Spacetime singularities and a novel formulation of indeterminism
SSpacetime singularities and a novel formulation of indeterminism
Feraz Azhar ∗ Department of Philosophy, University of Notre Dame, Notre Dame, IN, 46556, USA &Black Hole Initiative, Harvard University, Cambridge, MA, 02138, USA
Mohammad Hossein Namjoo † School of Astronomy, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (Dated: January 27, 2021)
Spacetime singularities in general relativity are commonly thought to be problematic, inthat they signal a breakdown in the theory. We address the question of how to interpretthis breakdown, restricting our attention to classical considerations (though our work hasramifications for more general classical metric theories of gravity, as well). In particular, weargue for a new claim: spacetime singularities in general relativity signal indeterminism .The usual manner in which indeterminism is formulated for physical theories can betraced back to Laplace. This formulation is based on the non-uniqueness of future (orpast) states of a physical system—as understood in the context of a physical theory—as aresult of the specification of an antecedent state (or, respectively, of a subsequent state).We contend that for physical theories generally, this formulation does not comprehensivelycapture the relevant sense of a lack of determination . And, in particular, it does not com-prehensively capture the sense(s) in which a lack of determination (and so, indeterminism)arises due to spacetime singularities in general relativity.We thus present a novel, broader formulation, in which indeterminism in the context ofsome physical theory arises whenever one of the three following conditions holds: future(and/or past) states are (i) not unique—as for Laplacian notions of indeterminism; (ii)not specified; or (iii) ‘incoherent’—that is, they fail to satisfy certain desiderata that areinternal to the theory and/or imposed from outside the theory. We apply this formulationto salient features of singularities in general relativity and show that this broader senseof indeterminism can comprehensively account for (our interpretation of) the breakdownsignaled by their occurrence.
Contents
I. Introduction II. Singularities in general relativity
III. A novel formulation of indeterminism
IV. Indeterministic features of singularities in general relativity
V. Transitions between sources of indeterminism VI. Envoi Acknowledgments References ∗ Email address: [email protected] † Email address: [email protected] a r X i v : . [ phy s i c s . h i s t - ph ] J a n . INTRODUCTION Singularities in general relativity (GR)—and in metric theories of gravity more broadly—remain largely mysterious and are commonly thought to signal the breakdown of such theories. Two notable circumstances in which they are thought to arise are in the final stages of theevolution of a sufficiently massive star, as for black holes, and in the very earliest momentsof our universe, in the form of a ‘big-bang singularity’. In the former case, the process ofgravitational collapse—a process that we believe has occurred many times in our universe—yields a class of objects that are not wholly described by the theory that posits their existence(namely, by GR). A similar claim holds in the latter case—in which the description of spacetimeas a whole, provided by GR, appears to break down. A notable feature of such circumstancesis that they are not necessarily a consequence of special, unrealistic circumstances, a claimencapsulated in a set of remarkable theorems proved and elaborated on in the five years from1965 to 1970—mainly by Penrose, Hawking, and Geroch—culminating in the Hawking-Penrosetheorem (Hawking and Penrose, 1970). And so, in analyzing such ubiquitous and mysteriousentities, a foundational question arises: in what sense does GR break down as a result ofsingularities?A new response to this question, which we develop in this paper, is that singularities signal indeterminism in GR. This is a thorny task, for the doctrine of determinism (and that ofindeterminism—which we will take to be its negation ) has a long and controversial history.Such controversy continues to manifest in our understanding of the two pillars of twentiethcentury physics, namely, GR and quantum mechanics. [See Salmon (1998, Ch. 2) for a broadoverview of determinism in science, and Earman (1986) for a more detailed treatise.] Here, wewill focus exclusively on classical considerations, though within this domain of applicability weexpect our remarks will apply relatively broadly (namely, to a broad class of metric theories ofgravity).The standard account of determinism in physics—and, indeed, the one with respect to whichaspects of GR are usually analyzed—can be traced back to Laplace (1814). This notion of‘Laplacian determinism’, as it applies to theories (as opposed to, say, the world itself), amountsto the following idea: a deterministic theory that describes some physical system is one inwhich, given a state of the physical system at some moment in time, the future (and possiblythe past) states of the physical system are uniquely prescribed . Of course, much remains unclearin this description, for we need to make precise what we mean by a ‘state’ of the ‘physicalsystem’, as well as a ‘moment in time’: for now, our intuitive notions of such concepts willsuffice. (We will elaborate on these concepts in Sec. III.A.) In this paper we will argue thatLaplacian determinism and recent closely related formulations [as in, for example, Butterfield(1989) and Doboszewski (2019)] are limited in their ability to describe salient features of physicaltheories that signal a lack of determination (on the part of the theory). In particular, we willargue that such formulations do not comprehensively capture the senses in which a lack ofdetermination (and so, indeterminism) arises as a result of spacetime singularities in GR. There are thus two parts to our project. The first is a description of how we understand Metric theories of gravity, as described by Will (2014), are those that satisfy three criteria: (i) there exists asymmetric metric tensor field on a manifold; (ii) test particles follow geodesics of this metric field; (iii) in localLorentz frames, the (non-gravitational) laws of physics employ special relativity. GR is, of course, one suchmetric theory, as are, for example, effective-field-theoretic generalizations of GR. In this paper we will focusprimarily on GR. Given this relationship between determinism and indeterminism, in this paper we will freely refer to one or theother concept as is appropriate in the relevant context. Note that neither Butterfield (1989) nor Doboszewski (2019) necessarily endorse their descriptions of deter-minism as those that are applicable to discussions about singularities; the more substantive claim in the maintext, which will again be touched upon in Sec. III.A, is that their descriptions are closely related to a sense ofLaplacian determinism. Our emphasis accords with the spiritof the following claim by Earman:Laplacian determinism and its close relatives are, to my knowledge, the only va-rieties which have received attention in the philosophical literature. The expla-nation cannot be that no other variety is relevant to the analysis of modern sci-ence . . . (Earman, 1986, p. 17).The formulation of determinism we describe is broader than the Laplacian notion and (roughly)corresponds to a disjunction of the uniqueness, existence, and what we term the ‘coherence’ ofa theory’s specification of states. (Indeterminism then arises whenever at least one of thesedisjuncts is not satisfied.) With regard to the second part of our project, we will enumeratesalient features of singularities in GR and show how the above sources of indeterminism arisefor each of the identified features. We thus aim to establish that there is, indeed, a sense inwhich singularities in GR signal a type of indeterminism and that this type of indeterminismcannot solely be understood as some variety of Laplacian indeterminism.To this end, in Sec. II, we describe how singularities are usually understood in GR, anddistinguish two classes of properties that apply to them: namely, geometrical properties and‘causal’ properties. In Sec. III, we describe and develop our formulation of indeterminism forphysical theories, which is broader than the usual Laplacian account. In Sec. IV, we providea description of salient features of singularities in GR, which, we argue, signal a failure ofdeterminism—as understood in the context of the formulation of (in)determinism developedin Sec. III. In Sec. V, we describe how certain physical processes in GR can change sourcesof indeterminism—in the case of certain black-hole solutions and for Cauchy horizons. Weconclude with a brief summary of our overall argument in Sec. VI.
