Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \,\varpi )\), where \\) is a 1-form with coefficients in the Lie algebra (...) of the Poincaré group and \ is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations. (shrink)

The massless Thirring model of a self-interacting ferinion field in a curved two-dimensional background spacetime is considered. The exact operator solution for the fields and the equation for the two-point function are given and used to examine the radiation emitted by a two-dimensional black hole. The radiation is found to be thermal in nature, confirming general predictions to this effect. We compute the particle spectrum of the Thirring fermions at finite temperature in Minkowski space and point out errors in (...) a previous attempt at this calculation. Finally we calculate the vacuum expectation value of the stress tensor in an arbitrary two-dimensional spacetime by exploiting the connection between the thermal Hawking radiation from a black hole and the value of the so-called conformal trace anomaly. The latter is also computed quite independently, using dimensional regularization, from general considerations of the renormalization group. (shrink)

In this paper I provide an ontology for the co‐variant vectors, contra‐variant vectors and tensors that are familiar from General Relativity. This ontology is developed in response to a problem that Timothy Maudlin uses to argue against universals in the interpretation of physics. The problem is that if vector quantities are universals then there should be a way of identifying the same vector quantity at two different places, but there is no absolute identification of vector quantities, merely a path‐relative one.My (...) solution to the problem is to use the mathematical characterization of vectors as differential operators on scalar fields. On the proposed hypothesis a scalar field is a conjunctive state of affairs, and vector and tensor fields are relations instantiated by scalar fields. (shrink)

A nonlinear partial differential equation is derived which admits plane solitary waves on a conformally flat Riemannian space-time. The metric is determined by the amplitude of these waves. By interpreting these solitary waves as particles we arrive at the following picture: these particles are confined to regions exhibiting singular (very large) amplitudes in an otherwise continuous wavetrain. There is, thus, no distinction between the notion of a particle and that of a wave.

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL c( $\dot x$ , x)=(M/2)gij(x) $\dot x$ i $\dot x$ j−v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point (...) transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian. (shrink)

We present a formal derivation of the Mino–Sasaki–Tanaka–Quinn–Wald (MSTQW) equation describing the self-force on a (semi-) classical relativistic point mass moving under the influence of quantized linear metric perturbations on a curved background space–time. The curvature of the space–time implies that the dynamics of the particle and the field is history-dependent and as such requires a non-equilibrium formalism to ensure the consistent evolution of both particle and field, viz., the worldline influence functional and the closed- time-path (CTP) coarse-grained effective (...) action. In the spirit of effective field theory, we regularize the formally divergent self-force by smearing the local part of the retarded Green’s function and employing a quasi-local expansion. We derive the MSTQW–Langevin equations describing the perturbations of the particle’s worldline about its semi-classical trajectory resulting from interactions with the quantum fluctuations of the linear metric perturbations. Finally, we demonstrate that the quantum fluctuations of the field could, in principle, leave imprints on gravitational waveforms expected to be observed by gravitational interferometers, thereby encoding information about tree-level perturbative quantum gravity. (shrink)

The history of the theory of general relativity presents unique features. After its discovery, the theory was immediately confirmed and rapidly changed established notions of space and time. The further implications of general relativity, however, remained largely unexplored until the mid 1950s, when it came into focus as a physical theory and gradually returned to the mainstream of physics. This essay presents a historiographical framework for assessing the history of general relativity by taking into account in an integrated narrative intellectual (...) developments, epistemological problems, and technological advances; the characteristics of post–World War II and Cold War science; and newly emerging institutional settings. It argues that such a framework can help us understand this renaissance of general relativity as a result of two main factors: the recognition of the untapped potential of general relativity and an explicit effort at community building, which allowed this formerly disparate and dispersed field to benefit from the postwar changes in the scientific landscape. (shrink)

The construction of the quantum-mechanical Hamiltonian by canonical quantization is examined. The results are used to enlighten examples taken from slow nuclear collective motion. Hamiltonians, obtained by a thoroughly quantal method (generator-coordinate method) and by the canonical quantization of the semiclassical Hamiltonian, are compared. The resulting simplicity in the physics of a system constrained to lie in a curved space by the introduction of local Riemannian coordinates is emphasized. In conclusion, a parallel is established between the result for various (...) coordinates and a proposed procedure for quantizing the semiclassical Hamiltonian for a single coordinate. (shrink)

A survey is presented of the essential principles for formulating relativistic wave equations in curved spacetime. The approach is relatively simple and avoids much of the philosophical debate about covariance principles, which is also indicated. Hypercomplex numbers provide a natural language for covariance symmetry and the two important kinds of covariant derivative.

