Asymptotically Flat Spacetimes

Asymptotic flatness is a collection of boundary structures tailored to isolated gravitating systems, not one universal estimate $g-\eta\to0$. Spatial infinity supports ADM data, null infinity supports radiation and BMS data, asymptotic simplicity controls endpoints of all null geodesics, and matching through $i^0$ connects incoming with outgoing charges only under additional regularity. This chapter develops those statements in six steps and keeps topology, differentiability, falloff, parity, completeness, and corner continuity logically separate.

Helpful background. The Riemann Curvature Tensor supplies Weyl data; Pseudo-Riemannian Metrics and Signature fixes null and spacelike conventions; Affine and Variational Geodesics supplies endpoint completeness; Causality and Global Structure provides conformal boundaries; Formulations of General Relativity supplies Cauchy and characteristic data; and Exact Solutions of Einstein’s Equations provides Minkowski, Schwarzschild, and Kerr tests.

Enter the asymptotic regimes

Throughout the chapter, $\widehat g_{ab}$ denotes the physical metric and

$$g_{ab}=\Omega^2\widehat g_{ab}$$

denotes an unphysical metric when a conformal completion is used. Near regular null infinity, $\Omega\sim r^{-1}$ and $d\Omega\ne0$. At the conformal point $i^0$, $\Omega\sim r^{-2}$ along spacelike directions, $d\Omega=0$, and the Hessian fixes the normalization. A single coordinate symbol $r$ therefore does not make the two limits equivalent.

The chapter uses four recurring tests:

• Where is the limit? Specify $\mathscr I^+$, $\mathscr I^-$, $i^0$, or timelike infinity, together with topology and number of ends.
• In what regularity class? Smooth, finite $C^k$, weighted Sobolev, direction-dependent, and polyhomogeneous completions support different curvature statements.
• Modulo which transformations? A conformal rescaling, an asymptotic diffeomorphism, or a supertranslation can change representatives without changing—or sometimes while changing—the physical charge.
• Complete in which sense? Completeness of null-boundary generators, endpoints of all physical null geodesics, and global geodesic completeness are distinct.

The standard conformal definitions and the sharp difference between a simple zero at $\mathscr I$ and a second-order zero at $i^0$ are presented in Ashtekar and Khera 2024, §§2.1–2.3, pp. 4–13.

Six pages through the chapter

1. What Asymptotic Flatness Must Specify turns the phrase “approaches Minkowski” into a checklist: end topology, limiting structure, derivative orders, matter decay, gauge group, parity, completeness, and corner data.
2. Asymptotic Flatness at Null Infinity gives a conformal definition of $\mathscr I^+$, derives that the boundary is null from the vacuum equation, and separates $S^2\times\mathbb R$ topology from generator completeness.
3. Asymptotic Flatness at Spatial Infinity develops asymptotically Euclidean ADM ends, allowed chart changes, and strict versus twisted parity. It also explains why the conformal metric has only direction-dependent $C^{\gt 0}$ regularity at a massive $i^0$.
4. Asymptotic Simplicity and Black-Hole-Compatible Variants shows why requiring both endpoints for every null geodesic forbids a black-hole region, then introduces weak asymptotic simplicity and exterior predictability.
5. Peeling, Smoothness, and Polyhomogeneous Infinity derives the classical Weyl hierarchy from conformal regularity and tracks how $r^{-p}[\log(r/r_0)]^q$ terms, with a fixed reference length $r_0$, preserve some leading orders while obstructing higher differentiability.
6. Matching Spatial and Null Infinity blows up directions at $i^0$, states the null-regularity and continuity assumptions behind antipodal matching, and separates supermomentum matching from the stronger hypotheses needed for Lorentz charges.

The order matters. One cannot assess peeling before fixing differentiability, declare a black-hole exterior asymptotically simple after checking only a neighborhood of $\mathscr I$, or infer a diagonal BMS group from separate definitions at $\mathscr I^-$ and $\mathscr I^+$.

