K_{\varphi}\left(x\right),E^{\varphi}\left(y\right)\right\\}=\delta\left(x-y\right)$
and
$\left\\{K_{x}\left(x\right),E^{x}\left(y\right)\right\\}=\delta\left(x-y\right)$.
(We use units such that Newton’s constant equals one.) In the classical
theory, the momenta $E^{x}$ and $E^{\varphi}$ are components of the densitized
triad, while the configuration variables are directly related to the
extrinsic-curvature components ${\cal K}_{\varphi}=K_{\varphi}$ and ${\cal
K}_{x}=2K_{x}$. Leaving the diffeomorphism constraint

$\displaystyle H_{x}=$ $\displaystyle
E^{\varphi}K_{\varphi}^{\prime}-K_{x}(E^{x})^{\prime}$ (7)

unmodified and building on [21], we consider the most general ansatz for the
Hamiltonian constraint up to second-order derivatives and quadratic first-
order derivative terms [6]

$\displaystyle\tilde{H}=$ $\displaystyle
a_{0}+((E^{x})^{\prime})^{2}a_{xx}+((E^{\varphi})^{\prime})^{2}a_{\varphi\varphi}+(E^{x})^{\prime}(E^{\varphi})^{\prime}a_{x\varphi}+(E^{x})^{\prime\prime}a_{2}$
$\displaystyle+(K_{\varphi}^{\prime})^{2}b_{\varphi\varphi}+(K_{\varphi})^{\prime\prime}b_{2}+(E^{x})^{\prime}K_{\varphi}^{\prime}c_{x\varphi}+(E^{\varphi})^{\prime}K_{\varphi}^{\prime}c_{\varphi\varphi}+(E^{\varphi})^{\prime\prime}c_{2}\,.$
(8)

Here, $a_{0}$, $a_{xx}$, $a_{\varphi\varphi}$, $a_{x\varphi}$, $a_{2}$,
$b_{\varphi\varphi}$, $b_{2}$, $c_{2}$, $c_{\varphi\varphi}$, $c_{x\varphi}$
are all functions of the phase-space variables, but not of their derivatives.
We have not included terms quadratic in the radial derivatives of $K_{x}$ in
(8) because it can be readily shown that the covariance condition (6) does not
allow them if
$\tilde{q}_{\vartheta\vartheta}=\tilde{q}_{\vartheta\vartheta}(E^{x})$ which
is our case of interest.

Demanding that this ansatz satisfies the anomaly-free brackets of hypersurface
deformation form and imposing the additional covariance conditions encoded in
(6), the general Hamiltonian constraint reduces to

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle-\sqrt{E^{x}}\;\frac{g}{2}\bigg{[}E^{\varphi}\left(f_{0}+\frac{K_{x}}{E^{\varphi}}f_{1}\right)+\frac{((E^{x})^{\prime})^{2}}{E^{\varphi}}\left(f_{2}-\frac{K_{x}}{2E^{\varphi}}\left(\frac{\partial\ln
g}{\partial K_{\varphi}}+C_{x\varphi}\right)\right)$ (9) $\displaystyle\hskip
56.9055pt+\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}+\frac{(E^{x})^{\prime}K_{\varphi}^{\prime}}{E^{\varphi}}C_{x\varphi}\bigg{]}\,.$

Now, $g$, $f_{0}$, $f_{1}$, $f_{2}$, and $C_{x\varphi}$ are functions of
$E^{x}$ and $K_{\varphi}$, and are related to one another by four differential
equations. Consequently, the structure function $\tilde{q}^{xx}$ is composed
of these functions plus an additional direct dependence on the phase-space
variables. More explicitly, it is given by

$\displaystyle\tilde{q}^{xx}=\left(\frac{\partial f_{1}}{\partial
K_{\varphi}}-f_{1}C_{x\varphi}-\frac{1}{2}\left(\frac{\partial^{2}\ln
g}{\partial K_{\varphi}^{2}}+C_{x\varphi}^{2}+\frac{\partial
C_{x\varphi}}{\partial K_{\varphi}}+\frac{\partial\ln g}{\partial
K_{\varphi}}C_{x\varphi}\right)\left(\frac{(E^{x})^{\prime}}{E^{\varphi}}\right)^{2}\right)\frac{g^{2}}{4}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$ (10)

The constraint brackets (3) and the covariance condition (6), which can be
formulated in terms of Poisson brackets as well, are both invariant under
canonical transformations. We can therefore apply canonical transformations in
order to further simplify the constraint and eliminate some of the free
functions. An interesting subset of canonical transformations is given by
those that preserve the diffeomorphism constraint (7):

$\displaystyle K_{\varphi}=$ $\displaystyle f_{c}(E^{x},\tilde{K}_{\varphi})\
,\hskip 142.26378ptE^{\varphi}=\tilde{E}^{\varphi}\left(\frac{\partial
f_{c}}{\partial\tilde{K}_{\varphi}}\right)^{-1}\ ,$ $\displaystyle K_{x}=$
$\displaystyle\frac{\partial(\alpha_{c}^{2}E^{x})}{\partial
E^{x}}\tilde{K}_{x}+\tilde{E}^{\varphi}\frac{\partial f_{c}}{\partial
E^{x}}\left(\frac{\partial f_{c}}{\partial\tilde{K}_{\varphi}}\right)^{-1}\
,\hskip 28.45274pt\tilde{E}^{x}=\alpha_{c}^{2}(E^{x})E^{x}\,,$ (11)

where the new variables are written with a tilde. It can be shown that there
is a canonical transformation with a function $f_{c}=f_{c}(K_{\varphi})$ and
$\alpha_{c}=1$, such that the transformed $C_{x\varphi}$ vanishes, at least
locally in the phase space.

This canonical transformation allows us to include $C_{x\varphi}=0$ among the
conditions for covariance, resulting in five equations for the five functions
in (9). The equations, which include partial differential ones, can be solved
exactly, yielding

$g=\chi\cos^{2}\left(\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)\
,$ (12a)
$gf_{1}=4\chi\left(c_{f}\frac{\sin(2\lambda(K_{\varphi}+\lambda_{\varphi}))}{2\lambda}+q\cos(2\lambda(K_{\varphi}+\lambda_{\varphi}))\right)\
,$ (12b)
$f_{2}=-\frac{\alpha_{2}}{4E^{x}}+\frac{\sin\left(2\lambda(K_{\varphi}+\lambda_{\varphi})\right)}{2\lambda\cos^{2}\left(\lambda(K_{\varphi}+\lambda_{\varphi})\right)}\left(\lambda\frac{\partial(\lambda\lambda_{\varphi})}{\partial
E^{x}}+\lambda K_{\varphi}\frac{\partial\lambda}{\partial E^{x}}\right)\ ,$
(12c) $\displaystyle gf_{0}$ $\displaystyle=$
$\displaystyle\chi\Bigg{(}\frac{\alpha_{0}}{E^{x}}+2\frac{\sin^{2}\left(\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{\lambda^{2}}\frac{\partial
c_{f}}{\partial
E^{x}}+4\frac{\sin\left(2\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{2\lambda}\frac{\partial
q}{\partial E^{x}}$ (12d)
$\displaystyle+4c_{f}\left(\frac{1}{\lambda}\frac{\partial(\lambda\lambda_{\varphi})}{\partial
E^{x}}\frac{\sin\left(2\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{2\lambda}+\left(\frac{\alpha_{2}}{4E^{x}}-\frac{\partial\ln\lambda}{\partial
E^{x}}\right)\frac{\sin^{2}\left(\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{\lambda^{2}}\right)$
$\displaystyle+8q\left(-\lambda\frac{\partial(\lambda\lambda_{\varphi})}{\partial
E^{x}}\frac{\sin^{2}\left(\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{\lambda^{2}}+\left(\frac{\alpha_{2}}{4E^{x}}-\frac{1}{2}\frac{\partial\ln\lambda}{\partial
E^{x}}\right)\frac{\sin\left(2\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{2\lambda}\right)$
$\displaystyle+4K_{\varphi}\frac{\partial\ln\lambda}{\partial
E^{x}}\left(c_{f}\frac{\sin\left(2\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)}{2\lambda}+q\cos\left(2\lambda\left(K_{\varphi}+\lambda_{\varphi}\right)\right)\right)\Bigg{)}\
,$

where $\chi,c_{f},\alpha_{0},\alpha_{2},\lambda,q,\lambda_{\varphi}$ are
undetermined functions of $E^{x}$. The classical constraint is recovered in
the limit $\chi,c_{f},\alpha_{0},\alpha_{2}\to 1$ and $\lambda,q\to 0$. (The
cosmological constant can be recovered by instead setting $\alpha_{0}\to
1-\Lambda E^{x}$, with $\Lambda>0$ corresponding to asymptotically de Sitter
space in the classical limit.)

Noting that the set of canonical transformations has not been exhausted, we
can use the residual canonical transformation with
$f_{c}=f_{x}K_{\varphi}-\tilde{\lambda}_{\varphi}$ and $\alpha_{c}=1$ (where
$f_{x}$ and $\tilde{\lambda}_{\varphi}$ are allowed to depend on $E^{x}$) to
further simplify the constraint. A convenient choice for such a simplification
is given by $\tilde{\lambda}_{\varphi}=\lambda_{\varphi}$:

$\displaystyle g$ $\displaystyle=$ $\displaystyle\chi
f_{x}\cos^{2}\left(\lambda f_{x}K_{\varphi}\right)\ ,$ (13a) $\displaystyle
gf_{0}$ $\displaystyle=$
$\displaystyle\frac{\chi}{f_{x}}\Bigg{(}\frac{\alpha_{0}}{E^{x}}+2\frac{\sin^{2}\left(\lambda
f_{x}K_{\varphi}\right)}{\lambda^{2}}\frac{\partial c_{f}}{\partial
E^{x}}+4\frac{\sin\left(2\lambda
f_{x}K_{\varphi}\right)}{2\lambda}\frac{1}{\lambda}\frac{\partial\left(\lambda
q\right)}{\partial E^{x}}$ (13b)
$\displaystyle+\left(\frac{\alpha_{2}}{E^{x}}-2\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}\frac{\sin^{2}\left(\lambda
f_{x}K_{\varphi}\right)}{\lambda^{2}}+2q\frac{\sin\left(2\lambda
f_{x}K_{\varphi}\right)}{2\lambda}\right)\Bigg{)}$
$\displaystyle+\frac{\partial\ln(\lambda f_{x})}{\partial
E^{x}}4\chi_{0}K_{\varphi}\left(c_{f}\frac{\sin(2\lambda
f_{x}K_{\varphi})}{2\lambda}+q\cos(2\lambda f_{x}K_{\varphi})\right)\ ,$
$\displaystyle gf_{1}$ $\displaystyle=$ $\displaystyle
4\chi\left(c_{f}\frac{\sin(2\lambda
f_{x}K_{\varphi})}{2\lambda}+q\cos(2\lambda f_{x}K_{\varphi})\right)\ ,$ (13c)
$\displaystyle gf_{2}$ $\displaystyle=$ $\displaystyle\chi
f_{x}\cos^{2}\left(\lambda
f_{x}K_{\varphi}\right)\Bigg{(}-\frac{\alpha_{2}}{4E^{x}}-\frac{\partial\ln
f_{x}}{\partial E^{x}}$ (13d) $\displaystyle+\frac{\sin\left(2\lambda
f_{x}K_{\varphi}\right)}{2\lambda\cos^{2}\left(\lambda
f_{x}K_{\varphi}\right)}\lambda^{2}f_{x}K_{\varphi}\frac{\partial\ln(\lambda
f_{x})}{\partial E^{x}}\Bigg{)}\ .$

The associated structure function (10) becomes

$\displaystyle\tilde{q}^{xx}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial f_{1}}{\partial
K_{\varphi}}-\frac{1}{2}\frac{\partial^{2}\ln g}{\partial
K_{\varphi}^{2}}\left(\frac{(E^{x})^{\prime}}{E^{\varphi}}\right)^{2}\right)\frac{g^{2}}{4}\frac{E^{x}}{(E^{\varphi})^{2}}$
$\displaystyle=$
$\displaystyle\left(\left(c_{f}+\lambda^{2}f_{x}^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\cos^{2}\left(\lambda
f_{x}K_{\varphi}\right)-2q\lambda^{2}\frac{\sin(2\lambda
f_{x}K_{\varphi})}{2\lambda}\right)\chi^{2}f_{x}^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$

If we choose $f_{x}=1$, then the resulting set of Hamiltonian constraints is
given by

