Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new class of obstructions to the smoothness of null infinity

J. A. Valiente-Kroon

TL;DR

The paper analyzes the smoothness of null infinity for asymptotically Euclidean, time-symmetric, conformally flat initial data by evolving the conformal field equations on a cylinder at spatial infinity. Using a detailed transport-equation framework and computer algebra, it demonstrates that obstructions to smoothness arise only at high order and are governed by Newman-Penrose constants $G^{(5)}_k$ and higher $G^{(6)}_k$, which are determined by the initial data. The main result shows that even with the regularity condition, smooth null infinity requires vanishing of these constants, and in fact implies Schwarzschild-like behaviour near spatial infinity to some order; a precise statement links the order of smoothness to how Schwarzschildean the data is. These findings suggest a deep connection between radiation content near spatial infinity and the global conformal structure, indicating that smooth null infinity likely cannot coexist with radiative data near $i^0$ and motivating extensions beyond conformally flat or time-symmetric settings.

Abstract

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if a time symmetric initial data which is conformally flat in a neighbourhood of spatial infinity yields a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.

A new class of obstructions to the smoothness of null infinity

TL;DR

The paper analyzes the smoothness of null infinity for asymptotically Euclidean, time-symmetric, conformally flat initial data by evolving the conformal field equations on a cylinder at spatial infinity. Using a detailed transport-equation framework and computer algebra, it demonstrates that obstructions to smoothness arise only at high order and are governed by Newman-Penrose constants and higher , which are determined by the initial data. The main result shows that even with the regularity condition, smooth null infinity requires vanishing of these constants, and in fact implies Schwarzschild-like behaviour near spatial infinity to some order; a precise statement links the order of smoothness to how Schwarzschildean the data is. These findings suggest a deep connection between radiation content near spatial infinity and the global conformal structure, indicating that smooth null infinity likely cannot coexist with radiative data near and motivating extensions beyond conformally flat or time-symmetric settings.

Abstract

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if a time symmetric initial data which is conformally flat in a neighbourhood of spatial infinity yields a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.

Paper Structure

This paper contains 13 sections, 7 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spacetime in a neighbourhood of spatial infinity
  3. Space 2-spinors
  4. Expansions of functions on $C_{a,\kappa}$
  5. The conformal evolution equations
  6. The initial data
  7. The transport equations
  8. A first analysis of the transport equations
  9. Solving the transport equations given time symmetric, conformally flat initial data
  10. The orders $p=0,\ldots,4$.
  11. The first obstructions to smoothness: orders $p=5,6$.
  12. Obstructions to smoothness and the Schwarzschild initial data.
  13. Conclusions and extensions

Key Result

Theorem 1

Necessary conditions for the development of initial data which are time symmetric, conformally flat in a neighbourhood $B_a(i)$ of (spatial) infinity to be smooth at the intersection of null infinity and spatial infinity are that the Newman-Penrose constants $G^{(5)}_k$, $k=0,\ldots,4$ and the highe

Figures (2)

  • Figure 1: Spacetime close to spatial and null infinities: to the left the standard representation of spatial infinity as a point $i^0$; to the right the representation where spatial infinity is envisaged as a cylinder.
  • Figure 2: Effect of the choice of the function $\kappa$ on the representation of null infinity near spatial infinity. To the left with the choice $\kappa=\rho$; to the right the choice $\kappa=\omega$ so that null infinity corresponds to the hypersurfaces $\tau=\pm 1$.

Theorems & Definitions (11)

  • Conjecture : Friedrich, 2002
  • Theorem : Main theorem
  • Conjecture : New conjecture
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1: Friedrich, 1997
  • Theorem : Main theorem, precise formulation
  • Corollary 1
  • Conjecture
  • ...and 1 more