Asymptotically flat spacelike end and derivative decay

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$. An end chart consists, for a specified $R>0$, of a spacelike embedded hypersurface $S_{\mathrm{e}\mathrm{x}\mathrm{t}}\subset M$ without manifold boundary and a specified diffeomorphism from the open exterior $\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R\}$ onto $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$. A finite-radius boundary sphere, if supplied as additional data, is omitted from this chart. All estimates concern $r=\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and are unchanged on restriction to $r>R^{\mathrm{\prime }}$ for any $R^{\mathrm{\prime }}>R$. Let $n$ be its unique smooth future unit normal, $\gamma =g{\mathrm{\mid }}_{T{S}_{\mathrm{e}\mathrm{x}\mathrm{t}}}$ its induced metric, and $k(v,w)=-g({\mathrm{\nabla }}_v^{g}n,w)$ its second fundamental form, with the tangential covariant derivative defined independently of smooth local extensions. The sign of $k$ is fixed here; reversing it does not change the decay condition.

For a scalar chart component $f$ and integer $m\ge 0$, write $f=O_m(r^{-b})$ if for each multi-index $\beta =({\beta }_1,{\beta }_2,{\beta }_3)$ with $\mathrm{\mid }\beta \mathrm{\mid }={\beta }_1+{\beta }_2+{\beta }_3\le m$ there are $C_{\beta },R_{\beta }$ such that $\mathrm{\mid }{\mathrm{\partial }}_x^{\beta }f(x)\mathrm{\mid }\le C_{\beta }{r}^{-b-\mathrm{\mid }\beta \mathrm{\mid }}$ for $r\ge R_{\beta }$. Here ${\mathrm{\partial }}_x^{\beta }={\mathrm{\partial }}_{x^1}^{{\beta }_1}{\mathrm{\partial }}_{x^2}^{{\beta }_2}{\mathrm{\partial }}_{x^3}^{{\beta }_3}$; tensor estimates apply componentwise in this specified chart. Call these end data asymptotically flat when, for some $\alpha >0$ and integer $m\ge 2$, ${\gamma }_{ij}-{\delta }_{ij}=O_m(r^{-\alpha })$ and $k_{ij}=O_{m-1}(r^{-1-\alpha })$, for $1\le i,j\le 3$. The chart and parameters are witnesses to this existential condition; no invariance under arbitrary changes of chart is asserted.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Metric compatibility, torsion freedom and the metric-selected connection class

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

Diffeomorphisms and local diffeomorphisms of manifolds

Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ

Given: The objects and specified data in the Definition.

Goal: Verify that the displayed conditions define unambiguous objects and predicates on these data.

1.1 The open exterior chart and the spacelike embedded hypersurface satisfy the boundaryless hypotheses of [F2]. That supplier gives the unique smooth normal and extension-independent tangential covariant derivative. It also gives smooth induced metric and second fundamental form; taking $k=-K$ preserves smoothness. By [F3] the inverse of the specified diffeomorphism is a smooth coordinate chart on $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$. Thus ${\gamma }_{ij}$ and $k_{ij}$ are well-defined smooth scalar functions in this chart and admit all the iterated coordinate partial derivatives of [F4] used below. [F1, F2, F3, F4]

2.1 For each of finitely many derivatives at each fixed $m$, the displayed inequality is a definite bound on a scalar function outside a sufficiently large ball. Quantifying over their constants, and then over $\alpha$ and $m$, gives the claimed predicate. Replacing $k$ by $-k$ leaves every absolute value unchanged. Restricting to $r>R^{\mathrm{\prime }}$ preserves the bounds after increasing each threshold radius above $R^{\mathrm{\prime }}$; conversely, bounds on that restricted chart already give the required bounds for all sufficiently large $r$ in the original chart. Hence omitting a separately supplied finite-radius boundary sphere has no effect on this decay predicate. [F1, F2, F4, step 1.1] ∎