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An invariant measure of deviation from Petrov type D at the level of initial data

Edgar Gasperin, Jarrod L. Williams

Abstract

In this article, we describe a simple covariant characterisation of initial data sets which give rise to Petrov type D vacuum spacetime developments. As an application, we derive an integral invariant which, when restricted to the appropriate class of asymptotically Euclidean initial data sets, vanishes if and only if the initial dataset is isometric to initial data for the Kerr spacetime. As such, the invariant can be considered a measure of non-Kerrness on such initial data sets. In contrast with other similar invariants constructed through the notion of 'approximate Killing spinors', the present invariant is algebraic in the sense that it is algorithmically computable directly from initial data without having to solve any PDEs on the initial data hypersurface.

An invariant measure of deviation from Petrov type D at the level of initial data

Abstract

In this article, we describe a simple covariant characterisation of initial data sets which give rise to Petrov type D vacuum spacetime developments. As an application, we derive an integral invariant which, when restricted to the appropriate class of asymptotically Euclidean initial data sets, vanishes if and only if the initial dataset is isometric to initial data for the Kerr spacetime. As such, the invariant can be considered a measure of non-Kerrness on such initial data sets. In contrast with other similar invariants constructed through the notion of 'approximate Killing spinors', the present invariant is algebraic in the sense that it is algorithmically computable directly from initial data without having to solve any PDEs on the initial data hypersurface.

Paper Structure

This paper contains 16 sections, 12 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and conventions
  4. Initial data sets and the Weyl spinor
  5. A covariant spacetime characterisation of Petrov type D spacetimes
  6. Killing spinor initial data
  7. Characterising propagating-type-D initial data
  8. Constructing an invariant
  9. Quantifying deviation from Kerr initial data
  10. A special case: boosted asymptotically–Schwarzschildean data
  11. The more general case
  12. Conclusion
  13. Normal derivative operators
  14. Asymptotic expansions of the Weyl tensor
  15. The canonical Killing vector
  16. ...and 1 more sections

Key Result

Lemma 1

(Penrose & Rindler, PenRin86) $(\mathcal{M},\bm g)$ is type D or more special at $p\in \mathcal{M}$ if and only if $\mathcal{H}_{ABCDEF}\vert_p=0$.

Figures (1)

  • Figure 1: Penrose--Petrov diagram. Here, the Petrov types in the blue region are characterised by $\mathcal{H}_{ABCDEF}=0$ (see equation \ref{['PetrovDZeroQuantity']}) and the Petrov types in the red region are characterised by $\Psi_{ABCD}\Psi^{ABCD}\neq 0$. At the intersection is Petrov type D.

Theorems & Definitions (27)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 3
  • ...and 17 more