Stemming from our energy localization hypothesis that energy in general relativity is localized in the regions of the energy-momentum tensor, we had devised a test with the classic Eddington spinning rod. Consistent with the localization hypothesis, we found that the Tolman energy integral did not change in the course of the motion. This implied that gravitational waves do not carry energy in vacuum, bringing into question the demand for the quantization of gravity. Also if information is conveyed (...) by the waves, the traditional view that information transfer demands energy is challenged. Later, we showed that the “body” angular momentum changed at a rate indicating that the moment of inertia increased to higher order, contrary to traditional expectations. We consider the challenges facing the development of a localized expression for the total angular momentum of the body including the contribution from gravity. We find that Komar's expression does not lead to an adequate formulation of localized angular momentum. (shrink)

We point out a fundamental problem that hinders the quantization of general relativity: quantum mechanics is formulated in terms of systems, typically limited in space but infinitely extended in time, while general relativity is formulated in terms of events, limited both in space and in time. Many of the problems faced while connecting the two theories stem from the difficulty in shoe-horning one formulation into the other. A solution is not presented, but a list of (...) desiderata for a quantum theory based on events is laid out. (shrink)

We consider an oscillator subjected to a sudden change in equilibrium position or in effective spring constant, or both—to a “squeeze” in the language of quantum optics. We analyze the probability of transition from a given initial state to a final state, in its dependence on final-state quantum number. We make use of five sources of insight: Bohr-Sommerfeld quantization via bands in phase space, area of overlap between before-squeeze band and after-squeeze band, interference in phase space, Wigner (...) function as quantum update of B-S band and near-zone Fresnel diffraction as mockup Wigner function. (shrink)

Weyl symmetry of the classical bosonic string Lagrangian is broken by quantization, with profound consequences described here. Reimposing symmetry requires that the background space-time satisfy the equations of general relativity: general relativity, hence classical space-time as we know it, arises from string theory. We investigate the logical role of Weyl symmetry in this explanation of general relativity: it is not an independent physical postulate but required in quantum string theory, so from a certain (...) point of view it plays only a formal role in the explanation. (shrink)

String theory reduces to general relativity in appropriate regimes. Huggett and Vistarini have given an account of this reduction that includes a deflationary thesis about symmetry: although the usual derivation of general relativity from string theory appeals to a premise about the theory’s symmetry, Huggett and Vistarini argue that this premise plays no logical role. In this article I disagree with their deflationary thesis and argue that their analysis is based on a popular but flawed conception (...) of the interaction between symmetry and quantization. I argue that a better conception recognizes a distinction between ordinary, broken, and anomalous symmetries. (shrink)

Based upon the proceedings of the First International Conference on the History of General Relativity, held at Boston University's Osgood Hill Conference Center, North Andover, Massachusetts, 8-11 May 1986, this volume brings together essays by twelve prominent historians and philosophers of science and physicists. The topics range from the development of general relativity (John Norton, John Stachel) and its early reception (Carlo Cattani, Michelangelo De Maria, Anne Kox), through attempts to understand the physical implications of the (...) theory (Jean Eisenstaedt, Peter Havas) and to quantize it (Peter G. Bergmann), to elaborations of the theory into a unified theory of electromagnetism and gravitation (Vladimir P. Vizgin, Michel Biezunski), and considerations of its cosmological extensions (Pierre Kerszberg, George F.R. Ellis). This is the first volume to survey many of the most important questions in the history of general relativity, with many of the contributions drawing upon such original resources as the Einstein Archive. It is hoped that it will stimulate much-needed further research in this hitherto neglected area. (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)

