Non-covariance of "covariant polymerization" in models of loop quantum gravity
aa r X i v : . [ g r- q c ] F e b Non-covariance of “covariant polymerization”in models of loop quantum gravity
Martin Bojowald ∗ Institute for Gravitation and the Cosmos,The Pennsylvania State University,104 Davey Lab, University Park, PA 16802, USA
Abstract
A new modification of spherically symmetric models inspired by loop quantumgravity has recently been introduced by Ben´ıtez, Gambini and Pullin, who claimedthat it preserves general covariance. This claim is shown here to be incorrect, basedon methods of effective line elements. The same methods imply that the only novelphysical effect introduced by the modification is the presence of time-reversal hyper-surfaces between classical space-time regions.
Models of loop quantum gravity attempt to implement quantum-geometry effects by usingcertain modifications of the classical equations of canonical gravity. The canonical nature,as usual, implies that general covariance is not manifest and must be tested by dedicatedmeans. Several no-go results for general covariance and slicing independence in such modelshave recently been derived, using setups relevant for cosmology [1] and black-holes [2, 3].The only known way to realize covariance in models of loop quantum gravity is througha deformed version [4, 5] that implies signature change at high density or curvature whenapplied to modifications commonly used in loop quantum cosmology or loop quantumblack holes [4, 6, 7, 8, 9, 10, 11, 12]. (Signature change may be avoided in some cases, butit would require non-standard modifications such as complex connections [13, 14, 15, 16],Euclidean-type gravity [17, 18] or non-bouncing background solutions [19].)It is therefore important to explore possible alternative modifications. In this context,[20] suggest to apply a non-bijective canonical transformation to the classical theory, hopingthat the modified model will be close enough to the classical system to preserve covariance,yet different enough to be considered a modification because the transformation is notbijective. As we will show in this paper, covariance is a subtle issue even in this case andmust be derived. Once this task has been completed, it can be seen that the modificationsare not compatible with general covariance or slicing independence in a global space-timestructure. The equations suggested in [20] therefore do not show how models of loopquantum gravity could be made consistent with general covariance, and they do not providecounter-examples to the no-go results of [1, 2]. ∗ e-mail address: [email protected] We present a detailed space-time analysis of the model introduced in [20]. Since the modelis canonical, we use methods of canonical gravity; see [22, 23] for details.
Canonical gravity of spherically symmetric models is described by line elements of the form[24] d s = − N d t + q xx (d x + M d t ) + q ϕϕ (d ϑ + sin ϑ d ϕ ) . (1)The spatial part is determined by two functions, q xx and q ϕϕ , depending on the radialposition x as well as time t , while the lapse function N and shift vector M , also dependingon x and t , describe its extension to space-time. In spherically symmetric models of loopquantum gravity [25, 26], one usually replaces metric components with components E x and E ϕ of a densitized triad, such that q xx = ( E ϕ ) | E x | , q ϕϕ = | E x | . (2)In what follows it will be sufficient to assume E x >
0, fixing the orientation of space.The triad components are, up to constant factors, canonically conjugate to componentsof extrinsic curvature, K x and K ϕ , such that { K x ( x ) , E x ( x ) } = 2 Gδ ( x , x ) , { K ϕ ( x ) , E ϕ ( x ) } = Gδ ( x , x ) (3)with Newton’s constant G . Extrinsic curvature depends on time and space derivatives ofthe densitized triad (as well as lapse and shift) in a way that may be modified in models ofloop quantum gravity. We will not need the precise relationships but only use the canonicalstructure.Depending on the time gauge, equations of motion for the basic phase-space variablesare generated by combinations of the Hamiltonian constraint, H [ N ], and the diffeomor-phism constraint, D [ M ]. We will not need the precise form of these expressions eitherbut only refer to their nature as gauge generators of deformations of spatial hypersurfacesin space-time. These transformations correspond to classical space-time [27] provided theconstraints obey Dirac’s hypersurface-deformation brackets [28], in particular { H [ N ] , H [ N ] } = − D [ E x ( E ϕ ) − ( N N ′ − N ′ N )] . (4)2he presence of a phase-space dependent structure function implies that the structure ofspace-time is sensitive to modifications of the constraints.As shown in [29], the structure function can be eliminated in an equivalent constrainedsystem obtained by suitable combinations of H and D . This construction has also beenused in the recent analysis of [20]. However, based on [27], the behavior of hypersur-face deformations and therefore of general covariance and slicing independence requires abracket of the form (4) for the generators of normal deformations of spatial hypersurfaces.Discussions of covariance therefore cannot avoid