Scope of the accessed small-angular-momentum Kerr work

For the mathematical terms, let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and $\iota :{\mathrm{\Sigma }}_0\hookrightarrow M$ a spacelike embedded three-manifold. Vacuum initial data mean $({\mathrm{\Sigma }}_0,h_0,K_0)$, where $h_0={\iota }^{\ast }g$ and $K_0(V,W)=g({\mathrm{\nabla }}_{V}n,W)$ for tangent vectors $V,W$ and the unique future-directed unit normal $n$; the data satisfy the matter-free, zero-cosmological-constant specialization of the initial-data constraints. Here the vacuum Einstein equation means $\mathrm{Ric}(g)=0$ with the curvature convention defining its Ricci tensor. A Cauchy hypersurface is a spacelike embedded hypersurface met exactly once, in parameter value, by every inextendible causal curve. A future globally hyperbolic development is a time-oriented Lorentzian solution with an embedding of ${\mathrm{\Sigma }}_0$ as a Cauchy hypersurface inducing $h_0$ and $K_0$. The local Kerr definition uses geometric parameters $M>0$ and $\mathrm{\mid }a\mathrm{\mid }<M$ for the subextremal exterior; when a physical interpretation is adopted, $M=G_N{m}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{c}^2$ and $a=J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}c)$, with $G_N$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in $\mathrm{k}\mathrm{g}$ and $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Thus $M$ and $a$ have units $\mathrm{m}$, and $a$ is the specific-angular-momentum length parameter. The paper's notation $\mathrm{Kerr}(a_0,m_0)$ corresponds to $(a,M)=(a_0,m_0)$, and its smallness condition $\mathrm{\mid }{a}_0\mathrm{\mid }\mathrm{/}{m}_0\ll 1$ is dimensionless.

For asymptotic terminology, an end is a component outside a compact subset of ${\mathrm{\Sigma }}_0$, equipped with coordinates $x$ onto $\{x\in {\mathbb{R}}^3:r=\mathrm{\mid }x\mathrm{\mid }>R\}$. A basic asymptotically flat initial-data convention is $h_{ij}-{\delta }_{ij}=O_1(r^{-1})$ and $K_{ij}=O_1(r^{-2})$: $T=O_k(r^{-q})$ means that each component satisfies $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }T\mathrm{\mid }\le C_{\alpha }{r}^{-q-\mathrm{\mid }\alpha \mathrm{\mid }}$ for all multiindices $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, uniformly in direction for sufficiently large $r$. These are the explicit end conditions in Gourgoulhon, §7.2.1, equations (7.1)–(7.4); they describe the asymptotic terminology, not a sufficient stability hypothesis. The local ADM-end definition gives the mass surface integral on its more specialized end class; it does not assert that its special expansion exhausts the data in the Kerr source. In a fixed asymptotically Cartesian chart, the geometric ADM mass is, whenever the finite limit exists,

$$m_{\mathrm{A}\mathrm{D}\mathrm{M}}=\frac{1}{16\pi }\underset{R\to \mathrm{\infty }}{\lim }{\int }_{\mathrm{\mid }x\mathrm{\mid }=R}({\mathrm{\partial }}_j{h}_{ij}-{\mathrm{\partial }}_i{h}_{jj})\frac{x^i}{R}d{S}_{\delta }\mathrm{.}$$

Here $d{S}_{\delta }$ is Euclidean sphere area, repeated indices are summed from $1$ to $3$, and the physical energy and mass are $E_{\mathrm{A}\mathrm{D}\mathrm{M}}=c^4{m}_{\mathrm{A}\mathrm{D}\mathrm{M}}\mathrm{/}{G}_N$ and $M_{\mathrm{A}\mathrm{D}\mathrm{M}}=c^2{m}_{\mathrm{A}\mathrm{D}\mathrm{M}}\mathrm{/}{G}_N$. For a chosen oriented rotation axis in that chart, let $\phi =x^1{\mathrm{\partial }}_2-x^2{\mathrm{\partial }}_1$ for the third axis, with cyclic variants for the other axes. With the present convention $K(V,W)=g({\mathrm{\nabla }}_{V}n,W)$, its geometric angular-momentum charge is, whenever this finite limit exists,

$$\mathcal{J}(\phi )=-\frac{1}{8\pi }\underset{R\to \mathrm{\infty }}{\lim }{\int }_{\mathrm{\mid }x\mathrm{\mid }=R}(K_{ij}-({\mathrm{tr}}_{h}K)h_{ij}){\phi }^i{\nu }_h^{j}d{S}_h\mathrm{.}$$

