PHYSICAL REVIEW D 102, 046006 (2020)
No-go result for covariance in models of loop quantum gravity
Martin Bojowald Institute for Gravitation and the Cosmos, The Pennsylvania State University, 104 Davey Lab, University Park, Pennsylvania 16802, USA
(Received 3 July 2020; accepted 30 July 2020; published 10 August 2020)
Based on the observation that the exterior space-times of Schwarzschild-type solutions allow two symmetric slicings, a static spherically symmetric one and a timelike homogeneous one, modifications of gravitational dynamics suggested by symmetry-reduced models of quantum cosmology can be used to derive corresponding modified spherically symmetric equations. Generally covariant theories are much more restricted in spherical symmetry compared with homogeneous slicings, given by 1 þ 1-dimensional dilaton models if they are local. As shown here, modifications used in loop quantum cosmology do not have a corresponding covariant spherically symmetric theory. Models of loop quantum cosmology therefore violate general covariance in the form of slicing independence. Only a generalized form of covariance with a non-Riemannian geometry could consistently describe space-time in models of loop quantum gravity.
DOI: 10.1103/PhysRevD.102.046006
I. INTRODUCTION space-time structure in models of loop quantum gravity [9]. Here, we elaborate on this application and use it to Models of black holes in quantum gravity are valuable demonstrate a no-go result that implies the noncovariance not only because their strong-field effects draw consider- of any model of loop quantum gravity, if covariance is able physical interest, but also because they are understood understood in the classical way related to slicing inde- as a consequence of nontrivial dynamical properties of pendence in Riemannian geometry. space-time. Given the incomplete status of all approaches to quantum gravity, the latter connection would, at present, seem even more important than the former. In this sense, II. SYMMETRIES IN SCHWARZSCHILD black-hole models in quantum gravity have a clear advan- SPACE-TIME tage over models of quantum cosmology because basic The construction utilized in Ref. [1] is an application of cosmological solutions work with simpler space-times minisuperspace results, originally derived for models of characterized by exact or perturbative spatial homogeneity quantum cosmology, to a black-hole context. The most with a preferred background time direction. common example of this form is based on the well-known The connection between black holes and space-time fact that the Schwarzschild solution, structure is particularly relevant in canonical, background- independent approaches, such as loop quantum gravity. In 2m dr2 such approaches, the structure of space-time is a derived ds2 ¼ − 1 − dt2 þ r 1 − 2m=r concept and not presupposed. Explorations in a physically 2 2 2 2 motivated context can therefore provide important insights þ r ðdϑ þ sin ϑdφ Þ; ð1Þ into the viability of any specific proposal. Even if implied quantum-gravity effects in black holes may not be realis- has a homogeneous spatial slicing in the interior, where tically observable, studying them in detail can often r
2470-0010=2020=102(4)=046006(7) 046006-1 © 2020 American Physical Society MARTIN BOJOWALD PHYS. REV. D 102, 046006 (2020) cosmology. Of major interest to Ref. [1] was the possibility We present these equations and our new derivations in a that a bounce in cosmological models, as sometimes claimed form based on metric variables, which are more common in models of loop quantum cosmology, might then be and therefore more easily accessible than the triad variables reinterpreted as a nonsingular transition through the black- used in models of loop quantum gravity. Our general result hole interior. Note, however, that most bounce claims in loop does not depend on the choice of variables because it is quantum cosmology are based on oversimplified models that invariant under canonical transformations. For an explicit do not capture the correct physics near a spacelike singularity derivation in triad variables, see Ref. [9]. [11–13]. Moreover, such