A spacetime singularity is a breakdown in spacetime, either in itsgeometry or in some other basic physical structure. It is a topic ofongoing physical and philosophical research to clarify both the natureand significance of such pathologies. When it is the fundamentalgeometry that breaks down, spacetime singularities are often viewed asan end, or “edge”, of spacetime itself. Numerousdifficulties, however, arise when one tries to make this notion moreprecise. Breakdowns in other physical structures pose other problems,just as difficult. Our current theory of spacetime, generalrelativity, not only allows for singularities, but tells us that theyare unavoidable in some real-world circumstances. Thus we apparentlyneed to understand the ontology of singularities if we are to graspthe nature of space and time in the actual universe. The possibilityof singularities also carries potentially important implications forthe issues of physical determinism and the scope of physical laws.

Black holes are regions of spacetime from which nothing, not evenlight, can escape. A typical black hole is the result of thegravitational force becoming so strong that one would have to travelfaster than light to escape its pull. Such black holes genericallycontain a spacetime singularity at their center; thus we cannot fullyunderstand a black hole without also understanding the nature ofsingularities. Black holes, however, raise several additionalconceptual problems and questions on their own. When quantum effectsare taken into account, black holes, although they are nothing morethan regions of spacetime, appear to become thermodynamical entities,with a temperature and an entropy. This seems to point to a deep andhitherto unsuspected connection among our three most fundamentaltheories, general relativity, quantum field theory and thermodynamics.It is far from clear, however, what it may mean to attributethermodynamical properties to black holes. At the same time, some ofthese thermodynamical properties of black holes now seem amenable todirect testing in terrestrial laboratories by observing the behaviorof “analogue” systems composed of ordinary material. Thisall raises problems about inter-theory relations, in particular aboutrelations between the “same” quantity as it appears indifferent theories. It also bears on the meaning and status of theSecond Law of thermodynamics, with possible implications forcharacterizing a cosmological arrow of time.

Finally, the entropy attributed to black holes seems to be a quantumgravitational property, since both the gravitational constant andPlanck’s constant appear in the standard formula for black holeentropy. Indeed, as purely gravitational entities with strikingquantum properties, what we know about black holes lies at the heartof and guides many attempts to formulate a theory of quantum gravity.This has led to a debate over what seemingly fundamental physicalprinciples are likely to be preserved in, or violated by, a fullquantum theory of gravity.

Because so few philosophers have worked on these issues, manyquestions and problems of great possible interest have not beeninvestigated philosophically at all; others have had only the bareststarts made on them; consequently, several sections discussed in thisarticle merely raise questions and point to problems that deservephilosophical attention. The field is wide open for expansive andintensive exploration.

All the technical material required to delve more deeply into thesubject of this entry can be found in any of a number of excellentclassic and recent sources, including: Hawking and Ellis (1973);Geroch and Horowitz (1979); Wald (1984, 1994); Broutet al.(1995); Malament (2007, 2012); and Manchak (2013, 2020). The readerunfamiliar with general relativity may find it helpful to review theHole Argument entry’sModern Spacetime Theories: A Beginner’s Guide, which presents a brief and accessible introduction to the concepts ofa spacetime manifold, a metric, and a worldline.

• 1. Spacetime Singularities
  • 1.1. Path Incompleteness
  • 1.2. Boundary Constructions
  • 1.3. Curvature Pathology
• 2. The Significance of Singularities
  • 2.1. Definitions and Existence of Singularities
  • 2.2. The Breakdown of General Relativity?
• 3. Black Holes
  • 3.1. Standard Definition and Properties
  • 3.2. Observations of Black Holes
  • 3.3. The Most Perfect Objects in the Cosmos
  • 3.4. Quasi-Local Black Holes
  • 3.5. Different Definitions of Black Holes
• 4. Naked Singularities, the Cosmic Censorship Hypothesis, and Indeterminism
• 5. Black Holes and Thermodynamics
  • 5.1. The Classical Laws of Black Holes
  • 5.2. Black Hole Thermodynamics
  • 5.3. What Is Black Hole Entropy?
  • 5.4. The Generalized Second Law of Thermodynamics
    • 5.4.1. A Dizzying Variety of Proofs
    • 5.4.2. Possible Violations
    • 5.4.3. Entropy Bounds and the Holographic Principle
    • 5.4.4. Possible Consequences of the Generalized Second Law
  • 5.5. General Gravitational Entropy
• 6. Black Holes and Quantum Theory
  • 6.1. Hawking Radiation
  • 6.2. Information Loss Paradox
  • 6.3. A Path to Quantum Gravity?
• 7. Analogue Black Holes and Hawking Radiation
• Bibliography
  • Bibliography: Philosophy
  • Bibliography: Physics
  • Bibliography: Philosophy Reference
  • Bibliography: Physics Reference
• Academic Tools
• Other Internet Resources
• Related Entries

1. Spacetime Singularities

General relativity, Einstein’s theory of space, time, andgravity, allows for the existence of singularities. Everyone agrees onthis. When it comes to the question of how, precisely, singularitiesare to be defined, however, there is widespread disagreement.Singularities in some way signal a breakdown of the geometry ofspacetime itself, but this presents an obvious difficulty in referringto a singularity as a “thing” that resides at somelocation in spacetime: without a well-behaved geometry, therecan be no location. For this reason, some philosophers and physicistshave suggested that we should not speak of “singularities”at all, but rather of “singular spacetimes”. In thisentry, the two formulations will generally be treated as equivalent,but the distinction will be highlighted when it becomessignificant.

Singularities are often conceived of metaphorically as akin to a tearin the fabric of spacetime. The most common attempts to definesingularities center on one of two core ideas that this image readilysuggests. The first is that a spacetime has a singularity if itcontains an incomplete path, one that cannot be continuedindefinitely, but draws up short, as it were, with no possibility ofextension. (“Where is the path supposed to go after it runs intothe tear? Where did it come from when it emerged from thetear?”) The second is that a spacetime is singular just in casethere are points “missing from it”. (“Where are thespacetime points that should be where the tear is?”)

Another common thought, often adverted to in discussion of the twoprimary notions, is that singular structure, whether in the form ofmissing points or incomplete paths, must be related to pathologicalbehavior of some sort in the singular spacetime’s curvature,that is, the fundamental deformation of spacetime that manifestsitself as “the gravitational field”. For example, somemeasure of the intensity of the curvature (“the strength of thegravitational field”) may increase without bound as onetraverses the incomplete path.

In recent years it was realized that there is another kind of singularbehavior that spacetimes may manifest, distinct conceptually andphysically from the idea that singularities come in the form ofincomplete curves or missing points. These are known as ‘suddensingularities’, and are particularly important in cosmologicalcontexts. Besides their intrinsic interest, they also call intoquestion much of the standard, traditional conceptions and claims madeabout singular structure in general relativity.

Finally, there is considerable disagreement over the significance ofsingularities. Many physicists believe that general relativity’sprediction of singular structure signals a serious deficiency in thetheory: singularities are an indication that the description offeredby general relativity is breaking down. Others believe thatsingularities represent an exciting new possibility for physicists toexplore in astrophysics and cosmology, holding out the promise ofphysical phenomena differing so radically from any that we have yetexperienced as to signal, in our attempt to observe, quantify andunderstand them, a profound advance in our comprehension of thephysical world.

Each of these issues will be considered in turn below.

The history of singular structure in general relativity isfascinating, with debate over it dating back to the earliest days ofthe theory, but discussion of it is beyond the scope of this article;the interested reader should consult Earman (1999), Earman andEisenstaedt (1999), Senovilla and Garfinkle (2015), and referencestherein.

1.1. Path Incompleteness

While there are competing definitions of spacetime singularities, themost central and widely accepted criterion rests on the possibilitythat some spacetimes contain incomplete, inextendible paths. Indeed,the rival definitions (in terms of missing points or curvaturepathology), as we will see, rely on the notion of pathincompleteness.

A path in spacetime is a continuous chain of events through space andtime. If you snap your fingers continually, without pause, then thecollection of snaps forms a path. The paths used in the most importantsingularity theorems represent possible trajectories of particles andobservers. Such paths are known as world-lines; they consist of thecontinuous sequence of events instantiated by an object’sexistence at each instant of its lifetime. That the paths beincomplete and inextendible means, roughly speaking, that, after afinite amount of time, a particle or observer following that pathwould “run out of world”, as it were—it would hurtleinto the tear in the fabric of spacetime and vanish. (SeeFigure 1.) Alternatively, a particle or observer could leap out of the tear tofollow such a path. While there is no logical or physicalcontradiction in any of this, it appears on the face of it physicallysuspect for an observer or a particle to be allowed to pop in or outof existence right in the middle of spacetime, so to speak—ifthat does not suffice for concluding that the spacetime is singular,it is difficult to imagine what else would. At the same time as thiscriterion for singularities was first proposed, the ground-breakingwork predicting the existence of such pathological paths (Penrose1965, 1968; Hawking 1965, 1966a, 1966b, 1966c, 1966d; Geroch 1966,1967, 1968b, 1970; Hawking and Penrose 1970) produced no consensus onwhat ought to count as a necessary condition for singular structureaccording to this criterion, and thus no consensus on a fixeddefinition for it.

Figure 1: a tear in spacetime

In this context, an incomplete path in spacetime is one that is bothinextendible and of finite proper length, which means that anyparticle or observer traversing the path would experience only afinite interval of existence that in principle cannot be continued anylonger. For this criterion to do the work we want it to, however, wewill need to limit the class of spacetimes under discussion.Specifically, we shall be concerned with spacetimes that are maximallyextended (or just ‘maximal’, for short). In effect, thiscondition says that one’s representation of spacetime is“as big as it possibly can be”. There is, from themathematical point of view, no way to treat the spacetime as being aproper subset of a larger, more extensive spacetime. (Seefigure 2.)

Figure 2: a non-maximal spacetime

If there is an incomplete path in a spacetime, goes the thinkingbehind the requirement, then perhaps the path is incomplete onlybecause one has not made one’s model of spacetime big enough. Ifone were to extend the spacetime manifold maximally, then perhaps thepreviously incomplete path could be extended into the new portions ofthe larger spacetime, indicating that no physical pathology underliesthe incompleteness of the path. The inadequacy would merely haveresided in the incomplete physical model we had been using torepresent spacetime.

An example of a non-maximally extended spacetime can be easily had,along with a sense of why they intuitively seem in some way or otherdeficient. For the moment, imagine spacetime is only two-dimensional,and flat, like a sheet of paper. Now, excise from somewhere onthis plane a closed set of any shape whatsoever. Any path that hadpassed through one of the points in the removed set is nowincomplete.

In this case, the maximal extension of the resulting spacetime isobvious, and does indeed fix the problem of all such incomplete paths:re-incorporate the previously excised set. (SeeFigure 3.) The seemingly artificial and contrived nature of such examples, alongwith the ease of rectifying them, seems to militate in favor ofrequiring spacetimes to be maximal. Also, inextendibility is sometimesargued for on the grounds that there is no known physical process thatcould cause spacetime to draw up short, as it were, and not continueon as it could have, were it to have an extension (Clarke 1975; Ellisand Schmidt 1977).

Figure 3: a non-maximal spacetime mademaximal by filling its holes

In recent important work, Manchak has questioned the need and even thereasonableness of requiring spacetimes to be maximal (i.e.,inextendible), pointing out problems with the condition’sepistemic status (Manchak 2011), its conceptual cogency (Manchak2016a), and its metaphysical character (Manchak 2016b). Becauseinextendibility is the most common assumption made in the physicsliterature when singular structure is discussed, however, we willcontinue to assume it for the purposes of this discussion,Manchak’s interesting arguments notwithstanding.(Manchak’s arguments will be discussed further insection 4 below.)

Once we have established that we are interested in maximal spacetimes,the next issue is what sort of path incompleteness is relevant forsingularities. Here we find a good deal of controversy. Criteria ofincompleteness typically look at how some parameter naturallyassociated with the path (such as its proper length) grows. Onegenerally also places further restrictions on the paths that oneconsiders—for example, one may rule out paths that could betraversed only by particles undergoing unbounded acceleration in afinite period of time. A spacetime, then, is said to besingular if it possesses a path such that the specifiedparameter associated with that path cannot increase without bound asone traverses the entirety of the maximally extended path. The idea isthat the parameter at issue will serve as a marker for some manifestlyphysical property, such as the time experienced by a particle orobserver, and so, if the value of that parameter remains finite alongthe whole path, then we have run out of path in a finite amount oftime, as it were. We have hit an edge or a “tear” inspacetime.

For a path that is everywhere timelike,i.e., that does notinvolve speeds at or above that of light, it is natural to take as theparameter the proper time a particle or observer would experiencealong the path, that is, the time measured along the path by a naturalclock, such as one based on the vibrational frequency of an atom.(There are also natural choices that one can make for spacelike paths,e.g., those that consist of points at a single“time”, and for null paths, those followed by lightsignals; however, because the spacelike and null cases add yet anotherlevel of technical complexity, we shall not discuss them here.) Thephysical interpretation of this sort of incompleteness for timelikepaths is more or less straightforward: a timelike path incomplete withrespect to proper time in the future direction would represent thepossible trajectory of a massive body that would never age beyond acertain point in its existence. (An analogous statement can be made,mutatis mutandis, if the path were incomplete in the pastdirection.)

We cannot, however, simply stipulate that a maximal spacetime issingular just in case it contains paths of finite proper time thatcannot be extended. Such a criterion would imply that even the flatspacetime described by special relativity is singular, which is surelyunacceptable. This would follow because, even in flat spacetime, thereare timelike paths with unbounded acceleration that have only a finiteproper time and are also inextendible.

The most obvious option is to define a spacetime as singular if andonly if it contains incomplete, inextendible timelike geodesics,i.e., paths representing the possible trajectories ofinertial observers, those in free-fall. This criterion, however, seemstoo permissive, in that it would count as non-singular some spacetimeswhose geometry seems otherwise pathological. For example, Geroch(1968c) describes a spacetime that is geodesically complete and yetpossesses an incomplete timelike path of bounded totalacceleration—that is to say, an inextendible path in spacetimetraversable by a rocket with a finite amount of fuel, along which anobserver could experience only a finite amount of proper time. Surelythe intrepid astronaut in such a rocket, who would never age beyond acertain point, would have just cause to complain that something wassingular about this spacetime.

When deciding whether a spacetime is singular, therefore, we want adefinition that is not restricted to geodesics. We need, however, someway of overcoming the fact that non-singular spacetimes includeinextendible paths of finite proper length that are notprimafacie pathological (e.g., flat spacetimes withinextendible paths of unbounded total acceleration). The most widelyaccepted solution to this problem makes use of a slightly different,technically complex notion of length, known as ‘generalizedaffine length’ (Schmidt 1971).^[1] Unlike proper time, this generalized affine length depends on somearbitrary choices. (Roughly speaking, the length will vary dependingon the coordinates one chooses to compute it; seenote 1.) If the length is infinite for one such choice, however, it will beinfinite for all other choices. Thus the question of whether a pathhas a finite or infinite generalized affine length is a well-definedquestion, and that is all we will need.

The definition that has won the most widespreadacceptance—leading Earman (1995, p. 36) to dub this thesemiofficial definition of singularities—is thefollowing:

A spacetime issingular if and only if it is maximal andcontains an inextendible path of finite generalized affine length.

To say that a spacetime is singular then is to say that there is atleast one maximally extended path that has a bounded (generalizedaffine) length. To put it another way, a spacetime is nonsingular whenit is complete in the sense that the only reason any given path mightnot be extendible is that it’s already infinitely long (in thistechnical sense).

The chief problem facing this definition of singularities is that thephysical significance of generalized affine length is opaque, and thusit is unclear what the physical relevance of singularities, defined inthis way, might be. It does nothing, for example, to clarify thephysical status of the spacetime described by Geroch (geodesicallycomplete but containing incomplete paths of bounded totalacceleration), which it classifies asnon-singular, as thecurve at issue indeed has infinite generalized affine length, eventhough it has only a finite total proper time (to the future). The newcriterion does nothing more than sweep the troubling aspects of suchexamples under the rug. It does not explain why we ought not to takesuchprima facie puzzling and troubling examples asphysically pathological; it merely declares by fiat that they arenot.

Manchak (2014a) proposed a condition spacetimes may satisfy,manifestly relevant to the issue of what characterizes singularbehavior, which he calls ‘effective completeness’. Theidea is to try to give what may be thought of as a quasi-localcharacterization of path incompleteness.^[2] Manchak (2014a, p. 1071) describes the intended physical significanceas follows: “If a space-time fails to be effectively complete,then there is a freely falling observer who never records someparticular watch reading but who ‘could have’ in the sensethat nothing in her vicinity precludes it.” This condition hasthe pleasant property of being logically intermediate between thecondition of geodesic incompleteness for spacetime, on the one hand,generally conceded to be too strong to capture the general idea ofsingular behavior (because of examples such that of Geroch 1968c,discussed above), and, on the other hand, the condition of beingextendible, generally conceded to be too weak, for effectivecompleteness is implied by geodesic completeness and in turn impliesinextendibility. While this new condition appears promising as a clearand useful characterization of singular structure (in the sense ofpath incompleteness), and does so in a way that avoids the problems ofphysical opacity plaguing the semi-official definition, one wants toknow, among other things, whether it can be used to prove noveltheorems with the same physical depth and reach as the standardsingularity theorems (Penrose 1965, 1968; Hawking 1965, 1966a, 1966b,1966c, 1966d; Geroch 1966, 1967, 1968b, 1970; Hawking and Penrose1970), and whether it can shed real light on the philosophical issuesdiscussed below insection 2. So far, “hole freeness” conditions were unable to providethat.

So where does all this leave us? The consensus seems to be that, whileit is easy in specific examples to conclude that incomplete paths ofvarious sorts represent singular structure, no entirely satisfactory,strict definition of singular structure in their terms has yet beenformulated (Joshi 2014). Moreover, spacetimes can evince entirelydifferent kinds of behavior that manifestly are singular in animportant sense, and yet which are independent of path incompleteness(as discussed in the Supplementary Document:Non-Standard Singularities). For a philosopher, the issues offer deep and rich veins for thosecontemplating, among other matters, the role of explanatory power inthe determination of the adequacy of physical theories, the role ofmetaphysics and intuition in the same, questions about the nature ofthe existence attributable to physical entities in spacetime and tospacetime itself, and the status of mathematical models of physicalsystems in the determination of our understanding of those systems asopposed to the mere representation of our knowledge of them. All ofthese issues will be touched upon in the following.

1.2. Boundary Constructions

We have seen that one runs into difficulties if one tries to definesingularities as “things” that have locations, and howsome of those difficulties can be avoided by defining singularspacetimes using the idea of incomplete paths. It would be desirablefor many reasons, however, to have a characterization of a spacetimesingularity in general relativity as, in some sense or other, aspatiotemporal “place”. If one had a precisecharacterization of a singularity based on points that are missingfrom spacetime, one might then be able to analyze the structure of thespacetime “locally at the singularity”, instead of takingtroublesome, perhaps ill-defined, limits along incomplete paths. Manydiscussions of singular structure in relativistic spacetimes,therefore, are premised on the idea that a singularity represents apoint or set of points that in some sense or other is missing from thespacetime manifold, that spacetime has a “hole” or“tear” in it that we could fill in, or patch, by attachinga boundary to it.

In trying to determine whether an ordinary web of cloth has a hole init, for example, one would naturally rely on the fact that the webexists in space and time. In this case one can point to a hole in thecloth by specifying points of space at a particular moment of time notcurrently occupied by any of the cloth, but which would complete thecloth were they so occupied. When trying to conceive of a singularspacetime, however, one does not have the luxury of imagining itembedded in a larger space with respect to which one can say there arepoints missing from it. In any event, the demand that the spacetime bemaximal rules out the possibility of embedding the spacetime manifoldin any larger spacetime manifold of any ordinary sort. It would seem,then, that making precise the idea that a singularity is a marker ofmissing points ought to involve some idea of intrinsic structuralincompleteness in the spacetime manifold rather than extrinsicincompleteness with respect to an external structure.

