Wesley C. Salmon (1977) has written a characteristically elegant and ingenious paper 'The Curvature of Physical Space'. He argues in it that the curvature of a space cannot be intrinsic to it. Salmon relates his view that space is affinely amorphous to Grunbaum's view (Grunbaum 1973, esp. Ch. 16 & 22) that it is metrically amorphous and acknowledges parallels between the arguments which have been offered for each opinion. I wish to dispute these conclusions on philosophical grounds quite (...) as much as on geometrical ones. Although I concentrate most on arguing for a well defined, intrinsic affinity for physical space the arguments extend easily to support a well defined, intrinsic metric. (shrink)

The Bianchi Identities relating asymmetric curvature to torsion are obtained as a new set of equations governing second-order curvature tensors. The usual contribution of symmetric curvature to the gravitational field is found to be a subset of these identities though with an added contribution due to torsion gradients. The antisymmetric curvature two-tensor is shown to be related to the divergence of the torsion. Using a model of particle-antiparticle pair production, identification of certain torsion components with electroweak (...) fields is proposed. These components obey equations, similar to Maxwell's, that are subsets of these linear Bianchi identities. These results are shown to be consistent with gauge and other previous analyses. (shrink)

The renormalization group is used to analyze the behavior of certain gravitationally significant renormalized coupling constants under a scaling of the spacetime curvature. After discussing a simple example, the results are summarized for a class of grand unified theories.

The internationally renowned physicist Harald Fritzsch deftly explains the meaning and far-flung implications of the general theory of relativity and other mysteries of modern physics by presenting an imaginary conversation among Newton, Einstein, and a fictitious contemporary particle physicist named Adrian Haller--the same device Fritzsch employed to great acclaim in his earlier book An Equation That Changed the World, which focused on the special theory of relativity. Einstein's theory of gravitation, his general theory of relativity, touches on basic questions of (...) our existence. Matter, according to Einstein, has no existence independent of space and time. It is even capable of bending the structure of space and changing the course of time--it introduces a "curvature." Gravity emerges not as an actual physical force but as a consequence of space-time geometry. Even the apple that drops from the tree follows the curvature of time and space. In this entertaining and involving account of relativity, Newton serves as the skeptic and asks the questions a modern reader might ask. Einstein himself does the explaining, while Haller explains the new developments that have occurred since the general theory was proposed. The result is an intellectual roller-coaster ride in which concepts that have entered the vernacular become clear for the first time: the Big Bang, "black holes," elementary particles, and much more. (shrink)

The physical, mathematical, and philosophical foundations of the quantum theory of free Bose fields in fixed general relativistic spacetimes are examined. It is argued that the theory is logically and mathematically consistent whereas semiclassical prescriptions for incorporating the back-reaction of the quantum field on the geometry lead to inconsistencies. Still, the relations and heuristic value of the semiclassical approach to canonical and covariant schemes of quantum gravity-plus-matter are assessed. Both conventional and rigorous formulations of the theory and of its principal (...) predictions, cosmological particle creation and horizon radiation, are expounded and compared. Special attention is devoted to spacetime properties needed for the existence or uniqueness of the relevant theoretical elements , renormalization of the stress tensor). The emergence of unitarily inequivalent representations in a single dynamical context is used as motivation for the introduction of the abstract $\rm C\sp{\*}$-algebraic axiomatic formalism. The operationalist and conventionalist claims of the original abstract algebraic program are criticized in favor of its tempered outgrowth, local quantum physics. The interpretation of the theory as a wave mechanics of classical field configurations, deriving from the Schrodinger representations of the abstract algebra, is discussed and is found superior, at least on the level of analogy, to particle or harmonic oscillator interpretations. Further, it is argued that the various detector results and the Fulling nonuniqueness problem do not undermine the particle concept in the ways commonly claimed. In particular, arguments are offered against the attribution of particle status to the Rindler quanta, against the physical realizability of the Rindler vacuum, and against the more general notion of observer-dependence as to the definition of 'particle' or 'vacuum'. However, the question of the ontological status of particles is raised in terms of the consistency of quantum field theory with non-reductive realism about particles, the latter being conceived as entities exhibiting attributes of discreteness and localizability. Two arguments against non-reductive realism about particles, one from axiomatic algebraic local quantum theory in Minkowski spacetime and one from quantum field theory in curved spacetime, are developed. (shrink)

A naive introduction of a dependency of the mass of a black hole on the Schwarzschild time coordinate results in singular behavior of curvature invariants at the horizon, violating expectations from complementarity. If instead a temporal dependence is introduced in terms of a coordinate akin to the river time representation, the Ricci scalar is nowhere singular away from the origin. It is found that for a shrinking mass scale due to evaporation, the null radial geodesics that generate the horizon (...) are slightly displaced from the coordinate singularity. In addition, a changing horizon scale significantly alters the form of the coordinate singularity in diagonal (orthogonal) metric coordinates representing the space-time. A Penrose diagram describing the growth and evaporation of an example black hole is constructed to examine the evolution of the coordinate singularity. (shrink)

This article contributes to the general literature on the relationship between corporate social performance and corporate financial performance, as well as to the emerging HRM–CSR literature, by exploring the curvature of the relationship between HRM–CSP and CFP. We advance conceptual arguments in favor of an inverted U-shaped relationship. Our results demonstrate a significant quadratic relationship between HRM–CSP and CFP. We provide evidence that this relationship is not linear or S-shaped but rather inverted U-shaped.