models are often in violation of general covariance [14], a conclusion that will be strength- A. Spherical symmetry and interior geometry ened by our derivations below. Specific predictions made in this context, in particular of a quantitative nature as in the We begin with the generic form of line elements subject example of the ratio of masses before and after the bounce, to the symmetries of Kantowski-Sachs models: therefore cannot be considered reliable. Nevertheless, it is justified to assume the modified dynam- ds2 ¼ −NðtÞ2dt2 þ aðtÞ2dx2 þ bðtÞ2ðdϑ2 þ sin2 ϑdφ2Þð2Þ ics implied by a quantum version of the Kantowski-Sachs model as a possible substitute for the dynamics of general with three free functions, N, a and b, depending on relativity in the Schwarzschild interior, and then to evaluate time. Such a line element can be used to describe the potential implications on qualitative features of the resulting spatially homogeneous Schwarzschild interior. Because model. An open question even in such less ambitious studies the line element is also spherically symmetric, it is of has been how to connect the Schwarzschild interior to a the general form possible inhomogeneous exterior geometry. Such a connec- tion has become possible by the useful suggestion of Ref. [1] ds2 ¼ −Nðt; xÞ2dt2 þ Lðt; xÞ2ðdx þ Mðt; xÞdtÞ2 to consider homogeneous timelike slicings in the exterior, þ Sðt; xÞ2ðdϑ2 þ sin2 ϑdφ2Þ; ð3Þ given by constant r in Eq. (1) even if r>RS, and apply modifications proposed in minisuperspace models. Using this method, the authors of Ref. [1] constructed a modified line specialized to r-independent coefficients as well as vanish- element that could possibly describe the exterior geometry of ing shift, M ¼ 0. a quantum-modified, nonsingular Schwarzschild black hole Since we will use spherically symmetric models later (or its Kruskal extension). on, we quote the dynamical equations implied for the In order to do so, Ref. [1] assumed that the modified coefficients of Eq. (3) by a local, generally covariant exterior is subject to the same space-time structure as the theory in which a line element of this form would indeed classical theory, given by Riemannian geometry and correctly describe the symmetries of solutions. In the 1 þ 1- described by a line element such as Eq. (1) but with a dimensional context in which spherically symmetric mod- modified r dependence in its coefficients. However, in els are placed, it is well known that this set of theories is background-independent models of quantum gravity, it is given by dilaton-gravity models [15–17] in which, up to not guaranteed that the structure of space-time as seen in field redefinitions, only a specific set of functions, includ- the classical theory remains intact. Space-time structure ing the dilaton potential VðSÞ, can be varied while all other should rather be derived from the theory, which would then terms in an action or Hamiltonian constraint are fully 2 a b show whether a line element of the form ds ¼ gabdx dx , determined by covariance. The equivalence of the gener- restricted to spherical symmetry in the present context, can alized dilaton models introduced in Ref. [17] with two- indeed describe the modified quantum dynamics. In dimensional Horndeski theories [18], and therefore with the canonical approaches to quantum gravity, such as loop most general two-dimensional local scalar-tensor theory quantum gravity used in Ref. [1], the task is to show that with second-order field equations, has recently been gauge transformations acting on components of gab, demonstrated in Ref. [19]. This general class of theories generated by modified constraints that generate the modi- also includes Palatini-fðRÞ models [20] through their fied dynamics used to obtain any kind of nonsingular equivalence with scalar-tensor theories with a nondynam- homogeneous model, are consistent with standard coor- ical scalar field [21]. dinate transformations of dxa assumed in the definition of a The action of any such theory can be written as line element. We will show that this is not the case for modifications suggested by loop quantum cosmology. 1 S½g; ϕ¼ d2x − det gðξðϕÞR þ kðϕ;XÞ 16πG Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi III. DYNAMICAL MODELS a b þ Cðϕ;XÞ∇ ϕ∇ ϕ∇a∇bϕÞð4Þ For our demonstration, we will need the relevant equations that describe the classical and modified dynamics with three free functions, ξðϕÞ, kðϕ;XÞ and Cðϕ;XÞ of the of the slicings involved in the construction of Ref. [1]. scalar field ϕ and