The most obvious route, especially in light of the previousdiscussion, and the one most often followed, is to define a spacetimeto have points missing from it if and only if it contains incomplete,inextendible paths, and then try to use these incomplete paths toconstruct in some fashion or other new, properly situated points forthe spacetime, the addition of which will make the previouslyinextendible paths extendible. These constructed points would then beour candidate singularities. Missing points on this view wouldcorrespond to a boundary for a singular spacetime—actual pointsof a (non-standard) extended spacetime at which paths incomplete inthe original spacetime would terminate. (We will, therefore, alternatebetween speaking ofmissing points and speaking ofboundary points, with no difference of sense intended.) Thegoal then is to construct this extended space using the incompletepaths as one’s guide.

Now, in trivial examples of spacetimes with missing points such as theone offered before, flat spacetime with a closed set excised from it,one does not need any technical machinery to add the missing pointsback in. One can do it by hand. Many spacetimes with incomplete paths,however, do not allow missing points to be attached in any obvious wayby hand, as that example does. For this program to be viable, which isto say, in order to give substance to the idea that there really arepoints that in some sense ought to have been included in the spacetimein the first place, we require a physically natural completionprocedure that can be applied to incomplete paths in arbitraryspacetimes. There are several proposals for such a construction(Hawking 1966c, Geroch 1968a, Schmidt 1971).^[3]

Several problems with this kind of program make themselves feltimmediately. Consider, for example, a spacetime representing the finalstate of the complete gravitational collapse of a sphericallysymmetric body resulting in a black hole. (Seesection 3 below for a description of black holes in general, andFigure 6 for a representation of a body collapsing to form a black hole.) Inthis spacetime, any timelike path entering the black hole willnecessarily be extendible for only a finite amount of propertime—it then “runs into the singularity” at thecenter of the black hole. In its usual presentation, however, thereare no obvious points missing from the spacetime at all. By anystandard measure, as a manifold in its own right it is as complete asthe Cartesian plane, excepting only the existence of incompletecurves, no class of which indicates by itself a place in the manifoldat which to add a point so as to make the paths in the class complete.Likewise, in our own spacetime every inextendible, past-directedtimelike path is incomplete (and our spacetime is singular): they allrun into the Big Bang. Insofar as there is no moment of time at whichthe Big Bang occurred (no moment of time at which time began, so tospeak), there is no point to serve as the past endpoint of such apath. We can speak of the cosmic epoch, the time after the Big Bang.That makes it easy to imagine that cosmic time zero is some initialevent. That, however, is an illusion of our labeling. Cosmic time“zero” is a label attached to no event. If instead we hadlabeled epochs with the logarithm of cosmic time, then the imaginarymoment of the Big Bang would be assigned the label of minus infinityand its fictional character would be easier to accept. (One can makethe point a little more precise: the global structure of our universe,as modeled by our best cosmological theories, is essentially the sameas a well known mathematical space, either \(\mathbb{R}^4\) or\(\mathbb{S}^3 \times \mathbb{R}\), which are both complete andinextendible as manifolds independent of any spacetime metricalstructure, in every reasonable sense of those terms.)

Even more troublesome examples are given by topologically compactregions of spacetimes containing incomplete, inextendible paths, as ina simple example due to Misner (1967). In a sense that can be madeprecise, compact sets, from a topological point of view,“contain every point they could possibly be expected tocontain”, one manifestation of which is that a compact manifoldcannot be embedded as an open submanifold of any other manifold, anecessary prerequisite for attaching a boundary to a singularspacetime. It is not only with regard to the attachment of a boundary,however, that compact sets already contain all points they possiblycould: every sequence of points in a compact set has a subsequencethat converges to a point in the set. Non-convergence of sequences isthe standard way that one probes geometrical spaces for“missing” points that one can add in by hand, as it were,to complete the space; thus, compact sets, in this natural sense,cannot have any missing points.

Perhaps the most serious problem facing all the proposals forattaching boundary points to singular spacetimes, however, is that theboundaries often end up having physically pathological properties(Gerochet al. 1982): in a sense one can make precise, theboundary points end up being arbitrarily “near” to everypoint in the interior of the spacetime. (The argument, however,might be circumvented; a boundary construction with the best knownseparation properties can be found in Floreset al. 2016.)Attaching boundary points to our own universe, therefore, to make theBig Bang into a real “place”, ends up making the Big Bangarbitrarily close to every neuron in my brain. Far from makingtractable the idea of localizing singular structure in a physicallyfruitful way, then, all the proposals only seem to end up making theproblems worse.

The reaction to the problems faced by these boundary constructions isvaried, to say the least, ranging from blithe acceptance of thepathology (Clarke 1993), to the attitude that there is no satisfyingboundary construction currently available while leaving open thepossibility of better ones in the future (Wald 1984), to not evenmentioning the possibility of boundary constructions when discussingsingular structure (Joshi 1993, 2007b, 2014), to rejection of the needfor such constructions at all (Gerochet al. 1982; Curiel1999).

Nonetheless, many eminent physicists seem convinced that generalrelativity stands in need of such a construction, and have exertedextraordinary efforts in trying to devise one. This fact raisesseveral philosophical problems. Though physicists sometimes offer asstrong motivation the possibility of gaining the ability to analyzesingular phenomena locally in a mathematically well-defined manner,they more often speak in terms that strongly suggest they suffer ametaphysical itch that can be scratched only by the sharp point of alocalizable, spatiotemporal entity serving as the locus of theirtheorizing. Even were such a construction forthcoming, however, whatsort of physical and theoretical status could accrue to these missingpoints? They would not be idealizations of a physical system in anyordinary sense of the term, since they would not represent asimplified model of a system formed by ignoring various of itsphysical features, as, for example, one may idealize the modeling of afluid by ignoring its viscosity. Neither would they seem necessarilyto be only convenient mathematical fictions, as, for example, are thephysically impossible dynamical evolutions of a system one integratesover in the variational derivation of the Euler-Lagrange equations. Tothe contrary, as we have remarked, many physicists and philosophersseem eager to find such a construction for the purpose of bestowingsubstantive and clear ontic status on singular structure. What sortsof theoretical entities, then, could they be, and how could they servein physical theory?

While the point of this project may seem at bottom identical to thepath-incompleteness account discussed insection 1.1, insofar as singular structure will be defined by the presence ofincomplete, inextendible paths, there is a crucial conceptual andlogical difference between the two. Here, the existence of theincomplete path does not constitute the singular structure, but ratherserves only as a marker for the presence of singular structure in thesense of missing points: the incomplete path is incomplete because it“runs into a hole” in the spacetime that, were it filled,would allow the path to be continued; this hole is the singularstructure, and the points constructed to fill it constitute its locus.Indeed, every known boundary construction relies on the existence ofincomplete paths to “probe” the spacetime, as it were,looking for “places” where boundary points should beappended to the spacetime; the characterization of singular structureby incomplete paths seems, therefore, logically, perhaps evenconceptually, prior to that by boundary points, at least, again, forall known constructions of boundary points.

Currently, there seems to be even less consensus on how (and whether)one should define singular structure based on the idea of missingpoints than there is regarding definitions based on pathincompleteness. Moreover, this project also faces even more technicaland philosophical problems. For these reasons, path incompleteness isgenerally considered the default definition of singularities. For theremainder of this article, therefore, singular structure will beassumed to be characterized by incomplete, inextendible paths.

There is, however, one special case in which it seems a boundary canbe placed on singular spacetimes in such a way as to localize thesingularity in a physically meaningful way: for so-called conformalsingularities. Their properties are discussed at the end ofsection 1.3, and their physical and philosophical significance explored in moredetail insection 5.5.

1.3. Curvature Pathology

While path incompleteness seems to capture an important aspect of theintuitive picture of singular structure, it completely ignores anotherseemingly integral aspect of it: curvature pathology. If there areincomplete paths in a spacetime, it seems that there should be areason that the path cannot go further. The most obvious candidateexplanation of this sort is that something is going wrong with thedynamical structure of the geometry of spacetime, which is to say,with the curvature of the spacetime. This suggestion is bolstered bythe fact that local measures of curvature do in fact blow up as oneapproaches the singularity of a standard black hole or the Big Bangsingularity. There is, however, one problem with this line of thought:no species of curvature pathology we know how to define is eithernecessary or sufficient for the existence of incomplete paths. (For adiscussion of foundational problems attendant on attempts to definesingularities based on curvature pathology, see Curiel 1999; for arecent survey of technical issues, see Joshi 2014.)

To make the notion of curvature pathology more precise, we will usethe manifestly physical idea of tidal force. Tidal force is generatedby the difference in intensity of the gravitational field atneighboring points of spacetime. For example, when you stand, yourhead is farther from the center of the Earth than your feet, so itfeels a (practically negligible) smaller pull downward than your feet. Tidal forces are a physical manifestation of spacetime curvature,and one gets direct observational access to curvature by measuring theresultant relative difference in accelerations of neighboring testbodies. For our purposes, it is important that in regions of extremecurvature tidal forces can grow without bound.

It is perhaps surprising that the state of motion of an object as ittraverses an incomplete path (e.g., whether it isaccelerating or spinning) can be decisive in determining its physicalresponse to curvature pathology. Whether an object is spinning or not,for example, or accelerating slightly in the direction of motion, maydetermine whether the object gets crushed to zero volume along such apath or whether it survives (roughly) intact all the way along it, asshown by examples offered by Ellis and Schmidt (1977). Indeed, theeffect of the observer’s state of motion on his or herexperience of tidal forces can be even more pronounced than this.There are examples of spacetimes in which an observer cruising along acertain kind of path would experience unbounded tidal forces and so betorn apart, while another observer, in a certain technical senseapproaching the same limiting point as the first observer,accelerating and decelerating in just the proper way, would experiencea perfectly well-behaved tidal force, though she would approach asnear as she likes to the other fellow who is in the midst of beingripped to shreds.^[4]

Things can get stranger still. There are examples of incompletegeodesics contained entirely within a well-defined, bounded region ofa spacetime, each having as its limiting point an honest-to-goodnesspoint of spacetime, such that an observer freely falling along such apath would be torn apart by unbounded tidal forces; it can easily bearranged in such cases, however, that a separate observer, whoactually travels through the limiting point, will experience perfectlywell-behaved tidal forces.^[5] Here we have an example of an observer being ripped apart byunbounded tidal forces right in the middle of spacetime, as it were,while other observers cruising peacefully by could reach out to touchhim or her in solace during the final throes of agony. This examplealso provides a nice illustration of the inevitable difficultiesattendant on attempts to localize singular structure in the sensesdiscussed insection 1.2.

It would seem, then, that curvature pathology as characterized basedon the behavior of tidal forces is not in any physical sense awell-defined property of a region of spacetimesimpliciter.When we consider the physical manifestations of the curvature ofspacetime, the motion of the device that we use to probe a region (aswell as the nature of the device) becomes crucially important for thequestion of whether pathological behavior manifests itself. This factraises questions about the nature of quantitative measures ofproperties of entities in general relativity, and what ought to countas observable, in the sense of reflecting the underlying physicalstructure of spacetime. Because apparently pathological phenomena mayoccur or not depending on the types of measurements one is performing,it seems that purely geometrical pathology does not necessarilyreflect anything about the state of spacetime itself, or at least notin any localizable way. What then does it reflect, if anything? Muchwork remains to be done by both physicists and by philosophers in thisarea,i.e., the determination of the nature of physicalquantities in general relativity and what ought to count as anobservable with intrinsic physical significance. See Bertotti (1962),Bergmann (1977), Rovelli (1991, 2001 inOther Internet Resources, henceforth OIR, 2002), Curiel (1999) and Manchak (2009a) for discussion of manydifferent topics in this area, approached from several differentperspectives.

There is, however, one form of curvature pathology associated with aparticular form of an apparently important class of singularities thathas been clearly characterized and analyzed, that associated withso-called conformal singularities, also sometimes called isotropicsingularities (Goode and Wainwright 1985; Newman 1993a, 1993b; Tod2002). The curvature pathology of this class of singularities can beprecisely pinpointed: it occurs solely in the conformal part of thecurvature; thus, what is singular in one spacetime will notnecessarily be so in a conformally equivalent spacetime.^[6] This property allows for a boundary to be attached to the singularspacetime in a way that seems to be physically meaningful (Newman1993a, 1993b; Tod 2002). Many physicists hold that, in a sense thatcan be made precise, all “purely gravitational degrees offreedom” in general relativity are encoded in the conformalstructure (Penrose 1979; Gomeset al. 2011). Theseproperties, along with the fact that the Big Bang singularity almostcertainly seems to be of this form, make conformal singularitiesparticularly important for the understanding and investigation of manyissues of physical and philosophical interest in contemporarycosmology.

Finally, we should mention that general relativity admits even morekinds of singularities than those discussed so far! See theSupplementary Document:Non-Standard Singularities for an introduction.

2. The Significance of Singularities

When considering the implications of spacetime singularities, it isimportant to note that we have good reasons to believe that thespacetime of our universe is singular. In the late 1960s, Penrose,Geroch, and Hawking proved several singularity theorems, using pathincompleteness as a criterion (Penrose 1965, 1968; Hawking 1965,1966b, 1966c, 1966d; Geroch 1966, 1967, 1968b, 1970; Hawking andPenrose 1970). These theorems showed that if certain physicallyreasonable premises were satisfied, then in certain circumstancessingularities could not be avoided. Notable among these conditions isthe positive energy condition, which captures the idea that energy isnever negative. These theorems indicate that our universe began withan initial singularity, the Big Bang, approximately 14 billion yearsago. They also indicate that in certain circumstances (discussedbelow) collapsing matter will form a black hole with a centralsingularity. According to our best current cosmological theories,moreover, two of the likeliest scenarios for the end of the universeis either a global collapse of everything into a Big Crunchsingularity, or the complete and utter diremption of everything, downto the smallest fundamental particles, in a Big Rip singularity. (SeeJoshi 2014 for a recent survey of singularities in general, and Berger2014 for a recent survey of the different kinds of singularities thatcan occur in cosmological models.)

Should these results lead us to believe that singularities are real?Many physicists and philosophers resist this conclusion. Some arguethat singularities are too repugnant to be real. Others argue that thesingular behavior at the center of black holes and at the beginning(and possibly the end) of time indicates the limit of the domain ofapplicability of general relativity. Some are inclined to take generalrelativity at its word, however, and simply accept its prediction ofsingularities as a surprising but perfectly consistent account of thepossible features of the geometry of our world. (See Curiel 1999 andEarman 1995, 1996 for discussion and comparison of these opposingpoints of view.) In this section, we review these and related problemsand the possible responses to them.

2.1. Definitions and Existence of Singularities

Let us summarize the results ofsection 1: there is no commonly accepted, strict definition of singularity;there is no physically reasonable characterization of missing points;there is no necessary connection between singular structure, at leastas characterized by the presence of incomplete paths, and the presenceof curvature pathology; and there is no necessary connection betweenother kinds of physical pathology (such as divergence of pressure) andpath incompleteness.

What conclusions should be drawn from this state of affairs? Thereseem to be two basic kinds of response, illustrated by the views ofClarke (1993) and Earman (1995) on the one hand, and those of Gerochet al. (1982) and Curiel (1999) on the other. The formerholds that the mettle of physics and philosophy demands that we find aprecise, rigorous and univocal definition of singularity. On thisview, the host of philosophical and physical questions surroundinggeneral relativity’s prediction of singular structure would bestbe addressed with such a definition in hand, so as better to frame andanswer these questions with precision, and thus perhaps find other,even better questions to pose and attempt to answer. The latter viewis perhaps best summarized by a remark of Gerochet al.(1982): “The purpose of a construction [of ‘singularpoints’], after all, is merely to clarify the discussion ofvarious physical issues involving singular space-times: generalrelativity as it stands is fully viable with no precise notion of‘singular points’.” On this view, the specificphysics under investigation in any particular situation should dictatewhich definition of singularity to use in that situation if, indeed,any at all.

In sum, the question becomes the following: is there a need for asingle, blanket definition of singularity or does the urge for onebetray only an old Aristotelian, essentialist prejudice? This questionhas obvious connections to the broader question of natural kinds inscience. One sees debates similar to those canvassed above when onetries to find, for example, a strict definition of biological species.Clearly, part of the motivation for searching for a singleexceptionless definition is the impression that there is some realfeature of the world (or at least of our spacetime models) that we canhope to capture precisely. Further, we might hope that our attempts tofind a rigorous and exceptionless definition will help us to betterunderstand the feature itself. Nonetheless, it is not clear why weshould not be happy with a variety of types of singular structure,taking the permissive attitude that none should be considered the“right” definition of singularities, but each has itsappropriate use in context.

Even without an accepted, strict definition of singularity forrelativistic spacetimes, the question can be posed: what would it meanto ascribe existence to singular structure under any of the availableopen possibilities? It is not far-fetched to think that answers tothis question may bear on the larger question of the existence ofspacetime points in general (Curiel 1999, 2016; Lam 2007). (See theentries onThe Hole Argument andabsolute and relational theories of space and motion for discussions of the question of the existence of spacetimeitself.)

It would be difficult to argue that an incomplete path in a maximalrelativistic spacetime does not exist in at least some sense of theterm. It is not hard to convince oneself, however, that theincompleteness of the path does not exist at any particularpoint of the spacetime in the same way, say, as this glass of beerexists at this point of spacetime. If there were a point on themanifold where the incompleteness of the path could be localized,surely that would be the point at which the incomplete pathterminated. But if there were such a point, then the path could beextended by having it pass through that point. It is perhaps this factthat lies behind much of the urgency surrounding the attempt to definesingular structure as missing points.

The demand that singular structure be localized at a particular placebespeaks an old Aristotelian substantivalism that invokes the maxim,“To exist is to exist in space and time” (Earman 1995, p.28).Aristotelian substantivalism here refers to the ideacontained in Aristotle’s contention that everything that existsis a substance and that all substances can be qualified by theAristotelian categories, two of which are location in time andlocation in space. Such a criterion, however, may be inappropriate forfeatures and properties of spacetime itself. Indeed, one need notconsider anything sooutré as incomplete, inextendiblepaths in order to produce examples of entities that seem undeniably toexist in some sense of the term or other, and yet which cannot haveany even vaguely determined location in time and space predicated ofthem. Several essential features of a relativistic spacetime, singularor not, cannot be localized in the way that an Aristoteliansubstantivalist would demand. For example, the Euclidean (ornon-Euclidean) nature of a space is not something with a preciselocation. (See Butterfield 2006 for discussion of these issues.)Likewise, various spacetime geometrical structures (such as themetric, the affine structure, the topology, etc.) cannot be localizedin the way that the Aristotelian would demand, whether that demand befor localization at a point, localization in a precisely determinateregion, or even just localization in a vaguely demarcated region. Theexistential status of such entitiesvis-à-vis moretraditionally considered objects is an open and largely ignored issue(Curiel 1999, 2016; Butterfield 2006). Because of the way the issue ofsingular structure in relativistic spacetimes ramifies into almostevery major open question in relativistic physics today, both physicaland philosophical, it provides a peculiarly rich and attractive focusfor these sorts of questions.

An interesting point of comparison, in this regard, would be thenature of singularities in other theories of gravity besides generalrelativity. Weatherall’s (2014) characterization ofsingularities in geometrized Newtonian gravitational theory,therefore, and his proof that the theory accommodates theirprediction, may serve as a possible testing ground for ideas andarguments on these issues.

