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, 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.     

Newton–Cartan theory and teleparallel gravity: The force of a formulation

It is well-known that Newtonian gravity, commonly held to describe a gravitational force, can be recast in a form that incorporates gravity into the geometry of the theory: Newton–Cartan theory. It is less well-known that general relativity, a geometrical theory of gravity, can be reformulated in such a way that it resembles a force theory of gravity; teleparallel gravity does just this. This raises questions. One of these concerns theoretical underdetermination. I argue that these theories do not, in fact, represent cases of worrying underdetermination. On close examination, the alternative formulations are best interpreted as postulating the same spacetime ontology. In accepting this, we see that the ontological commitments of these theories cannot be directly deduced from their mathematical form. The spacetime geometry involved in a gravitational theory is not a straightforward consequence of anything internal to that theory as a theory of gravity. Rather, it essentially relies on the rest of nature (the non-gravitational interactions) conspiring to choose the appropriate set of inertial frames.     

Einstein Equations and Hilbert Action: What is missing on page 8 of the proofs for Hilbert's First Communication on the Foundations of Physics?

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible.     

Heuristics versus norms: On the relativistic responses to the Kaufmann experiments

The aim of this article is to provide a historical response to Michel Janssen’s (2009) claim that the special theory of relativity establishes that relativistic phenomena are purely kinematical in nature, and that the relativistic study of such phenomena is completely independent of dynamical considerations regarding the systems displaying such behavior. This response will be formulated through a historical discussion of one of Janssen's cases, the experiments carried out by Walter Kaufmann on the velocity-dependence of the electron's mass. Through a discussion of the different responses formulated by early adherents of the principle of relativity (Albert Einstein, Max Planck, Hermann Minkowski and Max von Laue) to these experiments, it will be argued that the historical development of the special theory of relativity argues against Janssen's historical presentation of the case, and that this raises questions about his general philosophical claim. It will be shown, more specifically, that Planck and Einstein developed a relativistic response to the Kaufmann experiments on the basis of their study of the dynamics of radiation phenomena, and that this response differed significantly from the response formulated by Minkowski and Laue. In this way, it will be argued that there were, at the time, two different approaches to the theory of relativity, which differed with respect to its relation to theory, experiment, and history: Einstein's and Planck's heuristic approach, and Minkowski's and Laue's normative approach. This indicates that it is difficult to say, historically speaking, that the special theory of relativity establishes the kinematical nature of particular phenomena. Instead, it will be argued that the theory of relativity should not be seen as a theory but rather as outlining an approach, and that the nature of particular scientific phenomena is something that is open to scientific debate and dispute.     

Einstein and Singularities

Except for a few brief periods, Einstein was uninterested in analysing the nature of the spacetime singularities that appeared in solutions to his gravitational field equations for general relativity. The existence of such monstrosities reinforced his conviction that general relativity was an incomplete theory which would be superseded by a singularity-free unified field theory. Nevertheless, on a number of occasions between 1916 and the end of his life, Einstein was forced to confront singularities. His reactions show a strange asymmetry: he tended to be more disturbed by (what today we would call) merely apparent singularities and less disturbed by (what we would call) real singularities. Einstein had strong a priori ideas about what results a correct physical theory should deliver. In the process of searching through theoretical possibilities, he tended to push aside technical problems and jump over essential difficulties. Sometimes this method of working produced brilliant new ideas—such as the Einstein–Rosen bridge—and sometimes it lead him to miss important implications of his theory of gravity—such as gravitational collapse.     

Poincaré's Contributions to Relativistic Dynamics

In this paper I concentrate on the dynamic aspects of the special theory of relativity (in the non-Minkowski formalism), and not on the kinematic part of the story as is usually done. Following up the dynamic story leads to a new point of view as to Poincaré's important role in the development of special relativity. Much of Poincaré's dynamic work did not enter into Einstein's 1905 theory, since Einstein was mainly occupied with kinematics. However, the dynamic part is most fundamental in the development of the special theory of relativity after 1905. In this paper I consider the main developments of relativistic dynamics in which I demonstrate that much response to Poincaré's dynamic research can be found. I argue that Poincaré's dynamic work assisted in departing from Einstein's electrodynamic theory towards relativistic dynamics (independent of electrodynamics).     

Can we trust Einstein’s accounts of the genesis of special relativity?

