Equality in the spacetime positive mass theorem II

Let us first explicitly state the definition of asymptotic flatness we will be using.

Note that asymptotic flatness of $(g,k)$ implies that the ADM energy-momentum $(E,P)$ of each end is well-defined. Our main result characterizes the equality case $E=|P|$ of the spacetime positive mass theorem, but we would like to emphasize that, unlike all previous results characterizing $E=|P|$, ours follows from the positive mass inequality $E\geq|P|$ itself and does not rely on any particular proof of this inequality. Consequently, we introduce the following definition.

We say the boundary $\partial M$ of an asymptotically flat manifold $M$ is weakly outer trapped if its outward null expansion $\theta^{+}:=H+\operatorname{tr}_{\partial M}k$ satisfies $\theta^{+}\leq 0$, with respect to the normal that points toward $M$.¹¹1Our convention is that the mean curvature $H$ is the tangential divergence of the chosen unit normal vector. For example, the mean curvature of a round sphere in Euclidean space with respect to the outward normal is positive.

It follows from E. Witten’s proof of the positive mass theorem [Wit81] (and [GHHP83, Her98] for the boundary case) that the positive mass theorem is true near any $(g,k)$ on a spin manifold. Building on the classic works of R. Schoen and S.-T. Yau [SY79, SY81], in joint work with M. Eichmair and Schoen we proved that the positive mass theorem is true²²2The definition of asymptotic flatness used in [EHLS16] is slightly different from the one stated here. By going through the proofs there with more care, one sees that Definition 1.1 is adequate for proving the positive mass theorem. See [LLU22] for details. near any $(g,k)$ on a manifold of dimension less than 8 [EHLS16], with the boundary case proved by the second author with M. Lesourd and R. Unger [LLU22]. J. Lohkamp has also announced a proof in all dimensions [Loh16].

Understanding the $E=|P|$ case of the spacetime positive mass theorem breaks down into two separate steps: The first step is to show that $E=|P|=0$, and the second step is to use the fact that $E=0$ to find an embedding of the initial data into Minkowski space. In the first paper in this series [HL20], we built on the work of R. Beig and P. Chruściel [BC96] (see also [CM06]) in the spin case to construct an argument that proves the first step without appealing to any particular proof of the $E\geq|P|$ inequality. We state this result below and note that it was generalized to include the possibility of a boundary in [LLU22].

Although the stronger asymptotic assumption (1.3) may seem undesirable in higher dimensions (note that (1.3) can be dropped for $n=3,4$ if $\alpha$ in (1.1) is close to $1$), we found pp-wave counterexamples showing that the theorem is false without it for $n>8$. See [HL, Example 7] for details.

Once one knows that $E=0$, the “second step” described above follows from work of Eichmair [Eic13] if $n<8$. More specifically, following Schoen and Yau’s proof for the $n=3$ case³³3When $n=3$, one must make an extra decay assumption on $\operatorname{tr}_{g}k$. [SY81], Eichmair used the Jang equation to prove the inequality $E\geq 0$, and then by analyzing the case of equality in that proof, he showed that the $E=0$ case can only occur when the Jang solution provides a graphical isometric embedding of $(M,g)$ into Minkowski space with $k$ as its second fundamental form. The main result of this article is to replace this argument for the “second step” with one that naturally extends the methods used to prove Theorem 1.3 and is self-contained in the sense that it does not depend on how one proves that $E\geq|P|$ (nor on how one proves $E\geq 0$).

Combining the two previous theorems gives the following result.

This fact was essentially already proved in dimension 3 by Beig and Chruściel [BC96] and for general spin manifolds by Chruściel and D. Maerten [CM06]. These results required a slightly stronger assumption than (1.3) but do not require $C^{5}\times C^{4}$ regularity. More recently, Sven Hirsch and Yiyue Zhang [HZ23] gave a new proof in dimension $3$ that avoids assuming any of (1.3), (1.4), or $C^{5}\times C^{4}$ regularity.

The basic outline of the proof of Theorem 1.4 is the following: As in the proof of Theorem 1.3, we use the fact that $(g,k)$ minimizes a modified Regge-Teitelboim Hamiltonian subject to a constraint, and then we invoke Lagrange multipliers to construct lapse-shift pairs that satisfy a Killing initial data type of equation. The difference is that when $E=0$, $(g,k)$ minimizes many different Regge-Teitelboim Hamiltonians, so we can actually construct an entire $(n+1)$-dimensional space of these lapse-shift pairs that can be thought of as being asymptotic to the $(n+1)$-dimensional space of translational Killing fields on Minkowski space as we approach spatial infinity. The essential new ingredient is that by invoking our recent work [HL], we can also make sure that these lapse-shift pairs satisfy what we call the “$J$-null-vector equation.” Moreover, having so many solutions implies that $(g,k)$ is vacuum. All of this is explained in Section 2, culminating in the statement of Lemma 2.2. Observe that proving that $(g,k)$ is vacuum is itself a difficult task. Note that the vacuum condition $\mu=|J|_{g}=0$ is a much stronger condition than the “borderline” $\mu=|J|_{g}$ case of the dominant energy condition. The fact that $E=|P|$ implies $\mu=|J|_{g}$ is a previously unstated direct consequence of [HL20, Theorem 5.2] and [HL, Theorem 6], but the aforementioned pp-wave counterexamples from [HL, Example 7] demonstrate that $E=|P|$ does not imply vacuum.