Many of these questions, in the end, turn upon the issue of whatconstitutes “physically reasonable” spacetime structure.General relativity admits spacetimes exhibiting a vast and variegatedmenagerie of structures and behaviors, even over and abovesingularities, that most physicists and philosophers would consider,in some sense or other, not reasonable possibilities for physicalmanifestation in the actual world. But what is to count as“reasonable” here: who is to decide, and on what basis(Curiel 1999)? Manchak (2011) has argued that there cannot be purelyempirical grounds for ruling out the seemingly unpalatable structures,for there always exist spacetimes that are, in a precise sense,observationally indistinguishable from our own (Malament 1977; Manchak2009a) that have essentially any set of properties one may stipulate.Norton (2011) argues that this constitutes a necessary failure ofinductive reasoning in cosmology, no matter what one’s accountof induction. Butterfield (2012) discusses the relation ofManchak’s results to standard philosophical arguments aboutunder-determination of theory by data.

The philosopher of science interested in the definition and status oftheoretical terms in scientific theories has at hand here a richpossible case-study, enlivened by the opportunity to watch eminentscientists engaged in fierce, ongoing debate over the definition of aterm—indeed, over the feasibility of and even need for definingit—that lies at the center of attempts to unify our mostfundamental physical theories, general relativity and quantum fieldtheory.

2.2. The Breakdown of General Relativity?

At the heart of all of our conceptions of a spacetime singularity isthe notion of some sort of failure: a path that disappears, pointsthat are torn out, spacetime curvature or some other physical quantitysuch as pressure whose behavior becomes pathological. Perhaps thefailure, though, lies not in the spacetime of the actual world (or ofany physically possible world), but rather in our theoreticaldescription of the spacetime. That is, perhaps we should not thinkthat general relativity is accurately describing the world when itposits singular structure—it is the theory that breaks down, notthe physical structure of the world.

Indeed, in most scientific arenas, singular behavior is viewed as anindication that the theory being used is deficient, at least in thesense that it is not adequate for modeling systems in the regime wheresuch behavior is predicted (Berry 1992). It is therefore common toclaim that general relativity, in predicting that spacetime issingular, is predicting its own demise, and that classicaldescriptions of space and time break down at black hole singularitiesand the Big Bang, and all the rest (Hawking and Ellis 1973; Hawkingand Penrose 1996). Such a view denies that singularities are realfeatures of the actual world, and rather asserts that they are merelyartifacts of our current, inevitably limited, physical theories,marking the regime where the representational capacities of the theoryat issue breaks down. This attitude is widely adopted with regard tomany important cases,e.g., the divergence of the Newtoniangravitational potential for point particles, the singularities in theequations of motion of classical electromagnetism for point electrons,the singular caustics in geometrical optics, and so on. No oneseriously believes that singular behavior in such models in thoseclassical theories represents truly singular behavior in the physicalworld. We should, the thought goes, treat singularities in generalrelativity in the same way.

One of the most common arguments that incomplete paths and non-maximalspacetimes are physically unacceptable, and perhaps the mostinteresting one, coming as it does from physicists rather than fromphilosophers, invokes something very like the Principle of SufficientReason: if whatever creative force responsible for spacetime couldhave continued on to create more of it, what possible reason couldthere have been for it to have stopped at any particular point(Penrose 1969; Geroch 1970)?^[7] An opponent of this view could respond that it implicitly relies on acertain picture of physics that may not sit comfortably with generalrelativity, that of the dynamical evolution of a system. An advocateof this viewpoint would argue that, from a point of view natural forgeneral relativity, spacetime does not evolve at all. It just sitsthere, once and for all, as it were, a so-called block universe(Putnam 1967; the entries ontime machines,time travel, andbeing and becoming in modern physics). If it happens to sit there non-maximally, well, so be it. This kindof response, however, has problems of its own, such as with therepresentation of our subjective experience, which seems inextricablytied up with ideas of evolution and change. Those sorts of problems,however, do not seem peculiar to this dispute, but arise from thecharacter of general relativity itself: “dynamicalevolution” and “time” are subtle and problematicconcepts in the theory no matter what viewpoint one takes (Stein 1968,1970, 1991).

One can produce other metaphysical arguments against the view thatspacetime must be maximal. To demand maximality may lead toBuridan’s Ass problems, for it can happen that global extensionsexist in which one of a given set of incomplete curves is extendible,but no global extension exists in whichevery curve in theset is extendible (Ellis and Schmidt 1977). Also, there may existseveral physically quite different global extensions: the spacetimecovered by the usual Schwarzschild coordinates outside theSchwarzschild radius, for instance, can be extended analytically toKruskal-Schwarzschild spacetime with a spacetime “tunnel”or “bridge” to an otherwise disconnected part of theuniverse (Hawking and Ellis 1973, sec. 5.5), or it can be extended toa solution representing the interior of a massive spherical body. Itis, in any event, difficult to know what to make of the invocation ofsuch overtly metaphysical considerations in arguments in this mosthard of all hard sciences. See Curiel (1999) and Earman (1996) forcritical survey of such arguments, and Doboszewski (2017) for a recentcomprehensive survey of all these issues, including discussion of themost recent technical results.

A common hope is that when quantum effects are taken into account inthe vicinity of such extreme conditions of curvature wheresingularities are predicted by the classical theory, the singularnature of the spacetime geometry will be suppressed, leaving only wellbehaved spacetime structure. Advocates of various programs of quantumgravity also argue that in such a complete, full theory, singularitiesof the kinds discussed here will not appear. Recent important work byWall (2013a, 2013b) shows that these hopes face serious problems. Wepick up these issues below, insection 5.4.4 andsection 6.3 respectively, for it is in those contexts that many of the explicitdebates play out over the limits of general relativity.

In any event, it is well to keep in mind that, even if singularitiesare observed one day, and we are able to detect regularity in theirbehavior of a sort that lends itself to formulation as physical law,it seems likely that this law will not be a consequence of generalrelativity but will rather go beyond its bounds in radical ways, for,as we have seen, general relativity by itself does not have anymechanism for constraining the possible behavior that singularstructure of various types may manifest. It is perhaps just thispossibility that excites a frisson of pleasure in those of thelibertine persuasion at the same time as it makes the prudish shudderwith revulsion.

For a philosopher, the issues mooted here offer deep and rich veinsfor those contemplating, among other matters: the role of explanatorypower in the assessment of physical theories; the interplay amongobservation, mathematical models, physical intuition and metaphysicalpredilection in the genesis of scientific knowledge; questions aboutthe nature of the existence attributable to physical entities inspacetime and to spacetime itself; and the role of mathematical modelsof physical systems in our understanding of those systems, as opposedto their role in the mere representation of our knowledge of them.

3. Black Holes

3.1. Standard Definition and Properties

The simplest picture of a black hole is that of a system whose gravityis so strong that nothing, not even light, can escape from it. Systemsof this type are already possible in the familiar Newtonian theory ofgravity. The escape velocity of a body is the velocity at which anobject would have to begin to travel to escape the gravitational pullof the body and continue flying out to infinity, without furtheracceleration. Because the escape velocity is measured from the surfaceof an object, it becomes higher if a body contracts and becomes moredense. (Under such contraction, the mass of the body remains the same,but its surface gets closer to its center of mass; thus thegravitational force at the surface increases.) If the object were tobecome sufficiently dense, the escape velocity could therefore exceedthe speed of light, and light itself would be unable to escape.

This much of the argument makes no appeal to relativistic physics, andthe possibility of such Newtonian black holes was noted in the late18th Century by Michell (1784) and Laplace (1796, part ii, p. 305).These Newtonian objects, however, do not precipitate the same sense ofcrisis as do relativistic black holes. Although light emitted at thesurface of the collapsed body cannot escape, a rocket with powerfulenough motors firing could still push itself free. It just needs tokeep firing its rocket engines so that the thrust is equal to orslightly greater than the gravitational force. Since in Newtonianphysics there is no upper bound on possible velocities, moreover, onecould escape simply by being fired off at an initial velocity greaterthan that of light.

Taking relativistic considerations into account, however, we find thatblack holes are far more exotic entities. Given the usualunderstanding that relativity theory rules out any physical processpropagating faster than light, we conclude that not only is lightunable to escape from such a body:nothing would be able toescape this gravitational force. That includes the powerful rocketthat could escape a Newtonian black hole. Further, once the body hascollapsed down to the point where its escape velocity is the speed oflight, no physical force whatsoever could prevent the body fromcontinuing to collapse further, for that would be equivalent toaccelerating something to speeds beyond that of light. Thus once thiscritical point of collapse is reached, the body will get ever smaller,more and more dense, without limit. It has formed a relativistic blackhole. Here is where the intimate connection between black holes andsingularities appears, for general relativity predicts that, underphysically reasonable and generic conditions, a spacetime singularitywill form from the collapsing matter once the critical point ofblack-hole formation is reached (Penrose 1965; Schoen and Yau 1983;Wald 1984).

For any given body, this critical stage of unavoidable collapse occurswhen the object has collapsed to within its so-called Schwarzschildradius, which is proportional to the mass of the body. Our sun has aSchwarzschild radius of approximately three kilometers; theEarth’s Schwarzschild radius is a little less than a centimeter;the Schwarzschild radius of your body is about 10^-27cm—ten times smaller than a neutrino and 10¹⁰ timessmaller than the scale characteristic of quark interactions. Thismeans that if you could collapse all the Earth’s matter down toa sphere the size of a pea, it would form a black hole.

It is worth noting, however, that one does not need an extremely highdensity of matter to form a black hole if one has enough mass. If allthe stars in the Milky Way gradually aggregate towards the galacticcenter while keeping their proportionate distances from each other,they will all fall within their joint Schwarzschild radius and so forma black hole long before they are forced to collide. Or if one has acouple hundred million solar masses of water at its standard density(1 gm/cm³)—so occupying in total a region of about10²⁷ cubic kilometers, the approximate size of the smallestsphere containing the orbit of Uranus—it will be containedwithin its Schwarzschild radius. (In this case, of course, the waterwould indeed eventually collapse on itself to arbitrarily highdensities.) Some supermassive black holes at the centers of galaxiesare thought to be even more massive than the example of the water, atseveral billion solar masses, though in these cases the initialdensity of the matter thought to have formed the black holes wasextraordinarily high.

According to the standard definition (Hawking and Ellis 1973; Wald1984), the event horizon of a black hole is the surface formed by thepoints of no return. That is, it is the boundary of the collection ofall events in the spacetime closest to the singularity at which alight signal can still escape to the external universe. Everythingincluding and inside the event horizon is the black hole itself. (Seesection 3.5 for a discussion of different ways to define a black hole, and theproblems these competing definitions raise.) For a standard(uncharged, non-rotating) black hole, the event horizon lies at theSchwarzschild radius. A flash of light that originates at an eventinside the black hole will not be able to escape, but will instead endup in the central singularity of the black hole. A light flashoriginating at an event outside of the event horizon will escape(unless it is initially pointed towards the black hole), but it willbe red-shifted strongly to the extent that it started near thehorizon. An outgoing beam of light that originates at an event on theevent horizon itself, by definition, remains on the event horizonuntil the temporal end of the universe.

General relativity tells us that clocks running at different locationsin a gravitational field will, in a sense that can be made precise,generally not agree with one another. In the case of a black hole,this manifests itself in the following way. Imagine someone falls intoa black hole, and, while falling, she flashes a light signal to usevery time her watch hand ticks. Observing from a safe distanceoutside the black hole, we would find the times between the arrival ofsuccessive light signals to grow larger without limit, because ittakes longer for the light to escape the black hole’sgravitational potential well the closer to the event horizon the lightis emitted. (This is the red-shifting of light close to the eventhorizon.) That is, it would appear to us that time were slowing downfor the falling person as she approached the event horizon. Theticking of her watch (and every other process as well) would seem togo ever more slowly as she approached ever more closely to the eventhorizon. We would never actually see the light signals she emits whenshe crosses the event horizon; instead, she would seem to be eternally“frozen” just above the horizon. (This talk of seeing theperson is somewhat misleading, because the light coming from theperson would rapidly become severely red-shifted, and soon would notbe practically detectable.)

From the perspective of the infalling person, however, nothing unusualhappens at the event horizon. She would experience no slowing ofclocks, nor see any evidence that she is passing through the eventhorizon of a black hole. Her passing the event horizon is simply thelast moment in her history at which a light signal she emits would beable to escape from the black hole. The concept of an event horizon isaglobal one that depends on the overall structure of thespacetime, and in particular on how processes physically evolve intothe indefinite future. Locally there is nothing noteworthy about thepoints on the event horizon. In particular, locating the event horizonby any combination of strictly local measurements is impossibleinprinciple, no matter how ingeniously the instruments are arrangedand precisely the measurements made. The presence of an event horizonin this global sense is a strictly unverifiable hypothesis. One neednot be a verificationist about scientific knowledge to be troubled bythis state of affairs (Curiel 2019). Indeed, the global nature of theevent horizon manifests in an even more striking way: they are“prescient”, in the sense that where the event horizon islocated today depends on what I will throw in the black hole tomorrow.How should a good empiricist feel about all of this?

The global and geometrical nature of black holes also raisesinteresting questions about the sense in which one may or should thinkof them as physical objects or systems (Curiel 2019). A black hole issimply a geometrically characterized surface in spacetime, with noordinary matter at the event horizon, and no other local feature thatwould allow one to detect it. The same questions as with singularities (section 2.1), therefore, force themselves on us here: in what sense, if any, shouldwe attribute existence to black holes, in so far as, consideredlocally, they are an undistinguished region of spacetime whosephysically important properties manifest only as global structure?

3.2. Observations of Black Holes

Because of the peculiar nature of black holes as physical systems, theattempt to observe them also raises interesting epistemic problemsabout,inter alia, under-determination of theoretical modelsby data, the way that theoretical assumptions play ineliminable rolesin the interpretation of data, and what it means at all to“observe” a physical system that is, in principle, able toemit no signal directly. Eckartet al. (2017) provide animportant survey of the issues in the context of determining whetherSagittarius A*, the center of our galaxy, is indeed a supermassiveblack hole. We also refer to Collmaret al. (1998) forthe record of a round-table discussion on these questions by a groupof theoreticians and observational astronomers. In recent yearsimportant novel lines of evidence involving black holes opened up,allowing tests of various aspects of classical black holes. Many ofthem involve long periods of data collection, sophisticatedcalibration and processing procedures. Moreover, modeling andtheoretical assumptions often play substantial roles in the generationof evidence. Many of these issues are explored in a recent collectedvolume on philosophy of astrophysics, see Boydetal. (2023).

In 2016 the LIGO collaboration “directly” detectedgravitational waves by precisely measuring variation in the strain ofthe interferometer’s arms. The signature indicated thegravitational waves were generated by a binary black-hole systemcoalescing (Abbottet al. 2016, and many subsequentdetections, at the moment about 90 total; see The Gravitational-waveTransient Catalog,OIR). An important aspect of these detections is their reliance ongravitational wave templates to match the expected signal with thedata, which in turns requires an extent of epistemic control of thedynamics of strong gravitational fields. This, currently, can only beinvestigated numerically, and requires patching together verydifferent modeling approaches and approximations. We refer to Elder(2023, 2025) for a philosophical discussion of the epistemicchallenges LIGO faces, and Patton (2020) for investigation of thenotions of confirmation and theory testing in this context. Anexcellent history of ups and downs in the searches for gravitationalwaves can be found in Kennefick (2007).

Figure 4: Plots of the first detectedgravitational wave event ever, GW150914. Top row shows strain data atthe two detectors; the signal arrived first at L1. Second row presentsnumerical relativity waveform with parameters matching those of theevent. Third row shows residuals after subtracting the numericalwaveform. Bottom row shows signal frequency increasing over time.Image source: Figure 1 of Abbott et al 2016, see p. 2 of that paperfor the complete description.

In 2019 and 2022 the Event Horizon Telescope collaboration presentedtwo highly publicized images of black holes, M87* in Messier 87 galaxy(Event Horizon Telescope Collaboration 2019) and Sagittarius A* (EventHorizon Telescope Collaboration 2022). These images are reconstructedfrom high frequency radio waves collected with a virtual Earth-sizedradio-telescope array. To appreciate the feat, the angular resolutionrequired for producing these images is comparable to taking anEarth-based photo of a hypothetical orange located on the surface ofthe Moon, assuming it emitted radio waves at 220 GHz. The central darkpart of EHT images is consistent with the “shadow”generated by a Kerr black hole. Future observations will allow formore precise tests of not only the black hole spacetimes, but also, orperhaps primarily, of associated astrophysical phenomena such asaccretion disks and astrophysical jets.

Figure 5: 2019 image of the supermassiveblack hole M87*. Shape and size of the central dark region can be usedto make inferences about the mass of the object. Many further tests ofgeneral relativity as well as of astrophysical frameworks can be donewith such images. Image source:EHT Press Release (April 10, 2019)

Similarly to gravitational wave astronomy, EHT methodology is a richand still underexplored area of philosophical study, in which manyphilosophy of science themes arise, including roles of simulations,relationship between reliability and opacity, distinction betweenexperiment and observation and their relative merits, and ways inwhich theory and measurement are intertwined. For example, EHT imagingprocedures invoke various forms of consilience or robustness, in whichconvergence of independent lines of evidence are seen as increasingconfidence in the conclusions (Doboszewski and Elder 2024). However,some aspects of both LIGO and EHT inferences are theory-laden or, ifyou wish, theory-mediated (Doboszewski and Elder 2025): numericalrelativity and simulations of relativistic plasma are, so far,essential for inferences about their targets. Philosophicaldiscussions of these topics often invoke themes from Smith’s (2014)magisterial account of theory testing in Newtonian gravity, but timewill tell whether this even more complex general relativistic looptruly closes. Skulberg and Sparre (2023) discuss the history of blackhole imaging with its roots in Jean-Pierre Luminet first hand-drawnimage of a realistic black hole from 1978; and Muhr (2024) provides afurther media studies perspective on these images. Moreover, some ofthe future extensions of the EHT observations will involve humanitiesscholars in various capacities (Galisonet al. 2023), fromweighing in on ethical factors in telescope site selection, throughways of structuring the collaboration, to explorations of moretraditional philosophy of science and foundations of physicsquestions.

Both present and future observations of black holes involvemulti-messenger astronomy, in which different types of signals emittedfrom the same region or event are combined. For example, just thegravitational wave event GW170817 and its electromagnetic follow upmeasurement strongly constrained many modified theories of gravity,and improved measurements of important physical parameters by aroundten orders of magnitude. Combining many lines of evidence raiseseven more philosophical questions concerning analogies betweenastrophysics and historical sciences, value of variety of evidence,falsification, and many others; see Abelson (2022) and Elder (2024)for a discussion of these issues.

Finally, a critical analysis of the term “direct”observation (or detection, evidence, etc.) was recently carried out bySkulberg and Elder (2025), who isolate six different notions ofdirectness used by astrophysicists in various contexts and for variouspurposes: 1. a strict view (observation of entity itself, rather thanits effect on some other physical system); 2. transmission of a signalwithout interference; 3. a closer-in view concerned with how close tothe entity a signal was generated; 4. the number of steps used in aninference; 5. unambiguous signatures, or placing tighter constraintson the source being observed; and 6. a notion relying on whether amodel of the target system was used (or whether the model of themeasurement was sufficient in making the detection claim).

And yet, much more remains to be done. To give just a few examples,whole lines of observations and modeling of black holes were notinvestigated by the philosophers at all, including: many of theinferences made in the X-ray astronomy; microlensing evidence forisolated stellar black holes; tidal disruption events when a stardramatically interacts with a supermassive black hole; and thewhole issue of primordial black holes.