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle-\sqrt{E^{x}}\frac{\chi}{2}\bigg{[}E^{\varphi}\bigg{(}\frac{\alpha_{0}}{E^{x}}+2\frac{\sin^{2}\left(\lambda
K_{\varphi}\right)}{\lambda^{2}}\frac{\partial c_{f}}{\partial
E^{x}}+4\frac{\sin\left(2\lambda
K_{\varphi}\right)}{2\lambda}\frac{1}{\lambda}\frac{\partial\left(\lambda
q\right)}{\partial E^{x}}$ (15)
$\displaystyle+\left(\frac{\alpha_{2}}{E^{x}}-2\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}\frac{\sin^{2}\left(\lambda
K_{\varphi}\right)}{\lambda^{2}}+2q\frac{\sin\left(2\lambda
K_{\varphi}\right)}{2\lambda}\right)$
$\displaystyle+4\left(\frac{K_{x}}{E^{\varphi}}+\frac{K_{\varphi}}{2}\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}+q\cos(2\lambda K_{\varphi})\right)\bigg{)}$
$\displaystyle+\frac{((E^{x})^{\prime})^{2}}{E^{\varphi}}\bigg{(}-\frac{\alpha_{2}}{4E^{x}}\cos^{2}\left(\lambda
K_{\varphi}\right)+\left(\frac{K_{x}}{E^{\varphi}}+\frac{K_{\varphi}}{2}\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\lambda^{2}\frac{\sin\left(2\lambda
K_{\varphi}\right)}{2\lambda}\bigg{)}$
$\displaystyle+\left(\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}\right)\cos^{2}\left(\lambda
K_{\varphi}\right)\bigg{]}\ ,$

with the associated structure function

$\displaystyle\tilde{q}^{xx}$ $\displaystyle=$
$\displaystyle\left(\left(c_{f}+\lambda^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\cos^{2}\left(\lambda
K_{\varphi}\right)-2q\lambda^{2}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}\right)\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\ .$ (16)

If we instead choose $f_{x}=\tilde{\lambda}/\lambda$, where $\tilde{\lambda}$
is a constant reference value of $\lambda(E^{x})$, the resulting set of
Hamiltonian constraints is given by

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{\lambda}}{\lambda}\chi\frac{\sqrt{E^{x}}}{2}\Bigg{[}E^{\varphi}\Bigg{(}\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\frac{\alpha_{0}}{E^{x}}+2\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\frac{\partial
c_{f}}{\partial
E^{x}}+4\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}\frac{\partial}{\partial
E^{x}}\left(\frac{\lambda}{\tilde{\lambda}}q\right)$ (17)
$\displaystyle+\left(\frac{\alpha_{2}}{E^{x}}-2\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}+2\frac{\lambda}{\tilde{\lambda}}q\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}\right)\Bigg{)}$
$\displaystyle+4K_{x}\left(c_{f}\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}+\frac{\lambda}{\tilde{\lambda}}q\cos(2\tilde{\lambda}K_{\varphi})\right)$
$\displaystyle-\frac{((E^{x})^{\prime})^{2}}{E^{\varphi}}\left(\left(\frac{\alpha_{2}}{4E^{x}}-\frac{1}{2}\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)-\frac{K_{x}}{E^{\varphi}}\tilde{\lambda}^{2}\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}\right)$
$\displaystyle+\left(\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}\right)\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)\Bigg{]}\
,$

with the structure-function

$\displaystyle\tilde{q}^{xx}$ $\displaystyle=$
$\displaystyle\left(\left(c_{f}+\left(\frac{(E^{x})^{\prime}\tilde{\lambda}}{2E^{\varphi}}\right)^{2}\right)\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)-2\frac{\lambda}{\tilde{\lambda}}q\tilde{\lambda}^{2}\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$ (18)

Note that the constraint (17), unlike (15), is bounded in $K_{\varphi}$ and
fully periodic with “frequency” $\tilde{\lambda}$. The two constraints,
however, are related by a simple canonical transformation rescaling the
component $K_{\varphi}$. Therefore, they describe the same system, just in
different phase-space coordinates. In the modified context, the phase-space
variable $K_{\varphi}$ has lost its direct classical relationship with
extrinsic curvature, which is instead determined by the emergent space-time
metric. Therefore, there is no canonical or geometrical preference in favor of
one of the two versions. (However, the periodic nature of (17) may suggest
that this version and the associated canonical variables are more relevant for
a fundamental loop quantization in terms of basic holonomy operators.)

The classical constraint is recovered from (15) in the limit
$\chi,c_{f},\alpha_{2}\to 1$, $\lambda\to 0$, and $\alpha_{0}\to 1-\Lambda
E^{x}$ with cosmological constant $\Lambda$. The constraint (17) cannot
reproduce the classical limit directly because the canonical transformation
$K_{\varphi}\to(\tilde{\lambda}/\lambda)K_{\varphi}$ does not exist in the
classical limit. Since the canonical transformation has introduced a new
constant $\tilde{\lambda}$, its limit behavior must be specified separately.
One possibility is the combination of $\lambda\to\tilde{\lambda}$ followed by
$\tilde{\lambda}\to 0$.

If we redefine

$\displaystyle\chi\to\bar{\chi}\frac{\lambda}{\tilde{\lambda}}\ ,\
q\to\bar{q}\frac{\tilde{\lambda}}{\lambda}\ ,\
\alpha_{0}\to\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\bar{\alpha}_{0}\ ,\
\alpha_{2}\to\bar{\alpha}_{2}+4E^{x}\frac{\partial\ln\lambda}{\partial E^{x}}\
,$ (19)

the free function $\lambda$ disappears from the constraint (17) by being
absorbed by the other parameters, and is completely replaced by the constant
$\tilde{\lambda}$. A possible interpretation of $\tilde{\lambda}$ is given in
the next subsection as the length of a reference curve along which the
holonomy is computed. Replacing the function $\lambda$ with a constant
$\tilde{\lambda}$ simplifies the constraint, but it obscures the effects of
non-constant $\lambda$ by combining it with the other modification functions
according to (19). Since we are specifically interested in non-constant
$\lambda$ for the purpose of this paper, we will not make much use of this
redefinition.

A modified angular metric component
$\tilde{q}_{\vartheta\vartheta}=\tilde{q}_{\vartheta\vartheta}(E^{x})$ may
also be considered. However, through a residual canonical transformation (11)
with $f_{c}=K_{\varphi}$ and the appropriate $\alpha_{c}$, the angular metric
component can always be rendered into its classical form
$\tilde{q}_{\vartheta\vartheta}=E^{x}$. The appearance of $\alpha_{c}$ in the
Hamiltonian constraint can thereby be absorbed in the undetermined functions
of $E^{x}$, obtaining the same result (17). Any covariant Hamiltonian
constraint of second order in spatial derivatives in the spherically symmetric
sector is therefore contained in (17).

It is important to note that canonical transformations cannot have an effect
on the physical system because they merely represent a change of phase-space
coordinates. The canonical transformations used here, and discussed in detail
in [6], serve the purpose of simplifying the covariance conditions such that
they can be solved exactly. Factoring out canonical transformations is also
important because a different choice of phase-space coordinates may lead one
to misclassify a modified Hamiltonian constraint different from (17) as a new
theory, even if it is in fact equivalent to a well-known one. For instance,
[22] argued that the non-invertibility of a canonical transformation of the
form $K_{\varphi}\to\sin(\lambda K_{\varphi})/\lambda$ applied to the
classical system could model holonomy modifications. This assertion is
incorrect since the construction merely amounts to using the classical theory
with phase-space coordinates that are valid in a bounded range. It is
therefore not surprising that the results on critical collapse of scalar
matter, reported in [22], did not show any deviation from GR. Similarly, [7]
partially attributed modified gravitational effects to non-invertible
canonical transformations, misidentifying the correct reason for singularity
removal. The relevant modifications instead originate in a linear combination
of constraints with phase-space dependent coefficients which result in
additional terms for the new Hamiltonian constraint, as explained in [6].

###  2.5 Effective loop quantum gravity

The covariant Hamiltonian defined in (17) contains various independent
modification functions in several of its terms. How does it relate to
effective loop quantum gravity, which is usually considered with a smaller
number of modifications? Phenomenologically, the parameter $\lambda$ is
special since it is responsible for the non-singular black-hole solution
obtained for constant $\lambda=\tilde{\lambda}$ in [7]. In the functional
behavior of the constraint, this parameter has an appearance related to the
holonomy length $\mu$ in LQG.

####  2.5.1 Traditional ingredients

The starting point of any loop quantization consists in replacing a direct
quantization of the classical phase space by operators in the holonomy-flux
representation. In spherically symmetric LQG [19, 20], the basic holonomies
are given by

$\displaystyle h^{x}_{e}[K_{x}]$ $\displaystyle=$
$\displaystyle\exp\int_{e}{\rm d}x\ iK_{x}\,,$ (20) $\displaystyle
h^{\varphi}_{v,\mu}[K_{\varphi}]$ $\displaystyle=$
$\displaystyle\exp\int_{\mu}{\rm d}\vartheta\ iK_{\varphi}=\exp\left(i\mu
K_{\varphi}(v)\right)\,,$ (21)

where $e$ stands for an arbitrary radial curve of finite coordinate length,
while $v$ stands for an arbitrary point in the radial line, and $\mu$ is the
coordinate length of an arbitrary longitudinal curve on the 2-sphere
intersecting the point $v$. The explicit integration in the angular holonomy
(21) is possible due to spherical symmetry, but the radial holonomy
integration must remain formal. Similarly, the fluxes are given by direct
integration of the densitized triads $E^{x}$ and $E^{\varphi}$ over finite,
2-dimensional surfaces with normals in the radial and angular direction,
respectively. Fluxes do not require exponentiation and are therefore closer to
the original phase-space variables. For the following argument, we need only
the expressions of holonomies.

Because loop quantization is based on the holonomy-flux variables, the
Hamiltonian constraint must be rewritten as a function of holonomies rather
than the bare (extrinsic) curvatures $K_{x}$ and $K_{\varphi}$, leading to a
modified constraint in the spirit of emergent modified gravity, as outlined in
the previous section. However, the radial holonomy (20) is essentially non-
local, and cannot be studied within this framework, which has thus far only
been formulated locally and up to second order in derivatives. On the other
hand, the angular holonomies (21) are indeed local when restricting the state
space to spherically symmetric states. Therefore, it is natural to expect that
quantum effects due to the angular holonomy components can be captured by the
local equations of emergent modified gravity. Furthermore, since the angular
holonomies can be integrated, they become simple complex exponentials of the
angular extrinsic curvature (21). Because the Hamiltonian constraint must be
Hermitian as an operator in the quantum theory, or simply real-valued prior to
quantization, the holonomy modifications in the Hamiltonian constraint will
appear as trigonometric functions of $K_{\varphi}$ with ‘frequency’ $\mu$.
This is indeed the case of the Hamiltonian constraint (17) derived in the
previous section. Therefore, the parameter $\lambda$ in the notation of
emergent modified gravity can be given the interpretation of an angular
holonomy length within LQG, and hence acquires the status of a quantum
parameter under this interpretation. However, it also inherits new features
from the enhanced symmetry properties of emergent modified gravity compared
with traditional models of LQG.

In LQG, different triangulations of space are allowed, and a specific choice
would determine the form of the quantum parameter $\mu(E^{x})$ appearing in
the holonomy modification function, using the phase-space variable $E^{x}$ as
the squared radius of our geometry. For example, we can choose a fine lattice
such that the spheres are triangulated by small regular polygons with $n$
sides of angular length $\mu$. (It is common to choose tetrahedra for the
three-dimensional lattice and hence triangles for the two-dimensional lattice
of the spheres, setting $n=3$ for this purpose.) Each polygon, also referred
to as a plaquette, at radius $\sqrt{E^{x}}$ then covers a geometrical area of
$nE^{x}\mu^{2}/(4\tan(\pi/n))$. Choosing the plaquettes at different radii to
preserve the same geometrical area, we obtain the formula

$\mu=\frac{\hat{r}}{\sqrt{E^{x}}}\hat{\mu}\ ,$ (22)

where $\hat{r}$ is a constant reference radius at which $\mu=\hat{\mu}$. This
function satisfies $\mu\to 0$ as $E^{x}\to\infty$, as desired for recovering
an asymptotically smooth geometry without triangulation effects at large
radii.

####  2.5.2 Shortcomings

Several assumptions are used in arriving at this form. An additional,
previously unrecognized issue is that the classical component $E^{x}$ of a
space-time metric is used in the geometrical construction, which in general
may deviate from the correct geometrical meaning provided by the emergent
space-time metric. As we have seen, in spherically symmetric models it is
always possible to choose phase-space variables such that the angular
component of the metric is still given by $E^{x}$. The construction of
holonomy length may therefore be applied as described here. However, strict
proposals such as the $\bar{\mu}$-scheme would require an application of these
arguments to any surface, even if it extends into the radial direction which
may still be homogeneous in the black-hole interior. In this case, an emergent
space-time metric may have to be taken into account, but this is available
only after the modified theory has been analyzed. In general, the
$\bar{\mu}$-scheme therefore does not provide a self-contained recipe for the
construction of holonomy modifications.