This paper presents a qualitative comparison of opposing views of elementary matter—the Copenhagen approach in quantum mechanics and the theory of general relativity. It discusses in detail some of their main conceptual differences, when each theory is fully exploited as a theory of matter, and it indicates why each of these theories, at its presently accepted state, is incomplete without the other. But it is then argued on logical grounds that they cannot be fused, thus indicating the need (...) for a third revolution in contemporary physics. Toward this goal, the approach discussed is one of further generalizing the theory of general relativity in a way that incorporates the inertial manifestations of matter in covariant fashion, with quantum mechanics serving as a low-energy, linear approximation. Such a theoretical extension of general relativity will be discussed, with applications in elementary particle physics, such as the appearance of mass spectra in the microdomain, as an asymptotic feature of matter, mass doublets (electron-muon and proton-heavy proton), the explanation of pair annihilation and creation from a deterministic field theory, charge quantization, and features of pions. (shrink)

It is demonstrated that in quantized general relativity one is led to Jordan-Fock type uncertainty relations implying the occurrence of cut-off lengths. We argue that these lengths (i) represent limitations on the measurability of quantum effects of general relativity and (ii) provide a cut-off length of quantum divergences.

We propose here a new symplectic quantization scheme, where quantum fluctuations of a scalar field theory stem from two main assumptions: relativistic invariance and equiprobability of the field configurations with identical value of the action. In this approach the fictitious time of stochastic quantization becomes a genuine additional time variable, with respect to the coordinate time of relativity. Thisintrinsic timeis associated to a symplectic evolution in the action space, which allows one to investigate not only asymptotic, i.e. (...) equilibrium, properties of the theory, but also its non-equilibrium transient evolution. In this paper, which is the first one in a series of two, we introduce a formalism which will be applied to general relativity in its companion work (Gradenigo, Symplectic quantization II: dynamics of space-time quantum fluctuations and the cosmological constant, 2021). (shrink)

We discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves. Firstly, we have the problem of the influence of gravitational fields on the de Broglie waves, which influence is in accordance with Eeinstein's weak principle of equivalence and the limitation of measurements given by Heisenberg's uncertainty relations. Secondly, the quantization of the gravitational fields is a “quantization of geometry.” However, classical and quantum gravitation have (...) the same physical meaning according to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies. (shrink)

The quantization of gravity represents an important attempt at reconciling the two seemingly incompatible frameworks that lie at the base of modern physics, quantum theory and general relativity. The dissertation begins by looking at the incompatibilities between the two frameworks. The incompatibility with quantum theory, it is argued, is rooted in the profound differences between general relativity and ordinary field theories. The dissertation goes on to look at how, in practice, these incongruities are treated in (...) the canonical quantization program, and argues that their treatment is such that it is unlikely to lead to a satisfactory physical theory. The dissertation concludes by arguing that quantum theory only makes sense when embedded in a classical theory. Thus the reconciliation of quantum theory and general relativity suggests that we look outside the bounds of ordinary quantum theory for the proper theoretical framework. (shrink)

Einstein Versus Bohr is unlike other books on science written by experts for non-experts, because it presents the history of science in terms of problems, conflicts, contradictions, and arguments. Science normally "keeps a tidy workshop." Professor Sachs breaks with convention by taking us into the theoretical workshop, giving us a problem-oriented account of modern physics, an account that concentrates on underlying concepts and debate. The book contains mathematical explanations, but it is so-designed that the whole argument can be followed (...) with the math omitted. Professor Sachs' story begins with classical and nineteenth century physics, describes the early discoveries in particle theory, and introduces the "old" quantum theory, which evolved into the quantum mechanics of the Copenhagen School. Such important ideas as the Einstein Photon Box experiment and the Einstein-Podolsky-Rosen Paradox, and Schrodinger's Cat Paradox are clearly expounded, followed by a completely fresh explanation of relativity in conceptual terms, showing how apparent paradoxes can be removed by Einstein's own interpretation, especially that of his later years. Professor Sachs gives a detailed comparison of the fundamentals of the quantum and relativity theories, suggesting how the contradictions might be resolved. In an epilogue, he makes suggestions, with reference to religious notions, Taoism, and Buber's theory of I-Thou, for generalizing Einstein's approach beyond physics. (shrink)