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ 1 is meagre yet there (...) are ℵ 1 rectifiable curves in R 3 whose union is not meagre. The consistency of this statement when the phrase "rectifiable curves" is replaced by "straight lines" remains open. (shrink)

I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, to of the equation exhibits an (...) asymmetry in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Moun—despite their acute similarities—exhibit an asymmetry in their mass is possible. The Moun is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible. (shrink)

I propose an account of the metaphysics of the expressions of a mathematical language which brings together the structuralist construal of a mathematical object as a place in a structure, the semantic notion of indexicality and Kit Fine's ontological theory of qua objects. By contrasting this indexical qua objects account with several other accounts of the metaphysics of mathematical expressions, I show that it does justice both to the abstractness that mathematical expressions have because they are mathematical objects and to (...) the element of concreteness that they have because they are also used as signs. In a concluding section, I comment on the pragmatic element that has entered ontology by way of the notion of indexicality and use it to give an answer to a question Stewart Shapiro has recently posed about the status of meta-mathematics in the structuralist philosophy of mathematics. (shrink)

By 1870, non-Euclidean geometry had been established as a mathematical research field but was yet to be considered relevant to the real space inhabited by stars and nebulae. It was of much less interest to physicists and astronomers than to mathematicians and philosophers. Although most astronomers took the age-old Euclidean geometry for granted, during the following decades a few of them such as K. F. Zöllner, S. Newcomb and K. Schwarzschild followed in the footsteps of the pioneer N. I. Lobachevsky (...) by seriously considering the possibility that cosmic space might be curved. From the H. Poincaré’s conventionalist perspective the discussion was meaningless since there was no way in which observations or other empirical data could determine the geometry of space. Poincaré’s view was shared by a few astronomers, but its impact on astronomy was limited. The paper examines the question as it was discussed prior to the advent of the general theory of relativity. (shrink)

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars ${\mathcal{R}}, {\mathcal{S}}$ which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the (...) construction of the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in $\mathit{momentum}$ space is very large, namely the small size of P is of the order of $( 1/R_{\mathit{Hubble}})$ . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the $\mathit{deformed}$ Quaplectic group that is given by the semi-direct product of U(1,3) with the $\mathit{deformed}$ (noncommutative) Weyl-Heisenberg group involving four $\mathit{noncommutative}$ coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented. (shrink)

Prompted by the "Panel Discussion of Grünbaum's Philosophy of Science" (Philosophy of Science 36, December, 1969) and other recent literature, this essay ranges over major issues in the philosophy of space, time and space-time as well as over problems in the logic of ascertaining the falsity of a scientific hypothesis. The author's philosophy of geometry has recently been challenged along three main distinct lines as follows: (i) The Panel article by G. J. Massey calls for a more precise and more (...) coherent account of the Riemannian conception of an intrinsic as opposed to an extrinsic metric, which the author has invoked as his basis for the distinction between non-conventional and convention-laden ascriptions of metrical equality and inequality; (ii) the latter distinction has been claimed to suffer from the liabilities of the so-called "standard conception" of scientific theories [36]; and (iii) pursuant to H. Putnam's "An Examination of Grünbaum's Philosophy of Geometry" [56], J. Earman [16, 17] and R. Swinburne [65] have contended that the difference between intrinsic and extrinsic metrics is scientifically unilluminating, and that the associated distinction between non-conventional and convention-laden metrical comparisons does not have the kind of relevance to extant scientific theories that the author has claimed for it. The essay consists of two installments. The present installment, comprising the Introduction and Part A, is devoted to the clarification, correction and further development of the author's prior writings on the philosophy of geometry. Its main objective is constructive elaboration rather than offering polemics. But rebuttals to Earman, [16, 17], Swinburne [65] and Demopoulos [13] are included, because their inclusion conduced to clarity in giving the new exposition. Part B is to appear in a subsequent issue and will be devoted to replies to critiques contained in the Panel Discussion and in other recent literature. It will range over issues in the philosophy of geometry and in the logic of ascertaining the falsity of a scientific hypothesis. By way of merely elliptical anticipation of much more precise statements given in Part A, section 3(ii), the Introduction dissociates the notion of convention-ladenness developed in Part A from the quite different notion integral to the so-called "standard conception" of scientific theories. Thereby, the Introduction prepares the ground for seeing, as a corollary to Part A, section 3(ii), that the notion advocated in the present essay has nothing to fear from the following fact, noted by C. G. Hempel ([36]; cf. also his 1970 Carus Lectures): "even though a sentence may originally be introduced as true by stipulation, it soon joins the club of all other member-statements of the theory and becomes subject to revision in response to further empirical findings and theoretical developments." Part A, which begins with a fairly detailed table of contents, endeavors to meet the aforementioned three-fold challenge to the author's philosophy of geometry. Massey's call for the provision of clear and detailed characterizations of intrinsic and extrinsic metrics is answered with the invaluable aid rendered personally by Massey himself. These characterizations are shown to permit a precise explication of the portions of Riemann's Inaugural Dissertation relevant to (1) Riemann's brilliant anticipation of Einstein's dynamical conception of physical geometry, and (2) the author's philosophical characterization of the metrics and geometries of space, time, and space-time {section 2(c)}. A byproduct of the analysis is to raise two major philosophical doubts concerning Clifford's sketch of a so-called "space-theory of matter" as elaborated in J. A. Wheeler's relativistic geometrodynamics. That theory's vision of understanding matter as a manifestation of empty curved space is questioned in regard to (1) the existence of an intra-geometrodynamic basis for individuating the metrically homogeneous world points of its space-time manifold {section 1(a)}, and (2) the compatibility of the theory with the Riemannian metrical philosophy apparently espoused by its proponents {section 2(c)(i)}. (shrink)

