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© Elizabeth Chiles 2009 
Although black holes are objects of central importance across many fields of physics, there is no agreed upon definition for them, a fact that does not seem to be widely recognized. Physicists in different fields conceive of and reason about them in radically different, and often conflicting, ways. All those ways, however, seem sound in the relevant contexts. After examining and comparing many of the definitions used in practice, I consider the problems that the lack of a universally accepted definition leads to, and discuss whether one is in fact needed for progress in the physics of black holes. I conclude that, within reasonable bounds, the profusion of different definitions is in fact a virtue, making the investigation of black holes possible and fruitful in all the many different kinds of problems about them that physicists consider, although one must take care in trying to translate results between fields.
Every physical theory has (at least) two different forms of mathematical equations to represent its target systems: the dynamical (equations of motion) and the kinematical (kinematical constraints). Kinematical constraints are differentiated from equations of motion by the fact that their particular form is fixed once and for all, irrespective of the interactions the system enters into. By contrast, the particular form of a system’s equations of motion depends essentially on the particular interaction the system enters into. All contemporary accounts of the structure and semantics of physical theory treat dynamics, i.e., the equations of motion, as the most important feature of a theory for the purposes of its philosophical analysis. I argue to the contrary that it is the kinematical constraints that determine the structure and empirical content of a physical theory in the most important ways: they function as necessary preconditions for the appropriate application of the theory; they differentiate types of physical systems; they are necessary for the equations of motion to be well posed or even just cogent; and they guide the experimentalist in the design of tools for measurement and observation. It is thus satisfaction of the kinematical constraints that renders meaning to those terms representing a system’s physical quantities in the first place, even before one can ask whether or not the system satisfies the theory’s equations of motion.
I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects (e.g., spacetimes) living in infinitedimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measuretheoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinitedimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of spacetimes is finitedimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems.
In the early 1970s it is was realized that there is a striking formal analogy between the Laws of blackhole mechanics and the Laws of classical thermodynamics. Before the discovery of Hawking radiation, however, it was generally thought that the analogy was only formal, and did not reflect a deep connection between gravitational and thermodynamical phenomena. It is still commonly held that the surface gravity of a stationary black hole can be construed as a true physical temperature and its area as a true entropy only when quantum effects are taken into account; in the context of classical general relativity alone, one cannot cogently construe them so. Does the use of quantum field theory in curved spacetime offer the only hope for taking the analogy seriously? I think the answer is ‘no’. To attempt to justify that answer, I shall begin by arguing that the standard argument to the contrary is not physically well founded, and in any event begs the question. Looking at the various ways that the ideas of “temperature” and “entropy” enter classical thermodynamics then will suggest arguments that, I claim, show the analogy between classical blackhole mechanics and classical thermodynamics should be taken more seriously, without the need to rely on or invoke quantum mechanics. In particular, I construct an analogue of a Carnot cycle in which a black hole “couples” with an ordinary thermodynamical system in such a way that its surface gravity plays the role of temperature and its area that of entropy. Thus, the connection between classical general relativity and classical thermodynamics on their own is already deep and physically significant, independent of quantum mechanics.
I examine the debate between substantivalists and relationalists about the ontological character of spacetime and conclude it is not well posed. I argue that the socalled Hole Argument does not bear on the debate, because it provides no clear criterion to distinguish the positions. I propose two such precise criteria and construct separate arguments based on each to yield contrary conclusions, one supportive of something like relationalism and the other of something like substantivalism. The lesson is that one must fix an investigative context in order to make such criteria precise, but different investigative contexts yield inconsistent results. I examine questions of existence about spacetime structures other than the spacetime manifold itself to argue that it is more fruitful to focus on pragmatic issues of physicality, a notion that lends itself to several different explications, all of philosophical interest, none privileged a priori over any of the others. I conclude by suggesting an extension of the lessons of my arguments to the broader debate between realists and instrumentalists.
An energy condition, in the context of a wide class of spacetime theories (including general relativity), is, crudely speaking, a relation one demands the stressenergy tensor of matter satisfy in order to try to capture the idea that “energy should be positive”. The remarkable fact I will discuss in this paper is that such simple, general, almost trivial seeming propositions have profound and farreaching import for our understanding of the structure of relativistic spacetimes. It is therefore especially surprising when one also learns that we have no clear understanding of the nature of these conditions, what theoretical status they have with respect to fundamental physics, what epistemic status they may have, when we should and should not expect them to be satisfied, and even in many cases how they and their consequences should be interpreted physically. Or so I shall argue, by a detailed analysis of the technical and conceptual character of all the standard conditions used in physics today, including examination of their consequences and the circumstances in which they are believed to be violated.