3.3. The Most Perfect Objects in the Cosmos

One of the most remarkable features of relativistic black holes isthat they are purely gravitational entities: the prominent black-holespacetime models, Schwarzschild and Kerr, contain no matterwhatsoever. They are vacuum solutions to the Einstein field equation,which just means a solution in which the matter density is everywherezero. (Of course, one can also consider a black hole with matterpresent, as standard astrophysical models do for the supermassiveblack holes that are believed to live at the center of most galaxies,which are thought to be surrounded by strong magnetic fields andaccretion disks of super-heated matter.) In pre-relativistic physicswe think of gravity as a force produced by the mass associated withsome matter. In the context of general relativity, however, we do awaywith gravitational force, and instead postulate a curved spacetimegeometry that produces all the effects we standardly attribute togravity. One of the most characteristic features of general relativitythat sets it apart from Newtonian gravitational theory is that itadmits the possibility of such curvature (“gravitationaleffects”) in the absence of matter, such as at the boundary of ablack hole. Thus a black hole is not a thingin spacetime; itis instead a feature of spacetime itself.

A careful definition of a relativistic black hole will therefore relyonly on the geometrical features of spacetime. We will need to be alittle more precise about what it means to be “a region fromwhich nothing, not even light, can escape”. First, there willhave to be someplace to escapeto if our definition is tomake sense. The most common method of making this idea precise andrigorous employs the notion of escaping to infinity. The idea is thatif a particle or light ray cannot travel arbitrarily far from adefinite, bounded region in the interior of spacetime but must remainalways in the region, then that region is one of no escape, and isthus a black hole. The boundary of the region is the event horizon.Once a physical entity crosses the event horizon into the black hole,it never crosses it again.

Second, we will need a clear notion of the kind of geometry thatallows for escape, or makes such escape impossible. For this, we needthe notion of the causal structure of spacetime. At any event in thespacetime, the possible trajectories of all light signals form a cone(or, more precisely, the four-dimensional analogue of the boundary ofa cone). Since light travels at the fastest speed allowed in thespacetime, these cones map out the boundaries of the propagation ofpossible causal processes in the spacetime. If an occurrence at aneventA is able to causally affect another occurrence ateventB, there must be a continuous trajectory in spacetimefrom eventA to eventB such that the trajectorylies in or on the light cones of every event along it. (For morediscussion, see the Supplementary Document:Light Cones and Causal Structure.)

Figure 6 is a spacetime diagram of a sphere of matter collapsing to form ablack hole. The curvature of the spacetime is represented by thetilting of the light cones away from 45 degrees. Notice that the lightcones tilt inwards more and more as one approaches the center of theblack hole. The jagged line running vertically up the center of thediagram depicts the central singularity inside the black hole. As weemphasized in Section 1, this is not actually part of the spacetime,but might be thought of as the “place” where the structureof spacetime breaks down. Thus, one should not imagine the possibilityof traveling through the singularity; this would be as nonsensical assomething leaving the diagram (i.e., the spacetime)altogether.

Figure 6: a spacetime diagram of blackhole formation

What makes this a black hole spacetime is the fact that it contains aregion from which it is impossible to exit while traveling at or belowthe speed of light. This region is marked off by the events at whichthe outside edge of the forward light cone points straight upward. Asone moves inward from these events, the light cone tilts so much thatone is always forced to move inward toward the central singularity.This set of points of no return is, of course, the event horizon; andthe spacetime region inside it is the black hole. In this region, oneinevitably moves towards the singularity; the impossibility ofavoiding the singularity is just the impossibility of preventingourselves from moving forward in time. (Again, seesection 3.5 for a discussion of other ways to define a black hole.)

Notice that, as represented inFigure 6, the matter of the collapsing star eventually disappears into theblack hole singularity. All the details of the matter are thencompletely lost; all that is left is the geometrical properties of theblack hole. Astonishingly, those properties can be identified with asmall, fixed set of physical quantities. Indeed, the remarkableNo-Hair Theorems (Israel 1967, 1968; Carter 1971, 1973, 1997 [OtherInternet Resources]; Robinson 1975; Mazur 1982; Heusler 1996;Chruścielet al. 2012) make rigorous the idea that ablack hole in equilibrium is entirely characterized by just threenumbers,viz., its mass, its angular momentum, and itselectric charge.^[8] This has the remarkable consequence that no matter what theparticulars may be of any body that collapses to form a blackhole—it may be as intricate, complicated and Rococo as onelikes, composed of the most exotic materials—the final resultafter the system has settled down to equilibrium will be identical inevery respect to a black hole that formed from the collapse ofanyother body having the same total mass, angular momentum andelectric charge (Carter 1997 [Other Internet Resources]). Because ofthis extremity of simplicity, Chandrasekhar (1983, Prologue, p. xxiii)called black holes “the most perfect macroscopic objects… in the universe.” (The fact that their physical stateis entirely characterized by only three numbers plays an importantrole in the ascription of thermodynamical properties to black holes,discussed in5.2 below. But one should also be aware that although the no hairtheorems are valid when the matter side of the Einstein field equationconsists of electromagnetic fields, a generalized no hair conjectureencounters various counterexamples already in the classical theory;see section 5 of Chruścielet al. 2012.)

Remarkably, not only are black holes in and of themselves objects ofthe utmost simplicity. They enforce simplicity on all else in theuniverse as well, no matter how far away from themselves. In a sensethat can be made precise, one of the most basic structures of thespacetime manifold itself, its topology, is as simple as possibleeverywhere outside a well behaved black hole.^[9] This is known as the Topological Censorship Theorem (Friedmanetal. 1983; Chruściel and Wald 1994; Galloway 1995). As itsname suggests, it bears on the larger question of the CosmicCensorship Hypothesis (Galloway and Woolgar 1997), discussed insection 4 below. In itself, though, it raises fascinating questions about therelation of topological to metrical structure in a spacetime,questions almost completely unexplored by philosophers. (See Gerochand Horowitz 1979 for a long list of conceptual and technical problemsand questions about this relation.) For a philosopher interested inthe nature of spacetime, however, the way that its differentstructures relate to and constrain each other must be of fundamentalimportance.

3.4. Quasi-Local Black Holes

For the reasons discussed insection 3.1, the standard definition of a black hole, based on the idea of aglobal event horizon, has limited application to the modeling of realastrophysical systems (except in so far as one can idealize them asessentially isolated). In an attempt to rectify this situation,Hayward (1994) offered a generalized definition of a black hole, notrequiring any of the special global structures that the traditionaldefinition relies on. Hayward defines a black hole based on what hecalls a trapping horizon. This is, roughly speaking, a surface onwhich all inward-directed light rays are converging, and to which alloutward-directed light rays are tangent. This definition tries tocapture the idea that a black hole is a surface at which thegravitational intensity is such that not even light can escape: anylight ray incident on the surface the smallest bit inward will getsucked in; otherwise, light rays can be only tangent to the surface.The surface does not admit incident light rays traveling away from itsinterior. This definition has the virtue that the boundary of theblack hole now has clear, local physical significance in principle: anobserver could determine when she crossed it by making only localmeasurements. (More precisely, a team of synchronized observers, whosecombined instrumental reach encompasses the entire surface at a givenmoment of time, could jointly make this determination, with enoughbackground knowledge of the spacetime geometry outside theboundary.)

Ashtekaret al. (1999, 2000) offer a different, relatedgeneralization of the idea of a black hole, based on what they callisolated horizons. This definition is somewhat more restrictive thanHayward’s in so far as, as the name suggests, it requires thatno stress-energy cross such a horizon. Subsequent work by Ashtekar andKrishnan (2003), Ashtekar (2007) and Hayward (2006, inOIR, 2009) clarified the relationship between the two, showing that theisolated horizon can be considered, in a sense, a special case of thetrapping horizon. (See Hayward 2013 for a recent comprehensive review,and Faraoni 2013 for one with special attention to its relevance tocosmology.) For lack of a better term, we shall call black holesdefined by trapping or isolated horizons “quasi-local blackholes”. (‘Local’ because they are not global objectsin the sense that black holes as traditionally defined are, and‘quasi’ because they still can extend arbitrarily farthroughout spacetime.)

The status of these competing definitions of a quasi-local black holeand of the differences among them, and what their respective virtuesand demerits may be, appear to be open questions, though both Haywardand Ashtekaret al., in the works just cited, go some waytowards answering some of them by using their respective definitionsto prove generalizations of the so-called laws of black hole mechanics (section 5.1 below). Hayward also demonstrates that analogues to some of theclassical singularity theorems hold for his definition as well. Still,many questions remain open. To take one example, it is not clearwhether or not the new definitions coincide with the traditionaldefinition in those spacetimes in which the traditional definition canbe formulated, or whether collateral conditions must be met for thetwo to coincide. It is also not clear whether the analogues to theclassical No Hair Theorems hold using the new definitions or even whatthose analogues may be.

Perhaps the most fascinating feature of quasi-local black holes is thefact that, in a sense that can be made precise, they are“clairvoyant”: they are aware of and respond to changes inthe geometry in spacetime regions that they cannot be in causalcontact with (Bengtsson and Senovilla 2011). Indeed, they canencompass regions whose entire causal past is flat! This subjectexemplifies the exuberant weirdness that causal structure in generalrelativity can manifest.

3.5. Different Definitions of Black Holes

Besides the standard definition of a black hole based on the presenceof a global event horizon, and the quasi-local definitions justdiscussed, there is an enormous and greatly variegated menagerie ofdifferent definitions and conceptions of a black hole that physicistsin different fields (and sometimes those in the same field) use intheir day to day work, none agreeing with the standard or quasi-localdefinitions, many of them manifestly inconsistent with each other(Curiel 2019). However one views this situation, it is clear, as abrute fact about the practice of physics today, that there is nosingle definition of “black hole” that will serve allinvestigative purposes in all fields in which black holes are objectsof study.Table 1 lists the core concepts most commonly used in definitions andcharacterizations of black holes across several different fields ofphysics, sketched with only the broadest and crudest of brushes. Itshould be kept in mind that many investigators in each of these fieldsdo not use, or even accept as reasonable, what is given in thetable.

What seems to be the most common practice today is, during the courseof an investigation, to fix a list of important, characteristicproperties of and phenomena associated with black holes required forone’s purposes in the context of interest, and then to determinewhich of the known definitions imply the members of that list. If noknown definition implies the list, one either attempts to construct anew definition that does (and is satisfactory in other ways), or elseone concludes that there is an internal inconsistency in one’slist, which may already be of great interest to learn. Examining theway the idea of black holes are used across physics—inastrophysics, cosmology, classical general relativity, semi-classicalgravity, particle physics, various programs in quantum gravity, fluidmechanics, condensed matter, and analogue gravity—yields a listof potentially characteristic properties and phenomena, some subset ofwhich may plausibly be required or wanted in a characterization of ablack hole in a given investigative context (Curiel 2019):

• possesses a horizon that satisfies the four laws of black holemechanics;
• possesses a locally determinable horizon;
• possesses a horizon that is, in a suitable sense, vacuum;
• is vacuum with a suitable set of symmetries;
• defines a region of no escape, in some suitable sense, for someminimum period of time;
• defines a region of no escape for all time;
• is embedded in an asymptotically flat spacetime;
• is embedded in a topologically simple spacetime;
• encompasses a singularity;
• satisfies the No-Hair Theorem;
• is the result of evolution from initial data satisfying anappropriate Hadamard condition (stability of evolution);
• allows one to predict that final, stable states upon settling downto equilibrium after a perturbation correspond, in some relevantsense, to the classical stationary black hole solutions(Schwarzschild, Reissner-Nordström, Kerr, Kerr-Newman);
• agrees with the classical stationary black hole solutions whenevaluated in those spacetimes;
• allows one to derive the existence of Hawking radiation from someset of independent principles of interest (seesection 6.1);
• allows one to calculate in an appropriate limit, from some set ofindependent principles of interest, an entropy that accords with theBekenstein entropy (i.e., is proportional to the area of arelevant horizon, with corrections of the order of ℏ;—seesection 5.2 andsection 5.3);
• possesses an entropy that is, in some relevant sense,maximal;
• has a lower-bound on possible mass;
• is relativistically compact.

This list is not meant to be exhaustive. There are many other suchproperties and phenomena that might be needed for a given purpose. Itis already clear from this partial list, however, that no singledefinition can accommodate all of them. It is also clear fromexamining the literature, moreover, that, even within the samecommunities, different workers will choose different subsets of theseproperties for different purposes in their thinking about blackholes.

As in the case of singularities, these alternative definitions ofblack holes raise philosophical questions about the relations amongthe different definitions that attempt to capture different aspectsof, intuitively speaking, the “same kind” of physicalobject. One can, for instance, view the standard definition of a blackhole, with its global event horizon, as an extreme idealization of anisolated system (one with no neighboring systems at all), and thedefinitions based on isolated or trapping horizons as trying tocapture a more general, less idealized representation of an isolatedsystem, one that has neighboring systems at a finite remove, or arepresentation of a system that may be non-trivially interacting withother systems. For the looser, less precise definitions used byastrophysicists, for example, and some of the gestures at definitionsproposed in some programs of quantum gravity, however, it is difficultto know how even to begin to compare them to the precise global andquasi-local ones. It is simply not clear that the same type ofphysical system is being characterized.

This situation provides a fascinating case study, from both a physicaland a philosophical point of view, for questions about the nature ofidealization and de-idealization, and the definition of theoreticalentities more generally. On what grounds,e.g., could oneascertain the relative merits of each type of definition on its own,and each as proposed for a particular sort of investigation, in theabsence of empirical data? In what sense do the different definitionscharacterize the “same” type of physical system, if theydo so at all? Is there a need to settle on a single canonicaldefinition of a black hole? What would be gained or lost with orwithout one? The situation is closely analogous to that of the lack ofa canonical definition of a singularity, except it is even moreextreme here: the different definitions of singularities used bydifferent physicists are (almost always) not actuallyinconsistent with each other.

For the remainder of this encyclopedia entry, unless explicitly statedotherwise, when we speak of a black hole it should be understood thatwe mean one as determined by the standard definition of a global eventhorizon, because this is the one most often used in currentfoundational work.

4. Naked Singularities, the Cosmic Censorship Hypothesis, and Indeterminism

While spacetime singularities in general are frequently viewed withsuspicion, physicists often offer the reassurance that, even if theyare real, we expect most of them to be hidden away behind the eventhorizons of black holes. Such singularities therefore could not affectus unless we were actually to jump into the black hole. A nakedsingularity, on the other hand, is one that is not hidden behind anevent horizon. Such singularities appear much more threatening becausethey are uncontained, freely accessible to the rest of spacetime.

The heart of the worry is that singular structure seems to signify soprofound a breakdown in the fundamental structure of spacetime that itcould wreak havoc on any region of the universe that it were visibleto. Because the structures that break down in singular spacetimes arein general required for the formulation of our known physical laws,and of initial-value problems for individual physical systems inparticular, one such fear is that determinism would collapse entirelywherever the singular breakdown were causally visible. InEarman’s (1995, pp. 65–6) evocative conceit, nothing wouldseem to stop the singularity from disgorging any manner of unpleasantjetsam, from TVs showing Nixon’s Checkers Speech to old lostsocks, in a way completely undetermined by the state of spacetime inany region whatsoever. As a result, there could be no reasonableexpectation of determinism, nor even just predictability, for anyregion in causal contact with what it spews out.

One form that such a naked singularity could take is that of awhite hole, which is a time-reversed black hole. Imaginetaking a film of a black hole forming from the collapse of a massiveobject, say, a star, with light, dust, rockets, astronauts and oldsocks falling into it during its subsequent evolution. Now imaginethat film being run backwards. This is the picture of a white hole:one starts with a naked singularity, out of which might appear people,rockets, socks—anything at all—with eventually a starbursting forth. Absolutely nothing in the causal past of such a whitehole would determine what would pop out of it, since, as follows fromthe No Hair Theorems (section 3.3), items that fall into a black hole leave no trace on the futureoutside of it. (This description should feel familiar to the cannyreader: it is the same as the way that increase of entropy in ordinarythermodynamics as embodied in the Second Law makes retrodictionimpossible; the relationship of black holes to thermodynamics isdiscussed insection 5.) Because the field equation of general relativity does not pick out apreferred direction of time, if the formation of a black hole isallowed by the laws of spacetime and gravity, then those laws alsopermit white holes.^[10]

Roger Penrose (1969, 1973) famously suggested that although nakedsingularities are compatible with general relativity, in physicallyrealistic situations they will never form; that is, any process thatresults in a singularity will safely ensconce that singularity behindan event horizon. This conjecture, known as the Cosmic CensorshipHypothesis, has met with some success and popularity; however, it alsofaces several difficulties. As in our previous discussions ofsingularities and black holes, there are questions about how exactlyto formulate the hypothesis, and, once formulated, about whether ornot it holds in general relativity as a whole, or at least in somephysically reasonable subset of spacetimes—where, again,“physically reasonable” will likely be a vague andcontroversial notion.

Penrose’s original formulation relied on black holes: a suitablygeneric singularity will always be contained in a black hole (and socausally invisible outside the black hole). This is sometimes calledtheweak cosmic censorship, investigated in the context ofgravitational collapse. (The termstrong cosmiccensorship is used for a conjecture about determinism, oruniqueness of past and future evolution of solutions to the Einsteinfield equation.) As the counter-examples to various ways ofarticulating the weak cosmic censorship hypothesis haveaccumulated over the years, however, it has gradually been abandoned(Geroch and Horowitz 1979; Krolak 1986; Penrose 1998; Christodoulou1994, 1999; Joshiet al. 2002; Joshi 2003, 2007a; Joshi andMalafarina 2011a, 2011b). More recent approaches either begin with anattempt to provide necessary and sufficient conditions for cosmiccensorship itself, yielding an indirect characterization of a nakedsingularity as any phenomenon violating those conditions, or else theybegin with an attempt to provide a characterization of a nakedsingularity without reference to black holes and so conclude with adefinite statement of cosmic censorship as the absence of suchphenomena (Geroch and Horowitz 1979). The variety of proposals madeusing both approaches is too great to canvass here; the interestedreader is referred to Ringström (2010) for a review of thecurrent state of the art for standard black holes, Nielsen (2012) forcosmic censorship regarding Hayward’s quasi-local black holes (section 3.4), Ringström (2010) for a review of the bearing of theinitial-value problem in general relativity on cosmic censorship, andto Earman (1995, ch. 3) and Curiel (1999) for philosophical discussionof many of the earlier proposals. Manchak (2011) gives reasons forthinking that the question of providing a completely satisfactoryformulation of the Cosmic Censorship Hypothesis may never be settled,on the grounds that the idea of what counts as “physicallyreasonable” is not an empirically determinable question. Still,the possibility may remain open that there be several different,inequivalent formulations of the Cosmic Censorship Hypothesis, eachhaving its own advantages and problems, none “canonical”in a definitive sense, as may be the case for definitions ofsingularities and black holes themselves.

Recent mathematical physics literature tends to sidestep many of theseissues and formulates Cosmic Censorship as a dynamical problem,investigating it in the context of the initial value formulation ofgeneral relativity and with the use of partial differential equationsmethods. This assumes the causal condition of global hyperbolicity,but instead of seeing it as a fundamental assumption, one couldperhaps interpret it as a precondition for the validity of a certainstyle of mathematically precise analysis, and view this body of workas exploring the extent to which the initial value formulation remainsvalid. The aim, then, is proving that for generic initialdatasets, their well-behaved developments are inextendible. Manysubtleties of this statement are discussed in Landsman (2021); arecent overview can be found in Van de Moortel (2025). In isolation,some of these results might not seem very compelling, for example,concerning decay rate of a linearized scalar wave equation on a fixedbackground spacetime. However, many of them would be better seen asparts of a larger program of understanding the dynamics of theEinstein field equation in successively de-idealized situations. Aswith many topics discussed here, this philosophically woefullyunderexplored area of sophisticated mathematical physics often opensmore conceptual questions than it closes, including the issue ofsignificance of weak extensions of spacetimes, many standards ofstability at play, their interpretation (as epistemic or physical),and, ultimately, possible justification for the requirement ofstability as such.

