2.1. LM assumption

LM articulates the usual broadest constraints in GR on the kinematics of classical gravity. ^{Footnote 5} What might these have to do with applications of semiclassical gravity, in the course of quantum gravity research? As I discuss in Schneider (Reference Schneider2020, 10), given our current physics that includes GR, modeling in semiclassical gravity is intended to provide “claims about what we might infer from contemporary physics about approximations of future physics.” In light of this pragmatic view of the applications of semiclassical gravity, it is reasonable to assume that the broadest constraints from our current best classical theory are carried over into the semiclassical.

On the other hand, the relevance of the future theory of quantum gravity to semiclassical modeling on this pragmatic view also means that independent reasons might be supplied that ultimately counsel in favor of kinematic constraints that are at odds with those familiar from the classical theory. But such independent reasons, supplied in the course of quantum gravity research, are grounded in relationships between our current physics and the future theory yet to be developed. This would suggest that, unlike the constraints carried over from GR, any other constraints would have the status of speculations in the course of quantum gravity research.

So, consider Manchak and Weatherall’s (Reference Manchak and Weatherall2018, 618–19) objection to Maudlin (Reference Maudlin2017), whom the former interpret as rejecting LM: ^{Footnote 6}

Two points are worth emphasizing here. First, it seems right to say that physicists involved in quantum gravity research are often free to speculate that the development of the future theory may ultimately supply resources relevant to interpreting our current physics, which were not initially available in consultation of our current physics alone. On such a view, Manchak and Weatherall may here be understood as outlining a common speculation about what will come of quantum gravity research: that there may eventually be reason supplied by the future theory to justify our deviating from LM when modeling what we are today comfortable recognizing as astrophysical black holes.

Second, it is crucial that, nonetheless, deviating from LM in our modeling astrophysical black holes in light of quantum gravity, even as a matter of speculation, is not what would solve the evaporation paradox. To solve the paradox, it must be that the speculation would, moreover, compel us to change how we see fit to model astrophysical black holes when we continue to have cause to do so semiclassically. That is, something about the modified description of an astrophysical black hole provided by the future theory of quantum gravity must, consequent to our embrace of that future theory, teach us something new about what amounts to an adequate description of the modeling target in a suitable semiclassical limit.

But what it might teach us—and how—is hard to say in advance. For instance, arguably, the fuzzball proposal in string theory, discussed later, presents one case of what Manchak and Weatherall (Reference Manchak and Weatherall2018) envision, wherein the description of space and time as a smooth manifold breaks down at a “fauxrizon” relevant to the quantum gravity treatment of an astrophysical black hole. (As Huggett and Matsubara [Reference Huggett and Matsubara2021; 14] observe of the fuzzball proposal, “certainly it falls into the category of ‘drama at the horizon’ ”—whereas the classical GR tolerates no drama.) Yet, in this case, LM appears to remain entirely intact, so far as concerns the semiclassical description of that black hole. Namely, as I will argue following Huggett and Matsubara, it is plausible that adequate semiclassical descriptions of the astrophysical black hole relative to an exterior “weak-field” observer, in light of the identification of the fauxrizon in the fuzzball construction, are provided by exterior black hole spacetimes. So if the breakdown of space and time as a smooth manifold at the fauxrizon is to teach us how the paradox is solved, it must be that the fuzzball construction within string theory, in virtue of that breakdown at the fauxrizon, gives us further reason to reject either GH or EG in the semiclassical limit.

2.2. GH assumption

GH ensures the predictability and retrodictability of the evolution of classical fields in the spacetime, including the metric itself when understood locally (i.e., as a dynamical field on some underlying smooth manifold, with distinguished hypersurfaces identified as Cauchy). As Manchak and Weatherall (Reference Manchak and Weatherall2018) note, this is a version of the cosmic censorship hypothesis originally due to Penrose. Also as they note, at least in the context of the classical GR, predictability and retrodictability provide a neat (classical) description of that which is at stake in standard talk about the conservation of “information” in contemporary theoretical physics (see also Wüthrich Reference Wüthrich, Dardashti, Dawid and Thébault2019, sec. 2).