II. SINGULARITIES IN GENERAL RELATIVITY
Singularities are arguably general relativity’s most problematic feature and they remain overa century’s old discontent of the cognoscenti. Perhaps the first more comprehensive statementabout singularities for metric theories of gravity—and in particular for GR—was the Hawking-Penrose theorem (Hawking and Penrose, 1970). This theorem provides conditions under whichsingularities arise, showing that they are not a feature of special initial conditions. Strikingly,the theorem does not provide detailed information about the nature of the singularities thatarise: just the conditions that lead to singularities, which are diagnosed via curves that ‘stopshort’ (in particular, via timelike or null geodesics). (We will say more about precisely what itmeans for a curve to stop short in the following subsection.) This lack of a specification of thenature of the singularity is, indeed, a general feature of singularity theorems in GR (includingthose that have been developed since the theorem by Hawking and Penrose). Questions thusremain as to the relationship between the manner in which singularities in these theorems arediagnosed and more detailed information about the nature of spacetime at or near singularities[see, for example, Senovilla (1998, 2012)].In this section, to prepare for what follows, we will distinguish two broad classes of propertiesthat can be ascribed to singularities in GR: (i) geometrical properties and (ii) what we term‘causal’ properties. An analogous problem arises in converting our intuitions about singularities in GR into a precise definition ofa singularity. [See, for example, Hawking and Ellis (1973), Wald (1984), and Earman (1995).] . Geometrical properties of singularities Singularities as understood from a geometric perspective—and not just those that arise viagravitational collapse—are perhaps (still) most comprehensively classified, in the context of GR,by Ellis and Schmidt (1977). In their classification scheme, for maximally extended spacetimes(roughly, spacetimes that cannot be embedded into a larger spacetime) , singularities fall intoone of two classes:(i) quasi-regular singularities—the mildest type of singularity, where certain components ofthe Riemann tensor are well-behaved —such as one encountered at the tip of a cone; or(ii) curvature singularities, where there are pathologies related to the components of theRiemann tensor mentioned in (i).This latter category is composed of either non-scalar singularities (where curvature scalars arenot pathological) or scalar singularities (where curvature scalars are pathological). We will saymore about the sense in which such pathologies may arise in Sec. IV.B, but a common wayin which such pathologies manifest is via a divergence of such scalars, as one approaches thesingular region.In this classification scheme, all such singularities are diagnosed via incomplete curves,namely, curves that ‘stop short’. Broadly, such a curve cannot be extended to infinitely largevalues of the parameter that distinguishes different points along the curve. Precisely whichtypes of curves are used to diagnose singularities in this way is somewhat contentious. The clas-sification scheme of Ellis and Schmidt (1977) includes curves that are a superset of those used todiagnose singularities according to the Hawking-Penrose theorem. In the Hawking-Penrose the-orem, singularities are diagnosed by incomplete timelike or null geodesics, namely, ‘freely falling’timelike or null paths that stop short. However, as exemplified by Geroch (1968), geodesic com-pleteness (understood either as timelike, null, or spacelike geodesic completeness) does not seemto be sufficient for the diagnosis of a singularity: there exist spacetimes that are geodesicallycomplete, but where there remain timelike curves of bounded acceleration with finite length.[See also Beem (1976) for two further examples of such scenarios.] Such curves—traversable byan idealized physical observer with a rocket (for example)—seem to be problematic, and theconsideration of such curves has indeed guided work on the nature of singularities in particu-lar metric theories of gravity. [See, for example, Olmo, Rubiera-Garcia, and Sanchez-Puente(2018).] The Ellis-Schmidt classification scheme covers such a case by diagnosing singularitiesby curves that satisfy a form of incompleteness that is more general than both geodesic incom-pleteness and ‘bounded-acceleration incompleteness’, namely, via curves that are b -incomplete.The condition of b -completeness implies both geodesic completeness and bounded-accelerationcompleteness. In this paper, we will assume that singularities are indeed diagnosed by (half) curves that are b -incomplete—both to avail ourselves of the classification scheme described by Ellis and Schmidt,as well as for the physical character of this definition. Note that, however, when we connectour discussion of singularities to the issue of indeterminism, the sense of the incompleteness ofcurves that arises—whether it be, for example, geodesic incompleteness, bounded-acceleration The focus on maximally extended spacetimes is to avoid classing as singular, an entity or region that would bedescribed as nonsingular if the spacetime could be extended, yielding a ‘larger’ one. For example, Minkowskispacetime with a single point removed is not maximally extended and so does not count as singular. Formaximally extended spacetimes, singularities that arise are thus intrinsic to the spacetime. We refer, here, to components of the Riemann tensor in an orthonormal frame that is parallely propagatedalong any curve that ends at the singularity [see Ellis and Schmidt (1977) for further details]. A (half) curve is b -complete just in case it has infinite generalized affine length. [See Hawking and Ellis (1973)for some background.] b -incompleteness—will be less important than the fact that curves, indeed,stop short. B. Causal properties of singularities