Once we know that $(g,k)$ is vacuum, it follows that each of these lapse-shift pairs is actually vacuum Killing initial data for $(g,k)$, and we can extend them to become actual Killing fields on the vacuum Killing development of $(g,k)$. The next step is to show that on an asymptotically flat Lorentzian manifold, having such a space of Killing fields that are asymptotic to the $(n+1)$ translational directions implies that the Lorentzian manifold must be Minkowski space. This observation is described in Theorem 1.6 below, which requires no curvature assumptions.

As a direct consequence, the above theorem also implies a version for asymptotically flat Riemannian manifolds, which is included in Corollary 3.2 below.

Note that we do not assume $(\mathbf{N},\mathbf{g})$ is complete in Theorem 1.6. If $(\mathbf{N},\mathbf{g})$ is complete without boundary, then it must be Minkowski space by the Killing-Hopf theorem. To rule out the possibility of having a weakly outer trapped boundary, we prove the following general result.

When the spatial dimension is less than $8$, Theorem 1.7 follows from an elegant argument of A. Carrasco and M. Mars (used to prove [CM08, Corollary 1]). The dimension restriction is needed because their proof relies on the existence theory of smooth MOTS by Eichmair [Eic09]. Our proof is valid in all dimensions and also applies to more general situations (e.g. the stationary assumption can be significantly relaxed). See Theorem 4.1.

For the special case of static spacetime metrics, which are of the form $\mathbf{g}=-f^{2}dt^{2}+g$ where $f>0$, the null energy condition becomes the assumption

Specializing even more to the case where $f=1$ reduces to a well-known classical theorem for manifolds with nonnegative Ricci curvature, stated as Theorem 4.2. Note that many examples, such as static vacuum metrics and electro-vacuum metrics, with any cosmological constant, satisfy condition (1.5).

The paper is organized as follows. In Section 2, we explore the Lagrange multiplier method carried out in [HL20] for the $E=0$ case using the new results from [HL] to construct an $(n+1)$-dimensional space of asymptotically translational vacuum Killing initial data. In Section 3, we prove Theorem 1.6 and then complete the proof of the main result, Theorem 1.4, for the case of no boundary. Finally, the possibility of having a nonempty weakly outer trapped boundary is ruled out by Theorem 1.7, whose proof is presented in Section 4 and can be read independently from all prior sections.

We follow closely the definitions and notations in [HL20]. In particular, we refer the reader to [HL20, Section 2] for the definitions of the weighted Hölder space $C^{k,\alpha}_{-q}(M)$ and the weighted Sobolev space $W^{k,p}_{-q}(M)$. Assume $p>n$, and let $\mathscr{M}^{2,p}_{-q}$ denote the set of symmetric $(0,2)$-tensors $\gamma$ such that $\gamma-g_{\mathbb{E}}\in W^{2,p}_{-q}(M)$ and $\gamma$ is positive definite at each point. By the inclusion $C^{k,\alpha}_{-q}\subset W^{k,p}_{-q^{\prime}}$, for any $q^{\prime}<q$, whenever we work on the weighted Sobolev spaces, we shall assume the decay rate $q$ in the weighted Sobolev spaces to be slightly smaller than the assumed asymptotic decay rate appearing in Definition 1.1.

Initial data $(g,k)$ can be equivalently described by a pair $(g,\pi)$ where $\pi$ is a symmetric $(2,0)$-tensor called the conjugate momentum tensor, which is related to the $(0,2)$-tensor $k$ via the equation

By slight abuse of vocabulary, we will refer to $(g,\pi)$ as initial data. We define the constraint map $\Phi$ on initial data by

where $\mu$ and $J$ are functions of $(g,\pi)$, defined by (1.2) combined with (2.1). Recall that the dominant energy condition (or DEC) is the condition $\mu\geq|J|_{g}$.

