Information loss, made worse by quantum gravity?

Quantum gravity is often expected to solve both the singularity problem and the information-loss problem of black holes. This article presents an example from loop quantum gravity in which the singularity problem is solved in such a way that the information-loss problem is made worse. Quantum effects in this scenario, in contrast to previous non-singular models, do not eliminate the event horizon and introduce a new Cauchy horizon where determinism breaks down. Although infinities are avoided, for all practical purposes the core of the black hole plays the role of a naked singularity. Recent developments in loop quantum gravity indicate that this aggravated information loss problem is likely to be the generic outcome, putting strong conceptual pressure on the theory.


Introduction
There is a widespread expectation that quantum gravity, once it is fully developed and understood, will resolve several important conceptual problems in our current grasp of the universe. Among the most popular ones of these problems are the singularity problem and the problem of information loss. Several proposals have been made to address these questions within the existing approaches to quantum gravity, but it is difficult to see a general scenario emerge. Given such a variety of possible but incomplete solution attempts, commonly celebrated as successes by the followers of the particular theory employed, it is difficult to use these models in order to discriminate between the approaches. In this situation it may be more fruitful to discuss properties of a given approach that stand in the way of resolving one or more of the big conceptual questions. Here, we provide an example regarding the information loss problem as seen in loop quantum gravity.
Loop quantum gravity [1,2,3] is a proposal for a canonical quantization of space-time geometry. It remains incomplete because it is not clear that it can give rise to a consistent quantum space-time picture (owing to the so-called anomaly problem of canonical quantum gravity). Nevertheless, the framework is promising because it has several technical advantages compared to other canonical approaches, in particular in that it provides a welldefined and tractable mathematical formulation for quantum states of spatial geometry. The dynamics remains difficult to define and to deal with, but there are indications that a consistent version may be possible, one that does not violate (but perhaps deforms) the important classical symmetry of general covariance. These indications, found in a variety of models, lead to the most-detailed scenarios by which one can explore large-curvature regimes in the setting of loop quantum gravity.
The word "loop" in this context refers to the importance attached to closed spatial curves in the construction of Hilbert spaces for geometry according to loop quantum gravity [4]. More precisely, one postulates as basic operators not the usual curvature components on which classical formulations of general relativity are based, but "holonomies" which describe how curvature distorts the notion of parallel transport in space-time. If we pick a vector at one point of a closed loop in curved space and move it along the loop so that each infinitesimal shift keeps it parallel to itself, it will end up rotated compared to the initial vector once we complete the loop. The initial and final vectors differ from each other by a rotation with an angle depending on the curvature in the region enclosed by the loop. Loop quantum gravity extends this construction to space-time and quantizes it: It turns the rotation matrices into operators on the Hilbert space it provides. An important consequence is the fact that (unbounded) curvature components are expressed by bounded matrix elements of rotations. Most of the postulated loop resolutions of the singularity problem [5,6,7,8,9,10,11] rely on this replacement.
Classical gravity, in canonical terms, can be described by a Hamiltonian H that depends on the curvature. If H is to be turned into an operator for loop quantum gravity, one must replace the curvature components by matrix elements of holonomies along suitable loops, because only the latter ones have operator analogs in this framework. One has to modify the classical Hamiltonian by a new form of quantum corrections. The classical limit can be preserved because for small curvature, the rotations expressed by holonomies differ from the identity by a term linear in standard curvature components [12,13]. At low curvature, the classical Hamiltonian can therefore be obtained. At high curvature, however, strong quantum-geometry effects result which, by virtue of using bounded holonomies instead of unbounded curvature, can be beneficial for resolutions of the singularity problem.
Given the boundedness, it is in fact easy to produce singularity-free models. But one of the outstanding problems of this framework is to show that the strong modification of the classical Hamiltonian can be consistent with space-time covariance. This question is not just one of broken classical symmetries (which might be interesting quantum effects). Covariance is implemented by a set of gauge transformations which eliminate unphysical degrees of freedom given by the possibility of choosing arbitrary coordinates on space-time. When these transformations are broken by quantum effects, the resulting theory is meaningless because its predictions would depend on which coordinates one used to compute them. Showing that there are no broken gauge transformations (or gauge anomalies) is therefore a crucial task regarding the consistency of the theory. The problem remains unresolved in general, but several models exist in which one can see how it is possible to achieve anomaly-freedom, constructed using operator methods [14,15,16,17] or with effective methods [18,19,20,21,22].