There are three kinds of sources available to reconstruct the reflections that led Einstein to special relativity: a few contemporary letters and documents, his impersonal accounts of the genesis of this theory, and recollections of his own path. At first glance, contradictions within and between these sources hamper the reliability of Einstein’s accounts. Yet, a closer analysis reveals much more consistency than foreseen and helps eliminate the dubious, contradictory elements. It then becomes possible to combine the three kinds of sources to produce a minimally speculative and yet fairly coherent account of the genesis of special relativity.     

It ain't necessarily so: Gravitational waves and energy transport

I review and critically examine the four textbook arguments commonly taken to establish that gravitational waves (GWs) carry energy-momentum: 1. the increase in kinetic energy that a GW confers on a ring of test particles, 3.Bondi/Feynman's Sticky Bead Argument of a GW heating up a detector, 3. nonlinearities within perturbation theory, construed as the gravity's contribution to its own source, and 4. the Noether Theorems, linking symmetries and conserved quantities. As it stands, each argument is found to be either contentious, or incomplete in that it presupposes substantive assumptions which the standard exposition glosses over. I finally investigate the standard interpretation of binary systems, according to which orbital decay is explained by the system's energy being dissipated via GW energy-momentum transport. I contend that for the textbook treatment of binary systems an alternative interpretation, drawing only on the general-relativistic equations of motions and the Einstein Equations, is available. It's argued to be even preferable to the standard interpretation. Thereby an inference to the best explanation for GW energy-momentum is blocked. I conclude that a defence of the claim that GWs carry energy can't rest on the standard arguments.     

`Nature is the Realisation of the Simplest Conceivable Mathematical Ideas': Einstein and the Canon of Mathematical Simplicity

Einstein proclaimed that we could discover true laws of nature by seeking those with the simplest mathematical formulation. He came to this viewpoint later in his life. In his early years and work he was quite hostile to this idea. Einstein did not develop his later Platonism from a priori reasoning or aesthetic considerations. He learned the canon of mathematical simplicity from his own experiences in the discovery of new theories, most importantly, his discovery of general relativity. Through his neglect of the canon, he realised that he delayed the completion of general relativity by three years and nearly lost priority in discovery of its gravitational field equations.     

狭义相对论与丁格尔时钟悖论 --对一场著名争论的回顾与再认识

1967年10月14日《自然》杂志发表了H 丁格尔教授反对狭义相对论的论证和W．H．麦克利教授对丁格尔论证的批评。本文指出麦克利对丁格尔的批评是不能成立的,因为麦克利为反对丁格尔观点而画出的时空图恰恰是支持丁格尔的,提出：丁格尔的论证就其反对爱因斯坦从狭义相对论引出的结论——“运动的钟比静止时走得慢”——是成立的,就其反对狭义相对论是不成立的：狭义相对论自身在逻辑上是无矛盾和完备的,而爱因斯坦从狭义相对论中得出“运动的钟变慢”的结论是不对的。文章阐明了狭义相对论与“钟慢效应实验证据”的正确关系。     

Nova Geminorum 1912 and the origin of the idea of gravitational lensing

Einstein’s early calculations of gravitational lensing, contained in a scratch notebook and dated to the spring of 1912, are reexamined. A hitherto unknown letter by Einstein suggests that he entertained the idea of explaining the phenomenon of new stars by gravitational lensing in the fall of 1915 much more seriously than was previously assumed. A reexamination of the relevant calculations by Einstein shows that, indeed, at least some of them most likely date from early October 1915. But in support of earlier historical interpretation of Einstein’s notes, it is argued that the appearance of Nova Geminorum 1912 (DN Gem) in March 1912 may, in fact, provide a relevant context and motivation for Einstein’s lensing calculations on the occasion of his first meeting with Erwin Freundlich during a visit in Berlin in April 1912. We also comment on the significance of Einstein’s consideration of gravitational lensing in the fall of 1915 for the reconstruction of Einstein’s final steps in his path towards general relativity.     

Time isotropy,Lorentz transformation and inertial frames

Homogeneity of Euclidean space and time, spatial isotropy, principle of relativity and the existence of a finite speed limit (or its variants) are commonly believed to be the only axioms required for developing the special theory of relativity (Lorentz transformations). In this paper, however, it is pointed out that the Lorentz transformation for a boost cannot actually be derived without the explicit assumption of time isotropy (viz. time-reversal symmetry) which is logically independent of the other postulates of relativity. Postulating time isotropy also restores the symmetry between space and time in the postulates of relativity (i.e. time and space share the same symmetries then). Time isotropy also helps explain naturally one key general feature of the fundamental physical laws, viz. their time-reversal symmetry. But inertial frames are defined in influential texts as frames having space-time homogeneity and spatial isotropy only. Inclusion of time isotropy in that definition is thus suggested.     