A map of the main structures

Structure	Boundary object	Typical data or observable	Extra condition that is often hidden
ADM spatial end	An end modeled on $\mathbb R^3\setminus B$ or an explicitly stated variant	ADM energy–momentum and Lorentz charges	Derivative falloff, constraints, chart class, and parity
Null asymptotic flatness	A null hypersurface $\mathscr I^\pm\cong S^2\times\mathbb R$	News, Bondi mass, BMS charges	Conformal differentiability, matter decay, and generator completeness
Asymptotic simplicity	The full conformal boundary $\mathscr I$	Endpoints of every inextendible null geodesic	A global endpoint condition, not just a regular collar
Polyhomogeneous infinity	A boundary with a controlled power–log index set	Leading radiation and modified subleading data	Exact logarithmic order and remainder stability under differentiation
Spatial-to-null matching	Direction hyperboloid and its two null boundary spheres	Antipodally related charges and flux laws	Common conformal weights, null regularity, parity, and corner continuity

No row subsumes all the others. Standard ADM falloffs do not construct $\mathscr I$. A smooth $\mathscr I^+$ does not choose spatial parity. Full asymptotic simplicity includes a global null-geodesic endpoint statement but still does not select a Hamiltonian phase space at $i^0$. Matching adds compatibility across regimes rather than merely strengthening one regime.

Implications and non-implications

With sufficient vacuum differentiability, a smooth conformal completion makes the unphysical Weyl tensor vanish at $\mathscr I$ and gives a regular rescaled Weyl field, hence classical peeling Frauendiener 2004, §2.3, pp. 17–18. The converse is not automatic: observing several Newman–Penrose falloffs along one tetrad does not build a smooth manifold-with-boundary.

At spatial infinity, $h_{ij}=\delta_{ij}+O(r^{-1})$ and $\pi^{ij}=O(r^{-2})$ do not alone make the symplectic form or Lorentz generators well behaved. Strict Regge–Teitelboim parity gives one controlled phase space; twisted parity gives another that retains nontrivial BMS supertranslations. The construction and charge consequences require the full boundary conditions in Henneaux and Troessaert 2019, §§III–VI, pp. 5–24.

At null infinity, topology does not imply completeness, and a local completion does not imply asymptotic simplicity. Schwarzschild has a regular asymptotic collar but null geodesics that hit the singularity or remain captured; it is black-hole-compatible precisely because the full endpoint clause is weakened. Conversely, a complete $\mathscr I^+$ is needed before the event-horizon definition is protected against artificial truncation.

Finally, separate BMS groups at $\mathscr I^-$ and $\mathscr I^+$ do not match themselves. Prabhu’s supermomentum theorem assumes an Ashtekar–Hansen structure, specific null-regular curvature and news falloffs, and a totally fluxless diagonal supertranslation subalgebra Prabhu 2019, §4, pp. 23–31. Matching Lorentz charges adds magnetic-Weyl parity, gauge fixing, and continuity assumptions. These results are powerful conditional theorems, not generic consequences of the Einstein equation currently known for every scattering solution.

A disciplined workflow for asymptotic claims

When reading or proving an asymptotic statement, begin by writing the physical domain and the boundary component. Record whether the spacetime is vacuum near that component or give the exact stress-energy decay. Then write either the conformal data $(M,g,\Omega)$ with topology and differentiability or the coordinate data with derivative-controlled falloffs and allowed transformations.

Next, isolate the quantity of interest. ADM energy needs less asymptotic structure than boosts. Bondi four-momentum flux needs less decay than every supermomentum flux. The leading radiative Weyl components need fewer derivatives than full peeling. A common Poincaré subgroup across $i^0$ needs weaker gluing than a diagonal full BMS group.

Finally, test three neighboring failure cases:

1. a coordinate transformation preserving zeroth-order decay but spoiling derivative or parity conditions;
2. a black-hole or excised spacetime that is regular near infinity but fails the global endpoint condition; and
3. a power–log term or incompatible corner limit that preserves leading data but destroys a claimed higher-order result.

This method turns “asymptotically flat” from an adjective into a theorem-ready hypothesis.

Scientific boundary of the chapter

The chapter does not derive waveforms from sources, prove nonlinear stability of Minkowski or Kerr, classify all admissible weighted initial data, or establish that the strongest matching hypotheses hold for a generic scattering solution. It also does not identify detector strain with an asymptotic Weyl scalar without a propagation and gauge analysis.