The operator form of the generalized canonical momenta in quantum mechanics is derived by a new, instructive method and the uniqueness of the operator form is proven. If one wishes to find the correct representation of the generalized momentum operator, he finds the Hermitian part of the operator —iħ ∂/∂q, whereq q is the generalized coordinate. There are interesting philosophical implications involved in this: It is like saying that a physical structure is composed of two parts, one which is real (...) (the measurable quantity) and one which is pure imaginary. However, in order to understand the theoretical generation of the physical structure, one must look at the imaginary part as well as the real part since the sum of these two parts gives the simplified physical theory. That is why we can choose the total generalized momentum operator as simply —iħ ∂/∂q, but in order to arrive at the “measurable” momentum operator, we must choose the real (Hermitian) part, the other part being anti-Hermitian (corresponding to pure imaginary eigenvalues). We also discuss the operator form of the generalized Hamiltonian and show that the primary focus in developing fundamental concepts and prescriptions in quantum mechanics should be on the generalized momenta rather than on the Hamiltonian. (shrink)

Here we briefly review the concept of "prediction" within the context of classical relativity theory. We prove a theorem asserting that one may predict one's own future only in a closed universe. We then question whether prediction is possible at all (even in closed universes). We note that interest in prediction has stemmed from considering the epistemological predicament of the observer. We argue that the definitions of prediction found thus far in the literature do not fully appreciate this predicament. (...) We propose a more adequate alternative and show that, under this definition, prediction is essentially impossible in general relativity. (shrink)

In the early phase of the new history of physics that emerged at about 1970 and was pioneered by John Heilbron, Thomas Kuhn, Paul Forman, and others, the quantum and atomic theories of the first three decades of the twentieth century played a central role. Since then, interest in the area has continued, but for the last few decades at a slower rate. While other areas of the new physics—such as the general theory of relativity—have attracted much attention, (...) only relatively little has been written about the quantum revolution. Given this situation, Suman Seth’s comprehensive and innovative Crafting the Quantum is a welcome publication that provides the field with fresh blood and new perspectives. The title derives from a letter to Einstein of January 1922, in which Arnold Sommerfeld wrote that “I can only advance the craft of the quantum, you have to make its philosophy.” In spite of its title, Seth’s book is neither restricted to the new quantum theory nor to Sommerfeld’s role in the tr. (shrink)

A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, "The Motion of Small Bodies in Space-Time", Comm. Math. Phys. ], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call "tracking". In the present paper, I situate the main (...) results of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpretational significance of the new approach. My main suggestion is that "tracking" provides the resources for eliminating "point particles"---a problematic notion in general relativity---from the geodesic principle altogether. (shrink)

The issue of energy and its potential localizability in general relativity has challenged physicists for more than a century. Many non-invariant measures were proposed over the years but an invariant measure was never found. We discovered the invariant localized energy measure by expanding the domain of investigation from space to spacetime. We note from relativity that the finiteness of the velocity of propagation of interactions necessarily induces indefiniteness in measurements. This is because the elements of actual physical (...) systems being measured as well as their detectors are characterized by entire four-velocity fields, which necessarily leads to information from a measured system being processed by the detector in a spread of time. General relativity adds additional indefiniteness because of the variation in proper time between elements. The uncertainty is encapsulated in a generalized uncertainty principle, in parallel with that of Heisenberg, which incorporates the localized contribution of gravity to energy. This naturally leads to a generalized uncertainty principle for momentum as well. These generalized forms and the gravitational contribution to localized energy would be expected to be of particular importance in the regimes of ultra-strong gravitational fields. We contrast our invariant spacetime energy measure with the standard 3-space energy measure which is familiar from special relativity, appreciating why general relativity demands a measure in spacetime as opposed to 3-space. We illustrate the misconceptions by certain authors of our approach. (shrink)