Here ${\mathrm{tr}}_{h}K=h^{ij}{K}_{ij}$, ${\nu }_h$ is the outward $h$-unit sphere normal, and $d{S}_h$ is its induced area. This is Gourgoulhon's equation (7.63) with the sign translated from his $K(V,W)=-g({\mathrm{\nabla }}_{V}n,W)$, equation (2.43). The geometric charge has units ${\mathrm{m}}^2$ and $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=c^3\mathcal{J}\mathrm{/}{G}_N$; for Kerr about its spin axis, $\mathcal{J}=Ma$. Thus the source's parameter $a_0$ denotes the spin length, not a total angular momentum with SI units. Neither existence nor coordinate invariance of angular momentum follows just from the displayed basic falloff: a fixed asymptotic frame and the source's charge/admissibility conditions are retained, rather than silently imposing one additional parity or gauge condition on its theorem; see Gourgoulhon §7.5.2 for this distinction.

In this report the stability assertion quantifies over precisely the source's own initial-data class. Definition 3.4.2 takes an initial-data layer $L_0(a_0,m_0)$ in the future of asymptotically flat data on a spacelike ${\mathrm{\Sigma }}_0$, with source parameters $m_0$ and $a_0$, and defines $({ϵ}_0,k)$-admissibility using its initial-layer norm $I_k$ from §3.3.6. The unnumbered Main Theorem (version 2) in §3.4.3 assumes $\mathrm{\mid }{a}_0\mathrm{\mid }\mathrm{/}{m}_0$ sufficiently small, $k_{\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}$ sufficiently large, ${ϵ}_0$ sufficiently small, and in particular $I_{k_{\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}+10}\le {ϵ}_0^2$. Definition 3.4.5 uses “admissible future null complete” for a future development of such a layer with complete ${\mathcal{I}}^{+}$ and its other listed geometric conditions.

For the completeness terminology, a null generator with affine parameter $\lambda$ has nonzero future null tangent $\mathrm{\ell }$, $g(\mathrm{\ell },\mathrm{\ell })=0$ and ${\mathrm{\nabla }}_{\mathrm{\ell }}\mathrm{\ell }=0$; future completeness means that its maximal future affine parameter is unbounded above. A change $\lambda \mapsto A\lambda +B$, $A>0$, preserves this property; complete future null infinity means that every one of its future generators is complete in this sense. Here the source's future null infinity ${\mathcal{I}}^{+}$ is the asymptotic boundary $r\to \mathrm{\infty }$ along its outgoing $u$-foliation, not an additional hypothesis of a smooth conformal completion. Section 3.8.1, pp. 152–153, identifies its limiting null generators and explains completeness using the leaves for all $u\ge 1$ and the generator normalization: the limiting generator has $du\mathrm{/}d\lambda =2+O({ϵ}_0\mathrm{/}{u}^{1+\delta })$ for the source's $\delta >0$ and small ${ϵ}_0$, hence this derivative is bounded above and below by positive constants and unbounded $u$ gives unbounded future affine parameter. This specifies the reported boundary completeness; it does not say that every null geodesic of the black-hole spacetime is future complete.

The arXiv v1 abstract says that the authors prove full unconditional nonlinear stability of this small-angular-momentum Kerr family in asymptotically flat solutions of the vacuum Einstein equations. Its simplified Theorem 1.2.1 states that the future globally hyperbolic development of a general initial-data set in that source class, sufficiently close in a suitable topology to $\mathrm{Kerr}(a_0,m_0)$ initial data and with sufficiently small $\mathrm{\mid }{a}_0\mathrm{\mid }\mathrm{/}{m}_0$, has complete future null infinity and converges in the source's set $J^{-1}({\mathcal{I}}^{+})$ to a nearby $\mathrm{Kerr}(a_{\mathrm{\infty }},m_{\mathrm{\infty }})$, with final parameters close to the initial ones. The precise convergence and parameter-closeness conclusions are stated in the authors' norms, including estimate (3.4.8).

This is a report of the authors' stated result and its source-defined scope, not a local proof or independent verification of the full series. The v1 abstract records that three companion papers had been released and the remaining papers were expected; §1.2.1 identifies estimates assigned to forthcoming papers. The 2008 review's discussion of then-known global constructions is historical only. This remark makes no current-status claim about whether the full subextremal or extremal Kerr stability problem is open or solved.

Klainerman and Szeftel, abstract, §1.2.1, precise version §3.4.3, and null infinity §3.8.1

Jaramillo, Valiente Kroon and Gourgoulhon, §§1.2.1 and 2.2

Subextremal Kerr exterior in length coordinates

Spacelike initial data, the constraint equations, developments and maximal developments

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

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity, §2.3.4 equation (2.43), §§7.2.1, 7.3.1 and 7.5.1–7.5.2

A specified asymptotically Euclidean ADM end

Future null geodesic completeness and incompleteness