Understanding cosmic censorship in terms of maximal domains of predictability and retrodictability follows a tradition in the philosophical foundations of GR inaugurated by Earman (Reference Earman1995). Earman considers this topic as closely allied with various commitments that classical relativistic field theories be deterministic. And although global hyperbolicity is not necessary for determinism in these theories, it does seem sufficient. This is (likely) because global hyperbolicity implies that the spacetime, per LM, is diffeomorphic to $\mathbb{R} \times S$ , with $S$ a smooth spacelike Cauchy surface (Bernal and Sánchez Reference Bernal and Sánchez2003). Consequently, the space of suitable initial data on $S$ for a relativistic field can be arranged in a one-to-one relationship with global field configurations that are consistent, given a local dynamics of the field, with the maximal evolution of those initial data off of any one such embedded hypersurface. ^{Footnote 7}

Two remarks are, at this point, in order. First, I have just included a caveat that global hyperbolicity is not necessary for determinism. Lesourd (Reference Lesourd2018) provides a theorem in support of the view that fully evaporating black holes imply a failure of causal continuity, above and beyond a failure of global hyperbolicity. Accompanying that result is a discussion of the relationship between causal continuity and predictability/determinism, which runs more or less the same as that just discussed (except that causal continuity does not imply a product structure $\mathbb{R} \times S$ , with $S$ a smooth spacelike Cauchy surface). As stated within that discussion, causal continuity has historically been proposed as a weaker condition than global hyperbolicity that is nonetheless sufficient to ensure predictability. Meanwhile, it is suspected that causal continuity is necessary for predictability, though the question remains open. If causal continuity does prove necessary, then the theorem would seem to secure the sense in which GH, indeed via a relaxation thereof, fits together with LM and EG in the manufacture of paradox concerning determinism in full black hole evaporation or (in turn) paradox concerning the conservation of information through the same.

Second, all of the preceding is a matter of classical GR. In the context of semiclassical gravity, whether one focuses on causal continuity or global hyperbolicity with regard to the causal structure of the spacetime, there is a further question about how we might proceed to think about the quantum fields one ought to be prepared to define there. And, although Manchak and Weatherall (Reference Manchak and Weatherall2018) do not stress this point in their discussion of GH, it is the product structure $\mathbb{R} \times S$ ensured in the case of global hyperbolicity that one may exploit, in the context of quantizing what is otherwise a classical relativistic field theory defined locally on the spacetime. Namely, the association of classical states of the field with their immediate, quasi-local evolution off of a particular Cauchy hypersurface embedded in the spacetime allows for an unambiguous means of constructing a corresponding quantum field theory on that spacetime (Wald Reference Wald1994). There may, of course, turn out to be other ways to proceed in quantization if the spacetime is not globally hyperbolic, but there would appear to be significant costs (Yurtsever Reference Yurtsever1994; Friedman Reference Friedman1997). Along these lines, it is worth noting that global hyperbolicity remains as a constraint on global spacetime structure, even in much more recent efforts to define “locally covariant” formulations of quantum field theory—that is, formulations that are “background independent” in a sense licensing talk about “quantum fields,” analogous to classical fields, as physical fields that may be relocated between arbitrary (allowed) spacetime backgrounds (Rédei Reference Rédei2014; Brunetti, Fredenhagen, and Rejzner Reference Brunetti, Fredenhagen and Rejzner2016). ^{Footnote 8}

This formal state of affairs is invaluable for delimiting semiclassical gravity. After all, semiclassical gravity is concerned with the gravitational self-interaction of material systems, precisely when the quantum descriptions accorded to those systems, that is, by an appropriately quantized theory, cannot be ignored. (Though, when quantum fluctuations are large in that description—such as in the case of macroscopic superpositions of matter—semiclassical gravity, too, is inadequate.) Together with LM, GH thereby ensures that we at least know how, in the first place, to associate the spacetime backgrounds for relativistic quantum field theories with arbitrary allowed gravitational field states. This paves the way for a perturbative approach to studying the back-reaction of the quantized fields on such gravitational field states, with the latter now treated semiclassically: as a spacetime expression of the mean-field effects relevant to a gravitationally coupled quantum system, which is otherwise modeled by a state within the (non–gravitationally coupled) quantum field theory defined on top.

