Studying smooth families of certain subspaces of a Banach space X entails a construction of a Gra... more Studying smooth families of certain subspaces of a Banach space X entails a construction of a Grassmann manifold defined over the similarity class of a projection in a Banach space. Standard principles of fiber bundle theory can be adapted to describe these families in terms' of smooth maps from a possibly infinite dimensional paracompact manifold to the Grassmannian.
Commencing from a monoidal semigroup š“, we consider the geometry of the space š(š“) of pseudoregul... more Commencing from a monoidal semigroup š“, we consider the geometry of the space š(š“) of pseudoregular elements. When š“ is a Banachable algebra we show that there exist certain subspaces of š(š“) that can be realized as submanifolds of š“. The space š(š“) contains certain subspaces constituting the Stiefel manifolds of framings for š“. We establish several embedding results for such subspaces, where the relevant maps induce embeddings of associated Grassmann manifolds.
In this essay, we introduce a new approach to energyāmomentum in general relativity. Spaceātime, ... more In this essay, we introduce a new approach to energyāmomentum in general relativity. Spaceātime, as opposed to space, is recognized as the necessary arena for its examination, leading us to define new extended spaceātime energy and momentum constructs. From local and global considerations, we conclude that the Ricci tensor is the required element for a localized expression of energyāmomentum to include the gravitational field. We present and rationalize a fully invariant extended expression for spaceātime energy, guided by Tolman's well-known energy integral for an arbitrary bounded stationary system. This raises fundamental issues which we discuss. The role of the observer emerges naturally and we are led to an extension of the uncertainty principle to general relativity, of particular relevance to ultra-strong gravity.
Studying smooth families of certain subspaces of a Banach space X entails a construction of a Gra... more Studying smooth families of certain subspaces of a Banach space X entails a construction of a Grassmann manifold defined over the similarity class of a projection in a Banach space. Standard principles of fiber bundle theory can be adapted to describe these families in terms of smooth maps from a possibly infinite-dimensional paracompact manifold to the Grassmannian.
We give a fully covariant energy momentum stress tensor for the gravitational field which is easi... more We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically and intuitively motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the source matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter fields must be divergence free, when spacetime is 4 dimensional. Moreover, if the total source matter fields are assumed to be divergence free, then either the spacetime is of dimension 4 or the spacetime has constant scalar curvature. Mathematics Subject Classification (2000) : 83C05, 83C40, 83C99.
We discuss the structure of local gravity theories as resulting from the idea that locally gravit... more We discuss the structure of local gravity theories as resulting from the idea that locally gravity must be physically characterized by tidal acceleration, and show how this relates to both Newtonian gravity and Einstein's general relativity.
We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in gen... more We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in general relativity are uniquely determined by their monomial functions on the light cone. Thus, for an observer to observe a tensor at an event in general relativity is to contract with the velocity vector of the observer, repeatedly to the rank of the tensor. Thus two symmetric tensors observed to be equal by all observers at a specific event are necessarily equal at that event.
We give a fully covariant energy momentum stress tensor for the gravitational field which is easi... more We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice
We give a derivation of the Einstein equation for gravity which employs a definition of the local... more We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivation for this choice using light
We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibilit... more We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns giving them plausible values. Our proof is enormously simpler than others that have recently appeared in the literature, yet completely rigorous. For example, we do
Most early twentieth century relativists ā Lorentz, Einstein, Eddington, for examples ā claimed t... more Most early twentieth century relativists ā Lorentz, Einstein, Eddington, for examples ā claimed that general relativity was merely a theory of the Ʀther. We shall confirm this claim by deriving the Einstein equations using Ʀther theory. We shall use a combination of Lorentz's and Kelvin's conception of the Ʀther. Our derivation of the Einstein equations will not use the vanishing of the covariant divergence of the stressāenergy tensor, but instead equate the Ricci tensor to the sum of the usual stressāenergy tensor and a stressāenergy tensor for the Ʀther, a tensor based on Kelvin's Ʀther theory. A crucial first step is generalizing the Cartan formalism of Newtonian gravity to allow spatial curvature, as conjectured by Gauss and Riemann. In essence, we shall show that the Einstein equations are a special case of Newtonian gravity coupled to a particular type of luminiferous Ʀther. Our derivation of general relativity is simple, and it emphasizes how inevitable general re...
In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras ... more In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algebras that extends results of Brown, Douglas, and Fillmore [BDF], Fillmore [F], and Pimsner, Popa, and Voiculescu [PPV]. Since the category of Hilbert C (X) -modules is equivalent to the category of Hilbert bundles over X [DD;DG], many questions of topological interest can be recast in terms of Hilbert C(X)-modules which then give rise to questions about general Hilbert modules. In particular, Kasparovās stability theorem [K] (which plays an essential part in the proof that inverses exist in the general theory of EXT) is the noncommutative extension of a triviality theorem of Dixmier and Douady [DD, Th.4] (which itself provides the existence of classifying maps for arbitrary separable Hilbert bundles over paracompact spaces).
By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theore... more By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausible values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory.
Infinite dimensional fiber spaces arise naturally in the theory of representations of C*-algebras... more Infinite dimensional fiber spaces arise naturally in the theory of representations of C*-algebras. Often there are cases where one has to deal with more general notions of dierentiability. In order to create a unified framework, we introduce the notion of a D -space and a D -group action within a given category D. Then we present a self-contained account of
It is shown that a form of the CauchyāLagrange formula for the evolution of vorticity in a barotr... more It is shown that a form of the CauchyāLagrange formula for the evolution of vorticity in a barotropic flow generalizes to the case of ideal fluid motion on higher-dimensional Riemannian or semi-Riemannian manifolds.