The development and deployment of artificial intelligence (AI) is and will profoundly reshape human society, the culture and the composition of civilisations which make up human kind. All technological triggers tend to drive a hype curve which over time is realised by an output which is often unexpected, taking both pessimistic and optimistic perspectives and actions of drivers, contributors and enablers on a journey where the ultimate destination may be unclear. In this paper we hypothesise that this journey is not (...) dissimilar to the personal journey described by the Kubler-Ross change curve and illustrate this by commentary on the potential of AI for drug discovery, development and healthcare and as an enabler for deep space exploration and colonisation. Recent advances in the call for regulation to ensure development of safety measures associated with machine-based learning are presented which, together with regulation of the rapidly emerging digital after-life industry, should provide a platform for realising the full potential benefit of AI for the human species. (shrink)

We introduce a suitable notion of eight-shaped curve in the product S × ℝ of a Suslin line S for the real line ℝ, and we prove that if S is dense in itself, then every collection of pairwise disjoint eight-shaped curves in S × ℝ is countable. This parallels a folklore result which holds for the real plane.

A perverted space-time geodesy results from the idea of variable rods and clocks, whose length and rates are taken to be affected by the gravitational field. By contrast, what we might call a concrete geodesy relies on the idea of invariable unit-measuring rods and clocks. Indeed, this is a basic assumption of general relativity. Variable rods and clocks lead to a perverted geodesy, in the sense that a curved space-time may be seen as a result of a departure from (...) the Minkowskian space-time as an effect of the gravitational field on the rate of clocks and the length of rods. In the case of a concrete geodesy, we have a curved space-time “directly”, the curvature of which can be determined using (invariable) unit-measuring rods and clocks. In this paper, we will make the case for the plausibility of the claim that Einstein’s views on geometry in relation to general relativity are permeated by a perverted geodesy. (shrink)

The mathematical and physical aspects of the conformal symmetry of space-time and of physical laws are analyzed. In particular, the group classification of conformally flat space-times, the conformal compactifications of space-time, and the problem of imbedding of the flat space-time in global four-dimensional curved spaces with non-trivial topological and geometrical structure are discussed in detail. The wave equations on the compactified space-times are analyzed also, and the set of their elementary solutions constructed. Finally, the implications of global compactified (...) space-times for cosmology are discussed. It is argued that the recent discovery of periodic structure of matter distribution on large distances strongly suggests that the global cosmological space-time should be close. Next we analyze the inflation scalar field in the inflationary model of universe evolution considered on the spatially compact Robertson-Walker space-time. It is shown that the energy distribution in this model is periodic and the periods and density decrease with increasing distance, in striking agreement with experimental data. Our model of the universe also provides a definite predictions for the energy distribution, polar and azimuthal, considered as a function of angles θ and φ. These predictions should be tested with the new astronomical data. (shrink)

