**1. Introduction**

Canonical gravity describes the 4-dimensional, generally covariant structure of spacetime by canonical fields defined on the slices of a spatial foliation. The evolution of these fields in time as well as transformations between different foliations are described by the geometrical structure of hypersurface deformations. In a canonical theory, these transformations are generated by certain phase-space functions, the diffeomorphism and Hamiltonian constraints. In spherically symmetric models, which will be considered here, the full set of constraints can be written as *D*[*M*] and *H*[*N*] with arbitrary spatial functions *M* (of density weight −1) and *N*. The constraint equations *D*[*M*] = 0 and *H*[*N*] = 0, valid for any *M* and *N*, restrict the phase-space degrees of freedom, given by the spatial metric and its momentum related to extrinsic curvature.

At the same time, the constraints generate (i) time evolution,

$$\mathcal{L}\_{t(N,M)}f = \{f, H[N] + D[M]\}\tag{1}$$

for a phase-space function *f* along a time-evolution vector field *ta* = *Nna* + *Msa* in spacetime with the unit normal *n<sup>a</sup>* to a spatial slice and the tangent vector field *sa* = (*∂*/*∂x*)*<sup>a</sup>* within the radial manifold (with coordinate *x*) of a spatial slice, and (ii) gauge transformations

$$\delta\_{\vec{\xi}(\eta,\mathfrak{c})}f = \{f, H[\eta] + D[\mathfrak{e}]\} \tag{2}$$

along a space-time vector field

$$
\zeta^a = \eta n^a + \epsilon s^a \tag{3}
$$

where , like *M*, has density weight −1.

**Citation:** Bojowald, M. Abelianized Structures in Spherically Symmetric Hypersurface Deformations. *Universe* **2022**, *8*, 184. https://doi.org/ 10.3390/universe8030184

Academic Editor: Steven Duplij

Received: 17 February 2022 Accepted: 11 March 2022 Published: 15 March 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The reference to normal and tangential directions relative to a foliation implies crucial differences between the mathematical formulation of hypersurface deformations in canonical gravity and the more common formulation of general covariance in terms of space-time tensors. In space-time, vector components *ξa* transform, by definition, in such a way that *ξ<sup>a</sup>∂*/*∂x<sup>a</sup>* determines a unique direction independent of coordinate choices. Similarly, the spatial vector *s<sup>a</sup>* = *∂*/*∂x* defines a coordinate-independent direction because a scalar of density weight −1 in one dimension transforms like a 1-form dual to *∂*/*∂<sup>x</sup>*. The normal deformation, however, cannot be introduced in this way because the canonical setting does not provide a time coordinate or the corresponding *∂*/*∂t*. Moreover, even if such a coordinate could be introduced by hand, for instance by using *t* merely as a parameter as it also appears in Hamilton's equations, it would be impossible to endow *η* with a density weight −1 in the time direction because, canonically, there is no time manifold. The only alternative is given by the procedure that has been used since [1,2] and formalized in [3]: The normalization of *n<sup>a</sup>* as a unit vector (with respect to the space-time metric, which is available in the canonical setting through the spatial metric on a slice as well as lapse *N* and shift *M*) associates a unique normal displacement to any given function *η* (without density weight).

The normal can be made unit only by reference to the metric, which provides some of the canonical degrees of freedom. The geometrical meaning of normal hypersurface deformations and their commutators depend on the spatial metric, resulting in structure functions in the canonical bracket relations. As a consequence, the canonical symmetries do not form a Lie algebra. This property is responsible for several complications well-known in attempts of canonical quantizations of the theory, starting with [4]. It also makes it harder to develop suitable mathematical structures for transformations generated by the constraints, in particular in an off-shell manner when one does not insist on solving the constraint equations. In [3], for instance, it was shown that a direct composition of transformations generated by the constraints is meaningful in the sense of path independence (a notion introduced in there) only on-shell.

The full structure of transformations is nevertheless required for general covariance to be implemented properly in the solutions of a canonical theory of gravity, in particular one that has been quantized, modified or deformed by new physical effects. While the restricted on-shell behavior may be easier to handle, the off-shell structure is important to make sure that the theory has a well-defined space-time structure, independently of the dynamics. Only in this case can the theory be considered a geometrical effective theory of some deeper and as ye<sup>t</sup> unknown quantum space-time, just as different dynamical versions of gravity given by higher-curvature effective actions make use of the same Riemannian form of space-time. Because of its importance for covariance and the classification of meaningful effective theories, we will review the structure of hypersurface deformations in the beginning of our first section below, combining classic results from gravitational physics with more recent mathematical developments [5,6].

