Static Vacuum Extensions With Prescribed Bartnik Boundary Data Near a General Static Vacuum Metric

Abstract

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik’s static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry.

Appendices

Appendix A. Formulas of (Linearized) Geometric Operators

Given a Riemannian manifold (U, g), the Bianchi operator $\beta _{ g}$ and its adjoint operator $\beta _{g} ^*$ are defined by, for a symmetric (0, 2)-tensor h and a vector field X,

$$\begin{aligned} \beta _{g} h&= -\text {div}_{g} h + \tfrac{1}{2} d \textrm{tr}_{g} h \end{aligned}$$

(A.1)

$$\begin{aligned} \beta ^*_{g} X&= \tfrac{1}{2} \big (L_{X} g - (\text {div}_{g} X )g \big ). \end{aligned}$$

(A.2)

We write the Lie derivative of the Riemannian metric g along a vector field X by

$$\begin{aligned} {\mathcal {D}}_{g} X&= \tfrac{1}{2} L_{X} g. \end{aligned}$$

(A.3)

We record the following basic identities:

$$\begin{aligned} \beta _{g} {\mathcal {D}}_{g} X&= -\tfrac{1}{2} \Delta _{g} X - \tfrac{1}{2} \textrm{Ric}(X, \cdot ) \end{aligned}$$

(A.4)

$$\begin{aligned} \beta _{g} (fh)&= f \beta _{g} h - h(\nabla f, \cdot ) + \tfrac{1}{2} (\text {tr}_{g} h) df\quad \text{ for } \text{ a } \text{ scalar } \text{ function } f. \end{aligned}$$

(A.5)

We frequently use the linearization of geometric quantities or operators. In a smooth manifold U, let g(s) be a smooth family of Riemannian metrics with $g(0) = g$ and $g'(0) = h$. We define the linearization of the Ricci tensor at g by $\textrm{Ric}'|_{g} (h):=\left. {{\,\mathrm{\tfrac{d}{d t}}\,}}\right| _{s=0} \textrm{Ric}_{g(s)}$. The other linearized quantities are denoted in the similar fashion. For example,

$$\begin{aligned} (\nabla ^{2})'|_{g}(h) := \left. {{\,\mathrm{\tfrac{d}{d t}}\,}}\right| _{t=0} \nabla ^{2}_{g(t)} \quad \text{ and } \quad \Delta '|_{g}(h) := \left. {{\,\mathrm{\tfrac{d}{d t}}\,}}\right| _{t=0} \Delta _{g(t)}. \end{aligned}$$

We often omit the subscripts $|_{g}$ when the context is clear.

Let $\Sigma \subset U$ be a hypersurface and $\nu$ be the unit normal to $\Sigma$. Consider a local frame $\{ e_0, e_1, \dots , e_{n-1}\}$ such that $e_0$ is the parallel extension of $\nu$ along itself near $\Sigma$. We list those linearized quantities in the local frame (where f is an arbitrary scalar function), see [9, Section 2.1]:

$$\begin{aligned} (\textrm{Ric}'|_{g}(h))_{ij}&=-\tfrac{1}{2} g^{k\ell } h_{ij;k\ell } + \tfrac{1}{2} g^{k\ell } (h_{i k; \ell j} + h_{jk; \ell i} ) - \tfrac{1}{2} (\text {tr}\, h)_{;ij} \nonumber \\&\quad + \tfrac{1}{2} (R_{i\ell } h^\ell _{j} + R_{j\ell } h^\ell _{i} )- R_{ik\ell j} h^{k\ell } \end{aligned}$$

(A.6)

$$\begin{aligned} R'|_{g}(h)&=-\Delta (\text {tr}\, h ) +\text {div}\, \text {div}\, h - h \cdot \textrm{Ric}_{g}\nonumber \\ \big ((\nabla ^{2})'|_{g}(h) f \big )_{ij}&= \tfrac{1}{2} g^{k\ell }f_{,\ell } \big ( h_{ij;k} - h_{jk;i} - h_{ik;j} \big ) \end{aligned}$$

(A.7)

$$\begin{aligned} \big ( \Delta '|_{g}(h) \big )f&= - g^{ik} g^{j\ell }f_{;ij}h_{k\ell } + g^{ij} \left( -(\text {div}_{g} h)_{i} + \tfrac{1}{2} (\text {tr}_{g} h)_{;i} \right) f_{,j}. \end{aligned}$$

(A.8)