One ingredient of a $\bar{\mu}$-type scheme that may be applied here is a
relationship between holonomy parameters and areas of interest, such as the
smallest non-zero eigenvalue of an area operator in a loop quantization. Also
this step is heuristic and not fully justified because holonomies of a gauge
connection do not depend on a metric, and therefore need not have any
relationship with geometrical areas. Nevertheless, such a relationship may be
used in order to fix free parameters. (In practice, using area eigenvalues
merely determines numerical factors multiplying suitable powers of the Planck
length, whose form could easily be obtained from dimensional arguments.) To be
specific, by choosing the plaquette size derived above to equal a certain
distinguished area, $n\hat{r}^{2}\hat{\mu}^{2}/(4\tan(\pi/n))=\Delta$, we
arrive at the relation

$\mu(E^{x})^{2}=\tilde{\Delta}/E^{x}$ (23)

with $\tilde{\Delta}=4n^{-1}\tan(\pi/n)\Delta$. The square of the quantum
parameter (23) is proportional to the solid angle of a single plaquette on a
sphere of radius $\sqrt{E^{x}}$. Since this expression equals
$\tilde{\Delta}/E^{x}$, the number of plaquettes triangulating the sphere of
radius $\sqrt{E^{x}}$ is given by

$\mathcal{N}_{\Delta}(E^{x})=4\pi E^{x}/\tilde{\Delta}\ .$ (24)

Loop quantization requires the Hamiltonian constraint to be explicitly
periodic in $K_{\varphi}$. Having $\mu$ depend on $E^{x}$ preserves
periodicity in $K_{\varphi}$ at any fixed value of $E^{x}$, but it turns the
holonomy $\exp(i\mu(\hat{E}^{x})\hat{K}_{\varphi})$ into a holonomy-flux
hybrid operator upon quantization. Such expressions do not obey the basic
holonomy-flux algebra, and they do not even provide a closed set of
commutators because an expression such as

$\displaystyle\left\\{h^{x}_{e}[K_{x}],h^{\varphi}_{v,\mu}[K_{\varphi}]\right\\}=-h^{x}_{e_{1}}[K_{x}]\frac{\partial\mu(\nu)}{\partial
E^{x}(\nu)}K_{\varphi}(v)h^{\varphi}_{v,\mu}[K_{\varphi}]h^{x}_{e_{2}}[K_{x}]\,,$
(25)

where $e=e_{1}\cup e_{2}$, such that $e_{1}$ and $e_{2}$ join at $x=\nu$,
violates the periodicity condition. In the basic holonmy-flux algebra, the
right-hand side of the Poisson bracket between two holonomy operators should
be zero since the connection components Poisson commute. The factor of
$K_{\varphi}(v)$ in (25) shows that trying to implement an $E^{x}$-dependent
holonomy length is not compatible with a closed algebra of basic operators.
Hence, the periodic version (17) of the Hamiltonian constraint is preferred
over (15) when carrying out the loop quantization because it resolves two
important problems: It depends on pure holonomies of the form
$\exp(i\tilde{\lambda}\hat{K}_{\varphi})$ with constant $\tilde{\lambda}$, and
can therefore be expressed in terms of the generators of a closed basic
algebra. At the same time, it allows for scale-dependent $\lambda$-effects
because a non-trivial $\lambda(E^{x})$ appears in the coefficients of (17)
after it has been removed as a factor of $K_{\varphi}$ in holonomies by a
canonical transformation. It is then possible to choose the holonomy length
that appears in trigonometric functions as being equal to the
$E^{x}$-independent reference value $\tilde{\lambda}=\hat{\mu}$. At the same
time, dynamical implications of non-constant holonomy parameters are realized
through modified coefficients of holonomy terms, as we demonstrate in detail
below.

A common motivation for an $E^{x}$-dependent $\mu$ in models of loop quantum
gravity is not based on properties of basic holonomy operators, but rather on
phenomenological questions because a connection or extrinsic-curvature
component such as $K_{\varphi}$ may be large in classical regimes, depending
on the space-time slicing. A holonomy length $\mu(E^{x})$ might then be used
to counteract growing holonomy modifications if it decreases at a suitable
rate as $E^{x}$ grows, implying large spherical areas. It is often argued that
this reason requires an explicit appearance of a non-constant $\mu(E^{x})$ in
holonomies. This reasoning seems to favor (15) with a corresponding
$\lambda(E^{x})$ in holonomy-like terms, even though covariance prevents this
constraint from being strictly periodic in $K_{\varphi}$. Moreover, a quantum
constraint operator for this version cannot be based on basic holonomies in a
closed holonomy-flux algebra.

####  2.5.3 New features

Our new discussion of canonical transformations overturns this established
understanding because it is possible to map any constraint of the form (15)
into an expression (17) that is strictly periodic in $K_{\varphi}$. The
canonical transformation implies that both versions generate exactly the same
dynamics and physical observables. Effects of a non-constant holonomy length
in (15) are realized in (17) by a modified $E^{x}$ dependence of coefficients
of holonomy terms, indicated by the non-constant ratios
$\tilde{\lambda}/\lambda$ in (17), in particular in the $K_{x}$-term of the
modified Hamiltonian constraint. This term is relevant for Hamilton’s equation

$\dot{E}^{x}=\frac{\tilde{\lambda}}{\lambda}\chi\frac{\sqrt{E^{x}}}{2}\left(4c_{f}+\tilde{\lambda}^{2}\frac{((E^{x})^{\prime})^{2}}{(E^{\varphi})^{2}}\right)\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}$
(26)

for the spherical area $E^{x}$ (assuming vanishing $N^{x}$ for the sake of
simplicity). As an equation of motion generated by the periodic constraint,
this result implies that the classical relationship between the canonical
variable $K_{\varphi}$ and the time derivative $\dot{E}^{x}$ receives an
additional factor of $\lambda/\tilde{\lambda}$, making $K_{\varphi}$ decrease
as $\sqrt{E^{x}}$ grows according to the modified dynamics. (For small
$\tilde{\lambda}$ and $\lambda/\tilde{\lambda}$ independent of
$\tilde{\lambda}$, we have
$K_{\varphi}\approx(\lambda/\tilde{\lambda})\dot{\sqrt{E^{x}}}/(\chi c_{f})$,
where $c_{f},\chi\rightarrow 1$ in the classical limit.) This behavior has the
same effect of suppressing holonomy modifications in classical regimes as an
explicitly $E^{x}$-dependent holonomy length would imply. It is only harder to
see this outcome immediately because it requires an analysis of equations of
motion.

By dismissing this option, the traditional arguments for holonomy
modifications suggest an unjustified preference for one of the possible
versions of canonically equivalent constraints, based on flawed heuristics
that implicitly relate the canonical variable $K_{\varphi}$ in a modified
theory to a classical component of extrinsic curvature: Classically,
$K_{\varphi}$ is not necessarily small enough for
$\sin(\tilde{\lambda}K_{\varphi})/\tilde{\lambda}$ to be close to
$K_{\varphi}$ in regimes of small space-time curvature. It is therefore
important to distinguish between small-$\lambda$ regimes (as a formal limit of
the constraint) and small-$\lambda K_{\varphi}$ regimes (as a dynamical
feature of solutions for $K_{\varphi}$). In a modified theory, one cannot use
classical heuristics in order to estimate the magnitude of $K_{\varphi}$,
interpreted as a component of extrinsic curvature, in order to determine
whether $\lambda K_{\varphi}$ is sufficiently small in a given semiclassical
regime. Since $\lambda K_{\varphi}$ appears in a modified term of the
Hamiltonian constraint when holonomies are used and since the emergent space-
time metric is not the same as the classical metric, there is no kinematical
relationship between the magnitude of $K_{\varphi}$ in a modified theory and
the magnitude of extrinsic curvature in a corresponding classical slicing. An
evaluation of suitable holonomy modifications can therefore be made only if
equations of motion as well as the emergent metric are considered. However,
traditional arguments in favor of certain holonomy modifications are held
entirely at a kinematical level.

The distinction between small-$\lambda$ and small-$\lambda K_{\varphi}$ is
relevant in these considerations even if one is interested only in leading-
order corrections to the classical behavior. The leading order of a
small-$\lambda$ expansion of the theory is simply the limit of $\lambda\to 0$,
resulting in the classical dependence of the constraint on $K_{\varphi}$ and
an emergent metric identical with the classical metric. (The overall theory
may still be modified in this limit if the remaining modification functions
are non-classical.) Holonomy terms can also be expanded on a subspace of the
solution space where $\lambda K_{\varphi}$ is small, even if $\lambda$ is not
taken to zero in a limit. This expansion is relevant for physical properties
of the semiclassical limit. In the low-curvature regime of small $\lambda
K_{\varphi}$, the quadratic order in this expansion determines near-classical
physics. The non-periodic version of the Hamiltonian constraint, (15), then
becomes

$\displaystyle\tilde{H}$ $\displaystyle\approx$
$\displaystyle-\sqrt{E^{x}}\frac{\chi}{2}\bigg{[}E^{\varphi}\bigg{(}\frac{\alpha_{0}}{E^{x}}+2K_{\varphi}^{2}\frac{\partial
c_{f}}{\partial
E^{x}}+4K_{\varphi}\frac{1}{\lambda}\frac{\partial\left(\lambda
q\right)}{\partial E^{x}}$ (27)
$\displaystyle+\left(\frac{\alpha_{2}}{E^{x}}-2\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}K_{\varphi}^{2}+2qK_{\varphi}\right)$
$\displaystyle+4\left(\frac{K_{x}}{E^{\varphi}}+\frac{K_{\varphi}}{2}\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\left(c_{f}K_{\varphi}+q\left(1-2\lambda^{2}K_{\varphi}^{2}\right)\right)\bigg{)}$
$\displaystyle+\frac{((E^{x})^{\prime})^{2}}{E^{\varphi}}\bigg{(}-\frac{\alpha_{2}}{4E^{x}}\left(1-\lambda^{2}K_{\varphi}^{2}\right)+\left(\frac{K_{x}}{E^{\varphi}}+\frac{K_{\varphi}}{2}\frac{\partial\ln\lambda^{2}}{\partial
E^{x}}\right)\lambda^{2}K_{\varphi}\bigg{)}$
$\displaystyle+\left(\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}\right)\left(1-\lambda^{2}K_{\varphi}^{2}\right)\bigg{]}\,,$

with structure function

$\displaystyle\tilde{q}^{xx}$ $\displaystyle\approx$
$\displaystyle\left(\left(c_{f}+\lambda^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\left(1-\lambda^{2}K_{\varphi}^{2}\right)-2q\lambda^{2}K_{\varphi}\right)\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\,.$
(28)

(A similar result follows for the periodic version.) Some holonomy effects
therefore survive in the low-curvature regime and may lead to unexpected
dynamical implications, unless the strict limit of $\lambda\to 0$ is taken in
this regime too. This approximation of the Hamiltonian constraint is valid
only in slices where $\lambda K_{\varphi}$ is sufficiently small. Physical
implications can only be analyzed if equations of motion are studied,
revealing through $\tilde{q}^{xx}$ the geometrical meaning and dynamical
behavior of $K_{\varphi}$ in suitable semiclassical regimes.

An important additional flaw in several discussions of holonomy modifications
is the lack of covariance. It is then possible for
$\sin(\tilde{\lambda}K_{\varphi})/\tilde{\lambda}$ to be close to
$K_{\varphi}$ in some space-time slicings but deviate strongly in others, even
in classical regimes. This problem is solved by our constructions, which
explicitly realize slicing independence together with the correct geometrical
meaning of $K_{\varphi}$. As already emphasized, covariance is also the main
reason why non-periodic terms are required in a Hamiltonian constraint with
$E^{x}$-dependent holonomy length. Our constructions not only implement full
covariance, they also relate different versions of holonomy modifications by
carefully considering canonical transformations. For effective equations and
their solutions, we may use either of the two, or perhaps others related by
further canonical transformations. One may simply prefer an option that
generates the simplest equations of motion for a given purpose which, however,
will in general be gauge dependent.

Covariant holonomy modifications therefore cannot be expressed in the
straitjacket of $\mu_{0}$ and $\bar{\mu}$-like schemes that have only a
heuristic basis motivated by expectation about the holonomy length combined
with a classical understanding of canonical variables in a modified theory.
The appearance of holonomy terms, classified in this way, depends on the
canonical variables used, and a $\mu_{0}$-like scheme can be transformed into
an equivalent $\bar{\mu}$-like scheme if only the arguments of trigonometric
functions are used in the definition of holonomy modifications. As an
important implication of our new $\lambda$-scheme, dynamical properties of
holonomy modifications and their suppression in classical regimes can be
realized with strictly periodic holonomy terms, provided their coefficients
are amended by suitable modification functions. (In traditional models of loop
quantum gravity, such modifications are usually understood as inverse-triad
corrections because these coefficients classically depend only on the
densitized triad and its inverse but not on the momenta.)

####  2.5.4 Self-consistent treatment of holonomy effects

Taking these lessons into account, we arrive at a novel understanding of
holonomy modifications in models of loop quantum gravity. The freedom of
performing canonical transformations can be fixed by requiring that all terms
in the Hamiltonian constraint depending on $K_{\varphi}$ are strictly periodic
in this variable. According to emergent modified gravity, covariance then
implies that only the combination $\tilde{\lambda}K_{\varphi}$ with a constant
$\tilde{\lambda}$ can appear in these periodic functions. At this point, the
choice of canonical variables is unique if it is required to preserve the
classical limit for $\tilde{\lambda}\to 0$, such that
$\tilde{\lambda}K_{\varphi}$ cannot be used as a new phase-space variable
conjugate to $E^{\varphi}/\tilde{\lambda}$. There is therefore an unambiguous
interpretation of $K_{\varphi}$-dependent functions as an effective
description of angular holonomies. Accordingly, we refer to the constant
$\tilde{\lambda}$ as the holonomy length of the theory. By definition, this
length is always constant and does not depend on $E^{x}$.