Singularities may also be described as either spacelike, timelike, or null (Penrose, 1974). Werefer to these properties as ‘causal’ as they can be informatively described through the behaviorof timelike or null curves (that is, ‘causal’ curves).Singularities that serve as end-points for timelike curves that would otherwise be futurecomplete are known as future spacelike singularities. (A similar definition holds for pastspacelike singularities.) An example of a future spacelike singularity is the singularity that isthought to arise at the center of a black hole formed from the gravitational collapse of matter:a timelike curve that enters the event horizon of such a black hole will eventually ‘hit’ thesingularity and will come to an end [see Fig. 1(a)]. Timelike singularities are those such that atimelike curve (as traversed, for example, by an idealized observer) may pass along-side themwithout falling into them. They can serve as both the past and future end-points of timelikecurves. Such singularities can be found, for example, deep inside a Reissner-Nordstr¨om blackhole, as depicted in Fig. 1(b). Finally, null singularities are singularities for which photons (forexample) may travel along-side them without falling into them. Such singularities might befound inside the interior of an old black hole—the ‘mass-inflation singularity’ of a Kerr blackhole, depicted in Fig. 1(c), corresponds to one such example.Thus singularities have geometrical properties (determined by how geometrical quantities,such as the Riemann tensor, behave in their vicinity) as well as causal properties (determined bytheir relation to causal curves, in their vicinity). In what follows, we will relate our formulationof indeterminism to aspects of these properties.
III. A NOVEL FORMULATION OF INDETERMINISM
In this section we build the main thrust of our argument. In particular, we establish a notionof indeterminism—broader than the standard Laplacian notion—that describes indeterminismas it arises in physical theories, considered generally. We then show that it is this broader notionthat comprehensively captures the senses in which indeterminism arises as a result of singular-ities in GR. As we alluded to in Sec. I, this limitation of the scope of Laplacian indeterminismas it applies to singularities in GR is not necessarily surprising, for such singularities are notusually associated with indeterminism. Earman (1986, p. 188) presents a reason for why thisassociation is not more prevalent:. . . singularities are an ugly stain on the success of determinism in general relativity.Focus on the subclass of models with Cauchy surfaces. Then by our definition ofdeterminism and the results of the gravitational initial value problem, Laplaciandeterminism holds. But for models with singularities the victory of determinismhas a Pyrrhic flavor, for at best the prediction of singularities is a prediction ofthe breakdown of the laws of the theory. That breakdown is not countenancedas a breakdown in determinism since the ‘places’ where the singularities occur are Note that the definition of a singularity adopted in the main text does include spacelike b -incomplete curves;though there is a question about whether spacelike b -incompleteness should be included in one’s definition ofa singularity. Hawking and Ellis (1973, p. 260) define a “space-time to be singularity-free if it is b-complete”;and recognize that one may wish to relax this definition so that it refers to non-spacelike b -completeness.(Ultimately they do not endorse this latter option.) FIG. 1 Penrose diagrams illustrating three different types of singularities as they arise in three differenttypes of black holes. In all three cases, regions denoted by I are outside the black hole, whereas regionsdenoted by II and III are inside the event/outer horizon of the black hole. The red jagged lines denotesingularities. Also, for all three diagrams, i − , i , and i + respectively denote past-timelike infinity, space-like infinity, and future-timelike infinity; whereas J − and J + respectively denote past-null infinity andfuture-null infinity. (a) This diagram depicts the spacetime of a black hole that forms from gravitationalcollapse and illustrates a spacelike singularity. This singularity serves as an endpoint for the timelikecurve O . (b) This diagram depicts a portion of the Penrose diagram of a charged black hole (viz. aReissner-Nordstr¨om black hole) and illustrates a timelike singularity. This singularity can serve as thepast endpoint of a timelike curve (as for O ) or as the future endpoint of a timelike curve (as for O ).(c) This diagram depicts an old rotating black hole [adapted from Scheel and Thorne (2014)] and illus-trates a mass-inflation singularity, which is a null singularity. A photon (denoted by γ ) can pass by thesingularity without falling into it. not countenanced as part of the arena where determinism wins or loses. The evermore clever means by which determinism avoids falsification are as impressive asits straightforward successes.We argue that the suspect nature of determinism’s “victory” (alluded to in the above quote)betrays a different state of affairs: namely, there really was no such victory in the first place—indeed, the types of breakdown associated with singularities should also be countenanced as abreakdown in determinism. A. Indeterminism `a la Laplace