Cognitive impenetrability (CI) of a large part of visual perception is taken for granted by those of us in the field of computational vision who attempt to recover descriptions of space using geometry and statistics as tools. These tools clearly point out, however, that CI cannot extend to the level of structured descriptions of object surfaces, as Pylyshyn suggests. The reason is that visual space – the description of the world inside our heads – is a nonEuclidean curved space. (...) As a consequence, the only alternative for a vision system is to develop several descriptions of space–time; these are representations of reduced intricacy and capture partial aspects of objective reality. As such, they make sense in the context of a class of tasks/actions/plans/purposes, and thus cannot be cognitively impenetrable. (shrink)

In this paper, we first define the vector product in Minkowski space $\mathbb{R}_{4}^{7}$, which is identified with the space of spatial split-octonions. Next, we derive the $G_{2}-$ frame formulae for a seven dimensional Minkowski curve by using the spatial split-octonions and the vector product. We show that Frenet–Serret formulas are satisfied for a spatial split octonionic curve. We obtain the congruence of two spatial split octonionic curves and give relationship between the $G_{2}-$ frame and Frenet–Serret frame. Furthermore, we present the (...) Frenet–Serret frame with split octonions in $\mathbb{R}_{4}^{8}$. Finally, we give illustrative examples with Matlab codes. (shrink)

Already in 1835 Lobachevski entertained the possibility of multiple geometries of the same type playing a role. This idea of rival geometries has reappeared from time to time but had yet to become a key idea in space-time philosophy prior to Brown's _Physical Relativity_. Such ideas are emphasized towards the end of Brown's book, which I suggest as the interpretive key. A crucial difference between Brown's constructivist approach to space-time theory and orthodox "space-time realism" pertains to modal scope. Constructivism takes (...) a broad modal scope in applying to all local classical field theories---modal cosmopolitanism, one might say, including theories with multiple geometries. By contrast the orthodox view is modally provincial in assuming that there exists a _unique_ geometry, as the familiar theories have. These theories serve as the "canon" for the orthodox view. Their historical roles also suggest a Whiggish story of inevitable progress. Physics literature after c. 1920 is relevant to orthodoxy primarily as commentary on the canon, which closed in the 1910s. The orthodox view explains the spatio-temporal behavior of matter in terms of the manifestation of the real geometry of space-time, an explanation works fairly well within the canon. The orthodox view, Whiggish history, and the canon have a symbiotic relationship. If one happens to philosophize about a theory outside the canon, space-time realism sheds little light on the spatio-temporal behavior of matter. Worse, it gives the _wrong_ answer when applied to an example arguably _within_ the canon, a sector of Special Relativity, namely, _massive_ scalar gravity with universal coupling. Which is the true geometry---the flat metric from the Poincare' symmetry group, the conformally flat metric exhibited by material rods and clocks, or both---or is the question faulty? How does space-time realism explain the fact that all matter fields see the same curved geometry, when so many ways to mix and match exist? Constructivist attention to dynamical details is vindicated; geometrical shortcuts can disappoint. The more exhaustive exploration of relativistic field theories in particle physics, especially massive theories, is a largely untapped resource for space-time philosophy. (shrink)

These are just some of the fundamental questions addressed in Time and Space. Writing for a primary readership of advanced undergraduate and graduate philosophy students, Barry Dainton introduces the central ideas and arguments that make space and time such philosophically challenging topics. Although recognising that many issues in the philosophy of time and space involve technical features of physics, Dainton has been careful to keep the conceptual issues accessible to students with little scientific or mathematical training. Surveying historical debates and (...) ideas at the forefront of contemporary thinking, the book is unrivaled in its coverage. Topics include McTaggart's argument for the unreality of change; static, tensed, and dynamic time; time travel and causal arrows; space as void, motion, and curved spac; as well as a non-technical introduction to the special theory of relativity and the key features of general relativity, spacetime, and strings. Dainton also addresses the relationship between the philosophy of time and broader human concerns involving actions, ethics, fatalism, and death. (shrink)