Many curious features of general relativistic spacetimes arise invarious toy models, seemingly unphysical situations, or in separateregions of the solution space. So it is of quite some importance thatthe Kerr spacetime displays most of them in a single solution(Doboszewski 2022). More specifically, subextremal (i.e. oneswhich do not rotate too fast) Kerr black holes contain in theirinteriors Cauchy horizons. They are defined as the boundary of thedomain of dependence of a spatial slice: intuitively, events on andpast the Cauchy horizon cannot be determined solely on the basis ofphysical data defined on that slice. Any extension through this Cauchyhorizon will have the Malament-Hogarth property. That is, a futurecomplete timelike curve (and in fact a whole infinite exterior regionof the black hole) is contained in the timelike past of a pointlocated on the Cauchy horizon (see entry onSupertasks for a discussion of these spacetimes). And the most natural extension(with the same symmetries as the Kerr spacetime) contains closedtimelike curves. In this sense, indeterminism, supertasks, and timetravel arise jointly in this class of spacetimes. And yet, theexterior Kerr solution is the most plausible model of every isolatedrotating black hole in our universe! One hopes that these Cauchyhorizons are unstable as an artifact of the perfect axisymmetry, andif the Strong Cosmic Censorship is true, then these properties mightbe constrained. But validity of SCC remains open, and partialresults suggest that some versions of it may in fact be false invarious contexts (Hintz and Vasy 2017, Dafermos and Luk 2017, Cardosoet al. 2018). Perhaps some form of quantum effects needs tobe invoked to restore it (Hollands et al 2020). In any case, in theabsence of any conceivable observational data about the interior ofblack holes, the only means of approaching these issues aremathematical methods and conceptual analysis.

Issues of determinism, from an epistemic perspective, are intimatelybound up with the possibility of reliable prediction. (See the entryoncausal determinism.) The general issue of predictability itself in general relativity,even apart from the specific problems that singular structure mayraise, is fascinating, philosophically rich, and very much unsettled.One can make aprima facie strong argument, for example, thatprediction is possible in general relativity only in spacetimes thatpossess singularities (Hogarth 1997; Manchak 2013)! SeeGeroch (1977), Glymour (1977), Malament (1977), and Manchak (2008,2013) for discussion of these and many other relatedissues. Another area of investigation intimately related toissues of Cosmic Censorship in general, and issues of determinism ingeneral relativity in particular: whether or not spacetime is“hole-free”. For further discussion, see the SupplementaryDocument:Spacetimes With Holes.

Here again, as with almost all the issues discussed up to this pointin this entry regarding singularities and black holes, is an exampleof a sizable subculture in physics working on matters that have noclearly or even unambiguously defined physical parameters to informthe investigations and no empirical evidence to guide or even justconstrain them, the parameters of the debate imposed by and large bythe intuitions of a handful of leading researchers. From sociological,physical, and philosophical vantage points, one may well wonder, then,why so many physicists continue to work on it, and what sort ofinvestigation they are engaged in. Perhaps nowhere else in generalrelativity, or even in physics, can one observe such a delicateinterplay of, on the one hand, technical results, definitions andcriteria, and, on the other hand, conceptual puzzles and evenincoherence, largely driven by the inchoate intuitions of physicists.Not everyone views the situation with excitement or even equanimity,however: see Curiel (1999) for a somewhat skeptical discussion of thewhole endeavor.

5. Black Holes and Thermodynamics

The challenge of uniting quantum theory and general relativity in asuccessful theory of quantum gravity has arguably been the greatestchallenge facing theoretical physics for the past eighty years. Oneavenue that has seemed particularly promising is the attempt to applyquantum theory to black holes. This is in part because, as purelygravitational entities, black holes present an apparently simple butphysically important case for the study of the quantization ofgravity. Further, because the gravitational force grows without boundas one nears a standard black-hole singularity, one would expectquantum gravitational effects (which should come into play atextremely high energies) to manifest themselves in the interior ofblack holes.

In the event, studies of quantum mechanical systems in black holespacetimes have revealed several surprises that threaten to overturnthe views of space, time, and matter that general relativity andquantum field theory each on their own suggests or relies on. Sincethe ground-breaking work of Wheeler, Penrose, Bekenstein, Hawking, andothers in the late 1960s and early 1970s, it has become increasinglyclear that there are profound connections among general relativity,quantum field theory, and thermodynamics. This area of intersectionhas become one of the most active and fruitful in all of theoreticalphysics, bringing together workers from a variety of fields such ascosmology, general relativity, quantum field theory, particle physics,fluid dynamics, condensed matter, and quantum gravity, providingbridges that now closely connect disciplines once seen as largelyindependent.

In particular, a remarkable parallel between the laws of black holesand the laws of thermodynamics indicates that gravity andthermodynamics may be linked in a fundamental (and previouslyunimagined) way. This linkage strongly suggests, among many things,that our fundamental ideas of entropy and the nature of the Second Lawof thermodynamics must be reconsidered, and that the standard form ofquantum evolution itself may need to be modified. While thesesuggestions are speculative, they nevertheless touch on deep issues inthe foundations of physics. Indeed, because the central subject matterof all these diverse areas of research lies beyond the reach ofcurrent experimentation and observation, they are all speculative in away unusual even in theoretical physics. In their investigation,therefore, physical questions of a technically sophisticated natureare inextricable from subtle philosophical considerations spanningontology, epistemology, and methodology, again in a way unusual evenin theoretical physics.

Because this is a particularly long and dense section of the article,we begin with an outline of it.Section 5.1 states the laws of black holes in classical general relativity, andexpounds the formal analogy they stand in with the laws of ordinarythermodynamics.Section 5.2 briefly describes how taking account of quantum effects in theneighborhood of a black hole leads to the prediction of Hawkingradiation and the subsequent conclusion that the analogy with the lawsof ordinary thermodynamics is more than just formal, but represents atrue and intimate physical connection.Section 5.3 discusses the puzzles that arise when trying to understand theattribution of entropy to a black hole.Section 5.4 consists of several subsections, each examining a different puzzlingaspect of the so-called Generalized Second Law: the hypothesis thatthe total entropy of the universe, that of ordinary matter plus thatof black holes, never decreases. We conclude inSection 5.5 with a brief account of attempts to extend the attribution of aphysical entropy to gravitational systems more general than just blackholes.

5.1. The Classical Laws of Black Holes

Suppose one observes a quiescent black hole at a given moment,ignoring any possible quantum effects. As discussed above insection 3.3, there are three physical quantities of dynamical interest the blackhole possesses that are, more or less, amenable to measurement, andthat completely characterize the physical state of the black hole: itsmass, its angular momentum, and its electric charge. These quantities,like those of systems in classical mechanics, stand in definiterelation to each other as the black hole dynamically evolves, which isto say, they satisfy a set of equations characterizing their behavior.^[11]

Zeroth Law

The surface gravity of a stationary black hole is constant overits entire surface.

First Law

A change in the total mass of the black hole is determined in afixed way by changes in its area, angular momentum, and electriccharge, so that the total quantity is conserved.

Second Law (The Area Theorem)

The total surface area of a black hole never decreases.

Third Law

No physical process can reduce the surface gravity of a black holeto zero.

(A black hole isstationary if, roughly speaking, it does notchange over time; more precisely, it is stationary if its eventhorizon is generated by an asymptotically timelike Killing field.)

On the face of it, the Zeroth, First and Third Laws arestraightforward to understand. The Second Law, however, is not so“obvious” as it may at first glance appear. It may seemthat, because nothing can escape from a black hole once it hasentered, black holes can only grow larger or, at least, stay the samesize if nothing further falls in. This assumes, however, thatincreased mass always yields increased surface area as opposed to someother measure of spatial extent. Surprising as it may sound, it is thecase that, although nothing that enters a black hole can escape, it isstill possible to extract energy (i.e., mass) from a spinningblack hole, by means of what is known as the Penrose process (Penroseand Floyd 1971). It is therefore not obvious that one could not shrinka black hole by extracting enough mass-energy or angular momentum fromit. It also seems to be at least possible that a black hole couldshrink by radiating mass-energy away as gravitational radiation, orthat the remnant of two colliding black holes could have a smallersurface area than the sum of the original two. (This last possibilitycan now be confronted with observations in gravitational waveastronomy, and the area increase theorem seems to pass the test, atleast in some individual events; see Caberoet al 2019 for ananalysis of GW150914).

It is most surprising, therefore, to learn that the Second Law is adeep, rigorous theorem that follows only from the fundamentalmathematics of relativistic spacetimes (Hawking 1971), and does notdepend in any essential way on the particulars of relativisticdynamics as encapsulated in the Einstein field equation (Curiel 2017).This is in strict opposition to the Second Law in classicalthermodynamics, which stands as a more or less phenomenologicalprinciple derived by empirical generalization, perhaps justified insome sense by a “reduction” to statistical mechanics, withthe temporal asymmetry of entropy non-decrease argued to hold based onthe likelihood of initial states in conjunction with the forms ofdynamical evolution “physically permissible” for matterfields. (See the entry on thephilosophy of statistical mechanics.)

For those who know classical thermodynamics, the formal analogybetween its laws and the laws of black hole as stated should beobvious. (For exposition and discussion of the laws of classicalthermodynamics, see,e.g.: Fermi 1937 for a less technical,more physically intuitive approach; Fowler and Guggenheim 1939 for amore technical and rigorous one; and Uffink 2007 for a morehistorically and philosophically oriented one.) One formulation of theZeroth Law of thermodynamics states that a body in thermal equilibriumhas constant temperature throughout. The First Law is a statement ofthe conservation of energy. It has as a consequence that any change inthe total energy of a body is compensated for and measured by changesin its associated physical quantities, such as entropy, temperature,angular momentum and electric charge. The Second Law states thatentropy never decreases. The Third Law, on one formulation, statesthat it is impossible to reduce the temperature of a system to zero byany physical process. Accordingly, if in the laws for black holes onetakes ‘stationary’ to stand for ‘thermalequilibrium’, ‘surface gravity’ to stand for‘temperature’, ‘mass’ to stand for‘energy’, and ‘area’ to stand for‘entropy’, then the formal analogy is perfect.

Indeed, relativistically mass justis energy, so at least theFirst Law seems already to be more than just a formal analogy. Also,the fact that the state of a stationary black hole is entirelycharacterized by only a few parameters, completely independent of thenature and configuration of any micro-structures that may underlie it(e.g., those of whatever collapsed to form the thing),already makes it sound more than just a little thermodynamical incharacter. (Recall the discussion of the No Hair Theorems insection 3.3 above.) Still, although the analogy is extremely suggestiveintoto, to take it seriously would require one to assign a non-zerotemperature to a black hole, which, at the time Bardeen, Carter andHawking first formulated and proved the laws in 1973, almost everyoneagreed was absurd. All hot bodies emit thermal radiation (like theheat given off from a stove, or the visible light emitted by a burningpiece of charcoal); according to general relativity, however, a blackhole ought to be a perfect sink for energy, mass, and radiation,insofar as it absorbs everything (including light), and emits nothing(including light). So it seems the only temperature one might be ableto assign it would be absolute zero. (Seesection 5.4.2 below for more detailed arguments to this effect.)

In the early 1970s, nonetheless, Bekenstein (1972, 1973, 1974) arguedthat the Second Law of thermodynamics requires one to assign a finiteentropy to a black hole. His worry was that one could collapse anyamount of highly entropic matter into a black hole—which, as wehave emphasized, is an extremely simple object—leaving no traceof the original disorder associated with the high entropy of theoriginal matter. This seems to violate the Second Law ofthermodynamics, which asserts that the entropy (disorder) of a closedsystem—such as the exterior of an event horizon—can neverdecrease. Adding mass to a black hole, however, will increase itssize, which led Bekenstein to suggest that the area of a black hole isa measure of its entropy. This conjecture received support in 1971when Hawking proved that the surface area of a black hole, like theentropy of a closed system, can never decrease (Hawking 1971). Still,essentially no one took Bekenstein’s proposals seriously atfirst, because all black holes manifestly have temperature absolutezero, as mentioned above, if it is even meaningful to ascribetemperatures to them in the first place.^[12]

Thus it seems that the analogy between black holes and thermodynamicalobjects, when treated in the purely classical theory of generalrelativity, is merely a formal one, without real physicalsignificance.

5.2. Black Hole Thermodynamics

The “obvious fact” that the temperature of a black holecan be, at best, only absolute zero was shown to be illusory whenHawking (1974, 1975) demonstrated that black holes are not completelyblack after all. His analysis of the behavior of quantum fields inblack-hole spacetimes revealed that black holes will emit radiationwith a characteristically thermal spectrum: a black hole generatesheat at a temperature that is proportional to its surface gravity. Fora nonrotating, uncharged black hole, that quantity is inverselyproportional to black hole’s mass. It glows like a lump of smolderingcoal even though light should not be able to escape from it! Thetemperature of this Hawking radiation is extremely low for stellar-and galactic-scale black holes, but for very, very small black holesthe temperatures would be high. (The Hawking temperature of the blackhole at the center of the Milky Way, Sagittarius A^*, havinga mass of approximately 4 million solar masses, is approximately10^-14 Kelvin; for a black hole to be room temperature, itwould have to have a mass of about 10¹⁸ kg—about 1000times the mass of Mt. Everest—and so be about 10^-7 macross, the size of a virus.) This means that a very, very small blackhole should rapidly evaporate away, as all of its mass-energy isemitted in high-temperature Hawking radiation. Thus, when quantumeffects are taken into account, black holes will not satisfy the AreaTheorem, the second of the classical laws of black hole mechanics, astheir areas shrink while they evaporate. (Haywardet al. 2009discuss the status of deriving a “local” flux of Hawkingradiation for quasi-local black holes; Nielsen 2009 discusses thisalong with the status of attempts to prove the laws of black holemechanics for quasi-local black holes.)

These results—now referred to collectively as the Hawkingeffect—were taken to establish that the parallel between thelaws of black holes and the laws of thermodynamics was not a mereformal fluke: it seems they really are getting at the same deepphysics. The Hawking effect establishes that the surface gravity of ablack hole can, indeed must, be interpreted as a physical temperature.(The surface gravity, therefore, is often referred to as the‘Hawking temperature’.) Connecting the two sets of lawsalso requires linking the surface area of a black hole with entropy,as Bekenstein had earlier suggested: the entropy of a black hole isproportional to the area of its event horizon, which is itselfproportional to the square of its mass for a nonrotating, unchargedblack hole. (The area, therefore, is often referred to as the‘Bekenstein entropy’.) Furthermore, mass in black holemechanics is mirrored by energy in thermodynamics, and we know fromrelativity theory that mass and energy are identical, so the blackhole’s massis its thermodynamical energy. Theoverwhelming consensus in the physics community today, therefore, isthat black holes truly are thermodynamical objects, and the laws ofblack hole mechanics just are the laws of ordinary thermodynamicsextended into a new regime, to cover a new type of physicalsystem.

We will return to discuss Hawking radiation in more detail insection 6.1 below, but for now we note that this all raises deep questions aboutinter-theoretic relations that philosophers have not yet come to gripswith: although it seems undeniable, what does it mean to say that apurely gravitational system is also “a thermodynamical object”?^[13] How can the concepts and relations of one theory be translated so asto be applicable in the context of a radically different one? (See theentries onscientific unity andintertheory relations in physics.)

Although it is still orthodoxy today in the physics community thatthere is no consistent theory of thermodynamics for purely classicalblack holes (Unruh and Wald 1982; Wald 1999, 2001),i.e.,when quantum effects are not taken into account, primarily because itseems that they must be assigned a temperature of absolute zero, ifany at all. Curiel (2017a, Other Internet Resources) argues, to thecontrary, that there is a consistent way of treating purely classicalblack holes as real thermodynamical systems, that they should beassigned a temperature proportional to their surface gravity, and, infact, that not to do so leads to the same kinds of inconsistencies asoccur if one does not do so for black holes emitting Hawkingradiation.

Dougherty and Callender (2019) challenge the orthodoxy from theopposite direction. They argue that we should be far more skeptical ofthe idea that the laws of black holes are more than just a formalanalogy, and that, indeed, there are strong reasons to think that theyare not physically the laws of thermodynamics extended into a newdomain. Their main argument is that the Zeroth Law of black holescannot do the work that the standard formulation of the Zeroth Lawdoes in classical thermodynamics. In classical thermodynamics, thestandard formulation of the Zeroth Law is transitivity of equilibrium:two bodies each in equilibrium with a third will be in equilibriumwith each other. They point out that this transitivity of equilibriumunderlies many of the most important constructions and structures inclassical thermodynamics, which mere constancy of temperature (surfacegravity) for a single system in equilibrium does not suffice for.Curiel (2018, Other Internet Resources), however, proposed astrengthened version of the Zeroth Law for black holes, based on acharacterization of transitivity of equilibrium among them, in anattempt to address this challenge. But it suffers from severalproblems, most importantly the fact that it relies on a notion of“approximate symmetry” in general relativity that is notwell defined. This is an area of active dispute.

Wallace (2018, 2019) provides a more comprehensive exposition anddefense of the claim that black holes truly are thermodynamicalobjects, attacking the problem from several different directions, andoffers specific rejoinders to several of the other arguments made byDougherty and Callender (2019).

5.3. What Is Black Hole Entropy?

The most initially plausible and promising way to explain what theentropy of a black hole measures, and why a black hole has such aproperty in the first place, is to point to the Hawking radiation itemits, and in particular the well defined temperature the radiationhas. Indeed, it is not uncommon to see such“explanations”, not only in popular accounts but even inserious research papers. There are, however, many technical andconceptual reasons why such an explanation is not viable (Visser1998b, 2003), summed up in the slogan that Hawking radiation is astrictly kinematical effect, whereas black hole entropy is a dynamicalphenomenon. (This fact is discussed in more detail insection 7 below.) What, then, is the origin and nature of the entropy weattribute to a black hole?

In classical thermodynamics, that a system possesses entropy is oftenattributed to the fact that in practice we are never able to give it a“complete” description (Jaynes 1967). When describing acloud of gas, we do not specify values for the position and velocityof every molecule in it; we rather describe it using quantities, suchas pressure and temperature, constructed as statistical measures overunderlying, more finely grained quantities, such as the momentum andenergy of the individual molecules. On one common construal, then, theentropy of the gas measures the incompleteness, as it were, of thegross description. (See the entry on thephilosophy of statistical mechanics.) In the attempt to take seriously the idea that a black hole has atrue physical entropy, it is therefore natural to attempt to constructsuch a statistical origin for it. The tools of classical generalrelativity cannot provide such a construction, for it allows no way todescribe a black hole as a system whose physical attributes arise asgross statistical measures over underlying, more finely grainedquantities. Not even the tools of quantum field theory on curvedspacetime can provide it, for they still treat the black hole as anentity defined entirely in terms of the classical geometry of thespacetime (Wald 1994). Any such statistical accounting, therefore,must come from a theory that attributes to the classical geometryitself a description based on an underlying, perhaps discretecollection of microstates, themselves describing the fine-graineddynamics of “entities”, presumably quantum in nature,underlying the classical spacetime description of the black hole. Notethat a program aimed at “counting black-hole microstates”need not accept a subjectivist interpretation of entropyàla Jaynes. In any event, on any view of the nature of entropy,there arises a closely related problem,viz., to locate“where” black hole entropy resides: inside, on, or outsidethe event horizon? See Jacobsonet al. (2005) for athoughtful dialogue among three eminent physicists with differentpoints of view on the matter.