2.3. EG assumption

EG concerns the dynamical process of semiclassical black hole evaporation, relative to an exterior “weak-field” observer: an observer who is schematized in the semiclassical theory as a maximal wordline in spacetime sufficiently far away from the strong-field gravitational system—that is, the black hole interior—so as to be adequately regarded as “near” null infinity, for all proper time. Assume, for simplicity, the case of black hole spacetimes with eventually zero angular momentum and charge. As Hawking originally noted, a quantized scalar field in its vacuum state associated with any such black hole spacetime background is thermal at future null infinity, that is, in the eventual “vicinity” of the observer. Although vacuum states are globally defined and utterly nonlocalizable in spacetime, mode-by-mode conservation arguments may be used to motivate the expectation that these thermal modes are dynamically produced by the black hole, to then escape to future infinity. This dynamical production process is called “Hawking radiation.” And because the Hawking radiation modes indeed manage to escape, one associates the origins of that radiation with physical processes that are restricted quasi-locally to a neighborhood of the event horizon of the black hole, according to the semiclassical description. Famously, this is a locus of concern in our semiclassical means of modeling black holes, in the course of quantum gravity research: according to classical GR, nothing whatsoever should be significant about the local physics present along that horizon—that is, no “drama.” Yet, merely by treating matter as a quantum system, vacuum interactions between that matter and the black hole background imply novel physics in the vicinity.

Pushing onward nonetheless, it is natural to try to model the back-reaction of Hawking radiation on the background gravitational field state. One thereby expects from semiclassical gravity dynamical loss in the black hole mass parameter, ^{Footnote 9} and with that loss, the event horizon dynamically shrinks (as assessed in terms of suitably chosen hypersurfaces, which foliate the neighborhood of the relevant exterior weak-field observer). Eventually, one expects the black hole to fully evaporate (relative to that observer). But further details will not be important here; EG is a statement about the peculiar global spacetime structure where such a complete evaporation process occurs.

Understandably, EG is where all substantive controversy about the status of the evaporation paradox in contemporary quantum gravity research lies. Namely, there is a sense in which this dynamical process simply lies beyond the scope of semiclassical gravity, so that it may be disregarded in the course of our efforts to infer from contemporary physics about approximations of future physics. Recall from the discussion of GH that the scope of semiclassical gravity is delimited by our knowing how to quantize field theories in the appropriate spacetime setting. If our understanding of the dynamical process of black hole evaporation, per EG, implies that GH is false, we therefore, arguably, have a breakdown of the applicability of the semiclassical theory to our modeling astrophysical black holes. So there is some cause to be skeptical that EG arises within applications of semiclassical gravity, carefully construed.

Indeed, one may suspect that semiclassical evaporation dynamics fail to be metastable as a physical description of astrophysical black holes in semiclassical gravity, in the course of quantum gravity research. Intuitively, we were led to identify evaporation as a dynamical process in semiclassical gravity because of radiation modes associated quasi-locally with the event horizon of the black hole, whereas in the black hole’s final moments, when the mass parameter runs down to 0, the event horizon abruptly vanishes. Put more formally, one reason to doubt the metastability of semiclassical black hole evaporation over the lifetime of an astrophysical black hole is the appearance of singular limits as the mass parameter runs down to 0 in a finite time, in a variety of equations that are relevant in motivating the evaporation dynamics in the first place.

On the other hand, I take it that this is just another way of framing that there is indeed a paradox here, upon one’s embrace of LM, GH, and EG in the context of semiclassical modeling. Skepticism specifically focused on EG is, then, an expression of the kind of speculation sketched earlier: that we may eventually find cause in the future quantum gravity theory to reject specifically EG in the semiclassical limit, thereby solving the paradox. For instance, Belot, Earman, and Ruetsche (Reference Belot, Earman and Ruetsche1999) note the possibility of a “thunderbolt evaporation” as one avenue of escaping the paradox. In this case, one readily accepts LM and GH and then hopes that, in virtue of some further reason to come (i.e., as a lesson from future physics), the process of complete semiclassical black hole evaporation—even as just described—simply does not entail the peculiar global spacetime structure otherwise claimed in EG, instead singling out some other global geometry that is not so causally pathological in the face of GH. Alternatively, one might hope that, in virtue of some further reason to come, the process of semiclassical black hole evaporation is simply incomplete, entailing a black hole remnant of some nontrivial mass (Bokulich Reference Bokulich2001). Or perhaps there may turn out to be a quantum gravity “bounce,” whereby shrinking black holes eventually transition into white holes—at a minor cost of apparently violating familiar energy conditions in the semiclassical description of that new physics (Malafarina Reference Malafarina2017). ^{Footnote 10}