A model of deformed canonical symmetries
As a simple, yet representative, example, we consider a model with one field-theoretic degree of freedom φ(x) and momentum p(x). There is no room for gauge degrees of freedom in this model, and therefore we use it only to consider the form of symmetries of gravity, not the way in which spurious degrees of freedom are removed.

Algebra of transformations
For the example, we postulate a class of Hamiltonians with a function f to be specified, and with the prime denoting a derivative by the one spatial coordinate x. As in general relativity, the Hamiltonian depends on a free function N(x) because there is no absolute time.  [18,19,20,21,22] follow from the structure of derivatives in (1) in combination with a function f (p) which modifies the classical momentum dependence. The Hamiltonian, as a generator of local time translations, is accompanied by a second generator of local spatial translations, the form of which is more strictly determined: It is given by D[w] = dxwφp ′ with another free function w(x). It generates canonical transformations given by as they would result from an infinitesimal spatial shift by −w(x): (The transformation of φ is slightly different owing to a formal density weight.) Of special importance is the algebra of symmetries, which can be computed by Poisson brackets (as a classical version of commutators). We obtain Two local time translations have a commutator given by a spatial shift. (The numerical coefficients chosen in (1) ensure that the algebra of symmetry generators H and D is closed.) Although our model is simplified, the result (3) matches well with calculations in models of loop quantum gravity, constructed for spherical symmetry [19,22] and for cosmological perturbations [20,21]. The same type of algebra has also been obtained for H-operators in 2 + 1-dimensional models [14]. Since our choice (1) extracts the main dynamical features of loop models, it serves to underline the genericness of deformed symmetry algebras when f (p) is no longer quadratic.

Geometry
For the classical case in which f (p) = p 2 is a quadratic function of p, half the second derivative in (3) as expected. Holonomy effects of loop quantum gravity can be modeled by using a bounded function f (p) instead of a quadratic one. (A popular choice in the field is f (p) = p 2 0 sin 2 (p/p 0 ) with some constant p 0 , such as Planck-sized curvature.) The number of classical symmetries remains intact because the relation (3) is still a closed commutator. But the structure of space-time changes: we can no longer think in terms of local Minkowski geometry because the spatial shift in (3) with 1 2 d 2 f /dp 2 = 1 violates the relation ∆x = v∆t found classically in (4). The deviation from classical space-time is especially dramatic at high curvature, near any maximum of the holonomy function f (p): Around a maximum, the second derivative is negative, d 2 f /dp 2 < 0. For the popular choice of f (p) = p 2 0 sin 2 (p/p 0 ), we have 1 2 d 2 f /dp 2 = cos(2p/p 0 ) which is equal to −1 at the maximum of f (p). The counterintuitive relation ∆x = −v∆t can be interpreted in more familiar terms: the change of sign means that the classical Lorentz boost is replaced by an ordinary rotation. (An infinitesimal rotation by an angle θ in the (x, y)-plane and a spatial shift by ∆y commute to ∆x = −θ∆y.) At high curvature, holonomy-modified models of general relativity replace space-time with pure and timeless higher-dimensional space, a phenomenon called signature change [23,24,25].

Field equations
At the level of equations of motion, signature change means that hyperbolic wave equations become elliptic partial differential equations (in all four dimensions, or two in the model). Indeed, if one computes equations of motion from the Hamiltonian (1), one obtains where d 2 f (p)/dp 2 is a function ofφ viaφ = Ndf (p)/dp. This partial differential equation, which is hyperbolic for 1 2 d 2 f (p)/dp 2 > 0, becomes elliptic for 1 2 d 2 f (p)/dp 2 < 0. In the latter case, the equation requires boundary values for solutions to be specified; it is not consistent with the familiar evolution picture implemented by an initial-value problem. Instead of specifying our field and its first time derivative at one instant of time, once curvature (orφ in the model) becomes large enough to trigger signature change we must specify the field on a boundary enclosing a 4-dimensional region of interestincluding a "future" boundary in the former time direction. We can no longer determine the whole universe from initial data given at one time.
Although our specific model is simplified, the main conclusion about signature change agrees with the more detailed versions cited above, which latter directly come from reduced models of loop quantum gravity combined with canonical effective techniques. Our model presented here shows that the main reason for signature change is the modified dependence of gravitational Hamiltonians on curvature components when holonomies are used to express them, together with the general structure of curvature terms. (Especially the presence of spatial derivatives seems crucial for derivatives of the modification function to show up in the symmetry algebra after integrating by parts.) The rest of our discussions does not rely on the specific model but rather on the general consequence of signature change.