We will focus on aspects of hypersurface deformations of importance for a suggested simplification of the hypersurface-deformation brackets in spherically symmetric models, given by a partial Abelianization [7], but our statements will apply also to a variety of other reformulations that rely on phase-space dependent lapse and shift. Analyzing a partial Abelianization in the context of hypersurface deformations, we will show that this construction captures only a certain subset of these transformations and, upon modification or quantization, does not guarantee that invariance under hypersurface deformations or general covariance are still realized. This conclusion may be surprising because, at first sight, a partial Abelianization appears to implement the same number of symmetry generators as standard hypersurface deformations and uses only a linear redefinition of the generators. However, the coefficients of these linear redefinitions are phase-space dependent, complicating their mathematical description [5,6]. (Heuristically, phase-space dependent linear redefinitions of the generators introduce new structure functions or modify existing ones.) It is then a non-trivial question whether the redefinitions can be inverted. If they cannot be

inverted, the redefined theory is not invariant under full hypersurface deformations and its solutions violate general covariance. An additional construction is therefore needed in a partially Abelianized model (or other reformulations of standard hypersurface deformations) in order to recover all space-time transformations. As shown by explicit examples, this is not always possible if the generators have been modified by quantum corrections.

A recent paper [8] claims that it may be possible to realize general covariance in partial Abelianizations of spherically symmetric models with different types of quantum modifications, such as a spatial discretization. The claim is not accompanied by a successful reconstruction of hypersurface deformations and instead relies on a technical and so far incomplete case-by-case study of quantities that should be invariant in a covariant theory. Using our results about general hypersurface deformation structures, we will explain why the covariance claims of [8] cannot hold.

#### **2. Hypersurface Deformations**

Space-time vector fields with their standard Lie bracket generate the Lie algebra of diffeomorphisms. Similarly, the transformations generated by the canonical constraints form an algebraic structure. They are labeled by the components *η* and of a vector field *ξ* used in (3) in a basis (*na*,*s<sup>a</sup>*) adapted to a spatial foliation, rather than a coordinate basis. Their commutators

$$\begin{aligned} &\delta\_{\xi\_2}(\delta\_{\xi\_1}f) - \delta\_{\xi\_1}(\delta\_{\xi\_2}f) \\ &= \{ \{f, H[\eta\_1] + D[\varepsilon\_1] \}, H[\eta\_2] + D[\varepsilon\_2] \} - \{ \{f, H[\eta\_2] + D[\varepsilon\_2] \}, H[\eta\_1] + D[\varepsilon\_1] \} \\ &= \{ f, \{H[\eta\_1] + D[\varepsilon\_1], H[\eta\_2] + D[\varepsilon\_2] \} \} \end{aligned} \tag{4}$$

are determined by Poisson brackets {*H*[*η*1] + *<sup>D</sup>*[1], *<sup>H</sup>*[*η*2] + *<sup>D</sup>*[2]} of the constraints (using the Jacobi identity). Because the unit normal *n<sup>a</sup>* is normalized by using the space-time metric, including the spatial components *qab* on a slice, the brackets of two canonical gauge transformations [1,2,9] turn out to depend on the metric. In spherically symmetric models, in which the radial part of the metric is determined by a single function, *q* (of density weight 2), we have

$$\left\{ H[\eta\_1] + D[\varepsilon\_1], H[\eta\_2] + D[\varepsilon\_2] \right\} = H[\varepsilon\_1 \eta\_2' - \varepsilon\_2 \eta\_1'] + D[\varepsilon\_1 \varepsilon\_2' - \varepsilon\_2 \varepsilon\_1' + \eta^{-1}(\eta\_1 \eta\_2' - \eta\_2 \eta\_1')].\tag{5}$$

In general, the metric components are spatial functions independent of the components *η* and that label different gauge transformations. Unlike the Lie bracket of two space-time vector fields, the bracket of two pairs *δξi* , *i* = 1, 2, implied by the Poisson bracket (5) does not form a Lie algebra because coefficients determined by spatial fields *qab* or *q* cannot be considered structure constants.