The standard account of determinism can be traced back to Laplace (1814). We will describethis account as it relates to theories—as opposed to, say, some external reality directly [see, forexample, Butterfield (2005), who discusses this distinction]. On this account, a theory T isdeterministic iff given a ‘state’, S ( t ), of a ‘physical system’, P , at some moment in time, t , thefuture (and possibly the past) states of P are uniquely prescribed . Here, P refers to the entire ‘time-evolved’ history of S as understood in the context of the theory T .This broad definition can be applied to GR under the following identifications.(i) T is GR (in four spacetime dimensions, say).6ii) The physical system, P , consists (in principle) of three items: P = (cid:104) M, g, T (cid:105) , where M is a four-dimensional manifold, g is a metric tensor, and T is a stress-energy tensor.(iii) A state of this physical system, S ( t ), at some moment in time, t , consists of two items: S ( t ) = (cid:104) Σ t , D t (cid:105) , where Σ t is a (three-dimensional) hypersurface (that is, for example,compact or else asymptotically flat) in M , and D t corresponds to suitable initial data.These initial data must satisfy constraint equations (a subset of the full Einstein fieldequations). They correspond to a three-dimensional metric on Σ t and derivatives (ineffect, with respect to time) of the metric, as well as matter fields on Σ t and correspondingderivatives.Laplacian determinism then holds for GR when given a state S ( t ) = (cid:104) Σ t , D t (cid:105) , at t , all states S ( t (cid:48) ) for t (cid:48) > t (and possibly also for t (cid:48) < t ) are uniquely prescribed.Note that states at times other than t are generated by evolving the initial data accordingto Einstein’s equations, so that this characterization of determinism can be understood as astatement about the uniqueness of solutions of the appropriate differential equations. Further-more, there is a redundancy in GR such that the union of all states for which t (cid:48) > t [namely, thefour-dimensional spacetime that results from evolving S ( t )] can be related via diffeomorphismsto other observationally indistinguishable four-dimensional spacetimes. The notion of Laplaciandeterminism in which we are interested demands uniqueness only up to diffeomorphisms. There are also related versions of Laplacian determinism, which address various choices thathave been made in the characterization above. One such version is provided by Butterfield(1989) [see also Doboszewski (2019)] where a broader notion of a ‘state’ is introduced: inparticular, the hypersurface Σ t is replaced by a ‘spacetime region’—which does not have tocorrespond to a ‘time-slice’. Indeterminism arises when two “models” (where a model consistsof a manifold and associated geometrical structures) that agree (in a certain sense) on such aspacetime region, cannot be identified with each other on the entire manifold. [See Butterfield(1989, p. 9) for further details.]In sum, more standard ways of describing indeterminism (that is, Laplacian indeterminism)refer, in essence, to the non-uniqueness of states of a physical system—despite one havingspecified another (sufficiently detailed and, for example, antecedent) state. In the followingsubsection, we will redefine indeterminism, taking a broader point of view. B. A broader understanding of indeterminism
Let us step back for a moment and consider, in a general physical setting, a less precisedescription of determinism [as compared to, for example, the version by Butterfield (1989),mentioned above]. In particular, consider the following:Determinism requires a world that (a) has a well-defined state or description, atany given time, and (b) laws of nature that are true at all places and times . . . if(a) and (b) together logically entail the state of the world at all other times (or,at least, all times later than that given in (a)), the world is deterministic. Logicalentailment, in a sense broad enough to encompass mathematical consequence, isthe modality behind the determination in “determinism” (Hoefer, 2016, Sec. 2.5).On this account, an indeterministic world is one in which if you specify its state at some timethen the (true) laws of nature fail to determine the state of the world at some other time(s). Ourconception of indeterminism takes this insight as a starting point. In fact, we need to amend Thus, we will set aside the type of indeterminism that GR is arguably subject to as a result of the ‘holeargument’. [For a recent historical account see, for example, Secs. 1 and 2 of Roberts and Weatherall (2020).] world ). Secondly, the above characterization of indeterminism crucially involves a lack ofa logical entailment of states at some other time(s). We understand such a lack of entailment(namely, a lack of determination) of a state, in the context of some physical theory, as a failureto ‘specify’ a state —where this failure can arise in one of three ways. That is, there are threeways in which indeterminism can arise for some physical theory. For ease of later reference, wewill denote each such ‘source’ of indeterminism by a mnemonic label.
Non-uniqueness :The theory provides multiple, distinct accounts of future (or past) states of aphysical system.
Non-existence :The theory provides no account of some possible future (or past) state of aphysical system.