General aspects of signature change
As shown in [26], the structure of constraint algebras or gauge transformations, of which (3) provides a model, is much less sensitive to details of regularization effects or quantum corrections than the precise dynamics implied. Even if there may be additional quantum corrections in (5) in a fully quantized model, structure functions of the algebra, such as 1 2 d 2 f /dp 2 in (3), provide reliable effects of a general nature. For details, the reader is referred to the above citation, but the crucial ingredient in this observation is the definition of effective constraints C I = Ĉ I as expectation values of constraint operators, and their brackets as {C I , C J } = [Ĉ I ,Ĉ J ] /i . A regularization of a constraint operatorĈ I leads to corresponding modifications of the effective constraint Ĉ I . For any consistent operator algebra, the bracket of effective constraints mimics the commutator of constraint operators. Even if Ĉ I , computed to some order in quantum corrections, may give a poor approximation to the quantum dynamics, the possible consistent forms of effective constraint algebras restrict the possible versions of quantum commutators. If effective constraints of a certain form, such as those obtained with holonomy modifications, always lead to a change of sign of structure functions, the same must be true for operator algebras.
As noted also in [24,27], equations of the form (5) sometimes appear for matter systems with instabilities, in cosmology but also in other areas such as transonic flow. An instability would normally not be interpreted as signature change as long as a standard Lorentzian metric structure remains realized, as is the case in all the known matter examples. The present context, however, is different, because the instability affects the geometry of spacetime itself, and not just matter propagating in space-time. (In models of loop quantum gravity, φ in (5) stands for metric inhomogeneities.) Such an instability is more severe, and at the same time more inclusive because it affects all excitations -matter and geometry -in the same way. Indeed, the most fundamental structure where it appears is not the equation of motion (5) but the symmetry algebra (3). If matter is present, its Hamiltonian would be added to the gravitational one, the resulting sum satisfying a closed algebra of the form (3). (If adding matter terms would break the algebra, there would be anomalies making the theory inconsistent.) Matter and geometry are then subject to the same modified symmetries, and correspondingly to a modified evolution picture with a boundary rather than initial-value problem at high density.
Solutions might exist for elliptic partial differential equations with an initial-value problem. However, such solutions are unstable and depend sensitively on the initial values; therefore, initial-value problems for elliptic partial differential equations are not well-posed. Sometimes, a physical model of this form may just signal a growing mode which is increasing rapidly in actual time. In quantum gravity and cosmology, however, instabilities from signature change in (3) or (5) are much more debilitating. In this context, one does not perform controlled laboratory experiments in which one can prepare or directly observe the initial values. When signature change is relevant, it happens in strong quantum-gravity regimes where the analogs of f (p) differ much from the classical behavior. Not only initial values but also the precise dynamical equations (subject to quantization ambiguities) are so uncertain that an initial-value formulation can give no predictivity. (In cosmological parlance, instabilities from signature change present severe versions of trans-Planckian and fine-tuning problems. For more information on the dynamics of affected modes see [28].) In contrast to some matter systems in which elliptic field equations may appear, quantumgravity theories do not allow initial-value formulations in such regimes but rather require 4-dimensional boundary-value problems.
Evolution in these models is no longer fully deterministic. In the remainder of this article, we apply this conclusion to black holes and show that even low-curvature regions, where observers have no reason to expect strong quantum-gravity effects, will be affected by indeterminism. In this context, consequences of signature change are therefore much more severe than their analogs in cosmological models.