As in [HL], given an initial data set $(M,g,\pi)$ and a smooth function $\varphi$ on $M$, we introduce the $\varphi$-modified constraint operator, which is defined on initial data $(\gamma,\tau)$ on $M$ by

where $J$ denotes the current density of $(g,\pi)$, and $(\gamma\cdot J)^{i}:=g^{ij}\gamma_{jk}J^{k}$ denotes the contraction of $\gamma$ and $J$ with respect to $g$. The crucial property of $\Phi^{\varphi}_{(g,\pi)}$ is that for any $(\gamma,\tau)$, if $(g,\pi)$ satisfies the DEC, $|\gamma-g|_{g}<1$, and

then $(\gamma,\tau)$ also satisfies the DEC (see Lemma 3.4 of [HL]). Note that this is a property that the ordinary constraint operator $\Phi$ does not have, but since $\Phi^{\varphi}_{(g,\pi)}$ only differs from $\Phi$ by an affine function of $(\gamma,\tau)$, it behaves similarly from a PDE perspective.

Throughout this article, a lapse-shift pair $(f,X)$ on $M$ will simply refer to a function $f$ and a vector field $X$ on $M$. Assuming that $(g,\pi)$ is locally $C^{5}\times C^{4}$, Theorem 4.1 of [HL], which was the main technical result of [HL], tells us⁴⁴4Take $Z=J$ in the statement of that theorem. that there exists a dense subset $\mathcal{D}\subset C^{3}(\operatorname{Int}M)$ such that if $\varphi\in\mathcal{D}$, then any lapse-shift pair $(f,X)$ solving

in the interior of $M$ must also satisfy the pair of equations

where the second equation is what we call the $J$-null-vector equation. Choosing any bounded $\varphi\in\mathcal{D}$ ensures that the terms in (2.2) involving $\varphi$ have the appropriate fall-off rates so that

We will use the method of [HL20] to construct solutions to (2.3) and hence also (2.4). The following definition simply generalizes Definition 5.1 of [HL20] from the case $\varphi=0$ to allow for general $\varphi$.

As explained in [HL20], although the terms in (2.5) are not individually well-defined for arbitrary $(\gamma,\tau)\in\mathscr{M}^{2,p}_{-q}\times W^{1,p}_{-1-q}$, the functional $\mathcal{H}$ extends to all of $\mathscr{M}^{2,p}_{-q}\times W^{1,p}_{-1-q}$ in a natural way.

In what follows, we will assume that $a$ and $b$ satisfy $2a\geq|b|$. Continuing to follow [HL20] (with input from [LLU22] for the boundary case), we define

We claim that that if $(g,\pi)$ satisfies the DEC with $\partial M$ weakly outer trapped (if nonempty) and has $E=|P|=0$, then $(g,\pi)$ is a local minimizer of $\mathcal{H}$ in $\mathcal{C}_{(g,\pi)}$. To see why, observe that for all nearby $(\gamma,\tau)\in\mathcal{C}_{(g,\pi)}$, $(\gamma,\tau)$ also satisfies the DEC (as mentioned above, due to [HL, Lemma 3.4]) and has weakly outer trapped boundary, so the assumption that the positive mass theorem is true near $(g,\pi)$ tells us that $E(\gamma,\tau)\geq|P(\gamma,\tau)|$. (Technically, since $(\gamma,\tau)$ only has Sobolev regularity, we must invoke [HL20, Theorem 4.1] and [LLU22, Lemma 4.2].) Therefore

where we used the assumptions that $2a\geq|b|$ and $(g,\pi)$ has $E=|P|=0$. Hence the claim is true.

Thus $(g,\pi)$ locally minimizes the functional $\mathcal{H}$ subject to the constraint $\mathcal{C}_{(g,\pi)}$. In the finite dimensional setting, as long as the constraint carves out a smooth submanifold of a linear space, we can use the method of Lagrange multipliers to gain information about the minimizer. In our more general Banach space setting, in order to construct Lagrange multipliers, all we need is for $\mathcal{C}_{(g,\pi)}$ to be the level set of a map on $\mathscr{M}^{2,p}_{-q}\times W^{1,p}_{-1-q}$ whose linearization is surjective (see [HL20, Theorem D.1] for details). For $\varphi=0$, this surjectivity was established in [HL20, Lemma 2.10] for the case of no boundary, and in [LLU22, Proposition 3.9] when boundary is present. Meanwhile, the introduction of $\varphi$ does not change anything in these arguments.

Invoking Lagrange multipliers exactly as in [HL20] (or [LLU22] for the boundary case) leads to the existence of a lapse-shift pair $(f,X)\in C^{2,\alpha}_{\mathrm{loc}}(\operatorname{Int}M)$ such that

where $O_{2+\alpha}(|x|^{-q})$ denotes an object that decays in $C^{2,\alpha}_{-q}$. Although we will not use this fact elsewhere in the paper, we remark that the existence of even one such lapse-shift pair implies that $M$ can only have one asymptotically flat end.⁵⁵5The same argument also applies to the $E=|P|$ case and leads to the following fact. If $(M,g,\pi)$ is a complete asymptotically flat initial data set satisfying the dominant energy condition with $E=|P|$, and if the positive mass theorem is true near $(g,k)$, then $M$ has only one end. Note that this is consistent with the pp-waves counterexamples having $E=|P|\neq 0$ from [HL, Example 7], which all have only one end. This is because [CÓM81, Theorem 3.3] implies that it is impossible for a nontrivial solution of (2.3) to asymptotically vanish in an end.