Incoherence :The theory does provide an account of future (or past) states of a physicalsystem, but that account fails to satisfy conditions (viz. desiderata), C , thatare internal and/or external to the theory.The criteria C provide a check on whether the specification of a state by the the-ory leads to: (i) an internal conflict, in which case we refer to the incoherenceas Internal incoherence ; and/or (ii) a conflict with some pre-establishedtheory (or principle) that enjoys broad support—a case we dub
Externalincoherence .Note that Laplacian determinism only captures
Non-uniqueness . The other two sources arethus responsible for the claim that our formulation is broader than the usual Laplacian one.We now turn to some salient aspects of the above characterization of indeterminism.As regards
Non-uniqueness , the sense in which future (or past) states are ‘distinct’ can,of course, be understood in the context of the particular theory (though possibly in multiple,prima facie different, ways). For example, in Newtonian dynamics,
Non-uniqueness can ariseas a result of the time-reverse of a scenario [described by Xia (1992)] in which a particularconfiguration of five point-masses moving in three-dimensional Euclidean space can escape tospatial infinity in a finite time. Here, distinct accounts of future states arise depending onwhether or not particles do, indeed, fly in from spatial infinity. [See, for example, Hoefer (2016)for a description of such ‘space-invaders’ and Werndl (2016) who describes the resulting lackof uniqueness in this way.] In the case of GR, one can diagnose
Non-uniqueness through, forexample, the lack of a Cauchy surface for a spacetime manifold. The sense in which accounts offuture states are distinct can then be understood in terms of the existence of non-diffeomorphicspecifications of spacetime regions beyond the Cauchy horizon.Perhaps the most contentious assertion we have made is due to
Non-existence , whereinwe claim that it is not consistent with determinism for a theory to leave the description of somesystem unspecified. Previewing our discussion in Sec. IV.A, such a situation arises in GR forsingularities, when end-states of particles traveling along curves are left unspecified—such aswhen one traces such curves back into the big-bang singularity. That this signals indeterminismbreaks with the common intuition to which Earman refers, quoted in the preamble to thissection, wherein ‘places’ where singularities arise aren’t included “as part of the arena wheredeterminism wins or loses”. In contrast, we assert that those places where singularities ‘arise’8hould be included as part of the arena where determinism wins or loses: leaving unspecifiedthe end state of a particle moving along an incomplete trajectory is a key (definitional) aspectof a lack of determination . (We will elaborate on this issue in Sec. IV.A.)As regards Internal incoherence , it has been argued by Vickers (2013) that ‘internalinconsistency’ is difficult to diagnose. We will not need to take a stance on this claim—butpoint out only that if the specification of a state by a theory is internally incoherent (or, indeed,internally inconsistent) then that should count as a failure of the theory to specify the state.Nevertheless, there are clear cases where Internal incoherence can arise: as in certainsolutions of GR that contain closed timelike curves. Here, the possibility arises of an explicitcontradiction (that is indeed internal to the theory): namely, of the existence of some state andthe nonexistence of that same state (as for the usual paradoxes that relate to time-travel, suchas the ‘grandfather paradox’). Two examples of
External incoherence —with differingfates for the proposed model—are: (a) Bohr’s model of the atom, which when proposed wasin conflict with Maxwellian electrodynamics (Vickers, 2013); and (b) the steady-state modelof the universe, with its need for the creation of matter (so that the density of the expandinguniverse would remain constant), which was in conflict with the principle of conservation ofenergy (McMullin, 1982). As we will elaborate on in the following section—both senses of
Incoherence will arise in describing singularities in GR.Finally, note that our formulation of indeterminism can be used to assess indeterministicfeatures of individual models themselves (where, by a ‘model’, we mean a specific solution ofthe equations that one derives from the theory of interest). So, for example, in the context of GRand in metric theories of gravity more generally,
Non-existence and
External incoherence can be applied to specific manifolds and the fields defined on these manifolds, to diagnose theexistence of indeterminism in these models. For example—and previewing a claim that will arisein the following section—a specific set of initial conditions that yields a region in which the end-state of a particle traveling along a timelike curve is unspecified reveals a specific model that isindeterministic, by virtue of it satisfying
Non-existence . For cases that arise exclusively as aresult of
Non-uniqueness , at least two distinct models are required to diagnose indeterminism,though, of course, one can label a single spacetime as ‘non-unique’.We now further exemplify and apply our formulation of indeterminism specifically to featuresof singularities as they arise in GR.
IV. INDETERMINISTIC FEATURES OF SINGULARITIES IN GENERAL RELATIVITY
There are a variety of features of singularities in GR that, we contend, exhibit indeterminism,in the style of that described in Sec. III.B. In this section we will provide an account of suchfeatures, including a discussion of why they indeed signal indeterminism. Furthermore, we willhighlight how this indeterminism relates to the Laplacian notion discussed in Sec. III.A. Note that we are not suggesting that singularities should necessarily be considered to be part of spacetime—that is, we are not necessarily enlarging our conception of spacetime to include singularities; we are advocatingenlarging the arena over which determinism wins or loses. We have chosen to generally use the term ‘incoherent’ instead of ‘inconsistent’ as we have in mind a notionthat is broader than something akin to logical contradiction (which is readily brought to mind by the latterterm). See Sklar (1990), for an interesting discussion of the impact, if any, of such paradoxes on the (initial) conditionsthat may give rise to such a situation. See Doboszewski (2019) for another account of indeterministic features of GR more generally. . Unspecified end-states As described in Sec. II.A, the defining property of singularities, as they are commonly un-derstood, is that they bring curves to an end—viz. such curves cannot be extended indefinitely:they are incomplete . These curves fall into one of three classes: timelike, null, or spacelike.For singularities that are diagnosed via the former two classes (that is, via causal curves) oneencounters future and/or past incompleteness. Consider the case in which a singularity is diag-nosed by a timelike or null future-incomplete curve. This can be realized by a test particle thattravels along the curve and eventually encounters the singularity. Importantly, a consequenceof the incompleteness of the curve is thus that, according to the theory, the end-state of a par-ticle moving along such a trajectory is left unspecified. Therefore, we contend, indeterminismarises as a result of