referring to this relationship, especially inattempted modifications.The main ingredient in models of loop quantum gravity is a substitution of (almost) pe-riodic functions of connection or extrinsic-curvature components for the classical quadraticdependence in the Hamiltonian constraint. If this substitution is done only in these places,and in a careful way relating different substitution functions to one another, the bracket(4) in vacuum is modified by a new factor of the structure function such that the struc-ture of space-time is non-classical [30, 31, 32]. (See [33, 34] for an analogous result in thecosmological context.) In the presence of a scalar field, no such substitution is known thatpreserves the form of (4) even if one accepts modifications of the structure function [35].The authors of [20] suggest that this difficulty may be overcome if one uses a canonicaltransformation instead of substitution. For the gravitational variables, they propose totransform from the pair ( K ϕ , E ϕ ) to a new pair ( ˜ K ϕ , ˜ E ϕ ) such that K ϕ = sin( δ ˜ K ϕ ) δ , E ϕ = ˜ E ϕ cos( δ ˜ K ϕ ) . (5)The pair ( K x , E x ) remains unchanged. There is a similar transformation for a scalarmatter field, which we do not use explicitly here because (5) is sufficient for a discussionof space-time structure: The scalar field does not appear in the structure function of (4).Expressed in terms of the new variables, the Hamiltonian constraint depends on ˜ K ϕ through a periodic function, as in standard modifications, while the dependence of E ϕ on˜ K ϕ leads to new modifications in metric functions not considered before. The hope is thatthese new modifications may preserve general covariance because the model is obtainedby a canonical transformation from a covariant theory. At the same time, only a boundedrange of K ϕ is realized for an infinite range of ˜ K ϕ , which could introduce new physicaleffects and help with the resolution of singularities. The modified theory has Hamiltonian constraints such that { H [ N ] , H [ N ] } = − D [cos ( δ ˜ K ϕ ) E x ( ˜ E ϕ ) − ( N N ′ − N ′ N )] . (6)with a modified structure function, obtained by simply applying the canonical transfor-mation to (4). Since the modification introduces new zeros of the structure function at δ ˜ K ϕ = (2 n + 1) π with integer n , it eliminates some contributions of the diffeomorphism3onstraint from the right-hand side. The presence of structure functions implies that gen-erators of hypersurface deformations form a Lie algebroid [36, 37, 38] over phase space, la-beling independent contributions from the constraints. New zeros in the structure functionintroduced by the transformation mean that the algebroid gains new Abelian subalgebroidsby restriction to the zero-level sets of the structure function. The algebraic structure istherefore inequivalent to its classical form. (The authors of [20] claim that the modification“preserves the constraint algebra,” which presumably refers to a partial Abelianization ofthe generators as in [29]. However, as shown in [35], such a reformulation of the constraintsis not sufficient for a discussion of general covariance.)An inequivalent algebraic structure of hypersurface deformations implies that covari-ance is non-trivial in the modified system. As the authors of [20] point out, the canonicaltransformation employed to obtain the modification is not bijective. This property is thereason why there are additional zeros in the modified structure function of hypersurface-deformation brackets. According to [20], the non-bijective nature of the transformationmight provide a chance for the modified theory to describe new physical effects, but it isalso the reason why covariance is no longer obvious even though the modification has beenobtained by canonically transforming a covariant theory. The claim “It has the advantagethat it is a canonical transformation from the original variables. That means that it pre-serves the constraint algebra and the covariance of the theory, which previous choices didnot.” of [20] is therefore incorrect. In the presence of modified hypersurface deformationswith an inequivalent algebraic structure, covariance has to be derived by a careful analysisof generic solutions and their geometrical meaning. Local solutions for ˜ E ϕ and ˜ K ϕ can be derived without explicitly solving modified equationsof motion because they can simply be obtained by applying a local (in phase space) inverseof the canonical transformation (5) to a classical solution in canonical form. Starting atsmall δK ϕ for the classical solution, any modified local solution ˜ K ϕ remains valid until δK ϕ reaches the values ±