Different schemes are implemented by choosing specific functions for
$\lambda(E^{x})$, which does not appear in holonomy terms but rather in their
coefficients in the Hamiltonian constraint. The choice of $\lambda(E^{x})$
changes the geometrical meaning and the dynamical behavior of $K_{\varphi}$ in
holonomy terms according to (26). As already mentioned, this modification
makes it difficult to assess the significance of holonomy modifications in
different regimes because this task would require an estimate of the values
taken by $K_{\varphi}$, which depends on the dynamics and the space-time
gauge. In fact, the equation of motion for $E^{x}$ in a modified theory does
not directly determine $K_{\varphi}$ but rather the holonomy function

$\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}=\frac{\lambda}{\tilde{\lambda}}\frac{2\dot{E}^{x}}{\chi\sqrt{E^{x}}}\frac{1}{4c_{f}+\tilde{\lambda}^{2}((E^{x})^{\prime})^{2}/(E^{\varphi})^{2}}$
(29)

obtained by inverting (26) and still asuming vanishing $N^{x}$ for the sake of
convenience. For $\chi$ and $c_{f}$ close to their classical values, the ratio
$\lambda/\tilde{\lambda}$ determines the strength of holonomy modifications,
defined as the deviation of
$(2\tilde{\lambda})^{-1}\sin(2\tilde{\lambda}K_{\varphi})$ from the classical
expression $2\dot{E}^{x}/\sqrt{E^{x}}$ of $K_{\varphi}$. For the classical
limit to be realized at $\tilde{\lambda}\to 0$, we need
$\lambda(E^{x})=\tilde{\lambda}h(E^{x})$ with a function $h(E^{x})$
independent of $\tilde{\lambda}$ (or finite and non-zero for
$\tilde{\lambda}\to 0$). Since this new function
$h(E^{x})=\lambda(E^{x})/\tilde{\lambda}$ describes the relative strength of
holonomy modifications on different scales, we call it the holonomy function.
Its appearance shows that it is independent of the holonomy length and instead
describes the dynamical behavior of holonomy modifications or the variable
$K_{\varphi}$.

Some choices of the holonomy function $h(E^{x})$ resemble properties of the
traditional $\mu_{0}$ or $\bar{\mu}$-type schemes, but in some cases the usual
heuristics turns out to be misleading. For instance, a constant $h(E^{x})=1$,
or $\lambda=\tilde{\lambda}$, could be related to a $\mu_{0}$-type scheme of
constant holonomy parameter $\mu$ in the traditional formulation. A common
argument against using such a constant is that the interpretation of
$\tilde{\lambda}$ as a holonomy length derived from coordinates would imply a
growing physical length $\tilde{\lambda}\sqrt{E^{x}}$ as measured by the
relevant spatial geometry. The growth in $E^{x}$ of this function then
suggests that holonomy modifications increase in asymptotic regimes of large
$E^{x}$, which are often expected to behave in a classical manner. However,
equation (29) shows that holonomy modifications do not grow in this case but
merely track the classical behavior of $K_{\varphi}$: The second term in the
denominator on the right-hand side of (29) is sub-dominant in this regime, and
thus the expression $(2\tilde{\lambda})^{-1}\sin(2\tilde{\lambda}K_{\varphi})$
on the left is identical with the classical behavior
$2\dot{E}^{x}/\sqrt{E^{x}}$ of $K_{\varphi}$ for $\tilde{\lambda}=\lambda$.
Modifications then appear only because one has to rewrite
$\sin(2\tilde{\lambda}K_{\varphi})$ in terms of
$\sin(\tilde{\lambda}K_{\varphi})$ for some terms of the constraint, using
trigonometric identities. Holonomy modifications therefore do not grow in an
unbounded manner in the case of a constant holonomy length, but they may not
decrease quickly enough so as to become subdominant in asymptotic regimes.
This behavior can be changed by a holonomy function $h(E^{x})$ that decreases
at a sufficient rate for growing $E^{x}$. We will see examples in our specific
cases to be discussed later.

###  2.6 Gravitational observable

Consider a scalar function on the phase space $\mathcal{M}$. This function is
a weak observable if
$\delta_{\epsilon}\mathcal{M}=\mathcal{D}_{H}H+\mathcal{D}_{x}H_{x}$, where
$\mathcal{D}_{H}$ and $\mathcal{D}_{x}$ are functions on the phase space and
$\epsilon$ is a gauge function. On-shell, the function $\mathcal{M}$ then
remains invariant under time evolution or arbitrary gauge transformations. In
this section, we will keep the most general form of our modification functions
in the Hamiltonian constraint, before restricting them to take values as
required in effective LQG in the next section. If one considers the dependence
$\mathcal{M}=\mathcal{M}(E^{x},K_{\varphi},(E^{x})^{\prime}/E^{\varphi})$,
using the periodic constraint (17), the weak observability condition
$\delta_{\epsilon}\mathcal{M}=\mathcal{D}_{H}H+\mathcal{D}_{x}H_{x}$
determines the functional dependence [23]

$\displaystyle\mathcal{M}$ $\displaystyle=$ $\displaystyle
d_{0}+\frac{d_{2}}{2}\left(\exp\int{\rm d}E^{x}\
\frac{\bar{\alpha}_{2}}{2E^{x}}\right)\left(c_{f}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}+2\bar{q}\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}-\cos^{2}(\tilde{\lambda}K_{\varphi})\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)$
(30) $\displaystyle+\frac{d_{2}}{4}\int{\rm d}E^{x}\
\left(\frac{\bar{\alpha}_{0}}{E^{x}}\exp\int{\rm d}E^{x}\
\frac{\bar{\alpha}_{2}}{2E^{x}}\right)\ ,$

which is unique up to the constants $d_{0}$ and $d_{2}$ with classical limits
$d_{0}\to 0$ and $d_{2}\to 1$. (There exists an anomaly-free and covariant
constraint more general than (17), but after imposing the existence of a weak
observable, the extra terms are restricted to vanish and we recover (17) with
(30) as the unique weak observable [23].) Here we use the barred functions as
defined by (19) for the sake of conciseness.

If we insert the classical values $\alpha_{2}=1$, and $\alpha_{0}=1-\Lambda
E^{x}$, and $d_{0}=0$, $d_{2}=1$ the weak observable simplifies to

$\displaystyle\mathcal{M}=\frac{\sqrt{E^{x}}}{2}\left(1-\frac{\Lambda}{3}E^{x}+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\left(c_{f}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}+2\frac{\lambda}{\tilde{\lambda}}q\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right)}{2\tilde{\lambda}}-\cos^{2}(\tilde{\lambda}K_{\varphi})\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\right)$
(31)

where we have chosen the integration constant $\tilde{\lambda}^{2}$ for the
exponential integrals. As already seen for the constraint, we must be careful
in taking the classical limit. If we first invert the canonical transformation
to the non-periodic variables, the classical limit is given by $\lambda\to 0$.
If this inversion is not performed, we must first take the limit
$\lambda\to\tilde{\lambda}$, followed by $\tilde{\lambda}\to 0$. The classical
limit of this observable is then the classical mass observable. The structure
function (18) can be written in terms of this observable as

$\displaystyle\tilde{q}^{xx}=\left(c_{f}+\lambda^{2}\left(1-\frac{2\mathcal{M}}{\sqrt{E^{x}}}-\frac{\Lambda}{3}E^{x}\right)\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$ (32)

In the non-periodic version, the mass observable takes the form

$\displaystyle\mathcal{M}=\frac{\sqrt{E^{x}}}{2}\left(1-\frac{\Lambda}{3}E^{x}+c_{f}\frac{\sin^{2}\left(\lambda
K_{\varphi}\right)}{\lambda^{2}}+2q\frac{\sin\left(2\lambda
K_{\varphi}\right)}{2\lambda}-\cos^{2}(\lambda
K_{\varphi})\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\,,$
(33)

and the structure-function is given by

$\displaystyle\tilde{q}^{xx}=\left(c_{f}+\lambda^{2}\left(1-\frac{2\mathcal{M}}{\sqrt{E^{x}}}-\frac{\Lambda}{3}E^{x}\right)\right)\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$ (34)

Note that the mass observable is periodic in $K_{\varphi}$ even in the non-
periodic version of the constraint.

###  2.7 Reflection symmetry of the transition surface

The Hamiltonian constraint (17), besides being periodic in $K_{\varphi}$ has
another interesting symmetry in the special case of $q=0$. In this case, the
structure function is symmetric around the surface
$K_{\varphi}=\pi/(2\tilde{\lambda})$ in the sense that $\tilde{q}^{xx}$ is
invariant under the reflection $\delta\to-\delta$ where $\delta$ is defined by
$K_{\varphi}=\pm\pi/(2\tilde{\lambda})+\delta$. The Hamiltonian constraint is
invariant under the combined operation $\delta\to-\delta$, $K_{x}\to-K_{x}$.
Thus, the two conditions $K_{\varphi}=\pm\pi/(2\tilde{\lambda})$ and $K_{x}=0$
define a surface of reflection symmetry in the phase-space. This property can
manifest itself in dynamical solutions as a reflection surface in space-time
[24]. If $q\neq 0$, this reflection-symmetry is broken unless $q$, like
$K_{x}$, changes sign in the dynamical solution, but this is unlikely to be
the case in general. Physical implications of breaking the reflection-symmetry
breaking effects of $q\neq 0$ will be studied elsewhere [25]. In the following
we set $q=0$, such that reflection symmetry is realized.

It will be useful in the following sections to evaluate the mass observable
(31) at the reflection symmetry surface $K_{\varphi}=-\pi/(2\tilde{\lambda})$,
or equivalently its incarnation in the non-periodic variables (33) at
$K_{\varphi}=-\pi/(2\lambda)$. After some simplifications, we find that
$E^{x}$, for a given value of the observable ${\cal M}$ obeys the equation

$c_{f}+\lambda^{2}\left(1-\frac{2\mathcal{M}}{\sqrt{E^{x}}}-\frac{\Lambda}{3}E^{x}\right)=0$
(35)

in any region of space-time where $K_{\varphi}=-\pi/(2\lambda)$. Since
$\lambda$ is an arbitrary function of $E^{x}$, this equation may have multiple
solutions according to the chosen modification function. Multiple reflection
surfaces may therefore exist in a space-time solution. As will be clear from
dynamical solutions derived later on, this reflection symmetry happens around
the hypersurface in space-time where the areal radius $\sqrt{E^{x}}$ reaches
an extremum. For the specific solution $x_{\lambda}^{(-)}$, which we will use
to denote the minimal value of $\sqrt{E^{x}}$ according to the dynamics
defined by a given $\lambda$, this will correspond to a spacelike transition
surface connecting a black hole to a white hole through a wormhole interior.
In later sections, we will discuss the physical interpretation of these
extremal values, including the case where there exists a maximal value at
$x_{\lambda}^{(+)}$.

##  3 Black-hole solution for general triangulation schemes

We continue to work with a general functional dependence of
$\lambda=\lambda(E^{x})$ and now demonstrate that, qualitatively, singularity
resolution of black holes in models of loop quantum gravity based on holonomy
modifications is a triangulation-independent feature. In order to isolate
effects of the holonomy modification $\lambda$, we set the functions $c_{f}$,
$\alpha_{0}$, $\alpha_{2}$, and $q$ to their classical values, while keeping a
nonclassical, nontrivial $\chi(E^{x})$ for rescaling freedom. For now, we keep
a cosmological constant $\Lambda$ and only set it to zero when we analyze
suitable conditions for asymptotic flatness. As will become clear later on, it
is necessary to scale the emergent metric using a suitable $\chi$ if we are
interested in asymptotic flatness, depending on the chosen $\lambda$. For
instance, a constant $\lambda=\tilde{\lambda}$ requires
$\chi(\infty)=1/\sqrt{1+\tilde{\lambda}^{2}}$, while
$\lambda=\sqrt{\Delta/E^{x}}$ requires $\chi(\infty)=1$. The scaling of the
spatial metric by a constant $\chi$ is not a simple coordinate transformation
of the radial coordinate because the latter would change other components non-
trivially if there is space-time curvature.

