Geometric Theory of Generalized Functions with Applications to General Relativity

Grosser, Michael; Kunzinger, Michael; Oberguggenberger, Michael; Steinbauer, Roland

doi:10.1007/978-94-015-9845-3

Cited by 298 publications

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“…(ii) Case of E L1 τ (R).-The proof is an improvement of the proof of Theorems 1.2.28 of [11], based on the ideas developed for the case of E L1 S (R) above.…”

Section: Fundamental Lemmamentioning

confidence: 99%

“…The results about embeddings concerning G τ R d can be summarized (Theorems 1.2.27 and 1.2.28 of [11]) in the following proposition:…”

Section: Temperate Generalized Functionsmentioning

confidence: 99%

“…During the three last decades, theories of nonlinear generalized functions have been developed by many authors (see [1,11,14,15],...), mainly based on the ideas of J.-F. Colombeau [3,4], which we are going to follow in the sequel. Those theories appear to be a natural continuation of the distributions' one [12,21,22], specially efficient to pose and solve differential or integral problems with irregular data.…”

Section: Introductionmentioning

confidence: 99%

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Kernel theorems in spaces of tempered generalized functions

Delcroix¹

2007

Math. Proc. Camb. Phil. Soc.

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In analogy to the classical isomorphism between L (S (R n ) , S ′ (R m )) and S ′ R n+m , we show that a large class of moderate linear mappings acting between the space GS (R n ) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n ) of temperate ones admits generalized integral representations, with kernels belonging to Gτ R n+m . Furthermore, this result contains the classical one in the sense of the generalized distribution equality. (2000) Mathematics Subject Classification IntroductionDuring the three last decades, theories of nonlinear generalized functions have been developed by many authors (see [1,11,14,15],...), mainly based on the ideas of J.-F. Colombeau [3,4], which we are going to follow in the sequel. Those theories appear to be a natural continuation of the distributions' one [12,21,22], specially efficient to pose and solve differential or integral problems with irregular data.In this paper, we continue the investigations in the field of generalized integral operators initiated by [18] (recently republished in [19,20]) and carried on by [2,5,9,10,23]. Let us recall that those operators generalize, in the Colombeau framework, the operators with distributional kernels in the space of Schwartz distributions [2]. More specifically, in [5], we proved that any moderate net of linear maps (, that is satisfying some growth properties with respect to the parameter ε, gives rise to a linear map L :and G C (R n ) denote respectively the space of generalized functions and the space of compactly supported ones.) The main result is that L can be represented as a generalized integral operator in the spirit of Schwartz Kernel Theorem.Going further in this direction, we study here the generalization of the classical isomorphism between S ′ (R n+m ) and the space of continuous linear mappings acting between S (R n ) and S ′ (R m ) [22]. Thus, the spaces of generalized functions considered in this paper are the space G S (R n ) of rapidly decreasing generalized functions [7,10,17] and the space G τ (R n ) of tempered generalized functions [4,11,18]: G S (R n ) plays here, roughly speaking, the role of S (R n ) (resp. G C (R n )) in the classical case (resp. in [5]), whereas G τ (R n ) plays the role of S ′ (R m ) (resp. G (R m )) in the classical case (resp. in [5]). 1The main results are the following. First, any kernel H ∈ G τ (R m+n ) gives rise to a new type of linear generalized integral operator acting between G S (R n ) and G τ (R m ) and defined bywhere (H ε ) ε (resp. (f ε ) ε ) is any representative of H (resp. f ) and [·] τ , the class of an element in G τ (R m ). Moreover, the linear map H → H from G τ (R m+n ) to the space of linear maps Proposition 14). This gives the first part of the expected result. Then, new regular subspaces of G S (R n ) and G τ (R m ) are introduced in the spirit of [6]. They are used to define the moderate nets of linear maps, which can be extended to act between G S (R n ) and G τ (R m ) (Proposition 16). Finally, our main result states that ...