John Earman's new book,World Enough and Space-Time, is a brisk account of the controversy between space-time absolutists and relationists. The book is intended, one is told, to be “appropriate for use in an upper-level undergraduate or beginning graduate course in the philosophy of science”, but Earman's no-holds-barred approach to the mathematics of space-time theories will have bludgeoned most philosophical readers, undergraduate or beyond, into submission long before it is revealed that Pirani and Williams “have studied the integrability conditions for Born-rigid (...) motions in curved space-times and have shown that space-times of Petrov types II, III, and N do not admit of nonrotating Born-rigid motions”. I say this sadly, because Earman's book is a discerning review of an important literature, and most of its main arguments can be grasped even if some technical details remain out of reach. The more you reach for those details, the more compelling the book will become. (shrink)

In a previous paper by Pollock and Singh, it was proven that the total entropy of de Sitter space-time is equal to zero in the spatially flat case K=0. This result derives from the fundamental property of classical thermodynamics that temperature and volume are not necessarily independent variables in curved space-time, and can be shown to hold for all three spatial curvatures K=0,±1. Here, we extend this approach to Schwarzschild space-time, by constructing a non-vacuum interior space with line element (...) ds 2=e2λ(r) dt 2−e−2λ(r) dr 2−r 2(dθ 2+sin2 θdϕ 2), where $\mathrm{e}^{2{\lambda }(r)}=-\frac{1}{2}(1-\frac{r^{2}}{R_{0}^{2}})$ , which matches onto the vacuum exterior Schwarzschild metric in such a way that e2λ and d(e2λ )/dr are both continuous at the Schwarzschild radius R 0=2M. Then we show that the volume entropy is equal to A/4, where $A\equiv 4\pi R_{0}^{2}$ is the area of the apparent horizon, as found by Hawking. (shrink)

Gödel’s contention that closed timelike curves (CTC’s) are a necessary consequence of the Einstein equations for his metric is challenged. It is seen that the imposition of periodicity in a timelike coordinate is the actual source of CTC’s rather than the physics of general relativity. This conclusion is supported by the creation of Gödel-like CTC’s in flat space by the correct choice of coordinate system and identifications. Thus, the indications are that the notion of a time machine remains exclusively an (...) aspect of science fiction fantasy. The element of the identification of spacetime points is also seen to be the essential factor in the modern creation of CTC’s in the Gott model of moving cosmic strings. (shrink)

This chapter contains sections titled: Coordinate Systems: Euclidean Space Invariant Quantities Classical Space‐times Special Relativity Consequences of the Lorentz Transformation Lorentz Invariant Quantities.

The fact that the space of states of a quantum mechanical system is a projective space (as opposed to a linear manifold) has many consequences. We develop some of these here. First, the space is nearly contractible, namely all the finite homotopy groups (except the second) vanish (i.e., it is the Eilenberg-MacLane space K(ℤ, 2)). Moreover, there is strictly speaking no “superposition principle” in quantum mechanics as one cannot “add” rays; instead, there is adecomposition principle by which a given ray (...) has well-defined projections in other rays. When the evolution of a system is cyclic, any representativevector traces out an open curve, defining an element of the holonomy group, which is essentially the (geometrical) Berry phase. Finally, for the massless case of the representations of the Poincaré group (the so-called “Wigner program”), there could be in principle arbitrarily multivalued representations coming from the Lie algebra of the Euclidean plane group. In fact they are at most bivalued (as commonly admitted). (shrink)

In "The Epistemology of Geometry" Glymour proposed a necessary structural condition for the synonymy of two space-time theories. David Zaret has recently challenged this proposal, by arguing that Newtonian gravitational theory with a flat, non-dynamic connection (FNGT) is intuitively synonymous with versions of the theory using a curved dynamical connection (CNGT), even though these two theories fail to satisfy Glymour's proposed necessary condition for synonymy. Zaret allowed that if FNGT and CNGT were not equally well (bootstrap) tested by the (...) relevant phenomena, the two theories would in fact not be synonymous. He argued, however, that when electrodynamic phenomena are considered, the two theories are equally well tested. We show that it is not FNGT and CNGT which are equally well tested when the electrodynamic phenomena are considered, but only suitable extensions of FNGT and CNGT. Thus, there is good reason to consider FNGT and CNGT to be non-synonymous. We further show that the two extensions of FNGT and CNGT which are equally well tested when electrodynamic phenomena are considered (and which could be considered intuitively synonymous) not only satisfy Glymour's original proposed necessary condition for the synonymy of space-time theories, they satisfy a plausible stronger condition as well. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized (...) and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

