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Journal Article

The Taylor expansion at past time-like infinity

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons20687

Friedrich,  Helmut
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1306.5626.pdf
(Preprint), 403KB

CMP324_263.pdf
(Any fulltext), 471KB

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Citation

Friedrich, H. (2013). The Taylor expansion at past time-like infinity. Communications in Mathematical Physics, 324, 263-300. doi:10.1007/s00220-013-1803-1.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-B411-F
Abstract
We study the initial value problem for the conformal field equations with data given on a cone ${\cal N}_p$ with vertex $p$ so that in a suitable conformal extension the point $p$ will represent past time-like infinity $i^-$, the set ${\cal N}_p \setminus \{p\}$ will represent past null infinity ${\cal J}^-$, and the freely prescribed (suitably smooth) data will acquire the meaning of the incoming {\it radiation field} for the prospective vacuum space-time. It is shown that: (i) On some coordinate neighbourhood of $p$ there exist smooth fields which satisfy the conformal vacuum field equations and induce the given data at all orders at $p$. The Taylor coefficients of these fields at $p$ are uniquely determined by the free data. (ii) On ${\cal N}_p$ there exists a unique set of fields which induce the given free data and satisfy the transport equations and the inner constraints induced on ${\cal N}_p$ by the conformal field equations. These fields and the fields which are obtained by restricting the functions considered in (i) to ${\cal N}_p$ coincide at all orders at $p$.