All of this is to say, set aside reservations that black hole evaporation may simply be a dynamical process beyond the scope of semiclassical gravity. Except as a matter of speculation about the future theory of quantum gravity, such a view is not dispositive of rejecting EG in particular as a solution to the paradox today.

3.1. Global structure in the aftermath of the fuzzball proposal

To begin, note that the fuzzball construction as sketched is irrelevant to the three assumptions that composed the evaporation paradox, because the latter, as stated, were restricted in focus to applications of semiclassical gravity. Here, by contrast, we are considering models of the same astrophysical systems, and presumably relative to the same observers, only now in the context of string theory. On the other hand, the fuzzball construction as sketched would become relevant to the three assumptions were we to equate the stringy model, intertheoretically, to some description in a suitable “semiclassical limit”: in particular, as relevant for some observer localized as a quantum clock in effective spacetime suitably far away from the fauxrizon so as to correspond, in the semiclassical theory, with the schematic of the exterior weak-field observer outlined earlier. In that case, because the fauxrizon quarantines the irreducibly stringy quantum hair from the effective spacetime region of the fuzzball that is coincident with the exterior of a semiclassical black hole, what is recovered in the semiclassical theory (relative to the chosen observer) is exactly the latter: an exterior black hole spacetime. This is an extendible spacetime that comprises only the region (without boundary) of a black hole spacetime that is retrodictable from future infinity. On this account, the fauxrizon itself would simply disappear, along with all the stringy quantum hair “beyond” it, in the course of taking the relevant limit. ^{Footnote 12}

Exterior black hole spacetimes readily satisfy LM, and we may just as well stipulate that they satisfy GH. That is, in the previous paragraph, exterior black hole spacetimes are presented in terms of retrodiction from future infinity. Although vaguely specified, this immediately ensures LM. Meanwhile, if, consistent with this vague presentation, we understand our models of astrophysical black holes in the semiclassical theory to idealize the relevant astrophysical systems as isolated strong-field gravitational systems in asymptotically Minkowski spacetime, GH follows from properties of the latter. ^{Footnote 13}

In fact, it may be that GH is ultimately difficult to secure, as a matter of stipulation. ^{Footnote 14} While exterior spacetimes for nonevaporating black holes can easily be globally hyperbolic, it is trickier to imagine what might ever suffice as a choice of Cauchy surface for an exterior portion of an evaporating black hole spacetime. In the absence of explicit argument for GH, it may be that fuzzballs would force us to confront failures of global hyperbolicity in the semiclassical limit.

Accepting such a failure of strong cosmic censorship in the semiclassical limit may or may not be alarming (see note 8 and surrounding discussion). Still, it seems to me that the promise of the fauxrizon (to the extent that one is optimistic) is that stringy physics will ultimately prove to be quarantined in black hole scenarios, relative to a distant exterior observer, from an otherwise mundane exterior spacetime in which they reside. Unitarity in the underlying string theory being associated, loosely, with a globally hyperbolic target spacetime, that promised fact of quarantining would seem to point to a globally hyperbolic exterior portion. On the other hand, thinking of the fauxrizon instead as a more sophisticated description of a timelike quantum membrane counsels in the other direction, so intuitions are mixed.

In the face of ambiguity, I suggest a choice made for definiteness in the argument that will follow: suppose that GH, like LM, is seen as reasonable in descriptions of a fuzzball obtained in the semiclassical limit, relative to a suitable choice of exterior weak-field observer. The upshot is that, to ultimately solve the evaporation paradox, something about the fauxrizon must supply a reason to discard EG. ^{Footnote 15} Does it? Here the subject gets muddy. It is (plausibly) true that the fauxrizon in the full quantum gravity description undermines the descriptive relevance of geometric structure relevant to EG in the semiclassical limit by doing away with the interior region of the black hole spacetime, as an effective description at and beyond the horizon relative to the choice of exterior weak-field observer at all moments in their proper time.