1, the local maxima of sin( δ ˜ K ϕ ) where the canonical transformationis no longer invertible.If one were to solve modified equations directly for ( ˜ K ϕ , ˜ E ϕ ), starting with some initialvalues, it would be possible to cross regions where δ ˜ K ϕ = ± π , again corresponding to thefirst local maxima of sin( δ ˜ K ϕ ) close to small δK ϕ . Such an extension of the local solutionis no longer a simple local inverse of the canonical transformation, and presumably givesrise to “novel phenomena” that are, according to [20], introduced by the modification.However, a solution in the range where δ ˜ K ϕ > π (the case of δ ˜ K ϕ < − π beinganalogous) and δ ˜ K ϕ < π , can again be interpreted as a local inverse of (5), but onethat makes use of a different branch of the arcsine compared with the initial region at | δ ˜ K ϕ | < π . The canonical transformation therefore provides a classical analog in anyrange of δ ˜ K ϕ that excludes the values (2 n + 1) π with integer n . While the analogous K ϕ is always bounded thanks to (5), there is no upper limit on δ ˜ K ϕ beyond which classical4nalogs would no longer exist.We have obtained a direct correspondence between local solutions in the classical andmodified theories. The next question we have to address is whether physics or geometryin the modified theory should be based on the field ˜ K ϕ and its conjugate ˜ E ϕ , or on theirlocal classical analogs K ϕ and E ϕ . This question is relevant for the application presentedin [20], in which critical collapse is studied numerically by evaluating a “black hole mass.”Unfortunately, [20] does not specify how this mass is obtained, but presumably it refersto a mass parameter extracted in the usual way from a line element, constructed from ˜ E ϕ rather than E ϕ in the modified theory. We therefore have to analyze how a meaningfulline element can be constructed in the modified theory. A meaningful line element requiresspecific transformation properties to hold for its coefficients. Using local inverses of the canonical transformation, we have obtained local solutions incanonical form, resulting in evolutions of ˜ K ϕ and ˜ E ϕ depending on some time coordinateimplicitly determined by lapse and shift. Such a solution of equations of motion in amodified theory is not necessarily geometrical, that is, one cannot simply assume thatinserting ˜ E ϕ instead of E ϕ in (2) results in a well-defined space-time line element of the form(1) with the same lapse N and shift M as used in the relevant equations of motion. Anyline element, by definition, has to be invariant with respect to a combination of coordinatetransformations of d x α and gauge transformations of the canonical metric components.While d x and d t still transform like standard coordinate differentials after applying acanonical transformation such as (5), the new field ˜ E ϕ does not have the same (gauge)transformation behavior as the classical E ϕ because K ϕ in (5) is not a space-time scalar.Therefore, using a modified ˜ E ϕ in q xx for (1) implies that modified metric componentsno longer transform in a way dual to coordinate differentials, and the line element is notinvariant. Geometrical derivations from such an expression are meaningless because theydepend on coordinate choices. (One could try to modify the transformations of d x andd t to compensate for the modified gauge transformations of ˜ E ϕ , for instance by usingnon-classical manifolds. However, no such manifold structure is known for the specificmodifications discussed here. For the example of non-commutative manifolds from theperspective of hypersurface deformations, see [39].)As shown in [21], it is sometimes possible to apply a field redefinition to canonical fieldsin a modified theory so as to bring their gauge transformations to a form required for aninvariant effective line element. In the present case, one can use methods introduced in[40] to find a suitable field redefinition of ˜ E ϕ . Not surprisingly, this field redefinition issimply an application of the canonical transformation (5), mapping ˜ E ϕ back to E ϕ whichclearly has the correct transformation behavior for a well-defined line element.Methods of effective line elements therefore show that physics and geometry in themodified theory should be based on the classical analogs found in the previous subsection,and not on the modified solutions ˜ K ϕ and ˜ E ϕ . In any region in which (5) is locallyinvertible, the modified theory simply describes a transformed version of classical gravity.5ny potential for new physical effects is restricted to subsets of measure zero in phase spaceand (generically) space-time. In order to understand their meaning, we have to determinehow different regions of classical analogs may be connected in an effective space-time pictureof global form. So far, we have obtained formal piecewise solutions for the canonical fields ˜ K ϕ and ˜ E ϕ aswell as effective line elements that faithfully describe their geometrical meaning, based onfield redefinitions. The final question is how these piecewise solutions can be glued backtogether to obtain a global space-time picture. Such a gluing cannot be based on classicalmatching conditions because they would simply lead to a global classical solution that doesnot respect the boundedness of K ϕ implied by (5).Given a solution for ˜ K ϕ and ˜ E ϕ , a classical analog and an effective line element isobtained by applying the canonical transformation (5). Since the transformation is notbijective, different ranges of ˜ K ϕ may correspond to the same classical geometry. If we firstrestrict ourselves to ranges of ˜ K ϕ in which the transformation is invertible, the correspond-ing phase-space region corresponds, via the effective line element, to