Given the classical choice for most of the modification functions, the non-
periodic constraint (15) simplifies to

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle-\chi\frac{\sqrt{E^{x}}}{2}\Bigg{[}E^{\varphi}\bigg{(}\frac{1}{E^{x}}-\Lambda+\frac{1}{E^{x}}\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}+4\left(K_{\varphi}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}-\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}\right)\frac{\partial\ln\lambda}{\partial
E^{x}}\bigg{)}$ (36) $\displaystyle+4K_{x}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}-\frac{((E^{x})^{\prime})^{2}}{4E^{\varphi}}\left(\frac{1}{E^{x}}\cos^{2}(\lambda
K_{\varphi})-4\lambda^{2}\left(\frac{K_{x}}{E^{\varphi}}+K_{\varphi}\frac{\partial\ln\lambda}{\partial
E^{x}}\right)\frac{\sin(2\lambda K_{\varphi})}{2\lambda}\right)$
$\displaystyle+\cos^{2}(\lambda
K_{\varphi})\left(\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}\right)\Bigg{]}\
.$

with structure-function

$\displaystyle\tilde{q}^{xx}$ $\displaystyle=$
$\displaystyle\left(1+\lambda^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\cos^{2}\left(\lambda
K_{\varphi}\right)\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}$ (37)
$\displaystyle=$
$\displaystyle\left(1+\lambda^{2}\left(1-\frac{2\mathcal{M}}{\sqrt{E^{x}}}-\frac{\Lambda}{3}E^{x}\right)\right)\chi^{2}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$

For comparison, the periodic version (17) becomes

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle-\chi\frac{\tilde{\lambda}}{\lambda}\frac{\sqrt{E^{x}}}{2}\bigg{[}E^{\varphi}\left(\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(\frac{1}{E^{x}}-\Lambda\right)+4\left(\frac{1}{4E^{x}}-\frac{\partial\ln\lambda}{\partial
E^{x}}\right)\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)+4K_{x}\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}$
(38)
$\displaystyle+\frac{((E^{x})^{\prime})^{2}}{E^{\varphi}}\left(\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)\left(-\frac{1}{4E^{x}}+\frac{\partial\ln\lambda}{\partial
E^{x}}\right)+\tilde{\lambda}^{2}\frac{K_{x}}{E^{\varphi}}\frac{\sin\left(2\tilde{\lambda}K_{\varphi}\right))}{2\tilde{\lambda}}\right)$
$\displaystyle+\left(\frac{(E^{x})^{\prime}(E^{\varphi})^{\prime}}{(E^{\varphi})^{2}}-\frac{(E^{x})^{\prime\prime}}{E^{\varphi}}\right)\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)\bigg{]}\
,$

with structure-function

$\displaystyle\tilde{q}^{xx}$ $\displaystyle=$
$\displaystyle\left(1+\tilde{\lambda}^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\cos^{2}\left(\tilde{\lambda}K_{\varphi}\right)\chi^{2}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{E^{x}}{(E^{\varphi})^{2}}$
(39) $\displaystyle=$
$\displaystyle\left(1+\lambda^{2}\left(1-\frac{2\mathcal{M}}{\sqrt{E^{x}}}-\frac{\Lambda}{3}E^{x}\right)\right)\chi^{2}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{E^{x}}{(E^{\varphi})^{2}}\
.$

Since the effective line element can be written as

$\displaystyle{\rm d}s^{2}=-N^{2}{\rm d}t^{2}+\tilde{q}_{xx}({\rm
d}x+N^{x}{\rm d}t)^{2}+E^{x}{\rm d}\Omega^{2}\,,$ (40)

we shall now solve for the phase-space variables, imposing different gauges,
in order to explicitly obtain the resulting space-time geometry.

###  3.1 Non-periodic phase-space coordinates

We will use the non-periodic constraint (36) in this subsection to compute
regions of a Schwarzschild-type static exterior and homogeneous regions in two
different gauges. We show that a third gauge results in a modified Gullstrand-
Painlevé space-time and is related to the two previous gauges by a standard
coordinate transformation in regions of overlap.

These three gauges possess a second coordinate singularity beyond the
horizons, but they all show regular Ricci and Kretschmann scalars at those
points. In Section 3.2 we will use the periodic version of the constraint and
obtain another gauge that is free of such a coordinate singularity and related
to the Schwarzschild-type homogeneous region by a standard coordinate
transformation.

Because the overall factor $\chi$ always multiplies the lapse function in the
equations of motion, it will be useful to define

$\bar{N}=\chi N\,.$ (41)

We encourage readers to consult Appendix A for a detailed exposition of the
equations of motion utilized in this section.

####  3.1.1 Schwarzschild gauge: Static region

We define the Schwarzschild gauge by

$\displaystyle N^{x}=0\ ,\hskip 28.45274ptE^{x}=x^{2}\ .$ (42)

The consistency condition $\dot{E}^{x}=0$ implies the equation

$\displaystyle\left(1+\lambda^{2}\left(\frac{(E^{x})^{\prime}}{2E^{\varphi}}\right)^{2}\right)\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}=0$ (43)

which, assuming consistency with the classical limit, is solved by imposing
$K_{\varphi}=0$. This result implies a second consistency equation
$\dot{K}_{\varphi}=0$, which we will address later on by solving it for the
lapse function.

The vanishing of the diffeomorphism constraint further implies that $K_{x}=0$.
Finally, the vanishing of the Hamiltonian constraint implies the equation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left(1-\Lambda
x^{2}\right)(E^{\varphi})^{2}-3x^{2}+x^{3}(\ln(E^{\varphi})^{2})^{\prime}\ ,$
(44)

which is solved by

$\displaystyle E^{\varphi}=\frac{x}{\sqrt{1-c_{\varphi}/x-\Lambda
x^{2}/3}}\,,$ (45)

with constant $c_{\varphi}$. Substituting all these results into the mass
observable (31), and rewriting it as $\mathcal{M}=M$ (interpreted as the mass
parameter of a line element rather than a phase-space function), determines

$\displaystyle c_{\varphi}$ $\displaystyle=$ $\displaystyle 2M\ .$ (46)

Lastly, $\dot{K}_{\varphi}=0$ implies that $\bar{N}$ acquires the expression
of the classical lapse function, up to a scaling constant $\alpha$, since
$K_{\varphi}=0$ eliminates all $\lambda$ dependence in the constraint and the
equations of motion it generates. Therefore,

$\displaystyle N=\frac{\sqrt{1-J(x)}}{\alpha\chi}\,,$ (47)

where we have defined

$J(x)=\frac{2M}{x}+\frac{\Lambda x^{2}}{3}$ (48)

for brevity.

The structure function is then given by

$\tilde{q}^{xx}=\chi^{2}\left(1+\lambda^{2}\left(1-J(x)\right)\right)\left(1-J(x)\right)\
,$ (49)

where $\lambda$ is an arbitrary function of $x$ through $E^{x}$. Space-time
solutions therefore depend on $\lambda$ through the emergent space-time
metric, while traditional models of loop quantum gravity always imply
$\lambda$-independent solutions in the static gauge. This property is one
example of covariance problems in this setting because an unmodified static
gauge makes it difficult to construct an equivalent non-static gauge for the
same model. From (40), the space-time line element is given by

${\rm d}s^{2}=-\left(1-J(x)\right)\frac{{\rm
d}t^{2}}{\alpha^{2}\chi^{2}}+\frac{1}{\left(1+\lambda^{2}\left(1-J(x)\right)\right)\left(1-J(x)\right)}\frac{{\rm
d}x^{2}}{\chi^{2}}+x^{2}{\rm d}\Omega^{2}\,,$ (50)

for arbitrary $\lambda(x)$ and $\chi(x)$, and a constant $\alpha$. For static
solutions as constructed here, $\alpha$ may always be absorbed by a
redefinition of the time coordinate.

The locations of the black-hole and cosmological horizons are still given by
their classical expressions since they solve the same classical equation,

$1-\frac{2M}{x}-\frac{\Lambda x^{2}}{3}=0$ (51)

implied by factors of $(1-J(x))^{\pm 1}$ in (50). Therefore, using a small
positive cosmological constant so as to match with observations for our
universe, the black-hole horizon appears at

$x_{\rm H}\approx 2M\,,$ (52)

and the cosmological one at

$x_{\Lambda}\approx\sqrt{\frac{3}{\Lambda}}\,.$ (53)

Just as for a classical black hole in a de Sitter background, the static
metric is valid only in the region $x_{\rm H}<x<x_{\Lambda}$, beyond which the
metric turns spatially homogeneous in an extension of the same gauge, just
flipping the roles of $t$ and $x$ as time and space coordinates.

However, now we have further coordinate singularities at the values of $x$
that solve the equation

$\displaystyle 1+\lambda^{2}\left(1-\frac{2M}{x}-\frac{\Lambda
x^{2}}{3}\right)=0$ (54)

implied by the new factor of ${\rm d}x^{2}$ in (50). This equation can have
multiple solutions depending on the chosen triangulation scheme
$\lambda=\lambda(x)$, but we can still formulate several general results.
These new coordinate singularities are closely related to non-classical
hypersurfaces of reflection symmetry implied by holonomy effects.

In particular, note that the term multiplying $\lambda^{2}$ is the expression
on the left-hand side of the horizons equation (51) and hence it vanishes at
$x_{\rm H}$ and $x_{\Lambda}$. Since $\lambda^{2}>0$, this term must turn
negative for (54) to hold, and hence a coordinate singularity of this form
cannot appear in the static region where this term is always positive. Thus,
the solutions $x=x_{\lambda}^{(i)}$ of this equation, with $i$ labeling the
different solutions, must be either below the Schwarzschild horizon,
$x_{\lambda}^{(-)}<x_{\rm H}$ denoting the largest solution less than the
Schwarzschild radius, or above the cosmological horizon,
$x_{\lambda}^{(+)}>x_{\Lambda}$ denoting the smallest solution greater than
the cosmological horizon. Therefore, solutions to (54) do not lie in the
static region, but rather in the homogenous ones.

Finally, we note that this equation corresponds precisely to (35) for the
reflection-symmetry surface. Therefore, the solutions $x_{\lambda}^{(-)}$ and
$x_{\lambda}^{(+)}$ correspond to extrema of the radius $\sqrt{E^{x}}$, around
which the spacetime is symmetric, and will have particularly interesting
consequences. Solutions of these two types are not guaranteed to exist for any
triangulation scheme $\lambda=\lambda(E^{x})$, and there could be more than
two solutions in some cases. Nevertheless, we will show that there exists an
$x_{\lambda}^{(-)}$, corresponding to the minimum of the areal radius, for any
well-behaved holonomy function, introducing the quantum hypersurface where the
geometry transitions from a black hole to a white hole through a wormhole
interior. In later sections we will come back to specific solutions
corresponding to the traditional $\mu_{0}$ and $\bar{\mu}$ schemes. First,
however, we must ensure that the homogeneous regions exist as solutions to the
equations of motion in their corresponding gauges, and identify the coordinate
transformations relating the different gauge choices in regions of overlap.

####  3.1.2 Schwarzschild gauge: Homogeneous regions

We now restrict the gauge by setting all spatial derivatives equal to zero. We
will use the label $t_{\rm h}$ for the time coordinate and $x_{\rm h}$ for the
radial coordinate in a homogeneous patch. The homogeneity condition implies
the partial gauge fixing

$\displaystyle N^{x}=0\ ,\hskip 28.45274ptN^{\prime}=0\,.$ (55)

The on-shell condition $\tilde{H}=0$ can be solved to obtain

$\displaystyle K_{x}$ $\displaystyle=$
$\displaystyle-\frac{E^{\varphi}}{4E^{x}}\frac{2\lambda}{\sin(2\lambda
K_{\varphi})}\left(1-\Lambda E^{x}+\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+2\left(K_{\varphi}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}-\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}\right)\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\,.$

Equations of motion for the remaining phase-space variables are

$\displaystyle\dot{E}^{x}$ $\displaystyle=$ $\displaystyle
2\bar{N}\sqrt{E^{x}}\;\frac{\sin(2\lambda K_{\varphi})}{2\lambda}\ ,$ (57)

as well as

$\displaystyle\frac{\dot{E}^{\varphi}}{E^{\varphi}}$ $\displaystyle=$
$\displaystyle\frac{\bar{N}}{2\sqrt{E^{x}}}\frac{2\lambda}{\sin(2\lambda
K_{\varphi})}\left(\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}-\left(1-\Lambda E^{x}\right)\cos(2\lambda
K_{\varphi})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-2\lambda^{2}\frac{\sin^{4}(\lambda
K_{\varphi})}{\lambda^{4}}\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\ .$

and

$\displaystyle\dot{K}_{\varphi}$ $\displaystyle=$
$\displaystyle-\frac{\bar{N}}{2\sqrt{E^{x}}}\left(1-\Lambda
E^{x}+\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}+2\left(K_{\varphi}\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}-\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}\right)\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\,.$

Combining these three time derivatives, we can pick $E^{x}$ as an evolution
parameter by using the chain rule, ${\rm d}K_{\varphi}/{\rm
d}E^{x}=\dot{K}_{\varphi}/\dot{E}^{x}$, obtaining the equation

$\displaystyle\frac{{\rm d}}{{\rm d}\ln E^{x}}\left(\frac{\sin^{2}(\lambda
K_{\varphi})}{\lambda^{2}}\right)=-\frac{1}{2}\left(1-\Lambda
E^{x}+\frac{\sin^{2}(\lambda K_{\varphi})}{\lambda^{2}}\right)\ ,$ (60)

which has the solution

$\displaystyle\frac{\sin^{2}(\lambda K_{\varphi})}{\lambda^{2}}$
$\displaystyle=$ $\displaystyle\frac{c_{x}}{\sqrt{E^{x}}}+\frac{\Lambda
E^{x}}{3}-1$ (61)

with constant $c_{x}$. Direct substitution into the mass observable (33) and
relabeling ${\cal M}=M$ determines $c_{x}=2M$. The equation of motion for the
structure function is given by

$\displaystyle\left(\ln\frac{\tilde{q}^{xx}}{\chi^{2}}\right)^{\bullet}$
$\displaystyle=$ $\displaystyle\left(\ln\left(\cos^{2}(\lambda
K_{\varphi})\frac{E^{x}}{(E^{\varphi})^{2}}\right)\right)^{\bullet}$ (62)
$\displaystyle=$ $\displaystyle 2\left(-\lambda^{2}\frac{\tan(\lambda
K_{\varphi})}{\lambda}\left(\dot{K}_{\varphi}+\frac{1}{2}K_{\varphi}\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\frac{\dot{E}^{x}}{E^{x}}\right)-\frac{\dot{E}^{\varphi}}{E^{\varphi}}\right)+\frac{\dot{E}^{x}}{E^{x}}$
$\displaystyle=$
$\displaystyle\frac{\bar{N}}{\sqrt{E^{x}}}\frac{\lambda}{\tan(\lambda
K_{\varphi})}\left(\frac{2M}{\sqrt{E^{x}}}+\frac{\Lambda}{3}E^{x}-\frac{2\lambda
E^{x}}{3}\right)\ .$