But it also might not be! The treatment of the fauxrizon in the opening paragraph of the section—as something that simply disappears in the semiclassical limit—is telling. Namely, why should its disappearance in the semiclassical limit preclude our alternatively representing the fuzzball, semiclassically from far away, as extending well into an interior? A fairly common operationalist stance in the foundations of GR would even seem to counsel the opposite: that, in having recovered in the semiclassical limit a description of an observer as located in an exterior black hole spacetime, one may then proceed to freely choose between distinct descriptions of that same observer’s surrounding circumstance—as either terminating outside a horizon or continuing on through it. Either choice is descriptively adequate, for the sake of our representing the astrophysical black hole—now understood to be a fuzzball—as a spacetime in the semiclassical limit (relative to the exterior weak-field observer).

In other words, the paradox remains so long as the operationalist stance familiar from the classical theory GR continues to be viable in a semiclassical modeling context. If nothing precludes a description of the astrophysical black hole relative to the exterior weak-field observer in the semiclassical theory as extending through a horizon into a black hole interior, then the physical reasoning underpinning EG persists in exactly the case when one has freely chosen to consider the observer’s surrounding exterior black hole spacetime as itself extended through a distant horizon and into a black hole interior. Consequently, adequate representations of the astrophysical black hole—now understood to be a fuzzball—as a spacetime in the semiclassical limit can exhibit the peculiar global geometric features associated with EG, in contradiction with what LM and GH entail. (Those peculiar global geometric features would, perhaps, indicate that there is something misleading about such a choice of semiclassical description—but it would still be appropriate as one such description, and so the paradox would formally persist.)

3.2. What constitutes a (quantum gravity) solution to the semiclassical paradox

Where we have wound up so far is that it fails to suffice to solve the paradox that the fuzzball construction be free of conceptual difficulties reminiscent of the evaporation paradox, now formulated with regard to astrophysical modeling applications of string theory. Instead, to solve the paradox (disregarding loopholes; see note 15), something about the stringy description of the astrophysical black hole as a fuzzball must supply us with cause to necessitate abandoning EG: grounding a claim that we are, ultimately, not free in applying the semiclassical theory to choose to represent a fuzzball, as considered in the semiclassical limit relative to an exterior weak-field observer, by means of a spacetime that includes a horizon and a black hole interior. In other words, something about the demarcation of effective spacetime from the fuzzball’s quantum hair relative to the exterior weak-field observer, that is, the fauxrizon itself, must force our hand in describing a fuzzball, semiclassically, just in terms of an exterior black hole spacetime. (This, even while we simultaneously understand the fauxrizon itself to disappear in that same limit!) Something about the fauxrizon must render the complete black hole spacetime in the semiclassical limit, relative to any such observer, inadequate as a description of that observer’s physical circumstance.

Unfortunately, what would demonstrate such a conclusion is obscure. But I can point to a toy version of the same punchline, in the hope of spurring further ideas along these lines. What I have in mind is a case in which detailed inspection of a quantized successor theory (with native structures that disappear in the classical limit) precludes from descriptive relevance certain states of affairs in the classical limit that might otherwise have been kinematically plausible as part of a classical predecessor theory. This is an ongoing project by Benjamin Feintzeig concerning systems of mechanics with finite degrees of freedom; here I will focus on the line of argument developed in just two recent articles (Feintzeig Reference Feintzeig2020a, Reference Feintzeig2020b).

Feintzeig draws on tools from quantization theory appropriate for systems with finite degrees of freedom to offer a precise sense in which two closely related statements hold. First, the kinematic structure of a quantized theory of mechanics (characterized by canonical commutation relations) constrains the classical theory in the limit, which we may thereby regard as suitable for providing approximate descriptions of a quantum system ultimately described by an application of the quantized theory (Feintzeig Reference Feintzeig2020a). Hence facts about a quantized theory may plausibly delimit the descriptive scope of a classical theory applied to any such quantum systems. Second, requiring that the quantized theory account for the descriptive successes of its classical predecessor theory constrains the activity of constructing the quantized theory as a successor theory, given an interpretation of the classical one (Feintzeig Reference Feintzeig2020b).