Am 6. Dezember 1915 und am 8. Januar 1916 legte Arnold Sommerfeld der Bayerischen Akademie der Wissenschaften zwei Abhandlungen im Umfang von 75 Druckseiten vor, mit denen er das Bohrsche Atommodell aus dem Jahr 1913 zur Bohr-Sommerfeldschen Atomtheorie erweiterte. In Sommerfelds Gesammelten Schriften findet sich nur die im Juli 1916 von Sommerfeld in den Annalen der Physik eingereichte Publikation darüber. "Meine Spektrallinien sind endlich in der Akademie in’s Unreine gedruckt. In den Annalen werden sie in geläuterter Form (...) erscheinen", so hatte Sommerfeld am 10. Februar 1916 dem Herausgeber der Annalen der Physik diese Publikation angekündigt. Aus wissenschaftshistorischer Sicht sind für die Erweiterung der Bohrschen Atomtheorie jedoch gerade die vor dem Läuterungsprozess publizierten Akademieabhandlungen von besonderem Interesse. Der Reproduktion dieser Akademieabhandlungen wird ein ausführlicher physikhistorischer Essay (ca. 100 Druckseiten) vorangestellt. (shrink)

The multiple detections of gravitational waves by LIGO (the Laser Interferometer Gravitational-Wave Observatory), operated by Caltech and MIT, have been acclaimed as confirming Einstein's prediction, a century ago, that gravitational waves propagating as ripples in spacetime would be detected. Yunes and Pretorius (2009) investigate whether LIGO's template-based searches encode fundamental assumptions, especially the assumption that the background theory of general relativity is an accurate description of the phenomena detected in the search. They construct the parametrized post-Einsteinian (ppE) framework (...) in response, which broadens those assumptions and allows for wider testing under more flexible assumptions. Their methods are consistent with work on confirmation and testing found in Carnap (1936), Hempel (1969), and Stein (1992, 1994), with the following principles in common: that confirmation is distinct from testing, and that, counterintuitively, revising a theory's formal basis can make it more broadly empirically testable. These views encourage a method according to which theories can be made abstract, to define families of general structures for the purpose of testing. With the development of the ppE framework and related approaches, multi-messenger astronomy is a catalyst for deep reasoning about the limits and potential of the theoretical framework of general relativity. (shrink)

The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} on an additional time variable, named intrinsic (...) time at variance with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative g˙μν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{g}}_{\mu \nu }$$\end{document} of the metric field with respect to intrinsic time, corresponding to the conjugated momentum πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document}. Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional A[gμν,πμν]=K[gμν,πμν]-S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$\end{document}, playing formally the role of a Hamilton function, where S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} is the standard Einstein–Hilbert action while K[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action A[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} and not of energy. Since the Einstein–Hilbert action S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document}. Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document} gives rise to a cosmological constant term in the path-integral approach. (shrink)

The paper discusses from a metaphysical standpoint the nature of the dependence relation underpinning the talk of mutual action between material and spatiotemporal structures in general relativity. It is shown that the standard analyses of dependence in terms of causation or grounding are ill-suited for the general relativistic context. Instead, a non-standard analytical framework in terms of structural equation modeling is exploited, which leads to the conclusion that the kind of dependence encoded in the Einstein field equations (...) is a novel one. (shrink)