046006-2 NO-GO RESULT FOR COVARIANCE IN MODELS OF LOOP … PHYS. REV. D 102, 046006 (2020)
1 1 ∂ 2 1 ∂2 1 ∂ ∂ ab∇ ∇ pS pSpL LpL S N S X ¼ − g aϕ bϕ: ð5Þ ¼ − þ − − 2 N ∂t S2 S3 L ∂x2 LN ∂x ∂x 1 ∂L ∂ðNSÞ S ∂2N þ − As a parametrization of the most general second-order L2N ∂x ∂x LN ∂x2 theories in two dimensions, the three functions ξðϕÞ, L dðSVðSÞÞ ∂ðMp Þ kðϕ;XÞ and Cðϕ;XÞ are not independent if field redefini- − þ S ð10Þ 4 dS ∂x tions of ϕ and gab are allowed. For instance, ξðϕÞ can be mapped to one by a suitable ϕ-dependent conformal and transformation of gab, adjusting also kðϕ;XÞ. This ambi- guity will not concern us here. It is only important that for 1 ∂p p2 S ∂S ∂N 1 ∂S 2 any local generally covariant theory for a two-dimensional L ¼ − L − − N ∂t 2S2 NL2 ∂x ∂x 2L2 ∂x metric and a scalar field with second-order field equations 1 ∂p there is a choice of ξðϕÞ, kðϕ;XÞ and Cðϕ;XÞ such that the − SVðSÞþM L ð11Þ action is of the form (4). 4 ∂x The canonical formulation of Eq. (4) in this general form ∂ ∂ ∂ ∂ (and without fixing the gauge) is rather involved because it while equations for S= t and L= t follow from the requires inversions of some of the free functions or their Eq. (6) for the momenta. For spherically symmetric general 2 derivatives. Since the models under consideration here are relativity, the dilaton potential is given by VðSÞ¼− =S. canonical, we will therefore begin with a restricted class of Since Kantowski-Sachs models are spherically symmet- spherically symmetric theories in which, compared with ric, we can derive the momenta, constraints, and equations Eq. (4),wehaveξ ¼ 1, C ¼ 0 and k linear in X. That is, we of motion of Eq. (2) by specializing the equations of 0 will first consider minimally coupled scalar-tensor theories spherical symmetry, also using M ¼ and x independence. with quadratic kinetic terms. In a second step, we will then We obtain the momenta show that our result does not depend on field redefinitions b ∂b 1 ∂ðabÞ that change ξðϕÞ, or on an introduction of nontrivial k pa ¼ − ;pb ¼ − ; ð12Þ and C. N ∂t N ∂t The most general covariant theory under these condi- with the Hamiltonian constraint tions can be derived directly at the canonical level; see – Refs. [22 24] for explicit derivations. This dynamics tells p p ap2 a us that, up to canonical transformations, the momenta H N N − a b a − : 13 hom½ ¼ þ 2 2 2 ð Þ canonically conjugate to S and L, respectively, are given by b b It implies the equations of motion 1 ∂ðSLÞ ∂ðMSLÞ pS ¼ − − ; ð6Þ N ∂t ∂x da ∂Hhom½N pb apa ¼ ¼ N − þ 2 ; ð14Þ dt ∂pa b b S ∂S ∂S ∂ p ¼ − − M : ð7Þ db Hhom½N pa L N ∂t ∂x ¼ ¼ −N ; ð15Þ dt ∂pb b
2 The Hamiltonian constraint dp ∂H ½N 1 p a ¼ − hom ¼ N 1 − a ; ð16Þ dt ∂a 2 b2 2 ∂2 ∂ ∂ pSpL LpL S S S S L 2 Hsph½N¼ dxN − þ 2 þ 2 − 2 dp ∂H ½N p p ap Z S 2S L ∂x L ∂x ∂x b ¼ − hom ¼ −N a b − a : ð17Þ dt ∂b b2 b3 1 ∂S 2 1 þ þ LSVðSÞ ð8Þ 2L ∂x 4 B. Timelike homogeneity In the Schwarzschild exterior, r>R , slices of constant and diffeomorphism constraint S r are still homogeneous but timelike. The resulting canoni- cal relationships can be derived from the Kantowski-Sachs ∂ ∂ S pL equations by a complex canonical transformation from a, Dsph½M¼ dxM pS − L ð9Þ Z ∂x ∂x pa and N to then generate the equations of motion A ¼ ia; pA ¼ −ipa;n¼ iN ð18Þ