We write $\nu = \nu _{g}$ for short. We let $\omega (e_{a}) = h(\nu , e_{a})$ be the one-form on the tangent bundle of $\Sigma$ and $\{\Sigma _{t}\}$ be the foliation by g-equidistant hypersurfaces to $\Sigma$. For tangential directions $a, b, c \in \{ 1, \dots , n-1\}$ on $\Sigma$, we have

$$\begin{aligned} \nu '|_{g}(h)&= - \tfrac{1}{2} h(\nu , \nu ) \nu - g^{ab} \omega (e_{a}) e_{b} \end{aligned}$$

(A.9)

$$\begin{aligned} A'|_{g}(h)&= \tfrac{1}{2} (L_{\nu } h)^\intercal - \tfrac{1}{2} L_{\omega } g^\intercal - \tfrac{1}{2} h(\nu , \nu ) A_{g} \nonumber \\&= \tfrac{1}{2}( \nabla _{\nu } h)^\intercal + A_{g}\circ h - \tfrac{1}{2} L_{\omega } g^\intercal - \tfrac{1}{2} h(\nu , \nu ) A_{g} \nonumber \\ H'|_{g}(h)&=\tfrac{1}{2} \nu (\text {tr}\, h^\intercal ) - \text {div}_{\Sigma } \omega - \tfrac{1}{2} h(\nu , \nu ) H_{g} \end{aligned}$$

(A.10)

where $(A\circ h)_{ab} = \frac{1}{2} (A_{ac}h^{c}_{b} + A_{bc} h_{a}^{c})$. In the special case that $h = L_{X} g$ along $\Sigma$ where the vector field $X = \eta \nu + X^\intercal$ for some tangential vector $X^\intercal$ to $\Sigma$, we get

$$\begin{aligned} H'|_{g}(L_{X} g)&= - \Delta _{\Sigma } \eta - (|A|^{2}+\textrm{Ric}(\nu , \nu ))\eta + X^\intercal (H). \end{aligned}$$

We also have the following “linearized” Ricatti equation:

$$\begin{aligned} \begin{aligned} \nu (H'|_{g}(h))&= - \tfrac{1}{2} R'(h) + \tfrac{1}{2} R'^\Sigma |_{g^\intercal } (h^\intercal ) + A_{g} \cdot (A_{g} \circ h) \\&\quad - A_{g} \cdot A'(h) -H_{g} H'(h) \\&\quad + \tfrac{1}{4}\big ( - R_{g} +R^\Sigma _{g} - |A_{g}|^{2} - H_{g}^{2}\big )h(\nu , \nu )\\&\quad - \tfrac{1}{2}\Delta _{\Sigma } h(\nu , \nu ) + g^{ab} \omega (e_{a}) e_{b}(H_{g}). \end{aligned} \end{aligned}$$

(A.11)

In the third equation, $\text {tr}\, h^\intercal$ denotes the tangential trace of h on $\Sigma _{t}$, defined in a collar neighborhood of $\Sigma$. For the last equation, the dot $\cdot$ means the g-inner product, $(A_{g}\circ h)_{ab} = \tfrac{1}{2} (A_{ac} h^{c}_{b}+A_{bc} h^{c}_{a})$, and $R^\Sigma _{g}, \Delta _{\Sigma }$ are respectively the scalar curvature and the Laplace operator of $(\Sigma , g^\intercal )$.

Appendix B. Spacetime harmonic gauge and analyticity

Let (M, g) be a Riemannian manifold and let $u>0$ be a scalar function on M. We define the spacetime metric ${\textbf{g}}$ on ${\textbf{N}}:={\mathbb {R}}\times M$ by (in either Riemannian or Lorentzian signature)

$$\begin{aligned} {\textbf{g}} = \pm u^{2} dt^{2} + g. \end{aligned}$$

(B.1)

Recall that we say the pair (g, u) is static vacuum if $S(g, u)=0$ as defined in (1.1). When $u>0$, the condition is equivalent to that the spacetime metric satisfies $\textrm{Ric}_\textbf{g}=0$. Thus, we may also refer such ${\textbf{g}}$ as a static vacuum (spacetime) metric.