Explaining what these microstates are that are counted by theBekenstein entropy has been a challenge that has been eagerly pursuedby quantum gravity researchers. In 1996, superstring theorists wereable to give an account of how non-perturbative stringtheory generates the number of microstates underlying a certainclass of classical black holes, and this number matched that given bythe Bekenstein entropy (Strominger and Vafa 1996). Recently,philosophers (De Haroet al. 2020) have given a conceptualanalysis of the derivations of black hole entropy in string theory,clarifying the assumptions and inter-theoretic relations on which itdepends, and the idealizations that are made in the derivations. In aseparate paper (Van Dongenet al. 2020) they assessed whetherthe black hole should be considered as emergent from thenon-perturbative string theory system, and discussed the role  ofthe quantum-to-classical correspondence principle in the context ofstring theory black holes. Around the same time as in string theory, acounting of black-hole states using loop quantum gravity alsorecovered the Bekenstein entropy (Rovelli 1996). It is philosophicallynoteworthy that this is treated as a significant success for theseprograms (i.e., it is presented as a reason for thinking thatthese programs are on the right track), even though no quantum effectin the vicinity of a black hole, much less Hawking radiation itself,has ever been experimentally observed. (Sadly, we have no black holesin terrestrial laboratories, and those we do have good reason to thinkwe indirectly observe are too far away for anything like these effectsto be remotely detectable, given their minuscule temperatures.) It isalso the case that all known derivations in string theory held onlyfor a very special class of black holes (“extremal” and“near-extremal” ones), which everyone agrees areunphysical. There are no convincing derivations for more general,physically relevant black holes.

Nonetheless, the derivation of the Bekenstein entropy by the countingof “microstates” has become something of asine quanon for programs of quantum gravity, even if only for the specialcase of extremal black holes: if one cannot do it from something likethe first principles of one’s program, no one will take themseriously. This is noteworthy because it poses aprima facieproblem for traditional accounts of scientific method, and underscoresthe difficulties faced by fundamental physics today, that in manyimportant areas it cannot make contact with empirical data at all. Howdid a theoretically predicted phenomenon, derived by combiningseemingly incompatible theories in a novel way so as to extend theirreach into regimes that we have no way of testing in the foreseeablefuture, become the most important touchstone for testing novel ideasin theoretical physics? Can it play that role? Philosophers have notyet started to grapple seriously with these issues.

In a thoughtful survey, Sorkin (2005) concisely and insightfullycharacterizes in ten theses what seems to be a popular view on thenature of black hole entropy when studied as an essentially quantumphenomenon, which is distilled into the essential parts for ourpurposes as follows. The entropy:

1. derives from degrees of freedom associated only with the geometryof the event horizon, not degrees of freedom associated with matter orspacetime geometry inside or outside the black hole;
2. is finite because the fundamental structure of spacetime isdiscrete;
3. is “objective” because there is a distinguishedcoarse-graining based on the fact that the horizon itself has adistinguished geometry;
4. and obeys the Second Law of thermodynamics because the effectivedynamics of stuff outside the black hole does not obey the rules forstandard quantum evolution.

These theses concisely capture how radically different black holeentropy is from ordinary thermodynamical entropy. The first, as isalready obvious from the Second Law of black hole mechanics,underscores the fact that black hole entropy is proportional to thesurface area of the system, not to the bulk volume as for ordinarythermodynamical systems. The second articulates the fact that theunderlying entities whose statistics are conjectured to give rise tothe entropy are the constituents of perhaps the most fundamentalstructure in contemporary physics, spacetime itself, not high-level,derivative entities such as atoms, which are not fundamental in ourdeepest theory of matter, quantum field theory. The third emphasizesthe fact that, contrary to the way that there is no“natural” coarse-graining of underlying micro-degrees offreedom in the statistical mechanics of ordinary matter, there is aunique natural one here, intimately related to the fact that thegeometry of the event horizon is unique, and the Planck scale providesa measure of units of area thought by many to be physically privileged(albeit in a sense never made entirely clear). The fourth states thatthe Second Law of black hole thermodynamics, generalized to includecontributions from both black holes and ordinary matter (as discussedinsection 5.4 below), is not a phenomenologically derived empirical generalization,as is the Second Law for ordinary matter; rather, it follows directlyfrom the most fundamental dynamical principle, quantum evolution, inconjunction with the basic geometry of spacetimes in generalrelativity. (This will be discussed further insection 6.2 below.) This is of a piece with the fact that the Second Law forblack holes in the classical regime is a theorem of pure differentialgeometry (section 5.1).

In so far as one takes Bekenstein entropy seriously as a truethermodynamical entropy, then, these differences strongly suggest thatthe extension of entropy to black holes should modify and enrich ourunderstanding not only of entropy as a physical quantity, buttemperature and heat as well, all in ways perhaps similar to what thatof the extension of those classical quantities to electromagneticfields did at the end of the 19th century (Curiel 2017a, OtherInternet Resources). This raises immediate questions concerning thetraditional philosophical problems of inter-theoretic relations amongphysical quantities and physical principles as formulated in differenttheories, and in particular problems of emergence, reduction, thereferential stability of physical concepts, and their possibleincommensurability across theories. One could not ask for a more novelcase study to perhaps enliven these traditional debates. (See theentries onscientific unity,scientific reduction, andintertheory relations in physics.)

Dougherty and Callender (2019) have challenged orthodoxy here, aswell, by arguing that the many ways in which the area of a black holedoes not behave like classical entropy strongly suggests that weshould be skeptical of treating it as such. Curiel (2017b, OtherInternet Resources) attempts to rebut them using exactly the idea thatany extension of a known physical quantity into a new regime willinevitably lead to modifications of the concept itself, andemendations in the relations it may enter into with other physicalquantities. Thus, we should expect that black hole entropy will notbehave like ordinary entropy, and it is exactly those differences thatmay yield physical and philosophical insight into old puzzles.

5.4. The Generalized Second Law of Thermodynamics

In the context of thermodynamic systems containing black holes, onecan easily construct apparent violations of both the ordinary laws ofthermodynamics and the laws of black holes if one applies these lawsindependently of each other. So, for example, if a black hole emitsradiation through the Hawking effect, then it will lose mass—inapparent violation of the classical Second Law of black holemechanics. Likewise, as Bekenstein argued, we could violate theordinary Second Law of thermodynamics simply by dumping matter withhigh entropy into a black hole: for then the outside of the blackhole, a causally isolated system, spontaneously decreases in entropy.The price of dropping matter into the black hole, however, is that itsevent horizon will increase in size. Likewise, the price of allowingthe event horizon to shrink by giving off Hawking radiation is thatthe entropy of external matter fields will increase. This suggeststhat we should formulate a combination of the two laws that stipulatesthat the sum of a black hole’s area and the entropy of externalsystems can never decrease. This is the Generalized Second Law ofthermodynamics (Bekenstein 1972, 1973, 1974).

The Second Law of ordinary thermodynamics has a long, distinguished,and contentious history in the Twentieth Century debates about thephilosophical foundations of physics, ramifying into virtually everyimportant topic in the philosophy of physics in particular, and intomany important topics in philosophy of science in general, including:the relation between thermodynamics and statistical mechanics; theMeasurement Problem of quantum mechanics, and the status and meaningof theories of quantum information and computation; the definition ofvarious arrows of time and the relations among them; the so-calledPast Hypothesis in cosmology; determinism; causation; predictionversus retrodiction; the nature of reasoning based on idealization andapproximation; emergence and reduction; and problems with theoryconfirmation.

That black holes and other purely gravitational and geometricalsystems possess an entropy naturally leads to the idea that the SecondLaw of thermodynamics ought to be modified in order to accommodatethat fact. It is an almost completely unexplored issue how thisGeneralized Second Law itself may require modifications to thetraditional questions about the Second Law, and possibly lead to newinsights about them. Thus the postulation of the Generalized SecondLaw and its broad acceptance by the physics community raises manyinteresting puzzles and questions.

In the remainder of this section, we will review the issues raised bythe Generalized Second Law that bear on those puzzles and questions,namely that: contrary to the case in classical thermodynamics, theGeneralized Second Law admits not only of proof, but of many kinds ofproof (Section 5.4.1); several different physically plausible mechanisms have been proposedthat seem to violate the Generalized Second Law (Section 5.4.2) under relatively benign conditions; the Generalized Second Law seemsto allow for the possibility of formulating and proving the existenceof a universal bound on the amount of entropy any physical system canhave, along with a related constellation of ideas known as‘holography’ (Section 5.4.3); and, contrary to the Second Law of classical thermodynamics, theGeneralized Second Law seems to imply novel and deep propositions ofinterest in their own right (Section 5.4.4).

5.4.1. A Dizzying Variety of Proofs

The ordinary Second Law of thermodynamics is, at bottom, an empiricalgeneralization based on observation of the behavior of ordinarymaterial systems, albeit one with confirmation and thus entrenchmentmore profound than probably any other single principle in all ofphysics. One of the most remarkable features of the Generalized SecondLaw, by contrast, is that it seems to admit of proof in ways much moremathematically rigorous than does the ordinary Second Law (such as,e.g., the proof of Flanaganet al. 2000, in thecontext of classical general relativity and theories of matter). Thatalready raises interesting philosophical questions about the relationsbetween what seemsprima facie to be the “same”fundamental principle as formulated, evaluated and interpreted indifferent physical theories.

At least as interesting, from both a physical and a philosophicalpoint of view, is the fact that the Generalized Second Law in factadmits a wide variety of different ways of being proven (Wall 2009).Some of those ways are more mathematically rigorous than others, somemore physically perspicuous and intuitive, some more general, andalmost all have their respective validity in different regimes thanthe others, using different types of physical systems, differentapproximations and idealizations, and different physical andmathematical starting points. “Proofs” have been given,for example, in the classical, hydrodynamic, semiclassical, and fullquantum gravity regimes of black holes. Currently, the mostgeneral and widely accepted proof of the Generalized Second Law insemiclassical gravity is by Wall (2012), which holds for rapidlychanging quantum fields and arbitrary horizon slices.

Although the results of all those proofs are called by the samename—the Generalized Second Law—they seemprimafacie to be different physical principles, just because of theextreme differences in the assumptions and content of their respectiveproofs. Here is just a sample of some of the many questions and issuesone must take a stand on in order to formulate a version of theGeneralized Second Law and attempt to prove it.

• Black holes have different physically significant horizons apartfrom the event horizon (e.g., the so-called apparenthorizon)—which horizon should one attribute entropy to?
• For statistical proofs, should one use the Gibbs or the Boltzmannentropy?
• What physical underpinning of black hole entropy should one use(quantum entanglement entropy, quantum “atoms” ofspacetime,etc.)?
• Should one assume an entropy bound? (SeeSection 5.4.3 below.)
• If the approximation or representation one uses to model the areaof the chosen horizon admits quantum fluctuations, should one use theaverage area, or some other way of massaging it so that it hassomething like a well defined classical area?
• What definition of “quasi-stationary” should oneuse?
• What definition of “adiabatic” should one use?

The dizzying variety of proofs on offer, which can be roughlyclassified by how each answers these (and other related) questions,thus prompts the question: what is the relation among all thedifferent principles actually derived by each proof? Do they representthe same physical principle as it manifests itself in differentregimes, and as it is viewed from different perspectives? Again, theanswer one gives to this question will depend sensitively on,inter alia, one’s views on inter-theoretic relations.Indeed, because different answers to these questions can lead to“proofs” that have, respectively, contradictoryassumptions, one may well worry that the derived principle, if it isto be the same in all cases, will turn out to be a tautology!

Even putting aside the contradictory assumptions used in differentderivations, one should, in any event, note that one cannot try tojustify the multifariousness of proofs by using an argument based onsomething like consilience, for it will not be consilience in anythinglike the standard form. (See the entry onscientific discovery.) This is not a case in which the same equations or relations or model,or values of quantities, are being derived for a given phenomenonbased on studies of different types of interactions among differenttypes of physical systems, as in the classic case of Perrin’scalculation of Avogadro’s number. This is rather a case in whichdifferent physical assumptions are made about the very same class ofphysical systems and interactions among them, and calculations andarguments made in very different physical and mathematical frameworks,with no clear relation among them.

5.4.2. Possible Violations

When Bekenstein first proposed that a black hole should possessentropy, and that it should be proportional to its area, difficultiesthat appeared insurmountable immediately appeared. In a colloquiumgiven at Princeton at 1970, Geroch proposed a mechanism that seemed toshow that, if one could attribute a temperature to a black hole atall, it should be absolute zero; an immediate consequence of theworking of the mechanism showed that to do otherwise would seem toallow arbitrarily large violations of what was to become known as theGeneralized Second Law.^[14] Far away from a black hole, prepare an essentially massless box to befull of energetic radiation with a high entropy; then the mass of theradiation will be attracted by the black hole’s gravitationalforce. One can use this weight to drive an engine to produce energy(e.g., to produce friction from the raising of acounter-weight) while slowly lowering the box towards the eventhorizon of the black hole. This process extracts energy, but notentropy, from the radiation in the box. One can then arrange for allthe mass-energy of the radiation to have been exhausted when the boxreaches the event horizon. If one then opens the box to let theradiation fall into the black hole, the size of the event horizon willnot increase (because the mass-energy of the black hole does notincrease), but the thermodynamic entropy outside the black hole hasdecreased. Thus we seem to have violated the Generalized Second Law.Many ways to try to defuse the problem have been mooted in theliterature, from entropy bounds (discussed below insection 5.4.3) to the attribution of an effective buoyancy to the object beinglowered due to its immersion in radiation generated by itsacceleration (Unruh and Wald 1982), a consequence of the so-calledUnruh effect (for an account of which, seenote 16). None of them is completely satisfying.

The question of whether we should be troubled by this possibleviolation of the Generalized Second Law touches on several issues inthe foundations of physics. The status of the ordinary Second Law isitself a thorny philosophical puzzle, quite apart from the issue ofblack holes. Many physicists and philosophers deny that the ordinarySecond Law holds universally, so one might question whether we shouldinsist on its validity in the presence of black holes. On the otherhand, the Second Law clearly capturessome significantfeature of our world, and the analogy between black holes andthermodynamics seems too rich to be thrown out without a fight.Indeed, the Generalized Second Law is the only known physical law thatunites the fields of general relativity, quantum mechanics, andthermodynamics. As such, it seems currently to be the most promisingwindow we have into the most fundamental structures of the physicalworld (for discussion of which, seesection 6.3 below).

5.4.3. Entropy Bounds and the Holographic Principle

In response to the apparent violation of the Generalized Second Lawconsequent on Geroch’s proposed process, Bekenstein postulated alimit to how much entropy can be contained in a given region ofspacetime in order to try to avoid such seeming violations, the limitbeing given by the entropy of a black hole whose horizon wouldencompass the region. Current physics imposes no such limit, soBekenstein (1981) postulated that the limit would be enforced by theunderlying theory of quantum gravity that, it is hoped, black holethermodynamics provides our best current insight into. There is,moreover, a further, related reason that one might think that blackhole thermodynamics implies a fundamental upper bound on the amount ofentropy that can be contained in a given spacetime region. Supposethat there were more entropy in some region of spacetime than theBekenstein entropy of a black hole of the same size. Then one couldcollapse that entropic matter into a black hole, which obviously couldnot be larger than the size of the original region (or the matterwould have already collapsed to form a black hole). But this wouldviolate the Generalized Second Law, for the Bekenstein entropy of theresulting black hole would be less than that of the matter that formedit. Thus the Generalized Second Law itself appears to imply afundamental limit on how much entropy a region can contain (Bekenstein1983; Bousso 1999a, 2006). If this is right, it seems to be a deepinsight into the fundamental structure of the world, and in particularit should provide an important clue to the nature of an adequatetheory of quantum gravity.

Arguments along these lines led ’t Hooft (1993, inOIR) to postulate the Holographic Principle (though the name is due toSusskind 1995). This principle claims that the number of fundamentaldegrees of freedom in any spherical spatial region is given by theBekenstein entropy of a black hole of the same size as that region.The Holographic Principle is notable not only because it postulates awell-defined, finite number of degrees of freedom for any region, butalso because this number grows in proportion to the area surroundingthe region, not the volume. This flies in the face of the standardpicture of the dynamics of all other known types of physical systems,whether particles or fields. According to that picture, the entropy ismeasured by the number of possible ways something can be, and thatnumber of ways increases as the volume of any spatial region. To thecontrary, if the Holographic Principle is correct then one spatialdimension of any physical system can, in a sense, be viewed assuperfluous: the fundamental “physical story” of a spatialregion is actually a story that can be told merely about the boundaryof the region (Luminet 2016).

Still, there are reasons to be skeptical of the validity of theproposed universal entropy bounds, and the corresponding HolographicPrinciple. Unruh and Wald (1982), in response to Bekenstein’spostulated entropy bound, argued convincingly that there is a lessad hoc way to save the Generalized Second Law, namely byexploiting the Unruh effect (for an explanation of which, seenote 16).^[15] Flanaganet al. (2000), moreover, have offered strongarguments that the validity of the Generalized Second Law isindependent of Bousso’s proposed entropy bound (widely thoughtto be superior to Bekenstein’s original one), thus removing muchof the primary historical and conceptual motivation for theHolographic Principle.

A more modern version of the Holographic Principle is given bygauge/gravity dualities. These are (conjectured) dualities between a quantum gravity theory in a bulk spacetime and a gauge theoryliving on the boundary of that spacetime. The best understood exampleof a gauge/gravity duality is the so-called Anti-de Sitter/ConformalField Theory correspondence, which was originally discovered byMaldacena (1998). Gauge/gravity dualities realize the HolographicPrinciple in the sense that one spatial dimension is superfluous todescribe a gravitational theory. One particularly interesting featureof these dualities is that black hole entropy can be computed in theboundary theory using standard techniques in statistical mechanics,and matched to the Bekenstein entropy. This has been understood bothqualitatively and quantitatively in various setups in the Anti-deSitter/Conformal Field Theory correspondence, thus providing evidencethat black hole entropy has a statistical mechanical underpinning (seee.g. Harlow 2016, Hartman 2015 and Wallace 2019).

Again, all these questions are of great interest in their own right inphysics, but there is strong reason to believe that their analysis mayshed new light on several ongoing philosophical discussions about thenature of spacetime, with which they have obvious direct connections,especially concerning the dimensionality of space and spacetime, andthe substantivalism-versus-relationalism debate. The interested readershould see de Haroet al. (2015) for a discussion of therelation of holography to gauge/gravity dualities in general, and areview of the philosophical issues that raises, and Castellani (2016)for philosophical discussion of the ontological issues raised by thesedualities.

5.4.4. Possible Consequences of the Generalized Second Law

The ordinary Second Law has profound philosophical implications. Itis, however, rarely if ever used to prove other physical principles orresults of real depth, all of its important consequences being more orless immediate. Once again, the Generalized Second Law stands incontrast to the ordinary Law, for, as has recently been realized, itcan be used to prove several physical results of deep interest, overand above heuristically motivating the Holographic Principle.

In atour de force of physical argument, Wall (2013a, 2013b)showed that assumption of the Generalized Second Law rules outtraversable wormholes, other forms of faster-than-light travel betweendistant regions, negative masses for physical systems, and closedtimelike curves. (See the entries ontime machines andtime travel, and Visser 1996.) Furthermore, if the Generalized Second Law is to besatisfied, then it is impossible for “baby universes” thateventually become causally independent of the mother universe to form.Such baby universes and their eventual independence, however,constitute the fundamental mechanism for currently popular“multiverse” scenarios in cosmology.

In the same work, Wall also shows that the Generalized Second Law hasa striking positive conclusion: a “quantum singularitytheorem”, which shows that, even when quantum effects are takeninto account, spacetime will still be geodesically incomplete insideblack holes and to the past in cosmological models (like the currentlymost well supported ones, which start with a Big Bang singularity).This flies directly in the face of the pious hopes of most physiciststhat quantum effects, and in particular the hoped-for theory ofquantum gravity, will efface singularities from spacetime. (See,e.g., Ashtekar and Bojowald 2006, Ashtekaret al.2006, and Kiefer 2010 for typical sentiments along these lines, alongwith typical arguments forwarded to support them, in the context ofcanonical quantum gravity, and Roiban 2006 and Das 2007 for the samein the context of string theory; it is noteworthy that Roiban alsodiscusses known cases where it appears that string theory doesnot necessarily efface singularities.)