Non-existence .One may agree with a claim in the quote of Earman’s (as discussed at the outset of Sec. III)that there is no ‘place’ (or ‘time’) for which GR needs to provide an account of the particle,when that particle’s worldline indeed comes to an end in a singularity; yet we maintain thatthe scenario manifests indeterminism (see fn. 10). There are questions one can naturally askfor which the theory does not provide an answer. More explicitly (but rather colloquially), onecan ask about the fate of the particle—which, prior to encountering the singularity, possesseda well-defined state. For example, does the particle indeed have a finite lifetime? If so, whatdoes it mean, according to the theory, for the particle to not exist: that is, is there some statedescribing this non-existence in the space of possible states? If not, then what happens tothe particle? Moreover, one may wonder about the process of ‘hitting the singularity’. Toreiterate: the theory does not provide an answer to such questions.Note that the situation here is very different from what happens, for example, in quantumfield theory (QFT). In the latter context it is common to consider scenarios in which particlesare annihilated (or are created) via a suitable interaction. In this sense, the particle’s worldlinemay “stop short”—but in contrast with what happens in the case of incomplete curves in GR,the annihilation and creation of particles can be described within QFT. In fact, the state of the“physical system” remains well-defined even after the disappearance of a particle.Analogous pathologies may arise in the case of past incompleteness (which, one might argue,are more problematic). For example, particles may enter spacetime from a singularity andinteract with other particles (Earman, 1995; Penrose, 1969; Shapiro and Teukolsky, 1991). Thisis a general-relativistic version of the ‘space invaders’ scenario, described in Sec. III.B, whicharguably provides a source for indeterminism in Newtonian mechanics. Indeterminism arisesbecause GR does not provide an account of the process or mechanism by which such mattermay form at/near singularities; thereby, again, satisfying conditions for
Non-existence . How is this
Non-existence related to a Laplacian notion of determinism? Indeed, it isnot that the end-state of the particle is specified in ways that are non-unique. Rather, theend-state is not specified at all. Laplacian determinism is not compromised by the existence ofsuch incomplete curves. [See also Doboszewski (2019).] For a timelike future-incomplete curve, the test particle can be understood, for example, as an idealizedobserver. More generally, the particle can be described as part of the evolving state of the physical system,where it does not back-react on the underlying geometry and where it has a sufficiently small size (so that onecan describe its trajectory via a curve). See, for example, Hawking and Ellis (1973, p. 152), Wald (1984, p. 193), and Carroll (2019, p. 227), where suchlanguage is used or implied. (We do not, of course, assume the authors imply, by the use of such language, theexistence of some GR-based account of what the process of ‘hitting the singularity’ entails.) Note that scenarios resulting from whether or not particles are emitted by singularities, could be understood interms of
Non-uniqueness (in much the same way that scenarios resulting from ‘space invaders’ in Newtoniandynamics, discussed in Sec. III.B, are usually understood). Here (that is, in the main text above) we haveattributed the indeterminism that arises to
Non-existence as, we contend, this source is ultimately responsiblefor the indeterminism. . Ill-behaved physical quantities ‘Curvature singularities’ in GR, described in Sec. II.A, are associated with the phenomenonthat, in their vicinity, certain physical quantities (i) oscillate without limit or (ii) grow withoutbound. (Note that this is not true of singularities categorized, in Sec. II.A, as ‘quasi-regularsingularities’.)With regard to (i), for a version of Taub-NUT spacetime, scalar quantities may remainbounded along some curve as one approaches the singularity, but their values oscillate with-out approaching a limit [see, for further details, Earman (1995, pp. 38–40)]. In such cases,differentiability constraints (such as the assumption that the underlying manifold is smooth,viz. C ∞ ) can be violated. The sense of indeterminism that arises is different to that describedin Sec. IV.A; namely, it is Internal incoherence that arises.With regard to (ii), perhaps the most well-known example is at the center of the simplesttype of black hole (a non-spinning, non-charged black hole) in which a quantity that charac-terizes properties of the curvature of the spacetime in a coordinate invariant way, namely, theKretschmann scalar ( R µνστ R µνστ ) diverges as 1 /r (here, r measures radial distance from the‘center’ of the black hole). The sense of indeterminism at play in such a scenario is againdifferent to that described in Sec. IV.A. In particular, at some point (for example, at someradial distance r ∗ from the ‘center’ of the black hole—or for certain sufficiently large values of aparameter that describes points along a curve headed into the singularity), the values taken byphysical quantities are not to be taken literally: the theory supplies values for physical quantitiesthat are ‘too large’ according to certain criteria that we demand of GR (that are not encodedin the theory itself). The theory is subject to a charge of External incoherence . Note thatthis charge of indeterminism is leveled at GR despite the fact that the theory can (and does)specify a unique value for quantities such as invariant scalars—again (as in Sec. IV.A) a senseof Laplacian determinism is not being compromised.Of course, one may wonder about the external standard(s) by which we judge whethercertain physical quantities should be taken literally. There are two points of view, one morephenomenological, the other more formal. In the former case, the claim is that such divergencesdo not appear to arise in nature and so when a theory predicts that they do, something hasgone wrong. The more formal response can be understood in one of two ways.(i) First, dimensional analysis can reveal where quantum gravity effects are expected to beimportant (that is, where the classical theory is expected to break down)—in particular,when values for the curvature approach those associated with the Planck scale [for exam-ple, | R | ∼ /l ≡ c / ( (cid:126) G ), where l Pl denotes the Planck length]. Such an analysis thusprovides external conditions that can diagnose External incoherence .(ii) A second response (that is, in principle, different from the first) is related to how wecan conceive of GR as the leading-order contribution to a more general effective theory.In certain situations, the effects of higher-order contributions can become as importantas those of GR: the effective theory expansion thus breaks down and GR is no longerdescriptively accurate as the leading-order contribution. Again, this situation manifests