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“…(ii) Case of E L1 τ (R).-The proof is an improvement of the proof of Theorems 1.2.28 of [11], based on the ideas developed for the case of E L1 S (R) above.…”

Section: Fundamental Lemmamentioning

confidence: 99%

“…The results about embeddings concerning G τ R d can be summarized (Theorems 1.2.27 and 1.2.28 of [11]) in the following proposition:…”

Section: Temperate Generalized Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kernel theorems in spaces of tempered generalized functions

Delcroix¹

2007

Math. Proc. Camb. Phil. Soc.

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“…See also (Grosser et al, 2001(Grosser et al, , 2002(Grosser et al, , 2008 and references therein for further work developing generally covariant algebras of generalized tensors.…”

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Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

Tamir

2012

Studies in History and Philosophy of Science Part B: Studies in

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In his initial formulation of the general theory of relativity, Einstein's proposal that freely falling gravitating massive bodies follow geodesic paths was submitted as an independent fundamental principle. By adopting this geodesic principle to supply the theory's law of motion, Einstein was immediately able to recover both the free-fall motion of bodies in non-relativistic regimes and the previously anomalous precession of the perihelion of Mercury. Over the last century numerous ostensible proofs claiming to have derived the geodesic principle from Einstein's eld equations have been developed. As a result physicists and philosophers of science alike frequently herald Einstein's theory for having the unique distinction of being able to derive its dynamical law of motion from its own eld equations.In this paper I critically survey the multiple attempts to derive the geodesic principle in the context of Einstein's theory. Grouping these results into three major families, which I refer to as (1) limit operation proofs, (2) 0 th -order proofs, and (3) singularity proofs, I argue that none of these strategies successfully demonstrates the geodesic principle, canonically interpreted as a dynamical law that massive bodies must actually follow geodesic paths in Einstein's theory.Speci cally, I argue for the following three claims: First, limit operation proofs fail to demonstrate that massive bodies are ever guaranteed to follow geodesic paths. Second, on the contrary 0 thorder proofs demonstrate that extended massive bodies generically deviate from uniformly geodesic paths. Moreover, the only potentially extended distributions of matter and energy that fail to avoid a uniform geodesic evolution are highly unstable, deviating from such motion under arbitrary perturbations of their angular momentum (or higher order moments). Third, thanks to certain mathematical theorems concerning distribution theory, alternative representations of massive bodies as unextended point particles must result either in precluding the possibility of coupling the particle to the spacetime metric in a way that is coherent with Einstein's eld equations or in having to excise the particle (and its would-be path) from spacetime entirely. This three pronged argument reveals that not only does the geodesic law of motion fail to be a deductive consequence of the eld equations, but also any attempt to canonically interpret the geodesic principle in such a way requires * In the following, M will be taken to be a smooth, orientable, four-dimensional manifold, and (M, g ab ) will be referred to as a Lorentzian spacetime if g ab is a smooth metric of signature (+, −, −, −) de ned on M. Excepting quoted material all further notational conventions follow that of (Wald, 1984).1 Forthcoming: Studies in History and Philosophy of Modern Physics that either the gravitating body is not massive, its existence violates Einstein's eld equations, or it does not exist within the spacetime manifold at all (let alone along a geodesic).Having rejected the canonical in...

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“…Introduction. Over the past thirty years, nonlinear theories of generalized functions have been developed by many authors [1,15,19,20] mainly inspired by the work of J. F. Colombeau [2]. They have proved to be a valuable tool for treating partial differential equations with singular data or coefficients [3,12,17,20].…”

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confidence: 99%

G- and G^∞-hypoellipticity of partial differential operators with constant Colombeau coefficients

Garetto

2010

Linear and Non-Linear Theory of Generalized Functions and Its Applications

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Geometric Theory of Generalized Functions with Applications to General Relativity

Cited by 298 publications

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Kernel theorems in spaces of tempered generalized functions

Kernel theorems in spaces of tempered generalized functions

Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

G- and G^∞-hypoellipticity of partial differential operators with constant Colombeau coefficients

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Geometric Theory of Generalized Functions with Applications to General Relativity

Cited by 298 publications

References 0 publications

Kernel theorems in spaces of tempered generalized functions

Kernel theorems in spaces of tempered generalized functions

Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

G- and G∞-hypoellipticity of partial differential operators with constant Colombeau coefficients

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G- and G^∞-hypoellipticity of partial differential operators with constant Colombeau coefficients