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∂ while b and pb remain unchanged. The line element (2) b db h pA ¼ − þ δb ; ð23Þ then takes the form n dt ∂pb 1 ∂ ∂ ds2 ¼ nðtÞ2dt2 −AðtÞ2dx2 þ bðtÞ2ðdϑ2 þ sin2 ϑdφ2Þð19Þ dA db h h pb ¼ − b þ A þ δ A þ b ð24Þ n dt dt ∂pb ∂pA t in which slices of constant are timelike. This line element of the previous equations. We have not explicitly replaced represents the symmetry of the exterior Schwarzschild the appearance of p and p in δ terms on the right-hand x time A b solution, where would be the Schwarzschild coor- sides. To first order in δ, these appearances merely radial dinate and t the Schwarzschild coordinate. represent the classical form of the momenta. We will derive the dynamics of Eq. (19) in a generalized In terms of time derivatives, the modified Hamiltonian form that takes into account possible modifications from therefore equals loop quantum cosmology, applied to this homogeneous model. These modifications are subject to a large number b dA db 1 1 db 2 of quantization ambiguities. Our result, however, will H ½n¼−n þ A − 1 timelike n2 dt dt 2 N2 dt be insensitive to ambiguities because it holds for any Hamiltonian constraint ∂h ∂h þ nδ −pA − pb þ h : ð25Þ ∂pA ∂pb p p Ap2 A H n n − A b A δh A; b; p ;p It is modified by a δ term if and only if h is nonlinear in timelike½ ¼ þ 2 2 þ 2 þ ð A bÞ b b momenta which, to repeat, is always the case in models of ð20Þ loop quantum gravity. Our main result will depend only on this general feature. with a nonlinear function hðA; b; pA;pbÞ of the canonical variables, multiplied by a parameter δ that vanishes in the C. Testing slicing independence classical limit. (In an explicit version, both δ and h would If space-time is of classical Riemannian form, the be obtained from so-called holonomy modifications of loop condition of homogeneity in a timelike direction is quantum cosmology, which always imply a nonlinear and equivalent to the existence of a static solution in a spacelike 0 even nonpolynomial h.) For δ ¼ , the classical terms in slicing. The direction in which the timelike slicing Eq. (20) are derived by applying the complex canonical “evolves” then corresponds to a direction of inhomogeneity transformation (18) to Eq. (13). in the spacelike slicing. In the present context, both slicings While A and b are still defined geometrically by their share rotational symmetry implied by the angle-dependent appearance in the line element (19), the new term δh in spherical line element. For this statement, we do not need Eq. (20) modifies the relationship between momenta and complete slices, and our derivations will therefore apply to time derivatives of A and b. The previous equations (14) local properties of space-time. They are insensitive to any and (15) are replaced by renormalization procedures that may have to be applied to parameters in h or δ in an effective theory if fields are ∂ ∂ dA Htimelike½n pb ApA h evolved over a wide range of scales. ¼ ¼ n − þ 2 þ δ ; ð21Þ dt ∂pA b b ∂pA The modification of Eq. (25) by δ terms does not change the symmetric nature in the geometrical interpretation of the solution as the dynamics of a timelike slicing of db ∂H ½n p ∂h timelike − A δ space-time. If it belongs to a generally covariant, slicing- ¼ ∂ ¼ n þ ∂ : ð22Þ dt pb b pb independent theory, it must therefore allow an equivalent description as a static solution in a spherically symmetric Deriving the momenta requires an inversion of these spacelike slicing. The Hamiltonian (25) is based on a equations, which is now nontrivial unless h is a low-order canonical formulation with the same phase space as the polynomial in pA and pb. For our purposes, however, it is classical theory; therefore, it is a homogeneous model of a sufficient to invert these equations perturbatively in δ. Since local gravitational theory in metric variables. Nonlocality our aim is to show that no modifications of this form are would imply additional degrees of freedom through aux- compatible with slicing independence, and since an effec- iliary fields that describe nonlocal terms, or higher time tive theory parametrized by some δ is covariant if and only derivatives in a derivative expansion, but no such fields are if it is covariant order by order in δ, a perturbative treatment implied by holonomy modifications in homogeneous to leading order in δ suffices to show that the theory violates models. Therefore, the corresponding spherically symmet- covariance. To first order in δ, we then have the simple ric theory should be local if covariance is realized. Here, it inversion is important that we are not just looking for an embedding