Must space be a unity? This question, which exercised Aristotle, Descartes and Kant, is a specific instance of a more general one; namely, can the topology of physical space change with time? In this paper we show how the discussion of the unity of space has been altered but survives in contemporary research in theoretical physics. With a pedagogical review of the role played by the Euler characteristic in the mathematics of relativistic spacetimes, we explain how classical general relativity (modulo (...) considerations about energy conditions) allows virtually unrestrained spatial topology change in four dimensions. We also survey the situation in many other dimensions of interest. However, topology change comes with a cost: a famous theorem by Robert Geroch shows that, for many interesting types of such change, transitions of spatial topology imply the existence of closed timelike curves or temporal non-orientability. Ways of living with this theorem and of evading it are discussed. (shrink)

The empirical study of visual space has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are (...) the spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space, it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting. (shrink)

In Kate Bush’s 1993 album, The Red Shoes, and her film, The Line, the Cross and the Curve, she engages with the symbolism of The Red Shoes fairytale as first depicted in Hans Christian Andersen’s 1845 fairy tale and later developed by the Powell and Pressburger film of the same name. In Bush’s versions of the tale she attempts to find a space of agency for the main female protagonist in a plot structure over-determined by patriarchal narrative and symbolic logic. (...) I will argue that it is through her own use of mystical symbolism — the Line, the Cross and the Curve — mediated through the deployment of ritual magick and kabbalistic ritual — that she breaks the ‘spell’ of the red shoes story where the main female character can escape the gender specific ‘curse’ of the red shoes. (shrink)

The number of adjustable parameters in a model or hypothesis is often taken as the formal expression of its simplicity. I take issue with this `definition´ and argue that comparative simplicity has a quasi-empirical measure, reflecting experts’ judgements who track past use of a model-type in or across domains. Since models are represented by restricted sets of functions in a suitable space, formally speaking, a general `measure of simplicity´ may be defined implicitly for the elements of a function space. This (...) paper sketches such a framework starting from intuitive constraints. It is shown how experts’ judgements feed into this framework and how the usual definition can be recovered. A theorem by H. Akaike in the theory of model-choice has recently been used to shine new light on the relationship between the demand for simplicity and empirical success, or even `truth´. The approach favored here permits an alternative answer based on a reliabilist account of justification: if judgements of simplicity track past successful use of a model-type comparative simplicity is evidential and inductive. (shrink)

This response to Evanset alencourages broader consideration of what constitutes disability, extending beyond a protagonist’s capabilities toward society’s fuller chorus. Three avenues are submitted to encourage this. First, Engel’s biopsychosocial paradigm of health can be helpfully applied to the question of identity in general, and disability in particular. Second, the philosophy of language (and of naming) gives useful insight into the pitfalls of trying to define disability via descriptions of capability. Third, Kennedy’s critique ‘Unmasking Medicine’ offers a sociopolitical view that (...) builds on Foucault and Illich allowing us to recognise that it matters who judges who, as disabled, and on what grounds. Alongside this, I suggest alternative views first, on the authors’ liberal use of bell curves in the depiction of disability and second, on their terminology of capacity spaces. (shrink)

We examine the time discontinuity in rotating space–times for which the topology of time is S1. A kinematic restriction is enforced that requires the discontinuity to be an integral number of the periodicity of time. Quantized radii emerge for which the associated tangential velocities are less than the speed of light. Using the de Broglie relationship, we show that quantum theory may determine the periodicity of time. A rotating Kerr–Newman black hole and a rigidly rotating disk of dust are also (...) considered; we find that the quantized radii do not lie in the regions that possess CTCs. (shrink)

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of $\mathcal{C}$ - and $\mathfrak{D}$ -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space (...) is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect. (shrink)

This paper discusses the physical effect of gravity on space, a rather treacherous topic that has not gained much attention in the literature, unlike the effect of gravity on time which has been clearly established from the beginning as a consequence of the Equivalence Principle and also experimentally tested. The difficulties encountered in analysing the effect of gravity on space can be represented by the need to compare vectors associated with different spatial points in a curved manifold, where the (...) parallel-transport of vectors depends on the chosen path. This same problem can also be seen from another perspective: the effect of gravity on space is embodied in the components of the metric tensor, which however are not uniquely determined as a consequence of the degrees of freedom due to the Bianchi identities. We here show that, however, there are circumstances under which these difficulties can be overcome, resulting in a clear representation of the effect of gravity on space. This effect is perfectly dual to the corresponding effect on time: while gravity slows down time, it expands space. This can clarify topics like the effects of gravity at the centre of a spherically symmetric system, where the literature does not provide clear answers. A final confirmation of the gravitational expansion of space is provided by the Equivalence Principle. The conclusions help fill a conceptual gap that has so far prevented gravitational effects on time and space from being viewed on an equal footing. (shrink)

