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Hadamard Regularization
General Relativity and Quantum Cosmology · Submitted 3 Apr 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative.
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The abstract, sentence by sentence
Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. | background 0.97 | 0.29 | 0.03 | asserted 0.87 |
| s1 | We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. | contribution 0.48 | 0.26 | 0.04 | asserted 0.91 |
| s2 | We present and investigate different expressions, useful in applications, for the latter partie finie. | contribution 0.99 | 0.89 | 0.08 | asserted 0.99 |
| s3 | To each singular function, we associate a partie-finie (Pf) pseudo-function. | contribution 0.78 | 0.85 | 0.08 | asserted 1.00 |
| s4 | The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. | contribution 0.56 | 0.60 | 0.05 | asserted 1.00 |
| s5 | We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. | contribution 0.95 | 0.92 | 0.10 | asserted 1.00 |
| s6 | We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. | contribution 0.99 | 0.93 | 0.10 | asserted 1.00 |
| s7 | This operator is uniquely defined up to an arbitrary numerical constant. | result 0.93 | 0.76 | 0.07 | asserted 1.00 |
| s8 | Time derivatives and partial derivatives with respect to the singular points are also investigated. | contribution 0.98 | 0.86 | 0.03 | asserted 0.93 |
| s9 | In the course of the paper, all the formulas needed in the application to the physical problem are derived. | contribution 0.64 | 0.93 | 0.07 | asserted 1.00 |
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The paper
Original abstract
Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived.
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How the literature uses it
37 arXiv papers cite this in 70 sentences
Papers citing it · 33 sentences without a confident label
Builds on it 1 paper
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Using the same method as in Ref. [ref] , we obtained the corresponding required expressions for the double derivative:
Next-to-next-to-leading order spin-orbit effects in the equations of motion of compact binary systems · Non-compact support terms · 0.81
Uses its method or tool 24 papers
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This evaluation is thus given a sense through the Hadamard “partie finie” regularization procedure explained in Ref. [ref] , which is of course to be performed after computing the derivatives of the potentials.
Next-to-next-to-leading order spin-orbit effects in the equations of motion of compact binary systems · Post-Newtonian metric and equations of motion · 1.00
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The ill-defined distributional terms arising in the Riemann tensor will be treated by borrowing a technique from gravitational-wave theory and compact binaries dynamics, i.e., the Hadamard partie finie regularization scheme [ref] .
Avoiding singularities in Lorentzian-Euclidean black holes: the role of atemporality · Introduction · 1.00
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To obtain it, one first evaluates the Hadamard partie finie (Pf) of the integral in 3 dimensions [ref] , which depends on two regularization scales $s_1$ and $s_2$, and can be given an explicit form based on a double analytic continuation with two complex parameters $α$, $β∈ℂ$:
Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order in harmonic coordinates · Computation of the multipole moments · 1.00
Compares with it 1 paper
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So the contribution of the scalar field to the energy flux is [equation] A straightforward but tedious calculation shows that Eq. ([ref]) reduces to Eq. (3.24) in Ref. [ref] for Brans-Dicke theory with sensitivities set to zero and the Hadamard regularization imposed [ref] .
Constraints on Horndeski Theory Using the Observations of Nordtvedt Effect, Shapiro Time Delay and Binary Pulsars · Gravitational Radiation for a Compact Binary System · 0.91
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