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On the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes

Tim-Torben Paetz

Abstract

We analyze the appearance of logarithmic terms at the critical sets of Friedrich's cylinder representation of spatial infinity. It is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions. Conversely, it is proved that, under the additional hypothesis that the spacetime has constant (ADM) mass aspect and vanishing dual (ADM) mass aspect, this condition is also necessary for a spacetime to admit a smooth representation at the critical sets.

On the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes

Abstract

We analyze the appearance of logarithmic terms at the critical sets of Friedrich's cylinder representation of spatial infinity. It is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions. Conversely, it is proved that, under the additional hypothesis that the spacetime has constant (ADM) mass aspect and vanishing dual (ADM) mass aspect, this condition is also necessary for a spacetime to admit a smooth representation at the critical sets.

Paper Structure

This paper contains 71 sections, 509 equations.

Table of Contents

  1. Introduction
  2. Asymptotically flat spacetimes
  3. Overview
  4. Notation
  5. General conformal field equations and conformal Gauss gauge
  6. General conformal field equations
  7. Conformal geodesics and conformal Gauss gauge
  8. Definition and properties of conformal geodesics
  9. Conformal Gauss gauge
  10. Construction of the conformal Gauss gauge from null infinity
  11. Gauge freedom associated to $\nabla_{\dot x}\Theta|_{{\mycal I}^-}$ and $\theta^-$
  12. Some crucial relations
  13. Evolution equations for Schouten tensor, connection and frame coefficients
  14. Bianchi equation
  15. Frame field and coordinates at ${\mycal I}^-$ for $\lambda=0$
  16. ...and 56 more sections

Theorems & Definitions (5)

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