In an accompanying paper Gomes, we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space \ of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on (...) the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, \, if the projection of extremal curves on \ onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to \, where \ is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality. (shrink)

The Schrödinger equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi \sim (...) \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski space–time , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations that differ in a curved background space–time $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational Lagrangian $L_\mathcal{D}$ which contains higher-derivative terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter space for $8 \le \mathcal {D} \le 10$. (shrink)

This paper begins by asking a simple question: can a farmer own and fully utilise precisely five tractors and precisely six tractors at the same time? Of course not. He can own five or he can own six but he cannot own five and six at the same. The answer to this simple question eventually led this author to Alfred Marshall's historically-ignored, linguistically-depicted 'cardboard model' where my goal was to construct a picture based on his written words. More precisely, in (...) this paper the overall goal is to convert Marshall's ('three-dimensional') words into a three-dimensional picture so that the full import of his insight can be appreciated by all readers. After a brief digression necessary to introduce Euclidean three-dimensional space, plus a brief digression to illustrate the pictorial problem with extant theory, the paper turns to Marshall's historically- ignored words. Specifically, it slowly constructs a visual depiction of Marshall's 'cardboard model'. Unfortunately (for all purveyors of extant economic theory), this visual depiction suddenly opens the door to all manner of Copernican heresy. For example, it suddenly becomes obvious that we can join the lowest points on a firm's series of SRAC curves and thereby form its LRAC curve; it suddenly becomes obvious that the firm's series of SRAC curves only appear to intersect because mainstream theory has naively forced our three-dimensional economic reality into a two-dimensional economic sketch; and it suddenly becomes obvious that a two-dimensional sketch is analytically useless because the 'short run' (SR) never turns into the 'long run' (LR) no matter how long we wait. (shrink)

We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics. This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, and those that employ smooth fields supported in small neighborhoods of a curve. This result also applies to "bodies" constructed from wave packets of Maxwell or Klein-Gordon fields. There follows a simple and precise formulation of the optical limit for Maxwell (...) fields. (shrink)

In this monograph the author presents the special and general theories of relativity from a geochronometrical viewpoint. The amount of mathematics demanded is not too great, and one can get quite far along on the material on vectors presented early in the book. The first three chapters especially derive from the work of A. A. Robb several decades ago: they treat foundations of geochronometry, one-plus-one geochronometry and its generalization to a one-plus-three system. Later chapters cover such staples as the Lorentz (...) transformation, the metric and dynamical tensors, dynamical relationships of particles, accelerating and curving events, geodesics, and applications to the red-shift of spectral lines and bending of light-rays. The best introduction to the study of the foundation of physics is physics itself; so those philosophers interested in the foundations of relativity theory will find here a lucid and logically respectable exposition of that theory, and in that way an understanding of its fundamentals and problems will be reached.—P. J. M. (shrink)

We derive the first analytical formula for the density of "Dark Matter" (DM) at all length scales, thus also for the rotation curves of stars in galaxies, for the baryonic Tully–Fisher relation and for planetary systems, from Einstein's equations (EE) and classical approximations, in agreement with observations. DM is defined in Part I as the energy of the coherent gravitational field of the universe, represented by the additional equivalent ordinary matter (OM), needed at all length scales, to explain classically, with (...) inclusion of the OM, the _observed coherent_ gravitational field. Our derivation uses both EE and the Newtonian approximation of EE in Part I, to describe semi-classically in Part II the advection of DM, created at the level of the universe, into galaxies and clusters thereof. This advection happens proportional with their own classically generated gravitational field g, due to self-interaction of the gravitational field. It is based on the universal formula ρ D = λgg′ 2 for the density ρ D of DM advected into medium and lower scale structures of the observable universe, where λ is a universal constant fixed by the Tully–Fisher relations. Here g′ is the gravitational field of the universe; g′ is in main part its own source, as implied in Part I from EE. We start from a simple electromagnetic analogy that helps to make the paper generally accessible. This paper allows for the first time the exact calculation of DM in galactic halos and at all levels in the universe, based on EE and Newtonian approximations, in agreement with observations. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)