The geometro-stochastic quantization of a gauge theory for extended objects based on the (4, 1)-de Sitter group is used for the description of quantized matter in interaction with gravitation. In this context a Hilbert bundle ℋ over curved space-time B is introduced, possessing the standard fiber ℋ $_{\bar \eta }^{(\rho )} $ , being a resolution kernel Hilbert space (with resolution generator $\tilde \eta $ and generalized coherent state basis) carrying a spin-zero phase space representation of G=SO(4, 1) belonging (...) to the principal series of unitary irreducible representations determined by the parameter ρ. The bundle ℋ, associated to the de Sitter frame bundle P(B, G), provides a geometric arena with built-in fundamental length parameter R (taken to be of the order of 10−13 cm characterizing hadron physics) yielding, in the presence of gravitation, a quantum kinematical framework for the geometro-stochastic description of spinless matter described in terms of generalized quantum mechanical wave functions, Ψ x ρ (ξ, ζ), defined on #x210B;. By going over to a nonlinear realization of the de Sitter group with the help of a section ξ(x) on the soldered bundle E, associated to P, with homogeneous fiber V′4⋍G/H, one is able to recover gravitation in a de Sitter gauge invariant manner as a gauge theory related to the Lorentz subgroup H of G. ξ(x) plays the dual role of a symmetry-reducing and an extension field. After introducing covariant bilinear source currents in the fields Ψ x ρ (ξ, ζ) and their adjoints determined by G-invariant integration over the local fibers in ℋ, a quantum fiber dynamical (QFD) framework is set up for the dynamics at small distances in B determining the geometric quantities beyond the classical metric of Einstein's theory through a set of current-curvature field equations representing the source equations for axial vector torsion and the de Sitter boost contributions to the bundle connection (the latter defining the soldering forms of the Cartan connection in P(B, G) in the nonlinear gauge). The presented bundle framework yields a theory for quantized material objects in interaction with gravitation, the long-range metrical part of which remains classical. (shrink)

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal (...) clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity. (shrink)

We highlight and resolve what we take to be three common misconceptions in general relativity, relating to the interpretation of the weak equivalence principle and the relationship between gravity and inertia; the connection between gravitational redshift results and spacetime curvature; and the strong equivalence principle and the local recovery of special relativity in curved, dynamical spacetime.

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, theory-confirmation 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. (shrink)

Conformal rescalings of spinors are considered, in which the factor Ω, inε AB ↦Ωε AB, is allowed to be complex. It is argued that such rescalings naturally lead to the presence of torsion in the space-time derivative▽ a. It is further shown that, in standard general relativity, a circularly polarized gravitational wave produces a (nonlocal) rotation effect along rays intersecting it similar to, and apparently consistent with, the local torsion of the Einstein-Cartan-Sciama-Kibble theory. The results of these deliberations (...) are suggestive rather than conclusive. (shrink)

The gravitational equations were derived in general relativity using the assumption of their covariance relative to arbitrary transformations of coordinates. It has been repeatedly expressed an opinion over the past century that such equality of all coordinate systems may not correspond to reality. Nevertheless, no actual verification of the necessity of this assumption has been made to date. The paper proposes a theory of gravity with a constraint, the degenerate variants of which are general relativity and (...) the unimodular theory of gravity. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space–time metric. The original equations of the theory of gravity with the constraint are formulated. On this basis, a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons. It is claimed that: physical time is anisotropic, the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR. The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated. The wave equation of the gravitational field is formulated, its solution is found, and the nonstationary wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random. (shrink)

Several authors have claimed that prediction is essentially impossible in the general theory of relativity, the case being particularly strong, it is said, when one fully considers the epistemic predicament of the observer. Each of these claims rests on the support of an underdetermination argument and a particular interpretation of the concept of prediction. I argue that these underdetermination arguments fail and depend on an implausible explication of prediction in the theory. The technical results adduced in these arguments (...) can be related to certain epistemic issues, but can only be misleadingly or mistakenly characterized as related to prediction. (shrink)

A new hypothesis for energy localization in general relativity is introduced which is based upon the fact that the energy-momentum conservation laws are devoid of content in vacuum. The vanishing of pseudotensor components forms the basis of coordinate conditions consistent with the above. The implication is that energy is localized where the energy-momentum tensor is nonvanishing. As a consequence, gravitational waves are not carriers of energy in vacuum. A detailed analysis of a Feynman detector interacting with a plane (...) gravitational wave is consistent with the hypothesis. The fact that there has never been a confirmed direct energy transfer to a detector via gravitational radiation is also consistent with the hypothesis. (shrink)