We now complete the gauge fixing by setting $E^{x}=t_{\rm h}^{2}$, yielding

$\displaystyle\frac{\sin^{2}(\lambda K_{\varphi})}{\lambda^{2}}$
$\displaystyle=$ $\displaystyle\frac{2M}{t_{\rm h}}+\frac{\Lambda}{3}t_{\rm
h}^{2}-1\ .$ (63)

The consistency equation $\dot{E}^{x}=2t_{\rm h}$ can be used to solve for the
lapse function as

$\displaystyle\bar{N}$ $\displaystyle=$
$\displaystyle\frac{2\lambda}{\sin(2\lambda K_{\varphi})}=\left(J(t_{\rm
h})-1\right)^{-1/2}\left(1-\lambda^{2}\left(J(t_{\rm
h})-1\right)\right)^{-1/2}$ (64)

with our previous definition of the function $J$, and the equation of motion
for the structure function is solved by

$\displaystyle\tilde{q}^{xx}=\alpha^{2}\chi^{2}\left(\frac{2M}{t_{\rm
h}}+\frac{\Lambda}{3}t_{\rm h}^{2}-1\right)^{-1}$ (65)

with an integration constant $\alpha$. We therefore obtain the space-time line
element

$\displaystyle{\rm d}s^{2}=-\frac{1}{\left(1-\lambda^{2}\left(J(t_{\rm
h})-1\right)\right)\left(J(t_{\rm h})-1\right)}\frac{{\rm d}t_{\rm
h}^{2}}{\chi^{2}}+\left(J(t_{\rm h})-1\right)\frac{{\rm d}x_{\rm
h}^{2}}{\alpha^{2}\chi^{2}}+t_{\rm h}^{2}{\rm d}\Omega^{2}\ ,$ (66)

for arbitrary $\lambda$ and $\chi$ as functions of $t_{\rm h}$, which matches
the static Schwarzschild metric (50) up to the label swap $x\to t_{\rm h}$,
$t\to x_{\rm h}$. This effective metric has a new (quantum) coordinate
singularity at the time coordinates solving the equation

$\displaystyle 1-\lambda^{2}\left(\frac{2M}{t_{\rm h}}+\frac{\Lambda t_{\rm
h}^{2}}{3}-1\right)=0\,,$ (67)

which is the same equation as the one obtained in the static region (54) up to
the label change $x\to t_{\rm h}$. As discussed in this context, solutions
$t_{\rm h}$ represent hypersurfaces of reflection symmetry. (The name quantum
coordinate singularity is motivated by the observation that this new singular
hypersurface appears only in the presence of holonomy modifications. As
mentioned earlier, on adopting suitable coordinates, we shall demonstrate
later that one of the solutions of (67) corresponds to the transition surface
where $\sqrt{E^{x}}$ acquires a minimum value.)

Therefore, homogeneous Schwarzschild-type coordinates are valid only in the
regions $x_{\lambda}^{(-)}<t_{\rm h}<x_{\rm H}$ where $x_{\lambda}^{(-)}$ is
the largest solution of (67) smaller than $x_{\rm H}$, and $x_{\Lambda}<t_{\rm
h}<x_{\lambda}^{(+)}$ where $x_{\lambda}^{(+)}$ is the smallest solution of
(67) greater than $x_{\Lambda}$. If there is no $x_{\lambda}^{(-)}$ obeying
the condition that defines it, we set $x_{\lambda}^{(-)}=0$. If there is no
$x_{\lambda}^{(+)}$ obeying the condition that defines it, we set
$x_{\lambda}^{(+)}=\infty$. If (67) has more than two distinct solutions,
additional regions can be introduced. However, we will show in Section 3.3
that the solutions $x_{\lambda}^{(\pm)}$ closest to $x_{\rm H}$ and
$x_{\Lambda}$, respectively, represent extremal radii, such that
$x_{\lambda}^{(-)}$ is a minimum radius, or a “bounce,” and
$x_{\lambda}^{(+)}$ is a maximum radius of a recollapsing homogeneous region.
Solutions $x_{\lambda}^{(i)}$ less than $x_{\lambda}^{(-)}$ or greater than
$x_{\lambda}^{(+)}$ are therefore never reached by dynamical solutions if we
start in the static Schwarzschild region and extend it across coordinate
singularities. Nevertheless, these alternative solutions might be used to
define independent space-time models that could perhaps be connected with the
static region by tunneling processes of a more complete quantum theory.

As already indicated, the homogenous solutions can formally be obtained from
the static one by a simple flip of coordinates. However, the corresponding
space-time regions are disjoint and separated by horizons. For a more precise
connection between these regions, we should construct a gauge that describes a
space-time region overlapping with a portion of the static solution and a
portion of a homogeneous one, crossing the horizon. Classically, an example
for such a gauge choice is given by the Gullstrand-Painlevé coordinates, which
we now construct for the modified equations.

####  3.1.3 From static Schwarzschild to Gullstrand–Painlevé

The metric (50) is static and thus has a timelike Killing vector
$\xi=\partial_{t}$. The quantity $\varepsilon=-\xi\cdot{\bf
u}=-u_{t}=-g_{t\nu}\,u^{\nu}$ is therefore conserved along geodesics, using
the 4-velocity ${\bf u}$. Compared with the classical Gullstrand–Painlevé
system, we are interested in geodesics for free fall starting at rest at a
finite radius, $x=x_{0}$, because the asymptotic regime may be non-static if
there is a cosmological horizon or non-classical asymptotic behavior implied
by certain types of holonomy modifications. The 4-velocity of such an object
has a time component
$u^{t}|_{x=x_{0}}=g^{tt}u_{t}|_{x=x_{0}}=-g^{tt}|_{x=x_{0}}\varepsilon$, using
the diagonal nature of the static metric in the second step, while all spatial
components vanish. Timelike normalization,
$-1=g_{\mu\nu}u^{\mu}u^{\nu}|_{x=x_{0}}=g_{tt}(u^{t})^{2}|_{x=x_{0}}$, then
implies

$\varepsilon^{2}=-\frac{1}{g^{tt}|_{x=x_{0}}}=\frac{1-J(x_{0})}{\alpha^{2}\chi(x_{0})^{2}}\,.$
(68)

For $x\not=x_{0}$, $u^{x}$ is non-zero and is related to $u^{t}$ by the
normalization condition. Since $u_{t}=-\varepsilon$ is conserved, it is
convenient to evaluate normalization in the form
$g^{\mu\nu}u_{\mu}u_{\nu}=-1$, such that

$\displaystyle u_{x}$ $\displaystyle=$ $\displaystyle
s\sqrt{g_{xx}(x)(-g^{tt}(x)\epsilon^{2}-1)}$ (69) $\displaystyle=$
$\displaystyle
s\sqrt{\frac{1}{(1-J(x))(1+\lambda(x)^{2}(1-J(x)))\chi(x)^{2}}\left(\frac{\chi(x)^{2}}{\chi(x_{0})^{2}}\frac{1-J(x_{0})}{1-J(x)}-1\right)}$
$\displaystyle=$
$\displaystyle\frac{s}{\chi(x)\chi(x_{0})}\frac{\sqrt{\chi(x)^{2}(1-J(x_{0}))-\chi(x_{0})^{2}(1-J(x))}}{|1-J(x)|\sqrt{1+\lambda(x)^{2}(1-J(x))}}$

with a sign choice $s$ that distinguishes outgoing from ingoing GP
coordinates.

The normalization condition also implies that we can obtain the proper-time
interval along our geodesics from the co-velocity components:

${\rm d}\tau=-\frac{g^{\mu\nu}{\rm d}x_{\mu}{\rm d}x_{\nu}}{{\rm
d}\tau}=-u_{\mu}g^{\mu\nu}{\rm d}x_{\nu}=-u_{\mu}{\rm d}x^{\nu}\,.$ (70)

Using proper time as the new GP coordinate, we obtain

$\displaystyle{\rm d}t_{\rm GP}$ $\displaystyle=$ $\displaystyle-
u_{\mu}dx^{\mu}$ (71) $\displaystyle=$ $\displaystyle\varepsilon{\rm
d}t-\frac{s}{\chi(x)\chi(x_{0})}\frac{\sqrt{\chi(x)^{2}(1-J(x_{0}))-\chi(x_{0})^{2}(1-J(x))}}{|1-J(x)|\sqrt{1+\lambda(x)^{2}(1-J(x))}}{\rm
d}x\,.$

The coordinate transformation $t_{\rm GP}(t,x)$ is defined by an explicit
integration of the 1-form (71), which depends on the choice of $\lambda(x)$
and $\chi(x)$ and therefore cannot be performed at a general level. However,
since $\varepsilon$ is constant and the coefficient of ${\rm d}x$ is
independent of $t$, (71) defines a closed 1-form, such that an integral
$t_{\rm GP}(t,x)$ exists as a local coordinate for any choice of $\lambda(x)$
and $\chi(x)$.

Substituting ${\rm d}t=\varepsilon^{-1}{\rm d}t_{\rm
GP}+\varepsilon^{-1}u_{x}{\rm d}x$ in (50), we obtain

$\displaystyle{\rm d}s^{2}_{\rm GP}$ $\displaystyle=$
$\displaystyle-N^{2}\varepsilon^{-2}({\rm d}t_{\rm GP}^{2}+2u_{x}{\rm d}t_{\rm
GP}{\rm d}x+u_{x}^{2}{\rm d}x^{2})+\tilde{q}_{xx}{\rm d}x^{2}+x^{2}{\rm
d}\Omega^{2}$ (72) $\displaystyle=$
$\displaystyle-\frac{N^{2}}{\varepsilon^{2}}{\rm d}t_{\rm
GP}^{2}-2u_{x}\frac{N^{2}}{\varepsilon^{2}}{\rm d}t_{\rm GP}{\rm
d}x+\left(\tilde{q}_{xx}-\frac{N^{2}}{\varepsilon^{2}}u_{x}^{2}\right){\rm
d}x^{2}+x^{2}{\rm d}\Omega^{2}$ $\displaystyle=$ $\displaystyle-{\rm d}t_{\rm
GP}^{2}+\frac{\tilde{q}_{xx}N^{2}}{\varepsilon^{2}}\left({\rm
d}x-u_{x}\tilde{q}^{xx}{\rm d}t_{\rm GP}\right)^{2}+x^{2}{\rm d}\Omega^{2}$
$\displaystyle=$ $\displaystyle-{\rm d}t_{\rm
GP}^{2}+\frac{\chi^{-4}}{\alpha^{2}\varepsilon^{2}}\left(1+\lambda^{2}\left(1-J(x)\right)\right)^{-1}$
$\displaystyle\times\left({\rm
d}x-s\chi\sqrt{J(x)-1-\frac{\chi^{2}}{\chi(x_{0})^{2}}(J(x_{0})-1)}\sqrt{1+\lambda^{2}\left(1-J(x)\right)}{\rm
d}t_{\rm GP}\right)^{2}$ $\displaystyle+x^{2}{\rm d}\Omega^{2}$

where we used $u_{x}^{2}=\tilde{q}_{xx}(\varepsilon^{2}/N^{2}-1)$ twice in the
third line, and then inserted the explicit $u_{x}$ from (69). The metric
remains regular at the horizon coordinates, but still diverges at the new
(quantum) singular coordinates $x=x_{\lambda}^{(i)}$.