External incoherence .In a little more detail, for such effective-field-theoretic extensions of GR, one typicallywrites down an action by identifying the relevant symmetries and ordering the action We make no claims about the severity of the pathologies outlined in Sec. IV.A compared with those in Sec. IV.B,though it is interesting to note the following claim, that relates to such a comparison: “Timelike geodesicincompleteness has an immediate physical significance in that it presents the possibility that there could befreely moving observers or particles whose histories did not exist after (or before) a finite interval of propertime. This would appear to be an even more objectionable feature than infinite curvature and so it seemsappropriate to regard such a space as singular” (Hawking and Ellis, 1973, p. 258). S = (cid:90) d x √− g (cid:0) Λ + c R + c R + c R µν R µν + . . . (cid:1) , (1)where { Λ , { c i }} are constants to be determined, for example, from experiment. (We haveleft out a Lagrangian for matter degrees of freedom as that will not be important inwhat follows.) The four terms displayed in parentheses in Eq. (1) have, respectively,zero, two, four, and four derivatives of spacetime coordinates and so—as is usual in EFT-energy expansions—these last two terms are less important for describing physics belowthe energy cutoff of the EFT expansion (which is often taken to be the Planck energy-density—but, indeed, does not have to be). Such an expansion provides conditions inwhich GR might be deemed to be externally incoherent (that is, it would be deemed to beincoherent with the above-defined effective-field-theoretic generalization). For example,if when traversing some curve in a manifold that comprises a solution to the Einsteinfield equations, the second term in the action above becomes comparable to either of thenext largest terms, viz. | c R | ∼ min {| c R | , | c R µν R µν |} , then the perturbative expansionbreaks down—and the claimed (external) incoherence arises. C. Lawlessness
GR is effectively ‘silent’ about occurrences at (or sufficiently close to) singularities, so thereare, as noted by Penrose (1969), no ‘laws’ of singularities. One cannot study singularitieswithin GR as entities in and of themselves, in the same way that one might use the standardmodel of particle physics to probe properties of yet undiscovered particles. As a consequence,a form of indeterminism arises. For example, GR is effectively silent about the possibility ofmatter/energy emanating from singular regions (a process not expressly forbidden by GR—butabout which GR does not make any positive claims). This renders undetermined other regionsof spacetime in causal contact with singular regions. A second perhaps more exotic possibilityalso presents itself. Namely, GR does not account for the possibility of the evolution of thenature of singularities themselves. That is, the theory is also silent about whether singularitiescan evolve in such a way as to change their geometric and/or causal properties ; or whethersingularities can cease to exist, giving rise to non-singular spacetimes. In this way, the theory In this way, singularities can be (at least) locally visible, namely, they can be locally naked . (Note that it ispossible, in principle, for such singularities to arise behind an event horizon.) The weak cosmic censorshipconjecture (WCCC), formulated by Penrose (1969), provides a well-known (potential) limitation on how visiblecertain singularities may be (in the context of GR). Containing the scope of the resulting putative indeterminismis the primary goal of the WCCC. Despite a large (and growing) literature, the WCCC remains—50 or so yearsafter it was first described—an open problem. [See the following select set of references: Christodoulou (1999);Earman (1995); Geroch and Horowitz (1979); Hod (2008); Joshi and Malafarina (2011); Wald (1997).] As we will discuss in Sec.V, certain physical arguments suggest that such evolution is indeed possible, thoughGR does not provide a detailed account of the actual process. As noted at the outset, we are focusing here on classical considerations. Taking quantum effects into accountmay result in the evaporation of black holes (as well as their singularities) via the emission of Hawking radiation.However, it is not clear that the emission of Hawking radiation persists when the mass of the black hole becomescomparable to the Planck mass, and thus whether black holes and their singularities evaporate completely.(Here, the semi-classical approximation is expected to break down and quantum gravitational effects areexpected to become important.) So, despite taking into account such considerations that generalize GR, wedo not have a definite answer to the question of whether singularities can cease to exist. ources of indeterminismFeatures of singularities Non-uniqueness Non-existence Incoherence
Unspecified end-states – (cid:88) –Ill-behaved physical quantities – – (cid:88)
Lawlessness – (cid:88) –Cauchy horizons (cid:88) – –
TABLE I Sources of indeterminism (as described in Sec. III.B) for the four features of singularitiesdiscussed in the main text (in Secs. IV.A–IV.D). The left-hand column lists these features and theremaining columns catalog whether the particular source of indeterminism is present ( (cid:88) ) or not (–). satisfies conditions for
Non-existence —in that it stops providing a description of physicalregions near and moreover ‘at’ singularities.
D. Cauchy horizons
The final source of indeterminism we will mention relates to Cauchy horizons in GR [asdepicted in, for example, Fig. 1(b)]. Generally, these correspond to boundaries of spacetimebeyond which the initial value formulation of GR (alluded to in Sec. III.A) does not uniquelyspecify the state of spacetime.Certain types of singularities lead to Cauchy horizons. In particular, they can arisewhen one has a timelike singularity or a null singularity. As far as we are aware, spacelikesingularities only arise at the boundaries of spacetime—in such a way that no Cauchy horizonarises [see Penrose (1974)]. (So, for example, there is no Cauchy horizon associated witha big-bang-type singularity.) The sense of indeterminism that arises corresponds to a clearviolation of determinism understood in a Laplacian sense: such a scenario satisfies conditionsfor
Non-uniqueness .In sum, as regards the four indeterministic features of singularities described above, threeof the features have aspects that are not consistent with a Laplacian notion of indeterminism(see Table I). Perhaps most notably, Laplacian indeterminism fails to capture what is arguablya key feature of singularities, namely, that they stop particles in their tracks in such a way asto leave the end-state of the particle undetermined. Laplacian determinism does not thereforeprovide a general way to describe indeterministic features of singularities in GR. [Note that thisassessment is consistent with Doboszewski (2019) who takes issue with Laplacian determinismas a way to provide a general account of determinism in GR.] V. TRANSITIONS BETWEEN SOURCES OF INDETERMINISM
An interesting feature of indeterminism, as it manifests due to singularities in GR, is thatthe source of the indeterminism in a particular physical setting can change.For instance, dropping a charged particle into a Schwarzschild black hole converts the blackhole into that of a Reissner-Nordstr¨om black hole. [So, for example, a Penrose diagram such asthat depicted in Fig. 1(a) is transformed into one such as that (partially) depicted in Fig. 1(b).]The central singularity is originally a spacelike singularity in which (i) the end-state of a particle The question of the existence of singularities as they may arise in more general theories of gravity, in particularfor general metric theories of gravity, is an interesting one. Specific analyses do exist, and these analysesdiagnose singularities via incomplete curves. According to our formulation of indeterminism, therefore, suchsettings also signal indeterminism. [See, for example: Alani and Santillan (2016); Bejarano, Olmo, and Rubiera-Garcia (2017); Horowitz and Myers (1995).] We leave a more detailed analysis of these issues for future work.