Preparing general relativity for quantization in the Hamiltonian approach leads to the `problem of time,' rendering the world fundamentally timeless. One proposed solution is the `thermal time hypothesis,' which defines time in terms of states representing systems in thermal equilibrium. On this view, time is supposed to emerge thermodynamically even in a fundamentally timeless context. Here, I develop the worry that the thermal time hypothesis requires dynamics -- and hence time -- to get off the ground, thereby (...) running into worries of circularity. (shrink)

Elementary particles, regarded as the constituents of quarks and leptons, are described classically in the framework of the general relativity theory. There are neutral particles and particles having charges±1/3e. They are taken to be spherically symmetric and to have mass density, pressure, and (if charged) charge density. They are characterized by an equation of state P=−ρ suggested by earlier work on cosmology. The neutral particle has a very simple structure. In the case of the charged particle there is (...) one outstanding model described by a simple analytic solution of the field equations. (shrink)

Change and local spatial variation are missing in canonical General Relativity's observables as usually defined, an aspect of the problem of time. Definitions can be tested using equivalent formulations of a theory, non-gauge and gauge, because they must have equivalent observables and everything is observable in the non-gauge formulation. Taking an observable from the non-gauge formulation and finding the equivalent in the gauge formulation, one requires that the equivalent be an observable, thus constraining definitions. For massive photons, the (...) de Broglie-Proca non-gauge formulation observable A_{\mu} is equivalent to the Stueckelberg-Utiyama gauge formulation quantity A_{\mu} + \partial_{\mu} \phi, which must therefore be an observable. To achieve that result, observables must have 0 Poisson bracket not with each first-class constraint, but with the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints, in accord with the Pons-Salisbury-Sundermeyer definition of observables. The definition for external gauge symmetries can be tested using massive gravity, where one can install gauge freedom by parametrization with clock fields X^A. The non-gauge observable g^{\mu\nu} has the gauge equivalent X^A,_{\mu} g^{\mu\nu} X^B,_{\nu}. The Poisson bracket of X^A,_{\mu} g^{\mu\nu} X^B,_{\nu} with G turns out to be not 0 but a Lie derivative. This non-zero Poisson bracket refines and systematizes Kuchař's proposal to relax the 0 Poisson bracket condition with the Hamiltonian constraint. Thus observables need covariance, not invariance, in relation to external gauge symmetries. The Lagrangian and Hamiltonian for massive gravity are those of General Relativity + \Lambda + 4 scalars, so the same definition of observables applies to General Relativity. Local fields such as g_{\mu\nu} are observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers Hamiltonian-Lagrangian equivalence. (shrink)

In effort to investigate how quantum physics might modify Einstein's Theory of Relativity at speeds v→c, the relationship between space-time coordinates of different reference frames is revisited by introducing only one new parameter xo, a fundamental constant for the quantization of space. The starting point is three criteria: (a) real space-time data are conditioned by standard quantum effects on measurements; (b) since currently used apparatus are only capable of probing the aggregate behavior of these quanta the relevant model (...) is one which maximizes the Entropy subject to certain defining constraints; and (c) the constraints simply involve fixed ensemble averages in the case of an inertial frame, or boundary conditions on running averages in the case of an accelerated frame. In this context it is found that both the Lorentz transformation and a simple scheme for the quantization of space-time which resembles identically Planck's photon picture of radiation are a direct consequence of the Principle of Relativity. Non-inertial behavior corresponds to local Entropy maxima, obtainable by solution of a diffusion equation which gives gradually varying ensemble averages across space-time, as demonstrated by the example of a profile which connects a central region of highly agitated quanta with an asymptotic ambient environment—the outcome is the Schwarzschild metric of General Relativity. Apart from the above, a new feature emerges from the theory: the space-time data of an observer, when referred to the frame of his moving partner, are subject to extra quantum fluctuations which increase indefinitely in severity as v→c, with the Lorentz transformation providing only the mean data values. Thus for fast moving bodies like cosmic rays or matter at the horizon of a black hole, physical processes which affect them may not always be perceived by us to occur at the expected length or time scales. (shrink)

Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints. But other definitions of observables have been proposed. In pursuit of Hamiltonian–Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints. Kuchař waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms (...) might use the gauge generator but permit non-zero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity. Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg–Utiyama electromagnetism, one finds that the usual definition fails while the Pons–Salisbury–Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR’s external gauge freedom of general relativity share with internal gauge symmetries the 0 Poisson bracket, or is covariance sufficient? A graviton mass breaks the gauge symmetry, but it can be restored by parametrization with clock fields. By requiring equivalent observables, one can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail. (shrink)

It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein's equation (or of the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein's (...) equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). (shrink)

I suggest that classical General Relativity in four spacetime dimensions incorporates a Principal of Maximal Tension and give arguments to show that the value of the maximal tension is $\frac{{c^4 }}{{4G}}$ . The relation of this principle to other, possibly deeper, maximal principles is discussed, in particular the relation to the tension in string theory. In that case it leads to a purely classical relation between G and the classical string coupling constant α′ and the velocity of light (...) c which does not involve Planck's constant. (shrink)

From Einstein's point of view, his General Relativity Theory had strengths as well as failings. For him, its shortcoming mainly was that it did not unify gravitation and electromagnetism and did not provide solutions to field equations which can be interpreted as particle models with discrete mass and charge spectra, As a consequence, General Relativity did not solve the quantum problem, either. Einstein tried to get rid of the shortcomings without losing the achievements of General (...) Relativity Theory. Stimulated by papers of Weyl 465) and Eddington 194), from 1923 onward, he believed that, to reach this goal, one has to transit to space–times which possess more comprehensive geometrical structures than the Riemann space–time. This was the beginning of a decade's lasting search for a unitary field theory. We describe this exciting part of the history of physics, discuss achievements and failures of this development, and ask how these early attempts of a unified theory strike us today. Taking into account the fact that the Equivalence Principle only speaks for a geometrization of gravitation, we consider an alternative way to give those non-Riemannian structures which were introduced by the unitary field approach a physical meaning, namely the meaning of a generalized gravitational field. This is interesting since there are arguments in favor of such a generalization of General Relativity Theory, e.g., the problems the latter theory meets with if one tries to quantize it and to unify gravitation with other interactions. (shrink)

The elucidation of the gauge principle ‘‘is the most pressing problem in current philosophy of physics’’ said Michael Redhead in 2003. This paper argues for two points that contribute to this elucidation in the context of Yang–Mills theories. (1) Yang–Mills theories, including quantum electrodynamics, form a class. They should be interpreted together. To focus on electrodynamics is potentially misleading. (2) The essential role of gauge and BRST symmetries is to provide a local field theory that can be quantized and would (...) be equivalent to the quantization of the non-local reduced theory. If this is correct, the gauge symmetry is significant, not so much because it implies ontological consequences, but because it allows us to quantize theories that we would not be able to quantize otherwise. Thus, in the context of Yang–Mills theories, it is essentially a pragmatic principle. This does not seem to be the case for the gauge symmetry in general relativity. (shrink)

Harvey Brown believes it is crucially important that the "geodesic principle" in general relativity is an immediate consequence of Einstein's equation and, for this reason, has a different status within the theory than other basic principles regarding, for example, the behavior of light rays and clocks, and the speed with which energy can propagate. He takes the geodesic principle to be an essential element of general relativity itself, while the latter are better seen as contingent facts (...) about the particular matter fields we happen to encounter. The situation seems much less clear and clean to me. There certainly is a sense in which the geodesic principle can be recovered as a theorem in general relativity. But one needs more than Einstein's equation to drive the theorems in question. Other assumptions are needed. One needs to put more in if one is to get the geodesic principle out. My goal in this note is to make this claim precise, i.e., that other assumptions are needed. (shrink)