1.1 Harmonic Gauge

Fix a general background triple $(M, {\bar{g}}, {\bar{u}})$ and thus the background spacetime metric $\bar{{\textbf{g}}} = \pm {\bar{u}}^{2} dt^{2} + {\bar{g}}$ on ${\textbf{N}}$. Recall the Bianchi operator $\beta _{\bar{{\textbf{g}}}}$ sends any symmetric (0, 2)-tensor ${{\varvec{k}}}$ on ${\textbf{N}}$ to the covector $\beta _{\bar{{\textbf{g}}}}{{\varvec{k}}}$, defined by

$$\begin{aligned} \beta _{\bar{{\textbf{g}}}} {{\varvec{k}}}= - \text {div}_{\bar{{\textbf{g}}}} {{\varvec{k}}}+ \tfrac{1}{2} {\varvec{d}} (\text {tr}_{\bar{{\textbf{g}}}}{{\varvec{k}}}). \end{aligned}$$

The following proposition explains the motivation behind the definition of the static-harmonic gauge in Definition 3.5. Note that this fact is not used anywhere else in the paper.

Proposition B.1

For ${\textbf{g}}$ taking the form (B.1),

$$\begin{aligned} \beta _{\bar{{\textbf{g}}}} {\textbf{g}} = \beta _{{\bar{g}}} g + {\bar{u}}^{-2} u du -{\bar{u}}^{-1} g(\nabla _{{\bar{g}}} {\bar{u}}, \cdot ). \end{aligned}$$

That is, in the coordinate t of ${\mathbb {R}}$ and a local coordinate chart $\{ x_1,\dots , x_{n}\}$ of M

$$\begin{aligned}&\beta _{\bar{{\textbf{g}}}} {\textbf{g}} (\partial _{t})=0\\&\beta _{\bar{{\textbf{g}}}} {\textbf{g}} (\partial _{a} ) =\beta _{{\bar{g}}} g(\partial _{a} ) + {\bar{u}}^{-2} u \partial _{a} u -{\bar{u}}^{-1} g(\nabla _{{\bar{g}}}{\bar{u}}, \partial _{a} )\qquad \text{ for } a=1,\dots , n. \end{aligned}$$

Proof

Let ${\varvec{\nabla }}, \nabla$ be the covariant derivatives of $\bar{{\textbf{g}}}, {\bar{g}}$, respectively. We have

$$\begin{aligned} {\varvec{\nabla }}_{\partial t} \partial _{t}&= {\mp } {\bar{u}} \nabla {\bar{u}}\\ {\varvec{\nabla }}_{\partial a} \partial _{t}&= {\varvec{\nabla }}_{\partial t} \partial _{a}= {\bar{u}}^{-1} {\partial _{a}} {\bar{u}}\, \partial _{t}\\ {\varvec{\nabla }}_{\partial a} \partial _{b}&= \nabla _{\partial _{a}} \partial _{b}. \end{aligned}$$

Then we compute $\text {div}_{\bar{{\textbf{g}}}} {{\textbf{g}}}$:

$$\begin{aligned} (\text {div}_{\bar{{\textbf{g}}}} {{\textbf{g}}})(\partial _{t} )&= 0\\ (\text {div}_{\bar{{\textbf{g}}}} {{\textbf{g}}})(\partial _{a} )&= {\bar{u}} ^{-1} g(\nabla {\bar{u}}, \partial _{a}) - {\bar{u}}^{-3} u^{2} \partial _{a} {\bar{u}} + (\text {div}_{{\bar{g}}} g)(\partial _{a}). \end{aligned}$$

Next we compute the trace term:

$$\begin{aligned} \text {tr}_{\bar{{\textbf{g}}}} {\textbf{g}}&= {\bar{u}}^{-2} u^{2} + \text {tr}_{{\bar{g}}} g\\ {\varvec{d}} (\text {tr}_{\bar{{\textbf{g}}}} {\textbf{g}} )&=-2 {\bar{u}}^{-3} u^{2}d{\bar{u}} + 2{\bar{u}}^{-2} u du + d (\text {tr}_{{\bar{g}}} g). \end{aligned}$$

Combining the above identities give

$$\begin{aligned} \beta _{\bar{{\textbf{g}}}} {\textbf{g}}&= -\text {div}_{\bar{{\textbf{g}}}} {\textbf{g}} + \tfrac{1}{2} {\varvec{d}} (\text {tr}_{\bar{{\textbf{g}}}} {\textbf{g}} )\\&=\beta _{{\bar{g}}} g -{\bar{u}} ^{-1} g( \nabla {\bar{u}}, \cdot ) + {\bar{u}}^{-3} u^{2} d {\bar{u}} - {\bar{u}}^{-3} u^{2} d{\bar{u}} + {\bar{u}}^{-2} u du \\&=\beta _{{\bar{g}}} g -{\bar{u}} ^{-1} g( \nabla {\bar{u}}, \cdot ) + {\bar{u}}^{-2} u du. \end{aligned}$$

$\square$

If M has nonempty boundary $\partial M$, then on the boundary $\partial {\textbf{N}} := {\mathbb {R}}\times \partial M$, the Cauchy boundary data $({{\textbf{g}}}|_{\partial {\textbf{N}}}, A_\textbf{g})$ of $\partial \textbf{N}\subset ({\textbf{N}}, {\textbf{g}})$ can be expressed in terms of $u, \nu ( u)$ and $(g^\intercal , A_{g})$ of $\partial M \subset (M, g)$.