046006-4 NO-GO RESULT FOR COVARIANCE IN MODELS OF LOOP … PHYS. REV. D 102, 046006 (2020) of a single solution or a class of solutions in a covariant An explicit transformation of Htimelike½n¼0 to a spheri- theory, but rather have to make sure that the complete cally symmetric model, using Eq. (28) together with a canonical description, including the phase-space substitution of t derivatives by X derivatives, shows which structure, can be realized in a generally covariant theory. spherically symmetric equations should be referred to. Similarly, the number of phase-space degrees of freedom Transforming Htimelike½n, we obtain the expression in holonomy-modified homogeneous models, with a single momentum per spatial metric or triad component, S dK dS 1 1 dS 2 H ½L¼− − KL − 1 implies that we are looking for a theory with second-order timelike L dX dX 2 L2 dX field equations. No higher-derivative terms are therefore ∂h ∂h δ − − allowed, even if they are local. þ L pA ∂ pb ∂ þ h ð29Þ Since all local, generally covariant theories with one pA pb inhomogeneous spatial dimension and second-order field where equations are, up to field redefinitions, given by 1 þ 1- dimensional dilaton-gravity models of the form (4), where S dS the scalar ϕ is now given by the function S which does p ¼ − þ OðδÞ; ð30Þ A L dX transform like a scalar under two-dimensional coordinate transformations of t and x, there must be functions ξðSÞ, 1 dK dS p ¼ − S þ K þ OðδÞð31Þ kðS; XÞ and CðS; XÞ such that all solutions of the dynamics b L dX dX generated by Eq. (25) can be mapped to solutions of this generalized dilaton-gravity theory. This condition allows us are implied by Eqs. (23) and (24). [We do not need to replace to test whether models of loop quantum cosmology with these expressions explicitly in the δ term of Eq. (29), allowing nonlinear δ terms in Eq. (25) can be consistent with slicing us to work with a more compact constraint.] independence in a generally covariant theory. As already In the form (29), the constraint of the timelike slicing mentioned, we will first evaluate this condition in the clearly cannot directly correspond to the spherically sym- restricted setting in which a single dilaton potential VðSÞ in metric Hamiltonian constraint because it depends on the Eq. (8) characterizes a given model. lapse function K of the spherically symmetric slicing not In order to determine a possible mapping that could just through K itself but also through its derivative, dK=dX. relate the two slicings, we compare the homogeneous line An additional condition is therefore required if Htimelike½L element (19) on timelike slices with a static spherically is to vanish for all solutions in a static spherically symmetric one, symmetric model. It turns out that staticity can be used 2 2 2 2 2 2 2 2 2 to eliminate the derivative dK=dX in favor of K itself and ds ¼ −KðXÞ dT þ LðXÞ dX þ SðXÞ ðdϑ þ sin ϑdφ Þ: the remaining fields, L and S and their derivatives. In 0 0 ð26Þ particular, evaluating Eq. (11) with pL ¼ and M ¼ , implied by staticity, (as well as N ¼ K) leads to the Staticity restricts the dependence of the metric components differential equation to X, while it implies zero shift vector. This comparison S dS dK 1 dS 2 1 uniquely determines the candidate mapping 0 ¼ − − − SVðSÞ: ð32Þ KL2 dX dX 2L2 dX 4 X ¼ t; T ¼ x ð27Þ [We need only one further condition, and therefore will not for coordinates, combined with use a second independent equation (10) in the present context. This equation is more complicated but would give A ¼ K; b ¼ S; n ¼ L ð28Þ equivalent results.] Solving this equation algebraically for dK=dX, we can therefore eliminate this derivative from for metric components. Eq. (29), such that Solutions in the homogeneous slicing must be such that the Hamiltonian constraint H ½n¼0 is satisfied. In a 1 1 timelike H ½L¼ KL 1 þ SVðSÞ þ OðδÞ: ð33Þ covariant theory, the same equation must hold true after timelike 2 2 applying the mapping (28), but it need not correspond to the spherically symmetric Hamiltonian constraint. (In fact, Disregarding δ terms, this expression vanishes in a it does not, as we will see soon.) For covariance, it would be spherically symmetric model provided the dilaton potential sufficient if it were a combination of all the equations indeed belongs to classical spherically symmetric gravity, available in the spherically symmetric slicing, including the VðSÞ¼−2=S. Classically, therefore, the Hamiltonian con- staticity condition in addition to the general spherically straint equation in the timelike homogeneous slicing symmetric constraints and equations of motion. amounts to a combination of the staticity condition and