In the case of a vanishing cosmological constant, $\Lambda=0$, it is useful to
take the limit $x_{0}\to\infty$, in which case $\alpha\varepsilon\to
1/\lim_{x\to\infty}\chi(x)=:1/\chi_{\infty}$ from (68) and the metric
simplifies to

$\displaystyle{\rm d}s^{2}_{\rm GP}$ $\displaystyle=$ $\displaystyle-{\rm
d}t_{\rm
GP}^{2}+\frac{\chi_{\infty}^{2}}{\chi^{4}}\left(1+\lambda^{2}\left(1-\frac{2M}{x}\right)\right)^{-1}$
(73) $\displaystyle\times\left({\rm
d}x+s\chi\sqrt{\frac{2M}{x}+\frac{\chi^{2}}{\chi_{\infty}^{2}}-1}\sqrt{1+\lambda^{2}\left(1-\frac{2M}{x}\right)}{\rm
d}t_{\rm GP}\right)^{2}$ $\displaystyle\,\;\;\;\;\;\;\;+x^{2}{\rm
d}\Omega^{2}\,.$

####  3.1.4 From homogeneous Schwarzschild to Gullstrand-Painlevé

We can now start with the GP metric (72) we just derived and extend it across
the horizons. Because we expect the causal meaning of time and space
coordinates to change in the interior of the black hole, let us define $T_{\rm
GP}=x$ and $X_{\rm GP}=t_{\rm GP}$ and rewrite the GP metric (72) as

$\displaystyle{\rm d}s^{2}_{\rm GP}$ $\displaystyle=$ $\displaystyle-{\rm
d}X_{\rm
GP}^{2}+\frac{\chi^{-4}}{\alpha^{2}\varepsilon^{2}}\left(1+\lambda^{2}\left(1-J(T_{\rm
GP})\right)\right)^{-1}\bigg{(}{\rm d}T_{\rm GP}$ (74)
$\displaystyle\qquad+s\frac{\chi}{\chi(x_{0})}\sqrt{\chi^{2}(1-J(x_{0}))-\chi(x_{0})^{2}(1-J(T_{\rm
GP}))}\sqrt{1+\lambda^{2}\left(1-J(T_{\rm GP})\right)}{\rm d}X_{\rm
GP}\bigg{)}^{2}$ $\displaystyle+T_{\rm GP}^{2}{\rm d}\Omega^{2}$
$\displaystyle=$ $\displaystyle-\left(1-\lambda^{2}\left(J(T_{\rm
GP})-1\right)\right)^{-1}\left(J(T_{\rm GP})-1\right)^{-1}\frac{{\rm d}T_{\rm
GP}^{2}}{\chi^{2}}$ $\displaystyle+\frac{J(T_{\rm
GP})-1}{\alpha^{2}\chi^{2}\varepsilon^{2}}\Bigg{(}{\rm d}X_{\rm
GP}+\frac{s}{\chi}\sqrt{\frac{J(T_{\rm
GP})-1+(\chi^{2}/\chi(x_{0})^{2})(1-J(x_{0}))}{1-\lambda^{2}(J(T_{\rm
GP})-1)}}\frac{{\rm d}T_{\rm GP}}{J(T_{\rm GP})-1}\Bigg{)}^{2}$
$\displaystyle+T_{\rm GP}^{2}{\rm d}\Omega^{2}\,.$

A comparison of (74) with (66) shows that the two are related by the
coordinate transformation $t_{\rm h}=T_{\rm GP}$ and

$\displaystyle{\rm d}x_{\rm h}$ $\displaystyle=$ $\displaystyle\frac{{\rm
d}X_{\rm GP}}{\varepsilon}$
$\displaystyle-\frac{s}{\varepsilon\chi}\sqrt{\frac{J(T_{\rm
GP})-1+(\chi^{2}/\chi(x_{0})^{2})\left(1-J(x_{0})\right)}{1-\lambda^{2}\left(J(T_{\rm
GP})-1\right)}}\frac{{\rm d}T_{\rm GP}}{J(T_{\rm GP})-1}$

for arbitrary $\lambda$ and $\chi$. The explicit integration can only be
completed after specifying the choice of $\lambda(t_{\rm h})$ and $\chi(t_{\rm
h})$, but such an integration always exists locally because ${\rm d}x_{\rm h}$
is a closed 1-form.

The metric in (74) remains regular at the horizon coordinates if we choose
$\alpha\varepsilon=\sqrt{1-J(x_{0})}/\chi(x_{0})$, but it diverges at the
reflection surfaces $T_{\rm GP}=x_{\lambda}^{(i)}$ for any $\alpha$.
Therefore, the GP chart is valid only in the region
$x_{\lambda}^{(-)}<x<x_{\lambda}^{(+)}$, where for more than two solutions to
(54) the $\pm$-versions of $x_{\lambda}^{(i)}$ are as defined at the end of
Section 3.1.2.

In the case of a vanishing cosmological constant, we can take $x_{0}\to\infty$
and $\alpha\varepsilon=1/\lim_{x\to\infty}\chi(x)=:1/\chi_{\infty}$,
simplifying the metric to

$\displaystyle{\rm d}s^{2}_{\rm GP}$ $\displaystyle=$
$\displaystyle-\left(1+\lambda^{2}\left(1-J(T_{\rm
GP})\right)\right)^{-1}\left(J(T_{\rm GP})-1\right)^{-1}\frac{{\rm d}T_{\rm
GP}^{2}}{\chi^{2}}$ (76)
$\displaystyle+\frac{\chi_{\infty}^{2}}{\chi^{2}}\left(J(T_{\rm
GP})-1\right)\left({\rm d}X_{\rm GP}-\frac{s}{\chi}\sqrt{\frac{J(T_{\rm
GP})+\chi^{2}/\chi_{\infty}^{2}-1}{1-\lambda^{2}(J(T_{\rm GP})-1)}}\frac{{\rm
d}T_{\rm GP}}{J(T_{\rm GP})-1}\right)^{2}$ $\displaystyle+T_{\rm GP}^{2}{\rm
d}\Omega^{2}\ .$

####  3.1.5 Gullstrand-Painlevé gauge

The GP metric (72) covers both the static and the homogeneous regions in a
single coordinate chart. We now perform a consistency check by showing that
the solution obtained above satisfies the canonical equations of motion in an
appropriate GP gauge.

Based on the metric (72), we define the GP gauge as

$\displaystyle N=1\ ,\hskip 28.45274ptE^{x}=x^{2}\ .$ (77)

The consistency equation $\dot{E}^{x}=0$ determines the shift vector

$\displaystyle N^{x}$ $\displaystyle=$
$\displaystyle-\chi\left(1+\lambda^{2}\frac{x^{2}}{(E^{\varphi})^{2}}\right)\frac{\sin(2\lambda
K_{\varphi})}{2\lambda}\ .$ (78)

The on-shell conditions $H_{x}=0$ and $\tilde{H}=0$, respectively, determine
$K_{x}=E^{\varphi}K_{\varphi}^{\prime}/(2x)$ and

$\displaystyle\frac{\sin^{2}(\lambda K_{\varphi})}{\lambda^{2}}$
$\displaystyle=$
$\displaystyle\left(1+\lambda^{2}\frac{x^{2}}{(E^{\varphi})^{2}}\right)^{-1}\left(\frac{c_{\varphi}}{x}+\frac{\Lambda
x^{2}}{3}+\frac{x^{2}}{(E^{\varphi})^{2}}-1\right)\ ,$ (79)

for arbitrary $\lambda$, where $c_{\varphi}$ is an integration constant.

Finally, the equation of motion for $E^{\varphi}$ reads

$\displaystyle\left(\ln\left(\chi^{2}\frac{(E^{\varphi})^{2}}{x^{2}}\right)\right)^{\bullet}$
$\displaystyle=$
$\displaystyle-\sqrt{\left(\frac{c_{\varphi}}{x}+\frac{\Lambda
x^{2}}{3}+\frac{x^{2}}{(E^{\varphi})^{2}}-1\right)\left(1+\lambda^{2}\left(1-\frac{c_{\varphi}}{x}-\frac{\Lambda
x^{2}}{3}\right)\right)}\left(\ln\left(\chi^{2}\frac{(E^{\varphi})^{2}}{x^{2}}\right)\right)^{\prime}$

and is solved by $E^{\varphi}=x/(\alpha\chi\varepsilon)$ with constant
$\alpha$ and $\varepsilon$. (We introduce two integration constants for the
sake of comparing with our previous line element, which will relate
$\varepsilon$ to the constant denoted earlier by the same letter. As a partial
differential equation, (3.1.5) has additional solutions that depend on the
specific choice of $\lambda(x)$ but will not be required here.) The resulting
metric can then be derived from (40), which matches exactly with (72) upon the
identification of $c_{\varphi}=2M$ and
$\varepsilon^{2}=(1-J(x_{0}))/(\alpha^{2}\chi(x_{0})^{2})$.

Therefore, the modified GP metric (74) is related by standard coordinate
transformations to both the static and homogeneous solutions in regions of
overlap. Furthermore, it is well-defined at the Schwarzschild horizon
$x=x_{\rm H}$ provided $x_{0}>x_{\rm H}$, and at the cosmological horizon
$x=x_{\Lambda}$ provided $x_{0}>x_{\Lambda}$. In the latter case, $x_{0}$
(defined as the reference position for the inertial observer) can no longer be
interpreted as the coordinate where the observer is at rest, but the
coordinate transformation still holds.

###  3.2 Internal-time gauge of homogeneous regions

As we have seen, the homogeneous Schwarzschild charts terminate at the quantum
hypersurfaces of reflection symmetry at $x_{\lambda}^{(i)}$ solving (54). As
we will show in Section 5, the Ricci scalar remains regular at those surfaces,
suggesting that another chart must exist for which the metric remains regular
there. Since the coordinates $x_{\lambda}^{(i)}$ always lie in the homogeneous
regions, we will formulate a new homogeneous gauge where the time coordinate
is not given by the radius in an extension of the Schwarzschild exterior, but
is instead strategically chosen to track the angular momentum $K_{\varphi}$.
To this end, it is useful to switch to the constraint (17) because it is
periodic in the variable of interest, $K_{\varphi}$.

We set all spatial derivatives to zero and label $t_{\varphi}$ as the time
coordinate in our new gauge as well as $x_{\rm h}$ as the spatial coordinate,
which agrees with the previous homogeneous coordinate. The partial gauge-
fixing

$\displaystyle N^{x}=0\ ,\hskip 28.45274ptN^{\prime}=0\ ,$ (81)

is used for the homogeneous gauge. The on-shell condition $\tilde{H}=0$ can be
solved for $K_{x}$:

$\displaystyle
K_{x}=-\frac{E^{\varphi}}{4E^{x}}\frac{2\tilde{\lambda}}{\sin(2\tilde{\lambda}K_{\varphi})}\left(\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(1-\Lambda
E^{x}\right)+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)\,.$
(82)

The relevant equations of motion are

$\displaystyle\frac{\dot{E}^{x}}{E^{x}}=\frac{\tilde{\lambda}}{\lambda}\frac{2\bar{N}}{\sqrt{E^{x}}}\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}$
(83)

and

$\displaystyle\frac{\dot{E}^{\varphi}}{E^{\varphi}}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}}{\lambda}\frac{\bar{N}}{2\sqrt{E^{x}}}\frac{2\tilde{\lambda}}{\sin(2\tilde{\lambda}K_{\varphi})}\Bigg{(}\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}-\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(1-\Lambda
E^{x}\right)\cos(2\tilde{\lambda}K_{\varphi})\Bigg{)}\ ,$

as well as

$\displaystyle\dot{K}_{\varphi}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{\lambda}}{\lambda}\frac{\bar{N}}{2\sqrt{E^{x}}}\left(\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(1-\Lambda
E^{x}\right)+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)\
.$ (85)

(Recall that $\bar{N}=\chi N$.)

Combining these equations of motion we obtain

$\displaystyle\frac{{\rm d}}{{\rm d}\ln
E^{x}}\left(\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\left(1-\Lambda
E^{x}+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)\
,$ (86)

which has a general solution

$\displaystyle\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}$
$\displaystyle=$ $\displaystyle\frac{c_{x}}{\sqrt{E^{x}}}+\frac{\Lambda
E^{x}}{3}-1\ ,$ (87)

for arbitrary $\lambda(E^{x})$. Inserting into the mass observable (31) and
relabeling $\mathcal{M}=M$ a constant (rather than phase-space function)
determines $c_{x}=2M$. A different combination of the time derivatives gives
us the equation of motion

$\displaystyle\left(\ln\frac{\tilde{q}^{xx}}{\chi^{2}}\right)^{\bullet}$
$\displaystyle=$
$\displaystyle-2\tilde{\lambda}\tan\left(\tilde{\lambda}K_{\varphi}\right)\dot{K}_{\varphi}-2\frac{\dot{E}^{\varphi}}{E^{\varphi}}+\left(1-\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\dot{E}^{x}}{E^{x}}$ (88) $\displaystyle=$
$\displaystyle\frac{\tilde{\lambda}}{\lambda}\frac{\bar{N}}{\sqrt{E^{x}}}\Bigg{[}\frac{\sin(2\tilde{\lambda}K_{\varphi})}{2\tilde{\lambda}}+\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(\tilde{\lambda}^{2}\frac{\tan\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}}+\frac{2\tilde{\lambda}}{\tan(2\tilde{\lambda}K_{\varphi})}\right)\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\bar{N}}{\sqrt{E^{x}}}\frac{\lambda}{\tilde{\lambda}}\frac{\tilde{\lambda}}{\tan\left(\tilde{\lambda}K_{\varphi}\right)}\Bigg{[}1-\Lambda
E^{x}+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}(\tilde{\lambda}K_{\varphi})}{\tilde{\lambda}^{2}}\Bigg{]}$

for the structure-function.