The relevance of the explicit toy example to the present case is not found in the technical procedures developed there, for the case of classical limit descriptions of quantum mechanical systems—just so, I have neglected to discuss any of the technical work that forms the backbone of Feintzeig’s project. Rather, what is important is the philosophical thesis staked out by means of that technical work, concerning intertheoretic relations as a sophisticated, nontrivial contribution to the full development of a theory, particularly in light of that theory’s genealogy: its being intended as a successor to some other interpreted, empirically apt predecessor that we take to be descriptive in application to (many of) the same physical systems.

Adopting Feintzeig’s perspective to the present case, my suggestion is that where certain paradoxes might otherwise spoil the descriptive adequacy of the (semi)classical predecessor theory in certain modeling applications on a standard interpretation, we might ultimately be constrained in our writing down the new quantized theory so that those paradoxes are circumvented in the appropriate limit. That is to say, in the present case, one might look forward to (1) the string theory description of the astrophysical black hole as a fuzzball constraining adequate descriptions of the same system in a semiclassical limit (such as forcing the global spacetime property of extendibility there, in virtue of facts about the fauxrizon; compare note 12) and (2) our having developed string theory so that the paradox will, in that limit, turn out to be solved, accordingly.

That fauxrizons in a quantum gravity description of an astrophysical black hole as a fuzzball may intertheoretically constrain our semiclassically modeling them is an interesting proposal, for at least two reasons. First, it sharpens our attention in quantum gravity research toward the ongoing status of global spacetime structure in GR, now construed as a subject to be understood in terms of a semiclassical limit. As Manchak and Weatherall (Reference Manchak and Weatherall2018) elaborated, building on themes familiar from Earman (Reference Earman1995), it is a mistake to think of the causal pathology involved with full evaporation as merely amounting to there being some “missing point” at which the metric cannot be smoothly, locally extended—that is, a “naked singularity” (see note 4). Indeed, the present discussion might suggest that we are learning about that naked singularity’s ultimately stringy structure, so that what we ordinarily have cause to think of as a formal, quasi-local breakdown of the theory in the semiclassical limit is, in fact, a kind of fundamental physical—namely, stringy—system in itself. One might even wonder, along these lines, whether the global spacetime description featuring a naked singularity just is a mistaken semiclassical treatment of what is, ultimately, the dynamical production of a stringy black hole remnant, relative to a choice of exterior weak-field observer. This is reminiscent of “Attitude 2a” in the recent taxonomy Crowther and De Haro (Reference Crowther, De Haro and Vassallo2022) provided, where singularities in GR are regarded as physically significant, in the sense of being informative of new physics.

Second, and along similar lines, it is curious that for the fuzzball proposal to give us cause to necessitate abandoning EG, it evidently must be in virtue of the fauxrizon—the quasi-local fade-out of an effective description of spacetime, which quarantines quantum hair associated with the black hole interior at what would otherwise be a horizon—that EG is rendered false. The claim that, in a string theory approach to quantum gravity research, the fuzzball proposal ultimately solves the evaporation paradox would therefore seem to be a claim that some quasi-local, fundamental degrees of freedom in string theory wind up constraining global spacetime structure in the choices we make in our representation of that same physics in the semiclassical limit. In some sense, this is unsurprising: even before introducing back-reaction to motivate evaporation dynamics, as noted, Hawking radiation associated global states of affairs with the quasi-local production of modes along a horizon, which would otherwise in GR be unremarkable, locally, as a surface in the spacetime. But it is nonetheless tantalizing that local dynamics at the level of fundamental description are sought, which would intertheoretically constrain the specification of the boundary in an observer-relative application of the limit theory (in the sense of constraints relevant to solving a local geometrodynamics in the bulk). Most discussions about intertheoretic relations in fundamental physics research have, rather, so far focused exclusively on the capacity of a fundamental theory to constrain effective descriptions of dynamics in the bulk (without attention to any changes in description of the boundary or to the role of the schematic of the observer carried through the intertheory relation).