One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often claimed that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is yes: I state and sketch proofs of two technical results, inspired by simple physical arguments about the generic properties of classical systems, to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none that Hamiltonian mechanics does. The argument not only clarifies the conceptual structure of the two systems of mechanics, their relations to each other, and their respective mechanisms for representing physical systems. It also shows why naively structural approaches to the representational content of physical theories cannot work.
I argue that, contrary to the recent claims of physicists and philosophers of physics, general relativity requires no interpretation in any substantive sense of the term. I canvass the common reasons given in favor of the alleged need for an interpretation, including the difficulty in coming to grips with the physical significance of diffeomorphism invariance and of singular structure, and the problems faced in the search for a theory of quantum gravity. I find that none of them shows any defect in our comprehension of general relativity as a physical theory. I conclude by comparing general relativity with quantum mechanics, a theory that manifestly does stand in need of an interpretation in an important sense. Although many aspects of the conceptual structure of general relativity remain poorly understood, it suffers no incoherence in its formulation as a physical theory that only an “interpretation” could resolve.
A review of the state of the art concerning philosophical investigation of issues pertaining to singular structure and black holes in relativistic spacetimes. Issues treated include: the definition of singular structure; the type of existence, if any, one can ascribe to singular structure; the laws of black hole mechanics; some attempts in programs in quantum gravity to derive the mechanical black hole laws.
In 1672, Isaac Newton published in the “Transactions of the Royal Society” his theory of the structure of light rays. It was not understood by even his most brilliant contemporaries, scientific luminaries such as Robert Hooke and Christian Huygens. The primary obstacle hindering their understanding was their conception of proper scientific methodology, the hypetheticodeductive method as applied solely in the theater of the mechanical philosophy. Neither Newton’s account of the derivation of his theory nor the theory itself conformed to this conception of science, much to the discredit of that conception. In the 1990s, quantum gravity came into its own as an accepted, even a sexy field of research in theoretical physics. Except for adherence to the mechanical philosophy, all the major research programs in quantum gravity today do conform to the conception of science championed by Newton’s contemporaries, much to the discredit of quantum gravity. In this paper I make the case for this claim, and discuss a few of its unfortunate corollaries. I also examine the scientific standing of various programs in quantum gravity as reported by proponents of those programs in order to criticize what I see as their immodesty—unwarranted in light of my arguments in the first part of the paper—which at times seems to cross the border into disingenuity. It is no mere Pauline gripe I have against this immodesty; I think it does real damage to science on many levels. I conclude with a few reflections on this matter and on the role, if any, that philosophers of science ought to play in the maintenance of working science’s integrity.
All accounts of causality that presuppose the propagation or transfer of some physical stuff to be an essential part of the causal relation rely for the force of their causal claims on a principle of conservation for that stuff. General Relativity does not permit the rigorous formulation of appropriate conservation principles. Consequently, in so far as General Relativity is considered a fundamental physical theory, such accounts of causality cannot themselves be considered fundamental. The continued use of such accounts of causality perhaps ought not be proscribed, but justification is due from those who would use them.
Much controversy surrounds the question of what ought to be the proper definition of “singularity” in general relativity, and the question of whether the prediction of such entities leads to a crisis for the theory. I argue that there is no single canonical definition of such a thing, and that none is required—various definitions present themselves, respectively suitable for different sorts of investigations. In particular, I argue that a definition in terms of curve incompleteness is adequate for most purposes, though the idea that singularities correspond to “missing points” has insurmountable problems. I conclude that singularities per se pose no serious problem for the theory, but their analysis does bring into focus several problems of interpretation at the foundation of the theory often ignored in the philosophical literature.
I neither attack nor defend Jarrett’s (1984) conclusions concerning adequacy constraints appropriate to (models of) physical theories mooted in discussions of Bell’s Theorem. Rather I attempt to clarify what sorts of arguments can and cannot coherently be made when Jarretttype premises are accepted (or are accepted at least for the sake of argument). In particular I show that certain recent construals of Jarrett’s 1984 argument that focus on the notion of causality not only are beside the point of Jarrett’s argument but, more importantly, obfuscate what is salutary that can be gleaned from his argument. Martin Jones and Rob Clifton among recent commentators on Jarrett not only are notable for their clarity, precision and thoroughness of argument, but they also are typical in what, I suspect, is the main culprit behind confused arguments that are concerned with issues of causality, viz., a careless deployment of the notion of “causality” itself. For this reason I have chosen them as a foil, precisely because theirs seems to me the clearest, best case for what I take to be at bottom a confused way both of thinking of Jarrett’s argument in particular, and of deploying causal arguments in general.
