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Transverse expansion of the metric at null infinity

Marc Mars, Gabriel Sánchez-Pérez

TL;DR

This work develops a comprehensive, coordinate-free framework for solving the conformal Einstein equations to infinite order at null infinity in arbitrary dimensions and topologies. By employing hypersurface data and a transverse expansion formalism, the authors identify the free data encoded on ${\mathscr I}$ that determine the asymptotic geometry and prove a uniqueness theorem: two asymptotically flat spacetimes with the same free data are isometric to infinite order at ${\mathscr I}$. They establish an existence theorem ensuring such data can be realized by a smooth ambient spacetime, provided obstructions vanish. The analysis reveals obstruction tensors (Coulombian and radiative) whose vanishing is essential for smooth realizations and links these obstructions to the Bach tensor in 4D and the Fefferman–Graham obstruction in higher dimensions, offering a path toward higher-dimensional Bondi-type quantities and asymptotic dynamics.

Abstract

In this paper we analyze the conformal Einstein equations to all orders at null infinity without imposing any restriction on the spacetime dimension, the topology of $\mathscr{I}$, or fall-off conditions for the Weyl tensor. In particular, we study how the equations constrain the geometry of null infinity when it is assumed to be foliated by cross-sections, not necessarily spheres. Our approach is coordinate-free and treats the conformal factor $Ω$ as a dynamical variable. After identifying the free data at $\mathscr{I}$, we show that any two asymptotically flat spacetimes sharing the same free data at null infinity are necessarily isometric to infinite order. In addition, we provide a detached definition of null infinity and prove an existence theorem for asymptotically flat spacetimes solving the field equations to infinite order at $\mathscr{I}$ realizing the prescribed initial data.

Transverse expansion of the metric at null infinity

TL;DR

This work develops a comprehensive, coordinate-free framework for solving the conformal Einstein equations to infinite order at null infinity in arbitrary dimensions and topologies. By employing hypersurface data and a transverse expansion formalism, the authors identify the free data encoded on that determine the asymptotic geometry and prove a uniqueness theorem: two asymptotically flat spacetimes with the same free data are isometric to infinite order at . They establish an existence theorem ensuring such data can be realized by a smooth ambient spacetime, provided obstructions vanish. The analysis reveals obstruction tensors (Coulombian and radiative) whose vanishing is essential for smooth realizations and links these obstructions to the Bach tensor in 4D and the Fefferman–Graham obstruction in higher dimensions, offering a path toward higher-dimensional Bondi-type quantities and asymptotic dynamics.

Abstract

In this paper we analyze the conformal Einstein equations to all orders at null infinity without imposing any restriction on the spacetime dimension, the topology of , or fall-off conditions for the Weyl tensor. In particular, we study how the equations constrain the geometry of null infinity when it is assumed to be foliated by cross-sections, not necessarily spheres. Our approach is coordinate-free and treats the conformal factor as a dynamical variable. After identifying the free data at , we show that any two asymptotically flat spacetimes sharing the same free data at null infinity are necessarily isometric to infinite order. In addition, we provide a detached definition of null infinity and prove an existence theorem for asymptotically flat spacetimes solving the field equations to infinite order at realizing the prescribed initial data.
Paper Structure (13 sections, 32 theorems, 250 equations)

This paper contains 13 sections, 32 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Review of hypersurface data formalism
  3. Quasi-Einstein manifolds
  4. Transverse expansion of the metric
  5. Existence and uniqueness results
  6. Obstruction tensors at $\mathscr{I}$. Four and six dimensional cases
  7. Coulombian obstruction in four dimensions
  8. Radiative obstruction in six dimensions
  9. Conclusions and future work
  10. Some pullbacks into a null hypersurface
  11. Auxiliary computations and proof of \ref{['dotR']}
  12. Quasi-Einstein equations at null infinity
  13. Higher order Raychaudhuri equation

Key Result

Theorem 1.1

Let $(\mathcal{M},g,\Omega)$ and $(\mathcal{M}',g',\Omega')$ be two asymptotically flat spacetimes both written in a conformal geodesic gauge. Suppose that their respective null infinities have the same free data $\mathcal{D}$. Then $(\mathcal{M},g,\Omega)$ and $(\mathcal{M}',g',\Omega')$ are isomet

Theorems & Definitions (62)

  • Theorem 1.1: Informal version, see Theorem \ref{['uniqueness']}
  • Theorem 1.2: Informal version, see Theorem \ref{['teorema']}
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Proposition 4.1
  • proof
  • Proposition 4.2: Mio3 and Prop. \ref{['propliericxitang']}
  • ...and 52 more