Regular non-degenerate stationary vacuum spacetime

PDF p. 3: Definition 1.1, conditions (1)–(2); PDF p. 4: Definition 1.1, conditions (3)–(4)

Work in ZFC, including the axiom of choice. Let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, and let $T$ be a smooth Killing field: ${\mathcal{L}}_{T}g=0$, where the Lie derivative is defined using the local flow of $T$. Write $\mathrm{Ric}(g)$ for the Ricci contraction of the curvature of the Levi–Civita connection, and assume $\mathrm{Ric}(g)=0$. A timelike vector has negative squared length; future direction is determined by the time orientation. In the stationary setting $T$ is future timelike near infinity.

The article calls these data a regular non-degenerate stationary vacuum spacetime when all four conditions below hold.

1. There is an open end $M^{\mathrm{e}\mathrm{n}\mathrm{d}}\subset M$ with a diffeomorphism onto $\mathbb{R}\times \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R_0\}$, for sufficiently large $R_0>0$. In the corresponding coordinates $(t,x^1,x^2,x^3)$, with $r=\mathrm{\mid }x\mathrm{\mid }$, $T={\mathrm{\partial }}_t$ and $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2})$, $g_{ij}={\delta }_{ij}+O_6(r^{-1})$, and $g_{0i}=O_6(r^{-2})$. Here $M_0$ is the constant denoted $M$ in the article, and $u=O_m(r^{-b})$ means that for each spatial multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le m$, $\mathrm{\mid }{\mathrm{\partial }}_x^{\alpha }u\mathrm{\mid }\le C_{\alpha }{r}^{-b-\mathrm{\mid }\alpha \mathrm{\mid }}$ for sufficiently large $r$, uniformly in angular direction. The metric coefficients are independent of $t$ because $T={\mathrm{\partial }}_t$ is Killing. Set $\mathcal{E}=I_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$.

2. The region $\mathcal{E}$, with the restricted metric and time orientation, is globally hyperbolic. Every orbit of $T$ in $\mathcal{E}$ is complete and meets a specified smooth spacelike hypersurface ${\mathrm{\Sigma }}_0$; ${\mathrm{\Sigma }}_0\cap \mathcal{E}$ is Cauchy in $\mathcal{E}$. There is a diffeomorphism ${\mathrm{\Sigma }}_0\simeq \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>1\mathrm{/}2\}$. The hypersurface ${\mathrm{\Sigma }}_0$ agrees with the $t=0$ slice in $M^{\mathrm{e}\mathrm{n}\mathrm{d}}$.

3. Put $S={\mathrm{\partial }}_M{I}_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap {\mathrm{\partial }}_M{I}_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$. It is a smooth embedded two-sphere contained in ${\mathrm{\Sigma }}_0$ and corresponding to $\mathrm{\mid }x\mathrm{\mid }=1$ under the specified identification. There is a neighborhood $\mathcal{O}$ of $S$ in $M$ such that ${\mathcal{H}}^{+}=\mathcal{O}\cap {\mathrm{\partial }}_M{I}_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ and ${\mathcal{H}}^{-}=\mathcal{O}\cap {\mathrm{\partial }}_M{I}_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ are smooth embedded hypersurfaces, each diffeomorphic to $S\times (-1,1)$. They are null, have vanishing null expansion, and intersect transversely in $S$.

4. The field $T$ is tangent to both ${\mathcal{H}}^{+}$ and ${\mathcal{H}}^{-}$.

Here $I_M^{+}(U)$, respectively $I_M^{-}(U)$, consists of points reachable from, respectively reaching, $U$ by future-directed timelike curves in $M$; boundaries are taken in $M$. A spacelike hypersurface has positive-definite induced metric. A Cauchy set in $\mathcal{E}$ meets every inextendible causal curve in $\mathcal{E}$ exactly once. Global hyperbolicity means strong causality and compact causal diamonds, with both notions taken in $\mathcal{E}$. An orbit is complete when its integral curve is defined for every real flow parameter.

For a smooth null hypersurface $H$, its induced metric has a null radical line. Locally choose a nonzero smooth section $L$ of that line. Its screen is $TH\mathrm{/}\mathbb{R}L$ with the induced positive metric $q$. Define the null expansion as the $q$-trace of the screen bilinear form $([X],[Y])\mapsto g({\mathrm{\nabla }}_{X}L,Y)$, where $X,Y$ are tangent to $H$. Vanishing of this trace is the non-expanding condition; it is independent of the local nonzero choice of $L$. Transversality at $S$ means $T_p{\mathcal{H}}^{+}+T_p{\mathcal{H}}^{-}=T_{p}M$ at every $p\in S$.

Given: The smooth geometric data and the specified end, hypersurface and neighborhood in the Definition.

Goal: Show that the conditions define an unambiguous predicate on those data.