The analysis of theoryconfirmation generally takes the form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure of the degree of theoryconfirmation this yields. The issue of confirmation for an entire framework (e.g., Newtonian mechanics en bloc, as opposed, say, to Newton’s theory of gravitation) either does not arise, or is dismissed in so far as frameworks are thought not to be the kind of thing that admits scientific confirmation. I argue that there is another form of scientific reasoning that has not received philosophical attention, what I call Newtonian abduction, that does provide confirmation for frameworks as a whole, and does so in two novel ways. (In particular, Newtonian abduction is not IBE, but rather is much closer to Peirce’s original explication of the idea of abduction.) I further argue that Newtonian abduction is at least as important a form of reasoning in science as the deductive form sketched above. The form is beautifully summed up by Maxwell (1876): “The true method of physical reasoning is to begin with the phenomena and to deduce the forces from them by a direct application of the equations of motion.”
General relativity poses serious problems for counterfactual propositions peculiar to it as a physical theory, problems that have gone unremarked on in the physics and in the philosophy literature. Because these problems arise from the dynamical nature of spacetime geometry, they are shared by all schools of thought on how counterfactuals should be interpreted and understood. Given the role of counterfactuals in the characterization of, inter alia, many accounts of scientific laws, theoryconfirmation and causation, general relativity once again presents us with idiosyncratic puzzles any attempt to analyze and understand the nature of scientific knowledge and of science itself must face.
I argue that an adequate semantics for physical theories must be grounded on an account of the way that a theory provides formal and conceptual resources appropriate for—that have propriety in—the construction of representations of the physical systems the theory purports to treat. I sketch a precise, rigorous definition of the required forms of propriety, and argue that semantic content accrues to scientific representations of physical systems primarily in virtue of the propriety of its resources. That propriety largely consists in the satisfaction of a subset of the relations a theory posits among the quantities it treats, viz., the theory’s kinematical constraints, rather than in the predictive accuracy of its equations of motion. In particular, the adequacy (soundness, accuracy, truth, ...) of a theory’s representations plays no fundamental role in the determination of a representation’s semantic content. One consequence is that anything like traditional Tarskian semantics is inadequate for the task.
The dispute over the viability of various theories of relativistic, dissipative fluids is analyzed. The focus of the dispute is identified as the question of determining what it means for a theory to be applicable to a given type of physical system under given conditions. The idea of a physical theory’s regime of propriety is introduced, in an attempt to clarify the issue, along with the construction of a formal model trying to make the idea precise. This construction involves a novel generalization of the idea of a field on spacetime, as well as a novel method of approximating the solutions to partialdifferential equations on relativistic spacetimes in a way that tries to account for the peculiar needs of the interface between the exact structures of mathematical physics and the inexact data of experimental physics in a relativistically invariant way. It is argued, on the basis of these constructions, that the idea of a regime of propriety plays a central role in attempts to understand the semantical relations between theoretical and experimental knowledge of the physical world in general, and in particular in attempts to explain what it may mean to claim that a physical theory models or represents a kind of physical system. This discussion necessitates an examination of the initialvalue formulation of the partialdifferential equations of mathematical physics, which suggests a natural set of conditions—by no means meant to be canonical or exhaustive—one may require a mathematical structure, in conjunction with a set of physical postulates, satisfy in order to count as a physical theory. Based on the novel approximating methods developed for solving partialdifferential equations on a relativistic spacetime by finitedifference methods, a technical result concerning a peculiar form of theoretical underdetermination is proved, along with a technical result purporting to demonstrate a necessary condition for the selfconsistency of a physical theory.
I do not think the notion of rigidity in designation can be correct, at least not in any way that can serve to ground a semantics that purports both to be fundamental in a semiotical sense and to respect the best science of the day. A careful examination of both the content and the character of our best scientific knowledge not only cannot support anything like what the notion of rigidity requires, but actually shows the notion to be, at bottom, incoherent. In particular, the scientific meaning of natural kind terms can be determined only within the context of a fixed scientific framework and not sub specie ŀternitatis. Along the way, I provide grounds for the rejection of essentialist views of the ontology of natural kinds.