All change involves temporal variation of properties. There is change in the physical world only if genuine physical magnitudes take on different values at different times. I defend the possibility of change in a general relativistic world against two skeptical arguments recently presented by John Earman. Each argument imposes severe restrictions on what may count as a genuine physical magnitude in general relativity. These restrictions seem justified only as long as one ignores the fact that genuine change (...) in a relativistic world is frame-dependent. I argue on the contrary that there are genuine physical magnitudes whose values typically vary with the time of some frame, and that these include most familiar measurable quantities. Frame-dependent temporal variation in these magnitudes nevertheless supervenes on the unchanging values of more basic physical magnitudes in a general relativistic world. Basic magnitudes include those that realize an observer's occupation of a frame. Change is a significant and observable feature of a general relativistic world only because our situation in such a world naturally picks out a relevant class of frames, even if we lack the descriptive resources to say how they are realized by the values of basic underlying physical magnitudes. (shrink)

An extended scale relativity theory, actively developed by one of the authors, incorporates Nottale's scale relativity principle where the Planck scale is the minimum impassible invariant scale in Nature, and the use of polyvector-valued coordinates in C-spaces (Clifford manifolds) where all lengths, areas, volumes⋅ are treated on equal footing. We study the generalization of the ordinary point-particle quantum mechanical oscillator to the p-loop (a closed p-brane) case in C-spaces. Its solution exhibits some novel features: an emergence of two (...) explicit scales delineating the asymptotic regimes (Planck scale region and a smooth region of a quantum point oscillator). In the most interesting Planck scale regime, the solution recovers in an elementary fashion some basic relations of string theory (including string tension quantization and string uncertainty relation). It is shown that the degeneracy of the first collective excited state of the p-loop oscillator yields not only the well-known Bekenstein–Hawking area-entropy linear relation but also the logarithmic corrections therein. In addition we obtain for any number of dimensions the Hawking temperature, the Schwarschild radius, and the inequalities governing the area of a black hole formed in a fusion of two black holes. One of the interesting results is a demonstration that the evaporation of a black hole is limited by the upper bound on its temperature, the Planck temperature. (shrink)

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate (...) mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators. Our "Principle of general tovariance" states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under so-called geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle's principle of the excluded third nor Zermelo's Axiom of Choice may be invoked. Subsequently, our "Equivalence principle" states that any algebra of observables (initially defined in the topos Sets) is empirically equivalent to a commutative one in some other topos. (shrink)

We investigate the novel problem of what happens in special relativity and in relativistic field theories whenthree-dimensional space is quantized. First we examine the equation for elastic waves on a linear chain, the simplest example of a quantized medium, and propose, on its analogy, a nonlinearp-k relationp=ħk(sinhkl)/kl for light and material waves. Here,kl is a new variable which represents the space-quantization effect on the plane wave of wave numberk=|k|. (Note thatkl=0 givesp=ħk.) This relation makes the light velocity in (...) vacuum dependent onkl. We postulate, however, that the phase and group velocities of each individual light wave are still invariant, and try to generalize special relativity to the case ofkl ≠ 0. This can be simply done if the invariance ofkl is assumed. Our results suggest that “localization” might be a relative concept. One interesting consequence of our proposal is that relativistic field theories become automatically finite. This comes out without violating unitarity or causality. A precise measurement of velocities of high-energy photons or massive particles is desirable for checking our proposal. (shrink)

The article investigates the relations between Hausdorff and non-Hausdorff manifolds as objects of general relativity. We show that every non-Hausdorff manifold can be seen as a result of gluing together some Hausdorff manifolds. In the light of this result, we investigate a modal interpretation of a non-Hausdorff differential manifold, according to which it represents a bundle of alternative space-times, all of which are compatible with a given initial data set.