As described above, as long as we select $\varphi$ so that Theorem 4.1 of [HL] applies, this $(f,X)$ also satisfies (2.4) on $\operatorname{Int}M$. Since we can construct such a solution for any $(a,b)\in\mathbb{R}^{n+1}$ satisfying $2a\geq|b|$, and the space of all $(f,X)$ satisfying (2.4) forms a vector space $\mathcal{N}$, it follows that for any $(a,b)\in\mathbb{R}^{n+1}$, we obtain a lapse-shift pair $(f,X)$ which is both asymptotic to $\left(a,\sum b^{i}\frac{\partial}{\partial x^{i}}\right)$ in the desired end and also satisfies (2.4). Moreover, the dimension of $\mathcal{N}$ is at least $n+1$.

But Corollary 6.6 of [HL] states that if the dimension of $\mathcal{N}$ is greater than 1, then $(g,\pi)$ must be vacuum initial data. Since $(g,\pi)$ is vacuum, $\Phi^{\varphi}_{(g,\pi)}=\Phi^{0}_{(g,\pi)}=\Phi$. The intuition behind Corollary 6.6 of [HL] is the following: $(f,X)$ lying in the kernel of the overdetermined Hessian-type operator $\left(\left.D\Phi^{0}_{(g,\pi)}\right|_{(g,\pi)}\right)^{*}$ already means that $(f,X)$ is determined by its $1$-jet at any point. If there is a point $p$ where $J\neq 0$, we can use (2.4) to write $\nabla f$ in terms of $f$, and hence the $0$-jet of $f$ at $p$ determines the $1$-jet of $f$ at $p$, which then determines the $1$-jet of $(f,X)$ at $p$ by the $J$-null-vector equation, which determines $(f,X)$ everywhere. For details, see [HL].

Putting together the arguments described in this section, we have proved the following lemma.

In this section, we prove Theorem 1.6, whose proof is completely unrelated to the material in the previous section. After that, we will explain how to combine Lemma 2.2 with Theorem 1.6 to prove our main theorem, Theorem 1.4.

But first, we recall how the Killing condition for vector fields on a spacetime restricts to a spacelike slice. This computation was made in [BC97] (written in different notation).⁶⁶6We provided details of this computation in lapse-shift coordinates in the proof of [HL, Corollary B.3], but this required the assumption that $U$ is transverse to $\mathbf{Y}$. Performing the computation in Gaussian coordinates as in [BC97] shows that this assumption is unnecessary.

We recall the statement of Theorem 1.6.

By taking $(\mathbf{N},\mathbf{g})$ to be a trivial product, we can easily obtain a corresponding theorem in Riemannian geometry.

Again, note that there is no completeness assumption here, but when $(M,g)$ is complete (without boundary), it follows from the Killing-Hopf Theorem (and a simple covering space argument as in the proof of Theorem 1.4 below) that $(M,g)$ is isometric to Euclidean space.

In order to connect Lemma 2.2 to Theorem 1.6, we will use the concept of a Killing development, first introduced by Beig and Chruściel [BC96].

Observe that $\mathbf{Y}_{0}=\frac{\partial}{\partial x^{0}}$ is a Killing field on $(\mathbf{N},\mathbf{g})$.

Now suppose that $(M,g,\pi)$ is a vacuum initial data set equipped with a vacuum KID $(f_{0},X_{0})$ with $f_{0}>0$ (recall Definition 2.3). In this case, one can verify that the Killing development $(\mathbf{N},\mathbf{g})$ is vacuum and contains the initial data set $(M,g,\pi)$ in the sense that the $x^{0}=0$ slice is just the Riemannian manifold $(M,g)$ with second fundamental form $k$ (corresponding to $\pi$ via (2.1)). For details, see [BC96] or [HL, Appendix B]. Furthermore, by (3.2), one can see that every Killing field $\mathbf{Y}$ on $(\mathbf{N},\mathbf{g})$ restricts to a vacuum KID $(f,X)$ on $(M,g,\pi)$. It turns out that this provides a one-to-one correspondence between Killing fields on $(\mathbf{N},\mathbf{g})$ and vacuum KIDs on $(M,g,\pi)$. This fact is a simple special case of a theorem of Beig and Chruściel [BC97] that describes more general non-vacuum circumstances under which this one-to-one correspondence holds. A similar correspondence also holds for the maximal vacuum development (rather than the Killing development) by a classical result of V. Moncrief [Mon75].