On the empirical equivalence between special relativity and Lorentz׳s ether theory

In this paper I argue that the case of Einstein׳s special relativity vs. Hendrik Lorentz׳s ether theory can be decided in terms of empirical evidence, in spite of the predictive equivalence between the theories. In the historical and philosophical literature this case has been typically addressed focusing on non-empirical features (non-empirical virtues in special relativity and/or non-empirical flaws in the ether theory). I claim that non-empirical features are not enough to provide a fully objective and uniquely determined choice in instances of empirical equivalence. However, I argue that if we consider arguments proposed by Richard Boyd, and by Larry Laudan and Jarret Leplin, a choice based on non-entailed empirical evidence favoring Einstein׳s theory can be made.     

Why Einstein did not believe that general relativity geometrizes gravity

I argue that, contrary to folklore, Einstein never really cared for geometrizing the gravitational or (subsequently) the electromagnetic field; indeed, he thought that the very statement that General Relativity geometrizes gravity “is not saying anything at all”. Instead, I shall show that Einstein saw the “unification” of inertia and gravity as one of the major achievements of General Relativity. Interestingly, Einstein did not locate this unification in the field equations but in his interpretation of the geodesic equation, the law of motion of test particles.     

The constitutive a priori and the distinction between mathematical and physical possibility

This paper is concerned with Friedman׳s recent revival of the notion of the relativized a priori. It is particularly concerned with addressing the question as to how Friedman׳s understanding of the constitutive function of the a priori has changed since his defence of the idea in his Dynamics of Reason. Friedman׳s understanding of the a priori remains influenced by Reichenbach׳s initial defence of the idea; I argue that this notion of the a priori does not naturally lend itself to describing the historical development of space-time physics. Friedman׳s analysis of the role of the rotating frame thought experiment in the development of general relativity – which he suggests made the mathematical possibility of four-dimensional space-time a genuine physical possibility – has a central role in his argument. I analyse this thought experiment and argue that it is better understood by following Cassirer and placing emphasis on regulative principles. Furthermore, I argue that Cassirer׳s Kantian framework enables us to capture Friedman׳s key insights into the nature of the constitutive a priori.     

Hobbes on natural philosophy as “True Physics” and mixed mathematics

In this paper, I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument shows that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience.     

Quantum Mechanics as a Principle Theory

I show how quantum mechanics, like the theory of relativity, can be understood as a ‘principle theory’ in Einstein's sense, and I use this notion to explore the approach to the problem of interpretation developed in my book Interpreting the Quantum World.     

The Principle of Equivalence

We start from John Norton's analysis (1985) of the reach of Einstein's version of the principle of equivalence which is not a local principle but an extension of the relativity principle to reference frames in constant acceleration on the background of Minkowski spacetime. We examine how such a point of view implies a profound, and not generally recognised, reconsideration of the concepts of inertial system and field in physics. We then reevaluate the role that the infinitesimal principle, if adequately formulated, can legitimately be claimed to play in general relativity. We show that what we call the ‘punctual equivalence principle’ has significant physical content and that it permits the derivation of the geodesic law.     

Minkowski spacetime and Lorentz invariance: The cart and the horse or two sides of a single coin?

Michel Janssen and Harvey Brown have driven a prominent recent debate concerning the direction of an alleged arrow of explanation between Minkowski spacetime and Lorentz invariance of dynamical laws in special relativity. In this article, I critically assess this controversy with the aim of clarifying the explanatory foundations of the theory. First, I show that two assumptions shared by the parties—that the dispute is independent of issues concerning spacetime ontology, and that there is an urgent need for a constructive interpretation of special relativity—are problematic and negatively affect the debate. Second, I argue that the whole discussion relies on a misleading conception of the link between Minkowski spacetime structure and Lorentz invariance, a misconception that in turn sheds more shadows than light on our understanding of the explanatory nature and power of Einstein׳s theory. I state that the arrow connecting Lorentz invariance and Minkowski spacetime is not explanatory and unidirectional, but analytic and bidirectional, and that this analytic arrow grounds the chronogeometric explanations of physical phenomena that special relativity offers.