I examine an extraordinary circumstance of the work as a whole, a circumstance that has gone largely, and oddly, unremarked in the secondary literature. At the conclusion of both Socrates’s antepenultimate (IV.445ab) and penultimate (IX.580bc) answers to the brothers’ challenges, he asks Glaukon to render judgement on the worth and intrinsic goodness of justice and the just man’s life. Glaukon without hesitation declares justice to be good in and of itself, and the just life to be the best and the happiest of lives. In both places, to drive the point home, Socrates makes sure to mention one of the primary terms of the challenges: that justice and the just life have this character whether the just man is known to be just or not (IV.445a), in either the eyes of god or men (IX.580c), which Glaukon readily grants. And yet, for the entirety of his construction of the just city, his account of justice itself, and most of all his characterization of the just man, the just man both has seemed and has been known by all to be just. Socrates has flagrantly flouted the most fundamental term of the challenge, and not only this term but all the rest as well, for the just man, in Socrates’s recounting of his life, has had accrue to him all the appurtenances, pleasures, rewards and good repute that Glaukon and Adeimantos demanded be stripped from him, so they could with surety conclude that justice is good in and of itself rather than on account only of its repute and rewards. Are Glaukon and Adeimantos simpletons, dupes, that they would readily accept as an answer to their challenges one that is, prima facie, an answer to no question they had asked? I do not think Plato wanted us to draw this conclusion. What else, then, can be the resolution of this puzzle? I attempt in this series of three papers to sketch one: Socrates has, in fact, answered each of their challenges to the letter. In working out the answers, Socrates and the sons of Ariston have discovered that the nature of true justice demands that the just man live a life that will accrue to itself many (though by no means all) of the concomitants and the rewards and much of the repute given, according to common belief, to the just man (or, at least, the seemingly just man) on account of his (seeming) justice, not, however, as consequences of justice, but rather as necessary constituents, in some way or other, of justice itself. It follows that justice is, by its nature, a virtue that demands that its agent inhabit a definite sort of place in a richly appointed and textured society—it is essentially a social virtue, without real substance or sense in isolation from the social roles the just man plays and the acts he performs. In this first paper of the three, I explicate the brothers’ challenges to demonstrate how, on the face of it, Socrates does not answer them.
General remarks on what it is to learn philosophy—the reading and the writing of it, and the production of philosophical arguments—with emphasis on the relation between teacher and student, the roles and responsibilities of each. Reference is made to two essays by Mark Twain, Fenimore Cooper’s Literary Offences and Fenimore Cooper’s Further Literary Offences: Cooper’s Prose Style.
We numerically solve Einstein’s equations coupled to a scalar field in the interior of Kerr black holes. We find shock waves form near the inner horizon. The shocks grow exponentially in amplitude and need not be axisymmetric. Our numerical results are consistent with the geometry inside the inner horizon exponentially collapsing to zero volume, meaning that the Kerr geometry effectively ends at the inner horizon.
The standard argument for the uniqueness of the Einstein field equation is based on Lovelock’s Theorem, the relevant statement of which is restricted to four dimensions. I prove a theorem similar to Lovelock’s, with a physically modified assumption: that the geometric object representing curvature in the Einstein field equation ought to have the physical dimension of stressenergy. The theorem is stronger than Lovelock’s in two ways: it holds in all dimensions, and so supports a generalized argument for uniqueness; it does not assume that the desired tensor depends on the metric only up secondorder partialderivatives, that condition being a consequence of the proof. This has consequences for understanding the nature of the cosmological constant and theories of higherdimensional gravity. Another consequence of the theorem is that it makes precise the sense in which there can be no gravitational stressenergy tensor in general relativity. Along the way, I prove a result of some interest about the second jetbundle of the bundle of metrics over a manifold.
In the early 1970s it is was realized that there is a striking formal analogy between the socalled laws of blackhole mechanics and the laws of classical thermodynamics. Before the discovery of Hawking radiation, however, it was generally thought that the analogy was only formal, and did not reflect a deep connection between gravitational and thermodynamical phenomena. It is still commonly held that the surface gravity of a stationary black hole can be construed as a true physical temperature and its area as a true entropy only when quantum effects are taken into account; in the context of classical general relativity alone, one cannot cogently construe them so. Does the use of quantum field theory in curved spacetime offer the only hope for taking the analogy seriously? I think the answer is ‘no’. To attempt to justify that answer, I shall begin by arguing that the orthodoxy is not physically well founded, and in any event begs the question. Looking at the various ways that the ideas of “temperature” and “entropy” enter classical thermodynamics then will suggest arguments that, I claim, show the analogy between classical blackhole mechanics and classical thermodynamics should be taken more seriously, without the need to rely on or invoke quantum mechanics. In particular, I construct an analogue of a Carnot cycle in which a black hole “couples” with an ordinary thermodynamical system in such a way that its surface gravity plays the role of temperature and its area that of entropy. This strongly suggests that the connection between classical general relativity and classical thermodynamics on their own is already deep and physically significant, independent of quantum mechanics.
The question of the existence of gravitational stressenergy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stressenergetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and handwaving but, worse, are beside the point and do not address the heart of the issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a “geometric object”, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stressenergy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily nonlocal. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the nonlocalizability of gravitational stressenergy.
I give a novel construction and presentation of the intrinsic geometry of a generic tangent bundle, in the terms of which the EulerLagrange Equation can be formulated in a geometric, illuminating way. I conclude by proving a result that shows that, in a strong sense, not only must Lagrangian Mechanics be formulated on tangent bundles (as opposed to Hamiltonian Mechanics, which can be formulated on any symplectic manifold, whether diffeomorphic to a cotangent bundle or not), but moreover the intrinsic geometry of the EulerLagrange Equation itself allows one to completely reconstruct the space on which one formulates it as a tangent bundle over a particular base space.