Black holes
Black holes in general relativity have singularities where space-time curvature diverges. Loop quantum gravity has given rise to models in which curvature is bounded, apparently resolving the singularity problem [29]. As in some other approaches [30,31,32,33], there is then no event horizon but only an apparent horizon which encloses large curvature but eventually shrinks and disappears. If there is no singularity and information can travel freely through high-curvature regions, there is no information loss, so this important problem seems to be resolved too. However, previous black-hole models of this type in loop quantum gravity did not consider the anomaly problem. In an anomaly-free version, curvature may still be bounded, but when it is large (Planckian, or near the upper bound provided by the models), there can be signature change, preventing information from travelling freely through this regime. It is no longer obvious that the information loss problem can be resolved in singularity-free models of black holes.
If the singularity is resolved, there are two scenarios for Hawking-evaporating black holes: The black-hole region enclosed by an apparent horizon could reconnect with the former exterior at the future end of high curvature, or it could split off into a causally disconnected baby universe. The latter case does not solve the information loss problem because information that falls into the black hole is sealed off in the baby universe. The former case resolves the information loss problem only if information can travel through high curvature. If signature change happens, nothing travels through the high-curvature region and the fate of information must be reconsidered.
The elliptic nature of field equations in the high-curvature core of black holes requires one to specify fields at the future boundary, which would evolve into the future space-time after black-hole evaporation. In Fig. 1, boundary values on the bottom line surrounding the hashed high-curvature region would be determined by evolving past initial values forward in time, but boundary data on the top line around the region would have to be specified, unrestricted by any field equations. Their values are not predicted by the theory, and yet they are essential for determining the future space-time. Once the high-curvature region is passed by an outside observer, space-time is no longer predictable. The black-hole's event horizon H extends into a Cauchy horizon C: The region above C is affected by undetermined boundary data. Even if there are no infinities, the classical black-hole singularity is, for practical purposes, replaced by a naked singularity, a place out of which unpredictable fields can emerge.
In terms of information loss, whatever infalling matter hits the high-curvature core of the black hole determines some part of the boundary conditions required for the elliptic region, and thereby influences part of the solution in the core. But it does not restrict our choice for the future boundary data, or anything that evolves out of it at lower curvature. Infalling information is therefore lost even if there is no black-hole singularity. Similar conclusions apply to the alternative of a baby universe: Infalling information cannot be retrieved in the old exterior, and it cannot be passed on to the baby universe. Figure 1: Acausality: Penrose diagram of a black hole with signature change at high curvature (hashed region). In contrast to traditional non-singular models, there is an event horizon (dashed line H, the boundary of the region that is determined by backward evolution from future infinity) and a Chauchy horizon (dash-dotted line C, the boundary of the region obtained by forward evolution of the high-curvature region). After an observer crosses the Cauchy horizon, space-time depends on the data chosen on the top boundary of the high-curvature region and is no longer determined completely by data at past infinity. Information that falls through H affects field values in the hashed region, but not on the top boundary or its future; it is therefore lost for an outside observer. Unrestricted boundary values at the top part of the hashed region influence the future universe even at low curvature (zigzag arrow), a violation of deterministic behavior.

Conclusions: A no-heir theorem?
We have presented here a mechanism which appears to be generic in loop quantum gravity and helps to resolve curvature divergence, but makes the information loss problem of black holes worse. Black-hole singularities can turn into naked singularities in this framework, which implies an end to predictivity. In classical general relativity, there is strong evidence that cosmic censorship applies: given generic initial data, singularities may form but are enclosed by black-hole horizons; no naked singularities appear that would affect observations made from far away. In loop quantum gravity, a stronger version of cosmic censorship would be required if signature change is confirmed to be generic. Naked singularities (Cauchy horizons) could be avoided only if black-hole interiors split off into baby universes. But even then, information could not be passed on to the baby universe. From the point of view of observers in this new universe, the former black-hole singularity would appear as a true beginning, just as the big bang appears to us in our universe.
The information loss problem has turned into a more-severe problem of indeterminism. Two options remain for loop quantum gravity to provide a consistent deterministic theory without Cauchy horizons. First, one might be able to show that signature change does not happen under general conditions in the full theory, a question which requires an understanding of the off-shell constraint algebra and the thorny anomaly problem. All current indications, however, point in the opposite direction and suggest that signature change is generic. With signature change, Cauchy horizons can be avoided only if the high-curvature regions of black holes always remain causally disconnected from the universe in which they formed, that is if black holes open up into new baby universes. In this scenario, information that falls in a black hole is still lost even for the baby universe, but at least the more-severe problem of a Cauchy horizon can be avoided. In either case, a detailed analysis of possible consistent versions of the constraint algebra of loop quantum gravity could lead to a "no-heir theorem" if deterministic evolution through the high-density regime of black holes turns out to be impossible under all circumstances. Black holes would have no heirs since everything possessed by a collapsing star, including the information carried along, would be lost even if space-time did not end in a curvature singularity.
So far, loop quantum gravity is not understood sufficiently well for a clear model of black holes to emerge from it, but the mechanism analyzed here shows that, at the very least, scenarios obtained from generalizations of simple homogeneous models, such as the one postulated in [29], are likely to be misleading. Inhomogeneity can change the picture drastically, not just because there may be back-reaction on a homogeneous background but also, and often more surprisingly, because the non-trivial nature of symmetry algebras such as (3) is much more restrictive for inhomogeneous models. (The right-hand side would just be identically zero with homogeneity, hiding the crucial coefficient and its sign which indicates signature change.) Our considerations of black-hole models provide a concrete physical setting in which loop quantum gravity and its abstract anomaly problem can be put to a clear conceptual test.