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Erik Curiel

Publications and source records attributed to Erik Curiel.

16 recordsLinked to original sources
arXiv2020

Two-Dimensional Area and Matter Flux in the Theory of Causal Fermion Systems

The notions of two-dimensional area, Killing fields and matter flux are introduced in the setting of causal fermion systems. It is shown that for critical points of the causal action, the area change of two-dimensional surfaces under a Killing flow in null directions is proportional to the matter flux through these surfaces. This relation generalizes an equation in classical general relativity due to Ted Jacobson to the setting of causal fermion systems.

math-ph↗
arXiv2020

Schematizing the Observer and the Epistemic Content of Theories

I argue that, contrary to the standard view, one cannot understand the structure and nature of our knowledge in physics without an analysis of the way that observers (and, more generally, measuring instruments and experimental arrangements) are modeled in theory. One upshot is that standard pictures of what a scientific theory can be are grossly inadequate. In particular, standard formulations assume, with no argument ever given, that it is possible to make a clean separation between, on the one hand, one part of the scientific knowledge a physical theory embodies, viz., that encoded in the pure mathematical formalism and, on the other, the remainder of that knowledge. The remainder includes at a minimum what is encoded in the practice of modeling particular systems, of performing experiments, of bringing the results of theory and experiment into mutually fruitful contact---in sum, real application of the theory in actual scientific practice. This assumption comes out most clearly in the picture of semantics that naturally accompanies the standard view of theories: semantics is fixed by ontology's shining City on the Hill, and all epistemology and methodology and other practical issues and considerations are segregated to the ghetto of the theory's pragmatics. We should not assume such a clean separation is possible without an argument, and, indeed, I offer many arguments that such a separation is not feasible. An adequate semantics for theories cannot be founded on ontology, but rather on epistemology and methodology.

physics.hist-ph↗
arXiv2020

Summing over spacetime dimensions in quantum gravity

Quantum-gravity corrections (in the form of a minimal length) to the Feynman propagator for a free scalar particle in $\mathbb{R}^D$ are shown to be the result of summing over all dimensions $D'\geq D$ of $\mathbb{R}^{D'}$, each summand taken in the absence of quantum gravity.

gr-qc↗
arXiv2019

Singularities in Reissner-Nordström black holes

We study black holes produced by the collapse of a spherically symmetric charged scalar field in asymptotically flat space. We employ a late time expansion and show decaying fluxes of radiation through the event horizon imply the black hole must contain a null singularity on the Cauchy horizon and a central spacelike singularity.

gr-qc↗
arXiv2019

On Geometric Objects, the Non-Existence of a Gravitational Stress-Energy Tensor, and the Uniqueness of the Einstein Field Equation

The question of the existence of gravitational stress-energy 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 stress-energetic 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 hand-waving 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, \emph{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 stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. 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 non-localizability of gravitational stress-energy. The main theorem proven underlying all the arguments is considerably stronger than the standard result in the literature used for the same purposes (Lovelock's theorem of 1972): it holds in all dimensions (not only in four); it does not require an assumption about the differential order of the desired concomitant of the metric; and it has a more natural physical interpretation.

physics.hist-ph↗
arXiv2019

Framework Confirmation by Newtonian Abduction

The analysis of theory-confirmation generally takes the deductive 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 theory-confirmation 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 inference to the best explanation, but rather is closer to Peirce's original idea of abduction.) I further argue that Newtonian abduction is at least as important a form of reasoning in science as standard deductive and inductive forms. 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."

physics.hist-ph↗
arXiv2019

The Many Definitions of a Black Hole

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.

physics.hist-ph↗
arXiv2019

Numerical evolution of shocks in the interior of Kerr black holes

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. Observers who pass through the shocks experience exponentially large tidal forces and are accelerated exponentially close to the speed of light.

gr-qc↗
arXiv2016

A Weyl-Type Theorem for Geometrized Newtonian Gravity

I state and prove, in the context of a space having only the metrical and affine structure imposed by the geometrized version of Newtonian gravitational theory, a theorem analagous to that of Weyl's for a Lorentz manifold. The theorem says that a projective structure and a suitably defined compatible conformal structure jointly suffice for fixing the metric structure of a Newtonian spacetime model up to constant factors, and for fixing its affine structure as well. The theorem allows one to give a natural, physically compelling interpretation of the spatiotemporal geometry of a geometrized Newtonian gravity spacetime manifold, in close analogy with the way Weyl's Theorem allows one to do in general relativity.

gr-qc↗
arXiv2016

Kinematics, Dynamics, and the Structure of Physical Theory

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.

physics.hist-ph↗
arXiv2016

A Simple Proof of the Uniqueness of the Einstein Field Equation in All Dimensions

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 stress-energy. 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 second-order partial-derivatives, that condition being a consequence of the proof. This has consequences for understanding the nature of the cosmological constant and theories of higher-dimensional gravity. Another consequence of the theorem is that it makes precise the sense in which there can be no gravitational stress-energy tensor in general relativity. Along the way, I prove a result of some interest about the second jet-bundle of the bundle of metrics over a manifold.

gr-qc↗
arXiv2015

If Metrical Structure Were Not Dynamical, Counterfactuals in General Relativity Would Be Easy

General relativity poses serious problems for counterfactual propositions peculiar to it as a physical theory. Because these problems arise solely 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, theory confirmation and causation, general relativity once again presents us with idiosyncratic puzzles any attempt to analyze and understand the nature of scientific knowledge must face.

physics.hist-ph↗
arXiv2015

Measure, Topology and Probabilistic Reasoning in Cosmology

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 infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are difficult to work with on their own as well in that context. Even in cases where an appropriate family of spacetimes is finite-dimensional, however, and so admits a measure of the relevant sort, 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.

gr-qc↗
arXiv2014

Classical Black Holes Are Hot

In the early 1970s it is was realized that there is a striking formal analogy between the Laws of black-hole 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 black-hole 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.

gr-qc↗
arXiv2014

A Primer on Energy Conditions

An energy condition, in the context of a wide class of spacetime theories (including general relativity), is, crudely speaking, a relation one demands the stress-energy 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 far-reaching 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.

physics.hist-ph↗
arXiv2012

On Tensorial Concomitants and the Non-Existence of a Gravitational Stress-Energy Tensor

The question of the existence of gravitational stress-energy 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 stress-energetic 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 hand-waving 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, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. 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.

gr-qc↗