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Conformal structures of static vacuum data

Friedrich, H. (2013). Conformal structures of static vacuum data. Communications in Mathematical Physics, 321(2), 419-482. doi:10.1007/s00220-013-1694-1.

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Creators:
Friedrich, Helmut1, Author
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012

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Free keywords: General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG
Abstract: In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$. We provide two criteria: If $h$ is real analytic, $\kappa$ is closed, and one of it integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.

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Dates: 2012-03-272012-09-2520132013
Publication Status: Published in print
Pages: 68 pages, typos corrected, references and details added
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Rev. Method: -
Identifiers: arXiv: 1203.6125
DOI: 10.1007/s00220-013-1694-1
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Title: Communications in Mathematical Physics
Source Genre: Journal
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Publ. Info: Heidelberg : Springer-Verlag Heidelberg
Pages: - Volume / Issue: 321 (2) Sequence Number: - Start / End Page: 419 - 482 Identifier: ISSN: 0010-3616
CoNE: /journals/resource/954925392313