RECORD
Global Hadamard form for the Green Function in Schwarzschild space-time
General Relativity and Quantum Cosmology · Submitted 9 Jun 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic. The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood. We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution. (The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{\'n}ski-Hacyan spacetime (the latter approach also yields another representation for the {\it full} Schwarzschild Green function, not just for its non-smooth part). Our representation is not valid in a neighbourhood of caustic points. We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. | background 0.87 | 0.05 | 0.03 | asserted 1.00 |
| s2 | In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood. | contribution 0.98 | 0.97 | 0.76 | asserted 1.00 |
| s3 | We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution. | interpretation 0.96 | 0.93 | 0.37 | asserted 1.00 |
| s4 | (The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{\'n}ski-Hacyan spacetime (the latter approach also yields another representation for the {\it full} Schwarzschild Green function, not just for its non-smooth part). | method 0.97 | 0.89 | 0.12 | asserted 1.00 |
| s5 | Our representation is not valid in a neighbourhood of caustic points. | limitation 1.00 | 0.89 | 0.08 | asserted 1.00 |
| s6 | We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime. | contribution 0.84 | 0.95 | 0.11 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic. The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood. We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution. (The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{\'n}ski-Hacyan spacetime (the latter approach also yields another representation for the {\it full} Schwarzschild Green function, not just for its non-smooth part). Our representation is not valid in a neighbourhood of caustic points. We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime.
Human review
References
How the literature uses it
9 arXiv papers cite this in 30 sentences
Papers citing it · 25 sentences without a confident label
Uses its method or tool 1 paper
-
For this purpose, it is very convenient to follow Ref. [ref] .
Regularized calculation of the retarded Green function in Schwarzschild spacetime · Regularized calculation of the Green function · 0.99
Compares with it 1 paper
-
We use that as a check of our results: we verify that the real part of our calculation of the FGF agrees with existing literature results for the RGF [ref] , for which the fourfold structure is already known [ref] .
Quantum correlator outside a Schwarzschild black hole · Introduction · 1.00
Cites it as support 1 paper
-
When calculating the GF via Eq. [ref], the Dirac-$δ$ divergence at $σ=0$ arises from a divergence in the infinite $ℓ$-mode sum [ref] .
Regularized calculation of the retarded Green function in Schwarzschild spacetime · Green function · 0.98
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.