Proposition B.2

On the boundary $\partial \textbf{N}$, we have

$$\begin{aligned} \textbf{g} |_{\partial \textbf{N}}&= \pm u^{2} dt^{2} + g^\intercal \\ A_\textbf{g}&= \pm u \nu ( u) dt^{2} + A_{g}. \end{aligned}$$

Consequently, the corresponding linearizations at $\bar{\textbf{g}}$ along the deformation $\textbf{h}$, which is generated by an infinitesimal deformation (h, v) at $({\bar{g}}, {\bar{u}})$, are given by

$$\begin{aligned} \textbf{h} |_{\partial \textbf{N}}&= \pm 2 {\bar{u}} v dt^{2} + h^\intercal \\ A'|_{\bar{\textbf{g}}} (\textbf{h})&= \pm \big ( v \nu ({\bar{u}}) + {\bar{u}} (\nu (u))' \big ) dt^{2} + A'|_{g}(h). \end{aligned}$$

Proof

The identity for the induced metric $\textbf{g}$ is obvious. For $A_\textbf{g}$, we compute in local frame $\{ \partial _{t}, \nu , e_1, \dots , e_{n-1}\}$ where $\nu$ is the unit normal to $\partial \textbf{N}$ (which coincides with the unit normal for $\partial M$ and $e_1, \dots , e_{n-1}$ are tangential to $\partial M$. Then

$$\begin{aligned} A_\textbf{g} (\partial _{t}, \partial _{t})&= - \textbf{g}( \nu , {{\varvec{\nabla }}}_{\partial _{t}} \partial _{t} ) = \pm u \nu (u)\\ A_\textbf{g} (e_{a}, e_{b})&= - \textbf{g}( \nu , {{\varvec{\nabla }}}_{e_{a}} e_{b}) = A_{g} (e_{a}, e_{b}) \quad \text{ for } a, b = 1, \dots , n-1 \end{aligned}$$

and all other components of $A_\textbf{g}$ are zero. $\square$

1.2 Analyticity

We say a scalar function f is (real) analytic in the coordinate chart $\{ x_1,\dots , x_{n} \}$ on a manifold M if for each $p\in M$, there is a neighborhood U of p such that

$$\begin{aligned} f(x) = \sum _{|I|=0, 1, \dots , } \frac{1}{|I|!} \partial ^{I} f(p) (x-p)^{I} \quad \text{ for } \text{ all } x\in U \end{aligned}$$

where I is a multi-index.

A tensor h is said to be analytic in the coordinate chart $\{ x_1,\dots , x_{n} \}$ if all of its components are analytic. A Riemannian manifold (M, g) is called analytic if it can be covered by coordinate charts $\{U_{i}\}$ where g is analytic in each chart; and similarly, a hypersurface embedded in (M, g) is called analytic if it is analytic in each $U_{i}$.

A classical result of Müller zum Hagen [32] says that if the spacetime $(\textbf{N}, \bar{\textbf{g}})$ is static vacuum, then $\bar{\textbf{g}}$ is analytic in harmonic coordinates. Below, we state and prove a version of the corresponding result for the time-slice $(M, {\bar{g}}, {\bar{u}})$.

Let $(M, {\bar{g}})$ be a Riemannian manifold with a scalar function ${\bar{u}}>0$ on M. A coordinate chart $(x_1,\dots , x_{n})$ on $(M, {\bar{g}}, {\bar{u}})$ is called static-harmonic if

$$\begin{aligned} \Delta x_{k} + {\bar{u}}^{-1} \nabla {\bar{u}} \cdot \nabla x_{k}=0 \text{ for } \text{ all } k =1, \dots , n. \end{aligned}$$