a region in space-timewhich generically is incomplete because it is cut off at fixed values of K ϕ . A global solutiontherefore requires an extension through the hypersurfaces on which δ ˜ K ϕ = (2 n + 1) π withinteger n .It is easy to see how different regions are connected if we first focus on two neighbors,such as the low-curvature region, called region I where | δ ˜ K ϕ | < π , and a region II where π < δ ˜ K ϕ < π . For a transition from region I to region II to happen, ˙˜ K ϕ > δ ˜ K ϕ = π , which by continuity extends to a region around the transition hypersurface.Since K ϕ is a continuous function of ˜ K ϕ , it approaches the same value at the transitionhypersurface from both regions, given by δK ϕ = 1. Applying (5), we see that the cor-responding analog solutions K ϕ behave like time reversed versions in a neighborhood ofthe transition hypersurface: ˙ K ϕ = δ cos( δ ˜ K ϕ ) ˙˜ K ϕ has opposite signs on the two sides ofthe transition hypersurface because cos( δ ˜ K ϕ ) has opposite signs in the two regions while˙˜ K ϕ > E ϕ has opposite signs on the two sides and, unlike K ϕ , is notcontinuous because it goes through infinity if ˜ E ϕ remains finite. (The classical equationsof motion imply that K ϕ is proportional to ˙ E x rather than ˙ E ϕ , such that it may remainregular while E ϕ grows without bounds.) Therefore, the time derivative of the absolutevalue | E ϕ | , which is relevant for q xx in (2), has opposite signs on the two sides: The secondterm in | E ϕ | • = sgn( E ϕ ) ˙˜ E ϕ cos( δ ˜ K ϕ ) + δ ˜ E ϕ cos ( δ ˜ K ϕ ) sin( δ ˜ K ϕ ) ˙˜ K ϕ ! ∼ δ sgn( E ϕ )cos ( δ ˜ K ϕ ) ˜ E ϕ K ϕ ˙˜ K ϕ (7)is dominant near the hypersurface and has opposite signs on the two sides. The geometryin region II can therefore be interpreted as a time-reversed classical solution compared with6he time direction in region I. (It is not necessarily a time reversal of the same solution asin region I because E ϕ is not continuous across the transition hypersurface.)Applying this result to all transitions, we see that a global solution of the modifiedtheory is a concatenation of infinitely many classical regions with alternating orientationsof time. In each region, the geometry is indistinguishable from a classical solution. Theonly new physics therefore resides in the time reversals, which make it possible that K ϕ can remain bounded. In each local region, the geometry is covariant and slicing independent, provided thechanges of coordinates and slicings are sufficiently “small” such that they do not leavethe range of K ϕ relevant for the region. (We can apply slicing independence only inthe classical analogs, where the correct version (4) of hypersurface deformations holds.)Globally, space-time in this model would be covariant only if the reversal surfaces werecovariantly defined, but this is not the case: They refer to fixed values of δK ϕ = ±
1, and K ϕ is not a space-time scalar.Choosing a different slicing in a classical analog in general shifts the positions of time re-versal surfaces. A complete solution for ˜ K ϕ and ˜ E ϕ therefore violates slicing independence,even after it has locally been mapped to a suitable effective line element. For instance, ina vacuum solution there would be no time reversals outside the horizon in a Schwarzschildslicing, but there are other exterior slicings in which δK ϕ can be large and trigger timereversal in the modified geometry. Even with minimal modifications introduced by themodel, general covariance is violated. Using a detailed construction of effective line elements that consistently describe the space-time geometry of solutions in the modified theory introduced in [20], we have shown thatthe only new physical effect is the introduction of time-reversal surfaces connecting classicalspace-time regions. This observation corrects the claim “As [the canonical transformation]is not-invertible in the whole of phase space it still allows to have the usual novel phenomenathat loop quantizations introduce in regions where one expects general relativity not tobe valid, like close to singularities.” made in [20]. Locally, general relativity is valid inall regions of the modified theory, without any novel phenomena that have been claimedpreviously in loop quantizations. Our constructions also show that effective geometriesdescribed by the model depend only on the local maxima of the function K ϕ ( ˜ K ϕ ). Thespecific sine function, usually motivated by expressions of holonomies used in loop quantumgravity, does not matter at all.Even though the modifications are obtained by a canonical transformation of a covarianttheory, their global solutions violate covariance precisely at those places where “novelphenomena” happen. This outcome heightens the covariance crisis of loop quantum gravity:7ven a minor modification of the classical equations, inspired by loop quantum gravity butimplemented by a canonical transformation, is in conflict with the requirement of generalcovariance. Acknowledgements
The author thanks Rodolfo Gambini and Jorge Pullin for sharing a draft of [20] andsubsequent discussions. This work was supported in part by NSF grant PHY-1912168.
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