This paper has a twofold objective. First, it engages with the interrelation of time, space, and matter in Kant, Heidegger, and Derrida and questions whether and how this interrelation effects the possibility of self-relation. In Kant and the Problem of Metaphysics Heidegger suggests that the very structure of subjectivity is constituted by what he calls the ‘pure self-affection’ of time and thus the possibility of self-relation is intimately bound up with the temporalizing of time. In his 1964–65 seminar, Heidegger: the (...) Question of Being and History, Derrida translates this pure affection of time into the more generic term ‘auto-affection,’ which will remain a pivotal reference point for his deconstruction of the metaphysical privileging of time as presence. Derrida shows how the (im)possibility of auto-affection is bound up not only with time but also with space, or rather with the ‘spacing of time’ that he also refers to as ‘the trace.’ Second, the paper moves across the frontiers of philosophy and physics posing anew the question concerning the interrelations of temporality, spatiality, and materiality. With reference to what in general relativity is called ‘the curvature of spacetime,’ the efficacy of materiality in the movement of auto-affection is called into question. (shrink)

James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, I (...) generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free and so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object : a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. (shrink)

James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, I (...) generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free and so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object : a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. (shrink)

James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, I (...) generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free and so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object (as Robert Geroch has observed recently): a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:On Dante, Hyperspheres, and the Curvature of the Medieval CosmosWilliam EggintonIn the course of his lectures on medieval literature at Oxford University in the 1950s C. S. Lewis would ask students to walk alone at night, gaze at the star-filled sky, and try to imagine how it might look to a walker in the Middle Ages. It would not likely have occurred to him that some forty years (...) later several astrophysicists would ask that their readers perform the same activity, at least figuratively—not, however, with the intention of experiencing medieval conceptions of the cosmos but rather in order to begin to imagine the shape of our own. One of the astrophysicists in question is Stanford Professor Emeritus Robert Osserman, whose 1995 book The Poetry of the Universe is an attempt to acquaint lay readers with what the most recent research suggests is the shape of the universe by leading them through a brief history of the mathematical and technological developments that made such boundary-breaking thought possible. Paradoxically, when it comes to trying to imagine that all-encompassing form which is derived from a four-dimensional mathematical model called a hypersphere, he turns not to ultra-modern computer-generated images nor to dry but technically accurate exegeses of mathematical formulae but rather to the verses of a fourteenth-century Italian poet, Dante Alighieri. His claim is that in the third canticle of the Divine Comedy, which details his ascent through the heavens, Dante describes the cosmos with astonishing precision as a hypersphere, exactly as modern physics has determined it to be.If the cosmology of the present makes use of Dante heuristically, however, it never asks the most obvious question: Why? Why does one need to go back to the fourteenth century for a poetic description of how the universe appears to modern science? Osserman assumes it to be a mere coincidence, and why should he not? What possible access would a thinker—a poet even—of the European Middle Ages have to the highest achievements of almost six centuries of subsequent thought? In answering this question, I am not proposing to naively interpret the past in our own image, claiming Dante to be a mathematical genius [End Page 195] before his time. I am, however, suggesting that there is an excellent explanation for why one must look to the fourteenth century for models of expression adequate to the challenges posed by nineteenth and twentieth-century advances in mathematics: namely, that in the course of the Scientific Revolution, wherein were laid the epistemological foundations for the discoveries of modern science, certain possibilities of thought and imagination were discarded, forgotten, and certain abilities were, if only temporarily, lost. The Middle Ages has long been painted as a period whose culture was hostile to free thought and new knowledge; perhaps it is now time to recognize that it was also a time in which phenomena could sustain mutually exclusive and contradictory explanations, when the battle between faith and science had not yet been conclusively won by either side, and when the efforts to bridge the stories they told could produce conceptual edifices whose mere possibility would be unthinkable only 200 years later. This essay is about one such conceptual edifice and how openness to its perspective allowed a poet to put into words what could not be imagined by his descendants, and what eludes us even today.The Shape of the Modern UniverseImagine that you are a completely flat creature, living happily in your two-dimensional world.1 As observers we are outside this world, and we can see you milling about with your flat friends, blissfully unaware of the uncharted regions one dimension away on either side of you. Now imagine that this plane is not flat—in the sense of rectilineal—as you might have created it in your mind, but rather is curved; in fact it is a sphere. Assuming that this sphere is large enough, there is no reason that you, an inhabitant of this surface, should be aware of its curvature. Even we, the three-dimensional observers, if close enough to the surface, would not recognize its curvature. One could conjecture, however... (shrink)

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature K. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann–Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and K. For hyperbolic spaces, we also obtain a global lower (...) bound \, which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by \, where \ is the Planck length. (shrink)

In the absence of new physics around \ GeV, the electroweak vacuum is at best metastable. This represents a major challenge for high scale inflationary models as, during the early rapid expansion of the universe, it seems difficult to understand how the Higgs vacuum would not decay to the true lower vacuum of the theory with catastrophic consequences if inflation took place at a scale above \ GeV. In this paper we show that the non-minimal coupling of the Higgs boson (...) to curvature could solve this problem by generating a direct coupling of the Higgs boson to the inflationary potential thereby stabilizing the electroweak vacuum. For specific values of the Higgs field initial condition and of its non-minimal coupling, inflation can drive the Higgs field to the electroweak vacuum quickly during inflation. (shrink)

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz (...) potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

We consider a quadratic Lagrangian, in both curvature and torsion, with the aim of exploring the possibility that torsion and curvature behave as conjugate variables satisfying the commutation relations. For that proposal we first show that torsion represents a quantum correction to the classical equations of motion. We then observe that we have to introduce the spin in the Einstein theory with two spaces: a real space-time where we describe the curvature with tensors; and a complex space-time, (...) where we describe torsion with spinors. This is not satisfactory, and to overcome this contradiction we describe both mass and spin in a unique manifold, the real space-time, with the use of geometric algebra and geometric product. In that manner we are able to introduce a torsion trivector and curvature trivector that satisfy the commutation relations, opening the road for a quantum description of gravity theory. (shrink)