Another striking positive consequence of the Generalized Second Law isthat it allows one to derive energy conditions in the context ofgeneral relativity. An energy condition is, crudely speaking, arelation one demands the properties of matter to satisfy in order totry to capture the idea that “mass-energy should bepositive”. Energy conditions play a central and fundamental rolein general relativity, since they are needed as assumptions inessentially every deep, major result proven in the last 60 years,especially those pertaining to singularities and black holes (Curiel2017). One thing that makes them unusual is the fact that, uniquelyamong the central and fundamental tenets of general relativity, theythemselves do not admit of derivation or proof based on other suchprinciples. At least, no such derivations or proofs were known untilWall (2010) argued that the Generalized Second Law implies one. Thereare several problematic aspects to Wall’s argument (Curiel2017), but the mere fact that he was able to produce aprimafacie decent one at all is remarkable, showing that theGeneralized Second Law may be a very deep physical principle indeed.One, however, may contrarily conclude that the argument shows ratherthat the Generalized Second Law is a contingent matter, dependingsensitively on the kinds of matter fields that actually exist—ifmatter fields were such as to violate the energy condition Wall arguedfor, then his argument would show that the Generalized Second Law isnot valid.

Boussoet al. (2016) showed that a form of the GeneralizedSecond Law applicable to generalized horizons strongly suggests thatcausal geodesics in the regime where quantum field theory effectsbecome important will focus and converge on each even when thestandard energy conditions are violated. This is significant becauseit is propositions about the focusing properties of geodesics that lieat the heart of all the standard singularity theorems and most otherresults about horizons of all kinds, and all of the propositions thatshow focusing assume a standard energy condition. If this conjectureis correct, it would provide further strong evidence that quantumeffects may not remove singularities from generic spacetimes. Anotherimportant recent extension of singularity theorems to at least asemi-classical domain is through Quantum Energy Inequalities, orbounded violations of classical energy conditions. In many cases QEIscan be derived from underlying quantum field theories, and replace theclassical energy conditions in some proofs of singularity theorems;see Kontou and Sanders (2020) for an introduction to this importanttopic.

Nevertheless, many approaches to singularity resolution in a quantumgravity domain are actively pursued in various theoretical frameworks.The resolution often takes some form of a spacetime bounce - either acosmological one, or in a form of black to white hole transition. Therecent physics literature is too large to summarize here; but, inphilosophy of physics context, Crowther and de Haro (2022) explorevarious attitudes towards GR singularities and their roles in thesearch for quantum gravity, and Thébault (2023) explicates andevaluates a number of approaches to Big Bang singularity resolution ina particular class of quantum cosmology models.

5.5. General Gravitational Entropy

That black holes, purely gravitational objects, possess a physicalentropy strongly suggests that the gravitational field itself ingeneral may possess entropy, as Penrose (1979) hypothesized. Indeed,there are a number of reasons to suspect that the thermodynamicalcharacter of gravity should extend to gravitational systems andstructures beyond just those provided by black holes. Becausegravitational “charge” (i.e., mass-energy) comeswith only one sign (as opposed,e.g., to electromagneticcharge, which can be of either positive or negative sign), bits ofmatter tend to accelerate towards each other, other things beingequal. This already suggests that gravity has a built-inthermodynamical character, since it provides an objective, invariantmeasure of a direction for time: it is characteristic of futuretime-flow that bits of matter tend to accelerate towards each other,and so become more inhomogeneous in the aggregate. From thisperspective, the Generalized Second Law and the corresponding idea ofgeneral gravitational entropy might be seen as introducing a newpossible arrow of time, the gravitational.  (See Zeh 2014 for athorough recent review; see also the entry onthermodynamic asymmetry in time.)

Since the work of Gibbons and Hawking (1977), Bekenstein (1981),Smolin (1984), Bousso (1999a), Jacobson and Parentani (2003), andPadmanabhan (2005), among others, it has been known that an entropyand a temperature can be attributed to spacetime horizons more generalthan just the event horizon of a black hole. This has led in recentyears to several interesting proposals for a completely generalmeasure of gravitational entropy, such as that of Cliftonetal. (2013). Indeed, Anastopoulos and Savvido (2012) have evenattempted to attribute entropy directly to non-cosmologicalsingularities, those associated with collapse phenomena. Pavónand Zimdahl (2012), in a similar spirit, attempt to provide athermodynamical analysis of future cosmological singularities and socharacterize them by their thermodynamical properties.

From another direction, Penrose (1979) did far more than just arguethat the gravitational field itself possesses a generalizedentropy.  He also proposed what has come to be known as theConformal Curvature Hypothesis, which states that an entropy should beattributed to the gravitational field proportional to some measure of“purely gravitational” degrees of freedom, with a lowentropy attributed to homogeneous and isotropic gravitational fields.It also suggests that the gravitational and cosmological arrows oftime are driven, if not determined, by this generalized gravitationalentropy. This Hypothesis also suggests that certain types ofcosmological singularities, such as the Big Bang, should themselves beattributed an entropy.

Some work in subsequent decades has been done, primarily based onGoode and Wainwright (1985) and Newman (1993a, 1993b), to try togeneralize Penrose’s proposal and make it rigorous. Almost allthis work has focused on the behavior of conformal singularities(characterized at the end ofsection 1.3) which are, in a natural sense, “early” cosmologicalsingularities, such as the Big Bang, and on the behavior of variousmeasures of gravitational degrees of freedom moving to the future awayfrom such singularities. (There has been some work, such as Rudjordet al. 2008, attempting to link the Conformal CurvatureHypothesis directly to black hole entropy.) The idea is that theinitial cosmological singularity, in accord with Penrose’sConformal Curvature Hypothesis, had extraordinarily low entropy, thuscompensating the high entropy of the homogeneous ordinary matterpresent then, making the early universe a state of low totalentropy. As the universe develops over time, and matter clumpsinto individual system (stars, galaxies, clusters and superclusters ofgalaxies,etc.), the entropy of ordinary matter seems todrop, but, again, that is more than compensated for by the enormousincrease in gravitational entropy, thus saving the Generalized SecondLaw. (And for competing perspectives about links between gravityand initial low entropy, or the Past Hypothesis, see Albert 2000,Earman 2006, Wald 2006, Callender 2010, and Wallace 2010.)

These facts raise several fundamental puzzles about the nature ofentropy as a physical quantity and the relations among the differenttheories that involve it. How can such a quantity, which hitherto hasbeen attributed only to material systems such as fluids and Maxwellfields, be attributed to simple regions of spacetime itself? How doesgeneral gravitational entropy relate to more standard forms ofentropy, and how may the nature of general gravitational entropyitself inform our understanding of the standard forms? Does it shednew light on traditional general philosophical topics of interest,such as questions about reduction and emergence of thermodynamics toand from statistical mechanics?

6. Black Holes and Quantum Theory

As discussed already inSection 5.2 andSection 5.3, it is the addition of quantum field theory to general relativity thatdefinitively settles the issue of the thermodynamical character ofblack holes. There are, however, many other fascinating phenomena thatarise when one adds quantum field theory to the mix of black holes andsingularities, and general relativity in general, than just that, anda concomitant broadening and deepening of the philosophical issues andpuzzles that confront us.

InSection 6.1, we discuss the Hawking effect (the predicted emission of thermalradiation by black holes) and its associated problems and puzzles indetail. One puzzle in particular that seems to follow the predictionof the Hawking effect has exercised physicists and philosophers themost, the so-called Information Loss Paradox: the evaporation of blackholes by the emission of Hawking radiation seems to lead in the end toa violation of one of the most fundamental tenets of quantummechanics. We discuss that inSection 6.2. We conclude inSection 6.3 with an examination of the claims that black hole thermodynamicsprovides the best evidence to guide us in the search for a theory ofquantum gravity.

6.1. Hawking Radiation

In light of the notorious difficulty of constructing a theory thatincorporates and marries quantum mechanics and generalrelativity—a theory of quantum gravity—it may come as asurprise to learn that there is a consistent, rigorous theory ofquantum fields posed on the background of a classical relativisticspacetime. (Wald 1994 is a standard text on the subject; Jacobson[2003, inOIR] gives a less rigorous overview, discussing possible relations toproposed theories of quantum gravity; Wald [2006b, inOIR] gives a synoptic history of the technical aspects of the entiresubject, and an exposition of the advances in the field subsequent tothe publication of Wald 1994; and Hollands and Wald 2015 provides atechnically sophisticated overview of the most recent results.)Quantum field theory on curved spacetime, however, differs fromstandard quantum field theory (set on the flat Minkowski spacetime ofspecial relativity) in one profound respect, that difference ramifyinginto every part of the theory: a generic relativistic spacetime has nogroup of symmetries comparable to the Poincaré Group forspecial relativity. There is correspondingly no distinguished vacuumstate and no natural notion of a particle. This means, for instance,that one cannot employ many familiar and useful techniques fromstandard quantum field theory, and one must take care in the use ofmost of the others.

One expects that such a framework would find its most naturalapplication in the treatment of problems in which, in some sense orother, the curvature of spacetime is well above the Planck length,insofar as there are some theoretical grounds for suspecting that inthis regime one can safely ignore any quantum properties of thespacetime geometry itself. (Hence, the framework is often called‘the semi-classical approximation’ or‘semi-classical gravity’.) In this vein, its most popularand successful applications have been to problems involving particlecreation in the early universe and in the vicinity of black holes.Now, according to general relativity a black hole ought to be aperfect sink for energy, mass and radiation, in so far as it absorbseverything (including light), and emits nothing (including light). Itwas therefore more than shocking when Hawking (1974, 1975) predictedthat, when quantum effects are taken into account, a black hole oughtto behave rather like a perfect black body, in the sense of ordinarystatistical thermodynamics: a stationary black hole should emitthermal radiation with the Planckian power spectrum characteristic ofa perfect blackbody at a fixed temperature. It glows like a lump ofsmoldering coal even though light should not be able to escape from it!^[16]

As with the Generalized Second Law, one of the most fascinatingaspects of Hawking radiation from a foundational point of view is themultiplicity and multifariousness of the derivations known for it.They also differ radically among themselves with regard to themathematical rigor of the framework they adopt and the mathematicalcharacter of the structures they assume, and almost all are valid indifferent regimes than the others, using different types of physicalsystems and different approximations and idealizations, basing theirarguments on different physical principles, with varying degrees ofphysical perspecuity and intuitiveness. In consequence, thesedifferent derivations seem to suggest different physicalinterpretations of Hawking radiation itself, both for its origin andfor its character (Broutet al. 1995). It is thus not evenclear, at a foundational level, what the physical content of theprediction of Hawking radiation is. Indeed, as in the case of theGeneralized Second Law, some of the derivations of Hawking radiationmake assumptions that seem to contradict some of the assumptions ofother derivations—but ifA impliesB andnot-A impliesB, thenB must be atautology. Since this is an unappealing attitude to take towardsHawking radiation, some other way must be found to reconcile thecontrary derivations. Again, standard consilience cannot be invokedhere, for the same reasons as discussed at the end ofsection 5.4.1 for different proofs of the Generalized Second Law.

Because the interpretation of quantum field theory itself, even in theflat spacetime of special relativity, is already so contested, fraughtwith problems, and just poorly understood in general (see the entry onquantum field theory), one may think that there is even less of a chance here to get a gripon such issues. Contrarily, one may also think that the very fact thatthe phenomena are so different here than in ordinary quantum fieldtheory may suggest or afford us new avenues of approach to thetraditional problems that have so long frustrated us.

6.2. Information Loss Paradox

The existence of Hawking radiation has a remarkable consequence: asHawking (1976) pointed out and Unruh (1976) elaborated, the fact thata black hole radiates implies that it loses mass-energy, and so willshrink, in seeming violation of the Area Theorem. (The Area Theorem isnot in fact violated; rather, one of its assumptions is,viz., that locally energy is always strictly positive.)Because there is no limit to this process except that imposed by thetotal initial mass of the black hole itself, eventually the black holewill radiate itself entirely away—it evaporates. This predictionclearly bears on the issue of cosmic censorship: if the end-state ofthe evaporation leaves the previously hidden singularity open for therest of the universe to see, all the potential problems raised insection 4 can arise. In particular, it is now acknowledged that unless somenovel physics kicks in, fully evaporating black holes violate a fairlyweak causal property of future reflectivity (Lesourd 2018). Thisresult generalizes what became known as the Kodama-Wald theorem aboutthe causal structure of evaporating black holes (see Kodama 1979,Manchak and Weatherall 2018); subsequently, Minguzzi (2020) used theseobservations in weakening causal assumptions in Penrose’s 1965singularity theorem.

There is, however, a seemingly even deeper problem posed by thepossibility of black-hole evaporation, one that raises doubts aboutthe possibility of describing black holes using any standardformulation of quantum theory. According to standard quantum theory,the entropy of a closed system never changes; this is capturedformally by the nature of the evolution of a quantum system, by thetechnical property of unitarity. Unitary evolution guarantees that theinitial conditions, together with the Schrödinger equation (theequation governing the temporal evolution of quantum systems), willfix the future state of the system. Likewise, a reverse application ofthe Schrödinger equation will take us from the later state backto the original initial state. In other words, the states at each timecontain enough information to fix the states at all other times, giventhe unitarity of dynamical evolution for quantum systems. Thus thereis a sense in which thecompleteness of the state ismaintained by the standard time evolution in quantum theory. (See theentry onquantum mechanics.)

It is usual to characterize this feature by the claim that quantumevolution “preserves information”. If one begins with asystem in a precisely known quantum state, then quantum theoryguarantees that the details about that system will evolve in such away that one can infer the precise state of the system at some latertime, and vice versa. This quantum preservation of details impliesthat if we burn a chair, for example, it would in principle bepossible to perform a complete set of measurements on all the outgoingradiation, the smoke, and the ashes, and reconstruct exactly what thechair looked like. If we were instead to throw the chair into a blackhole, however, then orthodoxy holds that as a consequence of the NoHair Theorems (discussed insection 3.3 above) it would be physically impossible for the details about thechair ever to escape to the outside universe. This might not be aproblem if the black hole continued to exist for all time, since onecould then assume the information encoded in the chair still existedbehind the event horizon, preserved by the unitary evolution in thatregion. The existence of Hawking radiation, however, tells us that theblack hole is giving off energy, and thus it will shrink down andpresumably will eventually disappear altogether, along with whateverstuff had fallen past the event horizon before that. At that point,the details about the chair will be irrevocably lost; thus suchevolution cannot be described by the standard laws of quantum theory.This is theInformation Loss Paradox of quantum black holes(Hawking 1976).^[17] Although the paradox is usually formulated in terms of“information”, we will often speak of the issue as beingthe maintenance of correlations between different systems, as this isa physically more perspicuous notion that lies at the bottom of theparadox, and is much less problematic than the notoriously vexed andnebulous concept of “information”.

The attitude that individual physicists adopt towards this problem isstrongly influenced by their intuitions about which theory, generalrelativity or quantum theory, will have to be modified to achieve aconsistent theory of quantum gravity. Spacetime physicists tend toview non-standard quantum evolution as a fairly natural consequence ofsingular spacetimes: one would not expect all the details of systemsat earlier times to be available at later times if they were lost in asingularity. Hawking (1976), for example, argued that the paradoxshows that the full theory of quantum gravity will be a theory thatdoes not obey the standard dynamical principles of quantum theory, andhe began working to develop such a theory very soon after firstpromulgating the paradox. (He has since abandoned this position.)Unruh and Wald (2017) develop an extended review and defense of thisposition. Particle physicists (including superstring theorists),however, tend to view black holes as being just another state of aquantum field. If two particles were to collide at extremely highenergies, they would form a very small black hole. This tiny blackhole would have a very high Hawking temperature, and thus it wouldvery quickly give off many high-energy particles and disappear.(Recall, as discussed insection 5.2 above, that Hawking temperature is inversely proportional to the massof black hole.) Such a process would look very much like a standardhigh-energy scattering experiment: two particles collide and theirmass-energy is then converted into showers of outgoing particles. Thefact that all known scattering processes obey the standard dynamicalprinciples of quantum theory, and above all unitarity, then, seems togive us some reason to expect that black hole formation andevaporation should also do so.

The reactions to the puzzle are legion. (A helpful overview of earlierstages of this debate can be found in Belotet al. 1999.) Itis useful to classify them as belonging to one of six broadgroupings:

1. the argument for information loss is valid, and black holeevaporation violates the standard principle of quantum evolution;
2. the quantum correlations between physical systems that fall intothe black hole and those that remain outside are not lost but arerather stored in a “remnant” of the black hole, whichfails to evaporate entirely;
3. the correlations are not lost, but come out (slowly) asnon-thermal correlations in the Hawking radiation itself;
4. the correlations are not lost, for one reason or another, but onlyappear to be lost depending on the state of the observer;
5. the conclusion of the argument for information loss is an artifactof the invalidity of the semi-classical approximation at relevantperiods and scales of the evaporation, and a full, non-perturbativecalculation will show that the correlations are not in fact lost;
6. the semi-classical calculation is valid, but its result does notbear on the issue of the nature of the fundamental equations ofevolution; one should remain agnostic about whether or notcorrelations are lost.

In particular, today there are four main ways of trying to address theproblem that have a fair amount of support in different segments ofthe physics community:

1. acceptance of the loss of unitarity (the first response);
2. black hole complementarity, an instance of the fourth kind ofresponse;
3. firewalls, an instance of the fifth;
4. Hawking radiation’s deviation from perfect thermality, aninstance of the third.

We will briefly sketch each of them, along with their pros and cons.Chakraborty and Lochan (2017), Bryan and Medved (2017), Marolf (2017),and Unruh and Wald (2017) provide recent reviews of the most popularapproaches, with Marolf’s emphasizing possible approaches thatsave unitarity, and Unruh and Wald’s emphasizing ones thatviolate it. (See Mathur 2009 and Chenet al. 2015 for recentdiscussions of approaches based on remnants, which we will not coverhere.)

The arguments that we should accept the calculations that predictfailure of unitarity at face value are straightforward (Unruh and Wald2017). The calculations represent a regime (the semi-classical one) inwhich we have good theoretical grounds for trusting our theoreticalmachinery, and nothing is required that deviates from standardapplications of quantum field theory and general relativity,respectively. Even though there is failure of unitarity, there is noviolation of conservation of probability—all quantumprobabilities sum to 1 over the course of the entireevolution—and there is no other manifest form of indeterminismpresent. Nor is there any violation of energy conservation attendanton the failure of unitarity, as some have alleged must happen. Unitaryevolution, moreover, is arguablynot a fundamental tenet ofquantum theory: so long as probability is conserved, one can calculatewith confidence. Indeed, there are examples of just such non-unitary,but probability-conserving and energy-conserving evolution in standardapplications of ordinary quantum theory, with no need for anything ashigh-falutin’ as quantum field theory on curved spacetime andblack hole thermodynamics (Unruh 2012).

The conclusion, however—that what many still take to be one ofthe most fundamental principles of quantum theory is violated—istoo distasteful for many physicists to swallow, especially thosetrained in the tradition of particle physics, where unitarity is takento be inviolate. The sanguine acceptance of the loss of unitarityseems to come mostly from the trust the physicists in question have ingeneral relativity. This raises the question why general relativityought to be trusted enough in this regime to conclude that unitaritywill fail in any deeper quantum theory, but not trusted enough when itcomes to the prediction of singularities (section 2.2)—on what grounds do they pick and choose when and when not to trust it?This question becomes especially piquant when one considers that lossof unitarity is, on its face, an extraordinarily strong constraint toplace on any proposed theory of quantum gravity, especially when itcomes from a calculation made in the context of a merely effective andnot a fundamental theory, and when it is exactly that still unknownfundamental theory that is supposed to efface singularities. In anyevent, Manchak and Weatherall (2018) have recently argued that, evenif one does accept loss of unitarity—what seems to be astraightforward conclusion of the standard calculations—thestate of affairs is still justly called paradoxical.