1.1 The Lorentzian and causal interfaces determine the equations, chronological sets and ambient boundaries, and specify which causal curves and compact diamonds enter the Cauchy and global-hyperbolicity predicates. The diffeomorphism interface makes each stated identification a smooth bijection with smooth inverse. The chart and the differentiated-bound convention determine the end conditions; the local flow of ${\mathrm{\partial }}_t$ translates $t$ while fixing $x$, so the flow definition of ${\mathcal{L}}_{{\mathrm{\partial }}_t}g=0$ gives ${\mathrm{\partial }}_t{g}_{\mu \nu }=0$ in that chart. The remaining clauses are explicit conditions on the supplied embeddings, flow and intersections. The ZFC convention supplies the inherited choice assumptions. [given, A1, F1, F2, F3, F4, F5, F7]

1.2 Near any point of a smooth null hypersurface $H$, take a slice coordinate $\rho$ with $H=\{\rho =0\}$ and $d\rho \mathrm{\ne }0$. Define $L^a=g^{ab}{\mathrm{\partial }}_b\rho$ along $H$, where $g^{ab}$ are the smooth inverse metric components. This is a smooth nonzero normal field. Nondegeneracy of $g$ makes $(T_{p}H{)}^{\perp }$ one-dimensional; the nonzero radical of $g{\mathrm{\mid }}_{T_{p}H}$ lies in this normal space and therefore equals it. Thus $L$ is tangent and null, and $T_{p}H=L_p^{\perp }$. [given, F1, F3, F7]

2.1 Fix $p\in H$ and use the orthonormal basis supplied by F8 to identify $T_{p}M$ with $\mathbb{R}\times {\mathbb{R}}^3$, with $g_p((s,x),(t,y))=-st+x\cdot y$. Write $L_p=(a,v)$. Since $L_p$ is nonzero and null, $\mathrm{\mid }v{\mathrm{\mid }}^2=a^2$ and $a\mathrm{\ne }0$. Put $e=v\mathrm{/}a$, so $\mathrm{\mid }e\mathrm{\mid }=1$. A vector $(s,x)$ lies in $L_p^{\perp }$ exactly when $s=e\cdot x$. For such a vector set $w=x-se$; then $e\cdot w=0$ and $(s,x)=(s\mathrm{/}a)L_p+(0,w)$. This decomposition is unique, since a vector in $\mathbb{R}{L}_p$ with zero time component is zero. The space $E_p=\{(0,w):e\cdot w=0\}$ has dimension two: some component $e_j$ is nonzero, and the equation $e\cdot w=0$ solves uniquely for $w_j$ in terms of the other two components. Moreover $g_p((0,w),(0,w))=\mathrm{\mid }w{\mathrm{\mid }}^2$, which is positive for $w\mathrm{\ne }0$, while $L_p$ is orthogonal to all of $L_p^{\perp }$. Thus $T_{p}H=L_p^{\perp }=\mathbb{R}{L}_p\oplus E_p$ has positive-semidefinite induced form of rank two and radical $\mathbb{R}{L}_p$. Its quotient by that radical has a well-defined positive metric, because adding multiples of $L_p$ changes no pairing. This pointwise argument uses only the Lorentz-basis interface and elementary coordinate algebra, not the screen lemma. [F8, step 1.2, algebra]

3.1 The local field constructed in step 1.2 is smooth and nowhere zero. Shrink its neighborhood and reverse its sign if necessary to make it future with respect to the given time orientation. Step 2.1 has established the positive-semidefinite rank-two hypothesis at every point of $H$, so the stated smooth-quotient assertion in F6 now applies. It supplies the smooth positive metric $q$ on $TH\mathrm{/}\mathbb{R}L$ used in the Definition. No affine parameter or congruence from a two-surface is needed. [A1, F1, F6, step 1.2, step 2.1]

4.1 Use tangential covariant derivatives along $H$. They are independent of ambient extension: the difference of two extensions has zero components on $H$, so tangential derivatives of those components vanish there, and the connection terms multiplying those zero components also vanish. For tangent $X,Y$, write $b(X,Y)=g({\mathrm{\nabla }}_{X}L,Y)$. Metric compatibility gives $b(X,L)=\frac{1}{2}Xg(L,L)=0$. To check the other slot explicitly, $0=Lg(L,Y)=g({\mathrm{\nabla }}_{L}L,Y)+g(L,{\mathrm{\nabla }}_{L}Y)$; torsion-freeness gives ${\mathrm{\nabla }}_{L}Y={\mathrm{\nabla }}_{Y}L+[L,Y]$, and both $g(L,{\mathrm{\nabla }}_{Y}L)=\frac{1}{2}Yg(L,L)$ and $g(L,[L,Y])$ vanish. Hence $b(L,Y)=0$. Also $b(X,Y)-b(Y,X)=-g(L,{\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X)=-g(L,[X,Y])=0$. The tangent-bracket property in F7 justifies both bracket vanishings. The connection rules make $b$ bilinear over smooth functions in $X,Y$, so it descends to a smooth symmetric bilinear form on the positive screen. [F1, F6, F7, step 1.2, step 3.1]

5.1 In a screen basis, let $Q$ and $B$ be the matrices of $q$ and the descended form. The trace is $\mathrm{tr}(Q^{-1}B)$, the convention in F4. Under a basis change $C$, these become $C^{\mathsf{T}}QC$ and $C^{\mathsf{T}}BC$, so their contracted matrix is $C^{-1}{Q}^{-1}BC$; summing the diagonal entries shows its trace equals $\mathrm{tr}(Q^{-1}B)$. If $L$ is replaced by $aL$ for a nowhere-zero smooth function $a$, then $g({\mathrm{\nabla }}_X(aL),Y)=ag({\mathrm{\nabla }}_{X}L,Y)$ because $g(L,Y)=0$. Thus the form and its trace are multiplied by $a$, while the radical line and quotient metric stay fixed. Any two local nonzero radical fields differ by such a smooth function, so vanishing of expansion is well defined on overlaps. These are conditions, not assertions that suitable data exist, and tangency of $T$ does not assert that either boundary is a Killing horizon for $T$. [F1, F3, F4, F6, step 1.1, step 1.2, step 3.1, step 4.1, algebra] ∎