We now complete the gauge fixing by choosing $K_{\varphi}=-t_{\varphi}$. The
consistency equation $\dot{K}_{\varphi}=-1$ determines the lapse function

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\tilde{\lambda}}\frac{2\sqrt{E^{x}}}{\chi}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)^{-1}\,,$
(89)

such that the equation of motion for the structure function becomes

$\displaystyle\left(\ln\frac{\tilde{q}^{xx}}{\chi^{2}}\right)^{\bullet}$
$\displaystyle=$ $\displaystyle
2\frac{\tilde{\lambda}}{\tan\left(\tilde{\lambda}K_{\varphi}\right)}\left(1-\Lambda
E^{x}+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}(\tilde{\lambda}K_{\varphi})}{\tilde{\lambda}^{2}}\right)$
(90) $\displaystyle\qquad\qquad\times\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)^{-1}\
.$

This equation is hard to solve directly, but we can perform the associated
coordinate transformation from $t_{\rm h}$ to $t_{\varphi}$ using (87) with
$E^{x}=t_{\rm h}^{2}$ and $K_{\varphi}=-t_{\varphi}$:

$\frac{\tilde{\lambda}^{2}}{\lambda(t_{\rm
h}^{2})^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}=\frac{2M}{t_{\rm
h}}+\frac{\Lambda t_{\rm h}^{2}}{3}-1$ (91)

such that

$\displaystyle\frac{\sin\left(2\tilde{\lambda}t_{\varphi}\right)}{2\tilde{\lambda}}{\rm
d}t_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\frac{\partial\ln\lambda^{2}}{\partial
t_{\rm h}}\left(\frac{2M}{t_{\rm h}}+\frac{\Lambda t_{\rm
h}^{2}}{3}-1\right)-\frac{1}{t_{\rm
h}}\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(\frac{2M}{t_{\rm
h}}-\frac{2\Lambda t_{\rm h}^{2}}{3}\right)\right){\rm d}t_{\rm h}$ (92)
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{E^{x}}}\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(-\frac{2M}{\sqrt{E^{x}}}+\frac{2\Lambda
E^{x}}{3}+2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right){\rm
d}t_{\rm h}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{E^{x}}}\frac{\lambda^{2}}{\tilde{\lambda}^{2}}\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}K_{\varphi}\right)}{\tilde{\lambda}^{2}}\right){\rm
d}t_{\rm h}$

or

${\rm d}t_{\rm
h}=-2\sqrt{E^{x}}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin\left(2\tilde{\lambda}t_{\varphi}\right)}{2\tilde{\lambda}}\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)^{-1}{\rm
d}t_{\varphi}$ (93)

The homogeneous Schwarzschild metric (66) then becomes

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-4E^{x}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)^{-2}\frac{{\rm
d}t_{\varphi}^{2}}{\chi^{2}}$ (94)
$\displaystyle+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\frac{{\rm
d}x_{\rm h}^{2}}{\alpha^{2}\chi^{2}}+E^{x}{\rm d}\Omega^{2}\,,$

for arbitrary functions $\lambda$ and $\chi$, where $E^{x}$ is implicitly
given in terms of $t_{\varphi}$ by the solution to (87). Notice that the
reference value $\tilde{\lambda}$, which is always constant, can be eliminated
completely by scaling the time coordinate to
$\tilde{t}_{\varphi}=\tilde{\lambda}t_{\varphi}$. One can now check that the
lapse function of (94) matches the one obtained canonically, (89). Finally, we
can take the inverse of the radial component of (94) as the structure
function, take its time derivative and confirm that it indeed satisfies the
canonical equation of motion (90) since

$\displaystyle\partial_{t_{\varphi}}\ln\left(\frac{1}{\alpha^{2}\chi^{2}}\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)$
$\displaystyle=$ $\displaystyle
2\frac{\tilde{\lambda}}{\tan\left(\tilde{\lambda}t_{\varphi}\right)}\left(1-\Lambda
E^{x}+\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)$
$\displaystyle\qquad\quad\times\left(1-\Lambda
E^{x}+\left(1-2\frac{\partial\ln\lambda^{2}}{\partial\ln
E^{x}}\right)\frac{\tilde{\lambda}^{2}}{\lambda^{2}}\frac{\sin^{2}\left(\tilde{\lambda}t_{\varphi}\right)}{\tilde{\lambda}^{2}}\right)^{-1}\,.$

###  3.3 Proof of extremality

We are now ready to prove our previous claim that any solution of (54) for the
new quantum coordinate singularities in homogeneous regions of the modified
Schwarzchild geometry (50) dynamically corresponds to an extremal radius (with
some restrictions in case of higher multiplicities of the roots). Inserting
$E^{x}=x^{2}$ in equation (54), we obtain the condition that the function
defined by

$b(E^{x})=1+\lambda^{2}\left(1-\frac{2M}{x}-\frac{\Lambda
x^{2}}{3}\right)=1+\lambda^{2}\left(1-\frac{2M}{\sqrt{E^{x}}}-\frac{\Lambda
E^{x}}{3}\right)$ (96)

vanishes at $E^{x}=(x_{\lambda}^{(i)})^{2}$. With our identification of
$c_{x}=2M$ and equation (87), it then follows that

$\sin^{2}(\tilde{\lambda}K_{\varphi})=1-b(E^{x})$ (97)

and $\sin^{2}(\tilde{\lambda}K_{\varphi})=1$ at any solution of (54).

The function $b(E^{x})$ is always positive for $E^{x}$ between $x_{\rm H}^{2}$
and $x_{\Lambda}^{2}$. Outside of this range, its first zeros on the two sides
are defined as $(x_{\lambda}^{(\pm)})^{2}$. If these solutions have odd
multiplicity, $b(E^{x})$ becomes negative for values of $E^{x}$ less than
$(x_{\lambda}^{(-)})^{2}$ or greater than $(x_{\lambda}^{(+)})^{2}$, such that
there is no corresponding $K_{\varphi}$ for which equation (97) could hold.
The only dynamical option is for $E^{x}$ to start increasing after it reaches
$(x_{\lambda}^{(-)})^{2}$, or to start decreasing after it reaches
$(x_{\lambda}^{(+)})^{2}$. These values are therefore dynamical extrema of
$E^{x}$, as well as of $\sin^{2}(\tilde{\lambda}K_{\varphi})$ as a measure of
curvature. The extremum of $\sin^{2}(\tilde{\lambda}K_{\varphi})$ also
characterizes these extrema of $E^{x}$ as hypersurfaces of reflection
symmetry. If there are solutions of (54) other than $x_{\lambda}^{(\pm)}$,
they will not play a role for dynamical solutions obtained by extending the
static region across coordinate singularities. Unlike in Schwarzschild or GP
coordinates, the metric (94) remains regular at any hypersurface of reflection
symmetry, given by $t_{\varphi}=\pi/(2\tilde{\lambda})$ and can be used for
this purpose. We discuss the global structure in Section 4 and perform a
detailed analysis of singularity resolution in Section 5.

For completeness, we mention why the previous conclusions are restricted to
$x_{\lambda}^{(\pm)}$ with odd multiplicity. If $x_{\lambda}^{(-)}$ or
$x_{\lambda}^{(+)}$ has even multiplicity, the function $b(E^{x})$ stays
positive in a neighborhood around such a solution, and dynamical values of
$\sqrt{E^{x}}$ beyond this solution are not ruled out by the condition (97). A
detailed dynamical analysis would be required for a complete understanding,
which we will not perform in this paper. Nevertheless, such cases, even though
they are not generic, might be of interest as new types of extremal black
holes, defined in the general sense that initially distinct coordinate
singularities coincide for some parameter choices. For extremal black holes in
classical general relativity, the inner and outer horizon coincide, implying a
double root of the defining equation. For new extremal solutions in emergent
modified gravity, the black-hole horizon retains its non-extremal form, but
the internal structure changes because some of the interior coordinate
singularities coincide. Such solutions could shed light on possible
connections, such as tunneling processes, between space-time regions within
$x_{\lambda}^{(\pm)}$ and values beyond these extrema that lie in disjoint
space-time regions in the non-extremal case.

In these discussions, we assumed that the areal gauge is always available in a
region around the extremal radius, which generically is the case for
physically acceptable solutions: This gauge gauge would not be available in a
given region if and only if $E^{x}$ is completely constant there in space and
time. (If $E^{x}$ is constant only spatially in a given slicing but depends on
time, one can always deform the slicing locally such that
$(E^{x})^{\prime}\not=0$.) The diffeomorphism constraint then implies that
$K_{\varphi}^{\prime}=0$ in this region because $E^{\varphi}$ is non-zero
around these new coordinate singularities. Maintaining $\dot{E}^{x}=0$ in the
region implies that $K_{\varphi}$ cannot change in time according to the
$E^{x}$-equation of motion, and stays at the value required for reflection
symmetry on all slices in the region. With these conditions, the equation of
motion (289) for $K_{\varphi}$ implies a first-order differential equation for
$\lambda$ which, for zero cosmological constant, has the solution
$\lambda=(c/\sqrt{E^{x}}-1)^{-1/2}$ with a constant $c$. This behavior is not
of the form usually considered for holonomy modifications, and could only be
realized in a finite range of $E^{x}$.

##  4 Global structure

The global structure of classical space-time solutions may change if
singularities are resolved or if there are large-scale effects from holonomy
modifications that could alter the asymptotic behavior. Different implications
are possible, depending on the form of holonomy modifications as determined by
the function $\lambda(E^{x})$.

###  4.1 Asymptotic flatness and zero-mass limit

If we set $\Lambda=0$, we may impose asymptotic flatness. For this purpose,
the metric (50) for the modified Schwarzschild exterior is the relevant one,
which takes the asymptotic form

${\rm d}s^{2}\approx-\frac{{\rm
d}t^{2}}{\alpha^{2}\chi(\infty)^{2}}+\left(1+\lambda(\infty)^{2}\right)^{-1}\frac{{\rm
d}x^{2}}{\chi(\infty)^{2}}+x^{2}{\rm d}\Omega^{2}\,.$ (98)

Therefore, asymptotic flatness partially determines the limiting behavior of
the two modification functions by the relations

$\displaystyle\alpha=\chi(\infty)^{-1}\quad,\quad\chi(\infty)=1/\sqrt{1+\lambda(\infty)^{2}}\,,$
(99)

which require $\chi$, and hence $\lambda$, to be asymptotically constant
because $\alpha$ is a constant, implied by one of our integrations. We will
impose this condition in the following.

We may independently ask that flat space-time be recovered in the zero-mass
limit $M\to 0$, in which case the line element takes the form

${\rm d}s^{2}\approx-\frac{{\rm
d}t^{2}}{\alpha^{2}\chi^{2}}+\left(1+\lambda^{2}\right)^{-1}\frac{{\rm
d}x^{2}}{\chi^{2}}+x^{2}{\rm d}\Omega^{2}\,.$ (100)

The radial component is rendered flat by the choice

$\chi=1/\sqrt{1+\lambda^{2}}\,,$ (101)

which is stronger than the asymptotic-flatness condition (99), but compatible
with it. On the other hand, the time component becomes flat only if

$\chi={\rm constant}=:\chi_{0}\quad,\quad\alpha=\chi_{0}^{-1}\,,$ (102)

in which case the metric is asymptotically flat too according to (99).
Therefore, the radial and time components can both be flat in the zero-mass
limit only if $\lambda$ is constant, or if $\lambda$ depends not only on
$E^{x}$ but also on the mass (for instance via renormalization) such that it
approaches a constant as $M\to 0$. Here, we will not consider the second
possibility.

If one wishes to recover flat space as part of the classical geometry in the
zero-mass limit, one needs a constant radial component. In this case, we
impose the conditions (99) and (101) such that the static region is described
by

${\rm
d}s^{2}=-\left(1-\frac{2M}{x}\right)\frac{1+\lambda^{2}}{1+\lambda_{\infty}^{2}}{\rm
d}t^{2}+\left(1-\frac{\lambda^{2}}{1+\lambda^{2}}\frac{2M}{x}\right)^{-1}\left(1-\frac{2M}{x}\right)^{-1}{\rm
d}x^{2}+x^{2}{\rm d}\Omega^{2}\,,$ (103)

where $\lambda_{\infty}=\lim_{x\to\infty}\lambda(x)$. The conditon of having
flat space in the zero-mass limit could therefore be used to determine one of
the modification functions. However, the spatial quantum geometry realized in
LQG on Planckian scales can be expected to leave traces even in an effective
description, such that a non-Euclidean 3-dimensional space could be acceptable
at zero mass. Because the focus of this paper is on black holes in effective
loop quantum gravity we will adopt this perspective and do not require (101).

Therefore, we impose only the asymptotic-flatness condition (99), such that
the static region is described by

${\rm d}s^{2}=-\left(1-\frac{2M}{x}\right)\frac{{\rm
d}t^{2}}{\chi^{2}(1+\lambda_{\infty}^{2})}+\left(1+\lambda^{2}\left(1-\frac{2M}{x}\right)\right)^{-1}\left(1-\frac{2M}{x}\right)^{-1}\frac{{\rm
d}x^{2}}{\chi^{2}}+x^{2}{\rm d}\Omega^{2}$ (104)

with the overall factor $\chi$ required to have the asymptotic limit
$\chi(\infty)=1/\sqrt{1+\lambda_{\infty}^{2}}=:\chi_{0}$, but otherwise
arbitrary. While several of the following analyses hold for arbitrary
$\chi(x)$, we will, for simplicity of the notation, assume a constant overall
factor, $\chi=\chi_{0}$. The metric then further simplifies to

${\rm d}s^{2}=-\left(1-\frac{2M}{x}\right){\rm
d}t^{2}+\left(1+\lambda^{2}\left(1-\frac{2M}{x}\right)\right)^{-1}\left(1-\frac{2M}{x}\right)^{-1}\frac{{\rm
d}x^{2}}{\chi_{0}^{2}}+x^{2}{\rm d}\Omega^{2}\,,$ (105)

This chart, and its homogeneous counterpart, will suffice for most of the
following applications.

A second argument to prefer (105) over (103) is the realization that the time