The idea of black-hole complementarity, initiated by Susskindetal. (1993), tries to resolve the paradox by pointing out that theself-description of the experience of an astronaut falling into ablack hole will differ from the description made by an externalobserver, and then playing the contrary descriptions off each other ina dialectical fashion. It has been the subject of philosophicalcontroversy because it includes apparently incompatible claims, andthen tries to reconcile them by appeal to a form of so-called quantumcomplementarity, or (so charge the critics) simple-mindedverificationism (Belotet al. 1999). An outside observer willnever see the infalling astronaut pass through the event horizon.Instead, she will seem to hover just above the horizon for all time(as discussed insection 3.1 above). But all the while, the black hole will also be giving offheat, shrinking, and getting ever hotter. The black holecomplementarian therefore suggests that an outside observer shouldconclude that the infalling astronaut gets burned up before shecrosses the event horizon, with the result that all the details abouther state will be returned in the outgoing radiation, just as would bethe case if she and her belongings were incinerated in a moreconventional manner; thus the information (and standard quantumevolution) is saved.

This suggestion, however, flies in the face of the fact that for aninfalling observer, nothing out of the ordinary would be experiencedat the event horizon (as discussed insection 3.1 above). Indeed, in general she could not even know that she waspassing through an event horizon at all, unless classical generalrelativity were very wrong in regimes where we expect no quantumeffects to show themselves. This obviously contradicts the suggestionthat she might be burned up as she passes through the horizon. Theblack hole complementarian tries to resolve this contradiction byagreeing that the infalling observer will notice nothingremarkable at the horizon, but then suggests that the account of theinfalling astronaut should be considered to be complementary to theaccount of the external observer, rather in the same way that positionand momentum are complementary descriptions of quantum particles(Susskindet al. 1993). The fact that the infalling observercannot communicate to the external world that she survived her passagethrough the event horizon is supposed to imply that there is nogenuine contradiction here. This solution to the informationloss paradox has been criticized for making an illegitimate appeal toverificationism (Belotet al. 1999). Bokulich (2005), to thecontrary, argues that the most fruitful way of viewing black holecomplementarity is as a novel suggestion for how a non-local theory ofquantum gravity will recover the local behavior of quantum fieldtheory while accommodating the novel physics of black holes.

Almheiriet al. (2013) have recently claimed that black holecomplementarity is not viable on different, more physically orientedgrounds. They argue that the following three statements, assumed byblack-hole complementarity, cannot all be true: (i) Hawking radiationis in a pure state; (ii) the information carried by the radiation isemitted from the region near the horizon, with low energy effectivefield theory (i.e., the standard semi-classicalapproximation) valid beyond some microscopic distance from thehorizon; and (iii) the infalling observer encounters nothing unusualat the horizon. Based on powerful grounds for believing the first twopropositions, they conclude that the appropriate response to theparadox is to posit that there is a “firewall” at theevent horizon: the flux of Hawking radiation from the black holebecomes in general so intense that highly accelerated infalling bodiesare themselves incinerated as soon as they enter the black hole. 

Perhaps the physically most conservative—and correlatively thephilosophically least thrilling—proposal is to deny the implicitassumption that during black-hole evaporation the deviations ofHawking radiation from exact thermality are negligible. Thus theproblemprima facie does not ever arise, because all thequantum information does manage to escape in those non-thermalcorrelations. This proposal faces the serious challenge of showingthat such non-thermal corrections are rich and large enough to carryaway all possible information encoded in all possible bodies fallinginto black holes. Hawkinget al. (2016) argue, in this vein,that black holes do indeed have hair, violating the No Hair theorems,which makes possible the maintenance of correlations between early andlate time Hawking radiation in such a way as to preserve information.Dvali (2015) argues that exact thermality of Hawking radiation, inconjunction with other well established results about black holethermodynamics and quantum field theory on curved spacetime, implythat the black hole entropy would be infinite; thus, he concludes,there must be large deviations from thermality.Any suchargument, note, will have to conclude that the deviations from perfectthermality are large—otherwise there would be no hope ofencoding enough information to record recovery data about everyphysical system that fell into the black hole before evaporation. 

The evaporation of black holes has another startling consequence thatraises far-reaching philosophical and physical problems for ourcurrent picture of quantum field theory and particle physics: itimplies that baryon and lepton number need not be conserved. Suppose aneutron star composed of ∼10⁵⁷ baryons collapses toform a black hole. After evaporation, the resultant baryon number isessentially zero, since it is overwhelmingly likely that the blackhole will radiate particles of baryon number zero. (The radiation isnot energetic enough to produce baryons, until, perhaps, the very latestages of the evaporation.) This issue seems not to have agitatedresearchers in either the particle physics or the general relativitycommunity so much as the idea of non-standard quantum evolution eventhough conservation of baryons and leptons are surely principles aswell entrenched as that of the unitarity of quantum evolution.^[18] One could perhaps argue that they are even more entrenched, since ourempirical evidence for the conservation principles is simple andimmediate in a way that our evidence for standard evolution is not:one simply counts particles before and after an observedinteraction—interpretational questions arising from theMeasurement Problem in quantum theory and a possible“collapse” of the wave function do not bear on it. (Seethe entry onquantum mechanics.)

Okon and Sudarsky (2017) have in fact recently argued that there is anintimate connection between the Information Loss Paradox and theMeasurement Problem in quantum mechanics. Their arguments raisefurther questions about the Information Loss Paradox. Why arephysicists so exercised by the possible violation of unitarityseemingly entailed by black-hole evaporation, when almost all of thoseself-same physicists do not worry at all about the Measurement Problemof quantum mechanics, and the seeming violations of unitarity thathappen every time a measurement is performed? One possible explanationis perhaps best described as “sociological”: mosttheoreticians, as the ones involved in this debate, never modelexperiments, and so do not face the Measurement Problem directly intheir work. Thus it is generally not an issue that is at the forefrontof their thought. Along the same lines, many theoreticians in thisarea also work in cosmology, in which one considers the “wavefunction of the universe”, an object that seems not to admit ofexternal observers making measurements on it, and so the issue ofcollapse does not arise in their work. Perhaps a more intriguingexplanation, one not discussed by Okon and Sudarsky, is that theInformation Loss Paradox provides an explicit physical mechanism forviolations of unitarity. It is perhaps easier to dismiss seemingviolations of unitarity during measurements as an artifact of our lackof understanding of quantum mechanics, not as a reflection of whathappens in the world. One cannot dismiss the possible violation ofunitarity in the Information Loss Paradox with such equanimity: itappears to be an integral, explicit part of a model of the behavior ofa physically possible system, with an articulated mechanism forbringing it about.

Further, Wallace (2020) has introduced philosophers to another puzzle,intimately related to information loss in the context of black-holeevaporation. For lack of a better term, and so as to distinguish itfrom the standard problem, we call this ‘Page-timeparadox’, as it was first formulated by Page (1993), and turnson calculation of a distinguished time in the evolution of anevaporating black hole, the so-called Page time, that time at whichhalf of the black hole’s original entropy has been radiatedaway. Page showed that there is a manifest inconsistency between atreatment of black hole evaporation that is wholly formulated in theterms of statistical mechanics, and the standard semi-classicaltreatment used in derivations of Hawking radiation. Wallace arguesforcefully that this puzzle is incontrovertibly paradoxical,completely divorced from the issue of whether or not unitarity fails,and raises deep philosophical problems of its own.

Recently, there was considerable progress in resolving the Page-timeparadox (see Almheiriet al. 2021 for a review). Inparticular, in various works a new gravitational formula forcalculating the fine-grained entropy of Hawking radiation has beenproposed (culminating in the work Engelhardt-Wall 2015) that seems toreproduce the Page curve (Pennington 2020). This formula involvesa generalized entropy,i.e., an area of a two-dimensionalsurface and the entropy of quantum matter fields outside the surface.The surface is roughly chosen such that the generalized entropy isminimized. For an evaporating black hole there are two such surfaces:a trivial surface with vanishing area, and a non-trivial surface justbehind the horizon. At early times the generalized entropy of thetrivial surface is lower than the entropy of the non-trivial surface,but around the Page time the entropy of the non-trivial surfacebecomes lower. By transitioning between these two contributions theentropy of Hawking radiation closely follows the Page curve. Thisconstruction has been made most explicit in two-dimensionaldilaton-gravity toy models, and it is qualitatively understoodin higher dimensions. Moreover, the gravitational formula wasderived from a gravitational path integral approach by two differentgroups (Peningtonet al. 2022 and Almheirietal. 2020).

In sum, the debate over the Information Loss Paradox highlights theconceptual importance of the relationship between different effectivetheories. At root, the debate is over where and how our effectivephysical theories will break down: where can they be trusted, andwhere must they be replaced by a more adequate theory? This hasobvious connections to the issue of how we are to interpret theontology of merely effective physical descriptions, and how we are tounderstand the problems of emergence and reduction they raise. (See,e.g., Williams 2017 for an interesting survey of such issuesin the context of quantum field theory on flat spacetime.) TheInformation Loss Paradox ramifies into questions of ontology in otherways as well. When matter forms a black hole, it is transformed into apurely gravitational entity. When a black hole evaporates, spacetimecurvature is transformed into ordinary matter. Thus black holes appearto be crucial for our understanding of the relationship between matterand spacetime, and so provide an important arena for investigating theontology of spacetime, of material systems, and of the relationsbetween them.

6.3. A Path to Quantum Gravity?

Black hole thermodynamics and results concerning quantum fields in thepresence of strong gravitational fields more generally are without adoubt the most widely accepted, most deeply trusted set of conclusionsin theoretical physics in which our current best, deepesttheories—general relativity and quantum field theory—worktogether in seemingly fruitful harmony. Indeed, that black holespossess a physical temperature and entropy, and correlatively thatthere is a hitherto unsuspected and profound connection among gravity,quantum field theory and thermodynamics, is now as widely accepted anidea in theoretical physics as an idea with no direct empiricalsubstantiation can be. As such, the study of black hole thermodynamicsprima facie holds out the most promise for guidance in oursearch for a deeper theory of quantum gravity, in which the two wouldbe intimately combined in a unified account of all known physicalphenomena, from the behavior of quarks at the scale of10^-17 cm, to the cosmological structure of superclusters ofgalaxies at the scale of 10³² cm. (See the entry onquantum gravity.) What is not widely shared is the vision of the path that thisguidance purportedly shows us.

We list only a small sample of the many foundational and philosophicalissues that arise here. A full discussion is beyond the scope of thisarticle.

1. Given the apparent need for an underlying statistical mechanics toground the thermodynamical behavior of black holes, does that implythat spacetime itself at the most fundamental levels must have adiscrete structure? (See,e.g., Sorkin 2005 and Oriti2014.)
2. If so, in what sense does classical, continuous spacetimestructure “emerge” from that underlying discretestructure? (See,e.g., Oriti 2014, Wüthrich 2017 and VanDongenet al. 2020.)
3. Does the thermodynamical character of gravity suggest that theEinstein field equation of general relativity ought to be considereditself only an effective equation of state (e.g., Jacobson1995), and so therefore not itself an appropriate candidate to serveas the basis for an attempt to “quantize” gravity?
4. If so, can gravity itself be derived as a purely thermodynamicalphenomenon (e.g., Verlinde 2011, Padmanabhan 2015 andJacobson 2016)?
5. How can a set of theoretical models derived by extending twoseemingly mutually inconsistent theories into regimes far beyond ourexperimental reach, based on nothing but our intuitions about whichfundamental physical principles to hold on to and which to let go,provide evidence foranything?

Wallace (2019) provides an overview of the relation of black holethermodynamics to a few programs in quantum gravity, especially thoserelated to string theory and the AdS/CFT correspondence, andassociated foundational problems.

7. Analogue Black Holes and Hawking Radiation

The Hawking temperature of a macroscopic black hole is unimaginablysmall. For the black hole at the center of the Milky Way (SagittariusA^*), approximately 4 million solar masses, it isapproximately 10^-14 Kelvin. Even a black hole of one solarmass would have a temperature of only about 60 billionths of a Kelvin.Direct experimental verification of its existence therefore seemsbeyond the realm of the imaginable, at least for macroscopic blackholes. (If nothing else, it would be utterly swamped just by theordinary cosmic microwave background radiation, itself approximately2.7 Kelvin, a raging inferno in comparison.)

In 1981, Unruh pointed out that a direct analogue of Hawking radiationshould occur in the most mundane and ordinary of physical systems,flowing water (under particular conditions). The physical basis forhis idea is almost ridiculously simple: if water is flowing past aboundary more rapidly than its speed of sound, then an effective eventhorizon forms, for any disturbances in the water, which will propagatewith the speed of sound, will necessarily be “trapped”behind the boundary. He then argued that the scattering of waterwavelets at the boundary will occur with a thermalized spectrum, inexact accord with Hawking radiation (Unruh 1981, 2008). Since then,analogue models for Hawking radiation in a wide variety of fluid,solid-state, optical and quantum systems have been found. (SeeBarcelóet al. 2011, Robertson 2012, Jacobson 2013,and Faccioet al. 2013 for reviews.)

The remarkable fact that is of most interest to us is that, becauseUnruh’s arguments relied only on simple physical properties ofno-escape boundaries and the low-energy behavior of thermalizedradiation caused by scattering of fields off of such boundaries, Unruhconcluded that these so-called “dumb holes” (dumb becausesilent) could serve as experimentally viable proxies for testing theexistence of Hawking radiation for black holes (Leonhardt and Philbin2008). In particular, the validity of the analogue models is arguedfor on the grounds that the essential features of Hawking radiationare due solely to a few simple, formal kinematical conditionssatisfied by a wide range of kinds of physical systems (Visser 1998a,2013; Unruh and Schützhold 2005; Unruh 2014). In particular, themanifestation of radiation-like behavior formally analogous to trueHawking radiation from a black hole has nothing to do with anyspecific, dynamical features of general relativity. Therefore, thethought goes, to detect the analogue of Hawking radiation in any ofthese systems provides indirect but strong confirmational support forthe existence of actual Hawking radiation. There are, moreover, nowseveral claims to have experimentally detected analogue Hawkingradiation: Belgiornoet al. (2010) based on ultrashort laserpulse filaments,i.e., intense laser pulses in a transparentKerr medium (those with a third-order optical nonlinearity);Weinfurtneret al. (2011) based on obstructed supersonicfluid flow; Steinhauer (2014) based on a “black-holelaser” composed of phonons in an Einstein-Bose condensate; andthe list goes on. So, has Hawking radiation been experimentallyconfirmed, even if only indirectly?

From a philosophical angle, Dardashtiet al. (2017) arguethat such analogue models of event horizons and Hawking radiation canprovide powerful confirmatory support for the existence of Hawkingradiation around actual black holes. Indeed, they argue that theseparticular kinds of analogue model and the concomitant support theypurport to provide are novel, both in the sense of being of a sort notinvestigated before in the philosophical literature and in the senseof representing an innovation in actual scientific practice. (See theentry onanalogy and analogical reasoning.) They base their claim on the fact that these are not only theoreticalmodels, but that they can be—and are—implemented as actualexperiments, and thus constitute not merely analogical reasoning, butexperimentally controlled physical simulation. If one accepts acertain kind of universality argument (Unruh and Schützhold2005), they claim, then it is this latter characteristic that lendsthe analogue models the possibility of strong confirmatory support ofactual Hawking radiation; and to the contrary, without acceptance ofthat universality argument—if the models were based merely onstandard analogical theoretical reasoning—no confirmatorysupport at all would be had.

Grybet al. (2021) compare the kinds of universality argumentseemingly needed in this case to the more standard, familiar form ofsuch arguments made in the context of renormalization group methods.They conclude that all available universality arguments made tosupport taking analogue experiments to confirm the existence ofHawking radiation are wanting in at least one of six categories thatthey collectively deem necessary for such arguments to work(robustness, degree of universality, physical plausibility,comprehensiveness, empirical support, and integration or consistencyof theoretical bases for both robustness and universality), withfailure of integration being the most serious problem.

There is room, moreover, for yet more skepticism here. The argumentsareprima facie strong that the analogue of Hawking radiationshould manifest in a wide range of systems, as a purely kinematicaleffect following directly from a few simple kinematical principlesthat all those systems satisfy (Unruh 2014). Nonetheless, truegravitational black holes are radically different from all theproposed analogue systems, in a variety of extensive and deep ways, asis general relativity as a physical theory from all the theoriesgoverning those other types of systems. In a similar vein, Crowtheret al. (2019) argue that the framework of confirmation byanalogue experiments is circular, in the sense that the validity ofthe semi-classical framework (in which Hawking radiation is derived)is presupposed by the universality argument, but it is precisely itsvalidity (about the astrophysical black holes) that should bedetermined through the experiments. Crowtheret al. argument,however, is seen by some (e.g. Field 2025) as a form ofinductive skepticism.

Field (2025) formulates two desiderata for a successful universalityargument: one knows (1) that the micro-physics of the systems inquestion is all of the same type, and/or (2) that one can empiricallytest the macro-behavior of the system to compensate for theuncertainty about its micro-physics. She argues that although somesmall degree of confirmation for astrophysical Hawking radiation canbe had on the basis of the existing analogue experiments, the limitedunderstanding of the quantum gravity micro-physics—specifically,in how it affects the physical parameters that can be modeled in theanalogue systems—blocks any significant degree of confirmation.(However, in a related work, Field (2021) also points out thatanalogue experiments serve many more roles than just confirming theexistence of Hawking radiation, including formation of a concept ofgeneralized Hawking radiation, detecting instances of that moregeneral concept, and facilitating exploratory experimentation.)Indeed, as the debate and dissension discussed insection 6.1 illustrates, the fundamental physics of Hawking radiation may not bewell enough understood to have confidence that some confoundingphysical factor cannot be present in purely gravitational systems thatis not present in any of the analogue systems, a factor that wouldblock production of Hawking radiation by true black holes. In otherwords, there seemsprima facie little reason to have faiththat the universality condition holds, except on the basis of purelytheoretical arguments pertaining to systems we have no empiricalexperience of nor access to whatsoever.

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Other Internet Resources

• Curiel, Erik, 2017a, “Are Classical Black Holes Hot or Cold?”, unpublished manuscript.
• –––, 2017b, “How Thermodynamical Are Black Holes?”, colloquium given at the Black Hole Initiative, Harvard University, 08February 2017, youtube video available online (talk begins at 4:30 andends at 52:30 of the video).
• –––, 2018, “A Strengthened Zeroth Law for Black-Hole Thermodynamics”, colloquium given at the Black Hole Initiative, Harvard University, 27February 2018, youtube video available online (talk begins at 41:30 ofthe video)
• Carter, Brandon, 1997, “Has the Black Hole Equilibrium Problem Been Solved?”, invited contribution to the Eighth Marcel Grossman Meeting,Jerusalem, 1997.
• Hayward, Sean A., 2006, “Conservation Laws for Dynamical Black Holes”, unpublished manuscript.
• ’t Hooft, Gerard, 1993, “Dimensional Reduction in Quantum Gravity”, unpublished manuscript.
• Jacobson, Ted,, 2003, “Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect”, unpublished manuscript.
• Norton, John D., 2008,Einstein for Everyone, online manuscript.
• Rovelli, Carlo, 2001, “A Note on the Foundation of Relativistic Mechanics. I: Relativistic Observables and Relativistic States”, unpublished manuscript.
• The Gravitational-wave Transient Catalog.
• Wald, Robert M., 2006b, “The History and Present Status of Quantum Field Theory in Curved Spacetime”, unpublished manuscript.

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Acknowledgments

Parts of Sections 1.1–1.3, 2, 3.1–3.2, 5.2, and 6.2 arebased on thefirst version of this entry, which was co-authored by Erik Curiel with Peter Bokulich. Erik Curielthanks Jeremy Butterfield for supererogatorily detailed, thorough, andelegant comments on a draft of the 2019 version of this entry. Wethank John Manchak for producing figures 1, 2, and 3.