046006-5 MARTIN BOJOWALD PHYS. REV. D 102, 046006 (2020) a condition on the dilaton potential in the spherically L through the Christoffel symbol required in the C term of symmetric slicing. It is independent of the spherically Eq. (4). Again, even with the freedom of choosing k or C it symmetric Hamiltonian constraint, which rather can be is impossible to absorb the dependence on K or nonlocally seen to correspond to one of the equations of motion in the on L that results from a δ term. timelike homogeneous slicing. (Recall that spatial deriva- Our no-go theorem is therefore complete: there is no tives in the spherically symmetric slicing correspond to generally covariant spherically symmetric theory that normal derivatives in the timelike homogeneous slicing.) could have a solution space corresponding to the modified The equivalence is no longer realized if we include δ timelike homogeneous model. Slicing independence is terms of the general form as shown in Eq. (25) with therefore violated by holonomy modifications in symmetry- nonlinear h. In particular, these terms depend on pb which, reduced models of loop quantum gravity. following Eq. (31) depends on the lapse function K of a spherically symmetric slicing. If h is nonlinear in pb, the δ IV. CONCLUSION terms are nonlinear in p , or nonlinear in K after the b We have shown that the model proposed in Ref. [1] transformation to spherically symmetric variables. After violates general covariance and therefore fails to describe factoring out a single factor of KL, as in Eq. (33), the space-time or black holes. This result has implications remaining terms H ½L=ðKLÞ therefore still depend on timelike even if one is not interested in black holes but only in K. If the dynamics in this slicing corresponds to a dilaton cosmological applications of models of loop quantum model with Hamiltonian (8), H L = KL can depend timelike½ ð Þ gravity. If such models are sufficiently general, it must only on S because the dilaton potential is restricted by the be possible to apply any proposed modification to models covariance condition to have only such a dependence. with the symmetries of Kantowski-Sachs space-times, H δ However, if timelike½L=ðKLÞ depends on K when terms including those with a timelike homogeneous slicing. If are included, it cannot just depend on S: while the staticity they belong to a generally covariant theory, it must then be condition (32) can be used to solve for K in terms of S and L,it possible to find a consistent mapping to a static spherically requires solving the differential equation; an algebraic sol- symmetric slicing. Our results show that this is never the ution for dK=dX as in the classical case is not sufficient. A case for holonomy modifications proposed in models of solution K of Eq. (32) depends on S and L nonlocally because loop quantum cosmology. This is our no-go result about integrations are required. No nonlocal S dependence, and no general covariance in this setting. (Our result is consistent dependence on L at all, can be absorbed in a local dilaton with the observation that all proposed analog actions that potential VðSÞ. Therefore, there is no local generally covar- could describe holonomy modifications by higher-curva- iant theory of the restricted form considered so far, that could ture terms in isotropic models [25,26], based on mimetic describe a spherically symmetric slicing corresponding to the gravity [27,28], fail to describe related effects in anisotropic timelike homogeneous one