Here and below, the covariant derivatives and curvatures are all with respect to ${\bar{g}}$. In coordinates $\{ x_1,\dots , x_{n}\}$ we denote the Christoffel symbols $\Gamma ^{k}:={\bar{g}}^{ij} \Gamma ^{k}_{ij}$ and compute

$$\begin{aligned} \Delta x_{k} + {\bar{u}}^{-1} \nabla {\bar{u}} \cdot \nabla x_{k} = -\Gamma ^{k} + {\bar{u}}^{-1} {\bar{g}}^{ik} \frac{\partial {\bar{u}}}{\partial x_{i}}. \end{aligned}$$

Therefore, a local coordinate chart $\{ x_1, \dots , x_{n}\}$ is static-harmonic if and only if

$$\begin{aligned} -\Gamma ^{k}+ {\bar{u}}^{-1} {\bar{g}}^{ik} \frac{\partial {\bar{u}}}{\partial x_{i}} =0. \end{aligned}$$

Theorem B.3

Let $(M, {\bar{g}}, {\bar{u}})$ be a static vacuum triple with ${\bar{u}}>0$. Then ${\bar{g}}$ and ${\bar{u}}$ are analytic in static-harmonic coordinates in $\text {Int}\, M$.

Remark B.4

As a direct consequence of [21, Theorem 2.1], ${\bar{g}}$ and ${\bar{u}}$ are analytic in harmonic coordinates and in geodesic normal coordinates in $\text {Int}\, M$.

Proof

Let $\{ y_1, \dots , y_{n}\}$ be an arbitrary local coordinate chart about the point $p\in \text {Int}\, M$. By standard elliptic theory [16, p. 228], in a neighborhood of p there are solutions $x_1, \dots , x_{n}$ of

$$\begin{aligned}&\Delta x_{j} - {\bar{u}}^{-1} \nabla {\bar{u}} \cdot \nabla x_{j}=0 \\&x_{j} (p)=0 \; \text{ and } \; \frac{\partial x_{j}}{\partial y_{i} } (p) = \delta _{ij}. \end{aligned}$$

Those functions $\{ x_1,\dots , x_{n}\}$ are the desired static-harmonic coordinates.

The static vacuum pair satisfies

$$\begin{aligned} \begin{aligned} - \textrm{Ric}+{\bar{u}}^{-1} \nabla ^{2} {\bar{u}}&=0\\ \Delta {\bar{u}}&=0. \end{aligned} \end{aligned}$$

(B.2)

Recall the well-known formula:

$$\begin{aligned} \textrm{Ric}_{ij} = -\frac{1}{2} {\bar{g}}^{rs} \frac{\partial ^{2} {\bar{g}}_{ij}}{\partial x_{r} \partial x_{s}} + \frac{1}{2} \left( {\bar{g}}_{ri} \frac{\partial \Gamma ^{r}}{\partial x_{j}} + {\bar{g}}_{rj} \frac{\partial \Gamma ^{r}}{\partial x_{i} } \right) + {\mathcal {O}}({\bar{g}},\partial {\bar{g}}). \end{aligned}$$

where ${\mathcal {O}}({\bar{g}},\partial {\bar{g}})$ denotes the terms involving at most one derivative of the metric ${\bar{g}}$. In particular, in the static-harmonic coordinates:

$$\begin{aligned}&- \textrm{Ric}_{ij}+{\bar{u}}^{-1} {\bar{u}}_{;ij}\\&= \frac{1}{2} {\bar{g}}^{rs} \frac{\partial ^{2} {\bar{g}}_{ij}}{\partial x_{i} \partial x_{j}} - \frac{1}{2} \left( {\bar{g}}_{ri} \frac{\partial }{\partial x_{j}} \left( {\bar{u}}^{-1} {\bar{g}}^{\ell r} \frac{\partial {\bar{u}}}{\partial x_{\ell }} \right) + {\bar{g}}_{rj} \frac{\partial }{\partial x_{i}} \left( {\bar{u}}^{-1} {\bar{g}}^{\ell r} \frac{\partial {\bar{u}}}{\partial x_{\ell }} \right) \right) \\&\quad + {\bar{u}}^{-1} {\bar{u}}_{;ij} + {\mathcal {O}}({\bar{g}},\partial {\bar{g}})\\&= \frac{1}{2} {\bar{g}}^{rs} \frac{\partial ^{2}{\bar{g}}_{ij}}{\partial x_{i} \partial x_{j}} + {\mathcal {O}}( {\bar{g}},\partial {\bar{g}}, {\bar{u}}, \partial {\bar{u}}). \end{aligned}$$

Thus, (B.2) is a quasi-linear elliptic system in static-harmonic coordinates, so the solutions ${\bar{g}}_{ij}, {\bar{u}}$ are analytic in static-harmonic coordinates. $\square$