Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

General Relativity and Quantum Cosmology

arXiv:1804.01941 (gr-qc)
[Submitted on 5 Apr 2018 (v1), last revised 17 Jul 2019 (this version, v2)]

Title:Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole

Authors:Grigorios Fournodavlos, Jan Sbierski
View PDF
Abstract:We study the behaviour of smooth solutions to the wave equation, $\square_g\psi=0$, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards $r=0$ and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface $\{r=0\}$. Our method is based on deriving weighted energy estimates in physical space and requires no symmetries of solutions. However, a key ingredient in our argument uses a precise analysis of the spherically symmetric part of the solution and a monotonicity property of spherically symmetric solutions in the interior.
Comments: Minor improvements, version accepted for publication
Subjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP)
Cite as: arXiv:1804.01941 [gr-qc]
  (or arXiv:1804.01941v2 [gr-qc] for this version)
  https://doi.org/10.48550/arXiv.1804.01941
Related DOI: https://doi.org/10.1007/s00205-019-01434-0

Submission history

From: Grigorios Fournodavlos [view email]
[v1] Thu, 5 Apr 2018 16:31:03 UTC (98 KB)
[v2] Wed, 17 Jul 2019 14:33:41 UTC (102 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
gr-qc
< prev   |   next >
new | recent | 2018-04
Change to browse by:
math
math.AP

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
IArxiv Recommender (What is IArxiv?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)