Non-existence and
External incoherence , respectively. After dropping the charged particleinto the Schwarzschild black hole, this singularity is converted into a timelike singularity (withan associated Cauchy horizon) in which indeterminism arises—in addition to
Non-existence and
External incoherence —as a result of
Non-uniqueness . Another example arises in considering Cauchy horizons. Note that one can have Cauchyhorizons ostensibly without the presence of singularities. But there are interesting cases inwhich such horizons are, in fact, conjectured to be related to singularities. For example, certainblack-hole solutions with Cauchy horizons, such as the Kerr and Reissner-Nordstr¨om solutions,are conjectured to exhibit a ‘blue-shift instability’ of these horizons [see, for example, Penrose(1968) and Poisson and Israel (1990)]. Namely, small perturbations in initial data are conjec-tured to build-up at those horizons to convert them into curvature singularities . In such aninstance, the usual source of indeterminism associated with Cauchy horizons (independent oftheir relation to singularities), namely,
Non-uniqueness is, under our formulation of indeter-minism, traded for
Non-existence (due to curves stopping short at curvature singularitieswhere an end-state is unspecified) and
External incoherence (due to physical quantitiesbeing ill-behaved). In short, the presence of singularities that arise as a result of the conjecturedblue-shift instability affects the nature of the indeterminism that is associated with the horizonregion. This consequence of our approach to determinism invalidates the usual stance on howsuch an instability impacts the deterministic nature of GR. For example, our approach callsinto question the following claim by Hollands, Wald, and Zahn (2020) (in the abstract of theirpaper), in which the satisfaction of the strong cosmic censorship hypothesis is associated withavoiding indeterminism:The strong cosmic censorship conjecture asserts that the Cauchy horizon does not,in fact, exist in practice because the slightest perturbation (of the metric itself orthe matter fields) will become singular there in a sufficiently catastrophic way thatsolutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmiccensorship holds, the Cauchy horizon will be converted into a ‘final singularity,’and determinism will hold (Hollands, Wald, and Zahn, 2020, p. 1).Under our formulation of indeterminism, this conversion of the Cauchy horizon into a ‘final sin-gularity’ does not save determinism: it trades one source of indeterminism (
Non-uniqueness )for another (a combination of
Non-existence and
External incoherence ). VI. ENVOI
For physical theories generally, we have described a notion of indeterminism that takes abroader point of view than the usual Laplacian notion. Under our formulation, as developed inSec. III.B, indeterminism arises when at least one of the following three conditions obtains: (i)a theory does not provide a unique account of states of the physical system under consideration(what we term
Non-uniqueness ); (ii) a theory does not specify a physical state (what we term
Non-existence ); (iii) a theory provides an account of physical states that does not satisfycertain desiderata (what we term
Incoherence ). These three sources provide a more com-prehensive characterization—compared to the usual Laplacian notion, which only instantiates
Non-uniqueness —of the relevant sense of a lack of ‘determination’ in physical scenarios. An analysis similar in spirit can be given, for example, for a particle dropped into a Schwarzschild black holewith angular momentum; thereby converting the Schwarzschild black hole into a Kerr black hole. Poisson and Israel (1990, p. 1797) phrase a consequence of the satisfaction of this conjecture rather memorably:“The Cauchy horizon is the ultimate brick wall at which the evolution of spacetime is forced to stop”.
14e have further argued that these sources of indeterminism capture the sense in which a lackof determination arises for a broad set of features associated with singularities in GR. Thereare four such features, described in Sec. IV, namely: (a) unspecified end-states of particles; (b)ill-behaved physical quantities in the vicinity of singularities; (c) a lack of ‘laws’ of singularities;and (d) Cauchy horizons, which arise as a result of certain types of singularities. In Table I, weprovide a summary of the associations for which we have argued, between the three sources ofindeterminism mentioned above and these four features of singularities. Notably, only for oneof these features—namely, for Cauchy horizons—is the source of indeterminism associated withthe usual Laplacian notion.Thus, we conclude that one way to interpret the type of breakdown that is usually attributedto the existence of singularities in GR is, indeed, as a failure of determinism . Acknowledgments
We thank Bahram Mashhoon and Mahdiyar Noorbala for discussions. FA acknowledgessupport from: the Black Hole Initiative at Harvard University, which is funded through a grantfrom the John Templeton Foundation and the Gordon and Betty Moore Foundation; and theFaculty Research Support Program (FY2019) at the University of Notre Dame.
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