which, by construction, is also models [29] or for perturbative inhomogeneity [30].) local. Any δ term with nonlinear h therefore violates slicing Such a general violation of covariance might be inter- independence and general covariance. preted as ruling out not only a specific model but also the So far, we have shown that there is no minimally coupled entire approach, based on loop quantization. Luckily, generally covariant theory quadratic in momenta which however, previous research had already independently could correspond to the modified dynamics of a timelike shown a possible way out of this damning conclusion. It homogeneous slicing. It is not difficult to see that non- is possible to evade our no-go result if one takes into 1 minimal coupling, leading to ξ ≠ in Eq. (4), does not account the possibility of generalized space-time structures change the result. Such a theory can always be obtained that may be considered covariant in the sense that the same from a minimally coupled one by a field redefinition, using number of gauge transformations is realized as in the an S-dependent conformal transformation of the two- classical theory, but in a way that no longer corresponds to dimensional metric. Such a transformation, formulated slicing independence in Riemannian geometry [22,31].In canonically, would rescale K and L by S-dependent fact, the constructions of Ref. [1] implicitly assumed that functions, such that there would be new terms in the space-time, even after modifying dynamical equations equations of motion with spatial derivatives of S,butno originally derived from general relativity, retains its new derivative terms of K or L. It is impossible for such Riemannian structure and can be described by line ele- terms to absorb a K dependence in a δ term as mentioned in ments. Our no-go result rules out this implicit assumption, the preceding paragraph, or a term nonlocal in L or with an but it may be evaded if the assumption is relaxed. entire derivative expansion of L if a solution of Eq. (32) for It is difficult to describe non-Riemannian space-time K is used. Our no-go result therefore extends to non- structures in general terms because most of our intuition in minimally coupled scalars. Similarly, allowing for terms general relativity is built on Riemannian properties. with nonlinear k or nonzero C in Eq. (4) leads, in the static Nevertheless, in canonical theories, there are systematic case which is relevant here, only to terms with additional methods that allow one to test whether gauge symmetries spatial derivatives of S or at most first-order derivatives of are respected by modifications or quantization, even while
046006-6 NO-GO RESULT FOR COVARIANCE IN MODELS OF LOOP … PHYS. REV. D 102, 046006 (2020) algebraic relations of the symmetries may be subject to remove the correct number of spurious degrees of freedom. modifications themselves. General covariance in canonical Line elements might then be applicable in certain regions of gravity is expressed as the condition of anomaly-freedom space-time, after a field redefinition of metric components of the constraints that generate hypersurface deformations that in certain cases can be derived from the consistent in space-time, given by the Hamiltonian and diffeomor- generators of modified hypersurface deformations [32,33]. phism constraints. If modified constraints still have closed These constructions are now being investigated [34–36], Poisson brackets, the theory is anomaly-free and enjoys but generalized covariance must be better understood the same number of gauge transformations as classical before it is possible to derive complete and reliable models general relativity, with hypersurface deformations realized of black holes in loop quantum gravity. in the classical limit. If this is the case, the theory may be considered covariant but in a generalized way. The modi- ACKNOWLEDGMENTS fied constraints then may no longer generate hypersurface deformations in space-time, but they do provide a well- This work was supported in part by NSF Grant No. PHY- defined set of gauge transformations that allow one to 1912168.