Polyhomogeneous null infinity is a controlled alternative to smooth infinity, but formal power–log consistency is not automatically a global existence theorem. Likewise, smoothness at Friedrich’s critical sets requires special no-log conditions whose necessity is known only in stated subclasses Paetz 2018, Theorem 1.1 and Remark 1.2, pp. 5–6.

The next chapter, Asymptotic Symmetries and Gravitational Charges, builds the BMS algebra, charge definitions, and balance laws on the boundary structures fixed here.

Chapter diagnostic

Consider a vacuum spacetime with these advertised properties:

For item 2, let $u^a$ be the future-directed unit normal to the Cauchy hypersurface, with $u^a u_a=-1$, and use

$$K_{ij}=-h_i{}^a h_j{}^b\nabla_a u_b =-\frac12\mathcal L_u h_{ij}.$$

1. Near $\mathscr I^+$ it has topology $S^2\times\mathbb R$, complete generators, and an unphysical metric containing a nonzero $\Omega^3\log\Omega$ term.
2. On one Cauchy end, $h_{ij}=\delta_{ij}+O_2(r^{-1})$ and $K_{ij}=O_1(r^{-2})$, but no parity condition or replacement is stated.
3. Some future-directed null geodesics enter a black-hole singularity.
4. Smooth structures are given separately near $\mathscr I^-$, $i^0$, and $\mathscr I^+$, but no corner continuity or null-regularity condition is stated.

Assess four claims: (a) the spacetime is fully asymptotically simple; (b) classical five-component peeling follows from smooth conformal regularity; (c) all ADM Lorentz charges are finite and unambiguous; and (d) the BMS charges at $\mathscr I^-$ and $\mathscr I^+$ are antipodally matched. Then state what additional information would address each failed claim.

Solution

(a) False. A null geodesic ending at the black-hole singularity has no future endpoint on $\mathscr I^+$, so full asymptotic simplicity fails. A weakly asymptotically simple or one-sided null-asymptotically-flat description may still apply if its local hypotheses are checked.

(b) Not established. The term $\Omega^3\log\Omega$ is $C^2$ but not $C^3$ at the boundary, so one cannot invoke a $C^4$ completion to obtain the full $\Psi_4$ through $\Psi_0$ hierarchy. Individual leading falloffs could still be proved directly in a polyhomogeneous class.

(c) Not established. The derivative falloffs are a start, but rotations and boosts are sensitive to leading parity and subleading constraints. One must impose strict parity, twisted parity with its constraint and gauge conditions, or another phase space with proven finite, integrable generators.

(d) Not established. Separate completions do not identify the two BMS groups. One needs a common spatial-to-null construction, compatible conformal frames, the relevant null-regular curvature/news falloffs, and a choice such as the totally fluxless diagonal subalgebra. Lorentz-charge matching additionally needs magnetic-Weyl and continuity hypotheses. A global total-flux equality would further require controlled charges at $i^\pm$ and any horizon flux.

References

• Ashtekar, Abhay, and Neev Khera. “Unified Treatment of Null and Spatial Infinity III: Asymptotically Minkowski Space-times.” Journal of High Energy Physics 2024, no. 2 (2024): 210. DOI.
• Frauendiener, Jörg. “Conformal Infinity.” Living Reviews in Relativity 7 (2004): 1. DOI.
• Henneaux, Marc, and Cédric Troessaert. “The Asymptotic Structure of Gravity at Spatial Infinity in Four Spacetime Dimensions.” arXiv:1904.04495 [hep-th] (2019). arXiv.
• Paetz, Tim-Torben. “On the Smoothness of the Critical Sets of the Cylinder at Spatial Infinity in Vacuum Spacetimes.” Journal of Mathematical Physics 59, no. 10 (2018): 102501. DOI.
• Prabhu, Kartik. “Conservation of Asymptotic Charges from Past to Future Null Infinity: Supermomentum in General Relativity.” Journal of High Energy Physics 2019, no. 3 (2019): 148. DOI.