Studying smooth families of certain subspaces of a Banach space X entails a construction of a Gra... more Studying smooth families of certain subspaces of a Banach space X entails a construction of a Grassmann manifold defined over the similarity class of a projection in a Banach space. Standard principles of fiber bundle theory can be adapted to describe these families in terms' of smooth maps from a possibly infinite dimensional paracompact manifold to the Grassmannian.
Commencing from a monoidal semigroup š“, we consider the geometry of the space š(š“) of pseudoregul... more Commencing from a monoidal semigroup š“, we consider the geometry of the space š(š“) of pseudoregular elements. When š“ is a Banachable algebra we show that there exist certain subspaces of š(š“) that can be realized as submanifolds of š“. The space š(š“) contains certain subspaces constituting the Stiefel manifolds of framings for š“. We establish several embedding results for such subspaces, where the relevant maps induce embeddings of associated Grassmann manifolds.
In this essay, we introduce a new approach to energyāmomentum in general relativity. Spaceātime, ... more In this essay, we introduce a new approach to energyāmomentum in general relativity. Spaceātime, as opposed to space, is recognized as the necessary arena for its examination, leading us to define new extended spaceātime energy and momentum constructs. From local and global considerations, we conclude that the Ricci tensor is the required element for a localized expression of energyāmomentum to include the gravitational field. We present and rationalize a fully invariant extended expression for spaceātime energy, guided by Tolman's well-known energy integral for an arbitrary bounded stationary system. This raises fundamental issues which we discuss. The role of the observer emerges naturally and we are led to an extension of the uncertainty principle to general relativity, of particular relevance to ultra-strong gravity.
Studying smooth families of certain subspaces of a Banach space X entails a construction of a Gra... more Studying smooth families of certain subspaces of a Banach space X entails a construction of a Grassmann manifold defined over the similarity class of a projection in a Banach space. Standard principles of fiber bundle theory can be adapted to describe these families in terms of smooth maps from a possibly infinite-dimensional paracompact manifold to the Grassmannian.
We give a fully covariant energy momentum stress tensor for the gravitational field which is easi... more We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically and intuitively motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the source matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter fields must be divergence free, when spacetime is 4 dimensional. Moreover, if the total source matter fields are assumed to be divergence free, then either the spacetime is of dimension 4 or the spacetime has constant scalar curvature. Mathematics Subject Classification (2000) : 83C05, 83C40, 83C99.
We discuss the structure of local gravity theories as resulting from the idea that locally gravit... more We discuss the structure of local gravity theories as resulting from the idea that locally gravity must be physically characterized by tidal acceleration, and show how this relates to both Newtonian gravity and Einstein's general relativity.
We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in gen... more We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in general relativity are uniquely determined by their monomial functions on the light cone. Thus, for an observer to observe a tensor at an event in general relativity is to contract with the velocity vector of the observer, repeatedly to the rank of the tensor. Thus two symmetric tensors observed to be equal by all observers at a specific event are necessarily equal at that event.
We give a fully covariant energy momentum stress tensor for the gravitational field which is easi... more We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice
We give a derivation of the Einstein equation for gravity which employs a definition of the local... more We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivation for this choice using light
We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibilit... more We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns giving them plausible values. Our proof is enormously simpler than others that have recently appeared in the literature, yet completely rigorous. For example, we do
Most early twentieth century relativists ā Lorentz, Einstein, Eddington, for examples ā claimed t... more Most early twentieth century relativists ā Lorentz, Einstein, Eddington, for examples ā claimed that general relativity was merely a theory of the Ʀther. We shall confirm this claim by deriving the Einstein equations using Ʀther theory. We shall use a combination of Lorentz's and Kelvin's conception of the Ʀther. Our derivation of the Einstein equations will not use the vanishing of the covariant divergence of the stressāenergy tensor, but instead equate the Ricci tensor to the sum of the usual stressāenergy tensor and a stressāenergy tensor for the Ʀther, a tensor based on Kelvin's Ʀther theory. A crucial first step is generalizing the Cartan formalism of Newtonian gravity to allow spatial curvature, as conjectured by Gauss and Riemann. In essence, we shall show that the Einstein equations are a special case of Newtonian gravity coupled to a particular type of luminiferous Ʀther. Our derivation of general relativity is simple, and it emphasizes how inevitable general re...
In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras ... more In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algebras that extends results of Brown, Douglas, and Fillmore [BDF], Fillmore [F], and Pimsner, Popa, and Voiculescu [PPV]. Since the category of Hilbert C (X) -modules is equivalent to the category of Hilbert bundles over X [DD;DG], many questions of topological interest can be recast in terms of Hilbert C(X)-modules which then give rise to questions about general Hilbert modules. In particular, Kasparovās stability theorem [K] (which plays an essential part in the proof that inverses exist in the general theory of EXT) is the noncommutative extension of a triviality theorem of Dixmier and Douady [DD, Th.4] (which itself provides the existence of classifying maps for arbitrary separable Hilbert bundles over paracompact spaces).
By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theore... more By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausible values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory.
Infinite dimensional fiber spaces arise naturally in the theory of representations of C*-algebras... more Infinite dimensional fiber spaces arise naturally in the theory of representations of C*-algebras. Often there are cases where one has to deal with more general notions of dierentiability. In order to create a unified framework, we introduce the notion of a D -space and a D -group action within a given category D. Then we present a self-contained account of
It is shown that a form of the CauchyāLagrange formula for the evolution of vorticity in a barotr... more It is shown that a form of the CauchyāLagrange formula for the evolution of vorticity in a barotropic flow generalizes to the case of ideal fluid motion on higher-dimensional Riemannian or semi-Riemannian manifolds.
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Papers by MAURICE DUPRE