[1] A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. Lett. 121, [21] G. J. Olmo, Phys. Rev. Lett. 95, 261102 (2005). 241301 (2018). [22] M. Bojowald and G. M. Paily, Phys. Rev. D 86, 104018 [2] N. Bodendorfer, F. M. Mele, and J. Münch, Classical (2012). Quantum Gravity 36, 187001 (2019). [23] M. Bojowald and S. Brahma, Phys. Rev. D 95, 124014 [3] N. Bodendorfer, F. M. Mele, and J. Münch, Classical (2017). Quantum Gravity 36, 195015 (2019). [24] M. Bojowald, Foundations of Quantum Cosmology (Institute [4] N. Bodendorfer, F. M. Mele, and J. Münch, arXiv: of Physics, London, UK, 2020). 1911.12646. [25] N. Bodendorfer, A. Schäfer, and J. Schliemann, Phys. Rev. [5] N. Bodendorfer, F. M. Mele, and J. Münch, arXiv: D 97, 084057 (2018). 1912.00774. [26] D. Langlois, H. Liu, K. Noui, and E. Wilson-Ewing, [6] M. Bouhmadi-López et al., J. Cosmol. Astropart. Phys. 07 Classical Quantum Gravity 34, 225004 (2017). (2020) 066. [27] A. H. Chamseddine and V. Mukhanov, J. High Energy Phys. [7] M. Bouhmadi-López et al., Phys. Dark Universe 30, 100701 11 (2013) 135. (2020). [28] L. Sebastiani, S. Vagnozzi, and R. Myrzakulov, Adv. High [8] V. Faraoni and A. Giusti, Symmetry 12, 1264 (2020). Energy Phys. 2017, 3156915 (2017). [9] M. Bojowald, arXiv:1906.04650. [29] N. Bodendorfer, F. M. Mele, and J. Münch, Classical [10] R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 Quantum Gravity 35, 225001 (2018). (1966). [30] J. Haro, L. Arest´eSaló, and S. Pan, Gen. Relativ. Gravit. 51, [11] M. Bojowald, J. Cosmol. Astropart. Phys. 01 (2019) 026. 49 (2019). [12] M. Bojowald, J. Cosmol. Astropart. Phys. 07 (2020) 029. [31] M. Bojowald, S. Brahma, and J. D. Reyes, Phys. Rev. D 92, [13] M. Bojowald, Universe 6, 36 (2020). 045043 (2015). [14] M. Bojowald, Phys. Rev. D 102, 023532 (2020). [32] M. Bojowald, S. Brahma, U. Büyükçam, and F. D’Ambrosio, [15] T. Strobl, arXiv:hep-th/0011240. Phys. Rev. D 94, 104032 (2016). [16] D. Grumiller, W. Kummer, and D. V. Vassilevich, Phys. [33] M. Bojowald, S. Brahma, and D.-H. Yeom, Phys. Rev. D 98, Rep. 369, 327 (2002). 046015 (2018). [17] G. Kunstatter, H. Maeda, and T. Taves, Classical Quantum [34] J. Ben Achour, F. Lamy, H. Liu, and K. Noui, Eur. Phys. L. Gravity 33, 105005 (2016). 123, 20006 (2018). [18] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974). [35] J. Ben Achour, F. Lamy, H. Liu, and K. Noui, J. Cosmol. [19] K. Takahashi and T. Kobayashi, Classical Quantum Gravity Astropart. Phys. 05 (2018) 072. 36, 095003 (2019). [36] D. Aruga, J. Ben Achour, and K. Noui, Universe 6,39 [20] H. A. Buchdahl, Mon. Not. R. Astron. Soc. 150, 1 (1970). (2020).
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