1 Introduction↩︎

General relativity is our foremost theory of gravity on classical scales, and its predictive power largely stems from the well-posedness of the initial value problem formulation: Given a global “instant in time" characterized by a complete Riemannian 3-manifold \((\Sigma, \tilde{g})\) equipped with a symmetric \((0,2)\)-tensor \(\widetilde{K}_{ab}\) subject to the Einstein constraint equations 7 8 , one seeks a globally hyperbolic Lorentzian 4-manifold \((\widehat M, \widehat g)\) satisfying Einstein’s gravitational field equation into which \(\Sigma\) embeds as a Cauchy surface with the induced metric \(\tilde{g}\) and the second fundamental form \(\widetilde{K}_{ab}\). Since Choquet-Bruhat’s seminal result, proven over half a century ago [1], [2], establishing the existence of such a (maximal) spacetime \((\widehat M, \widehat g)\) under general hypotheses, the mathematical relativity community has been broadly interested in characterizing the geometric structure of \((\widehat M, \widehat g)\) satisfying various conditions on the initial data set \((\Sigma, \tilde{g}, \widetilde{K})\), with particular emphasis on the asymptotically flat case modeling an isolated system. Features of particular interest include the existence of past and future null infinity \(\mathscr I^\pm\) with particular prescribed curvature behaviors there, and much work has been done on establishing the stability of these behaviors under asymptotically flat perturbations of initial data sets extracted from fundamental explicit solutions such as the Minkowski, Schwarzschild, and Kerr spacetimes. Such works invoke different notions of asymptotic flatness to different effects, and there is some interest in the question of how stringent one’s notion of asymptotic flatness on \((\Sigma,\tilde{g}, \widetilde{K})\) must be to ensure that desirable physical features are realized within \((\widehat M, \widehat g)\).

The first work which established the broad existence of solutions to the initial value problem for Einstein’s equation admitting a complete notion of infinity, outside of the few known explicit spacetimes, was Christodoulou and Klainerman’s proof of the nonlinear stability of Minkowski space [3]. They proved the stability of key geometric features of the maximal vacuum Cauchy development of sufficiently “small" initial data sets that are asymptotically flat in the sense that there exists a coordinate system \((x_1,x_2,x_3)\) on the complement of a compact set in \(\Sigma\) in which⁶ \[\begin{align} \tilde{g}_{ij} & = \left(1+\frac{2M}{r}\right)\delta_{ij} + o_4(r^{-3/2}), \tag{1} \\ \widetilde{K}_{ij} & = o_3(r^{-5/2}). \tag{2} \end{align}\] We define these falloff conditions as type (CK). This conclusion was later generalized by Bieri [4], [5] to initial data sets with the relaxed falloff conditions (type (B)) \[\begin{align} \tilde{g}_{ij} & = \delta_{ij} + o_3(r^{-1/2}), \tag{3} \\ \widetilde{K}_{ij} & = o_2(r^{-3/2}). \tag{4} \end{align}\] In each case, one has different degrees of control on the induced falloff rates of various curvature components as one approaches \(\mathscr I^\pm\). As also investigated by Bieri [6]–[8], different control still is afforded by the intermediate falloff conditions (type (A)) \[\begin{align} \tilde{g}_{ij} & = \delta_{ij} + h_{ij} + o_3(r^{-3/2}), \tag{5} \\ \widetilde{K}_{ij} & = o_2(r^{-5/2}), \tag{6} \end{align}\] where each component of \(h_{ij}\) is homogeneous of degree \(-1\), representing an anisotropic mass.

To better understand the asymptotic behavior of the geometry of spacetimes compatible with these three falloff conditions, and to assess the status of physically-motivated conjectures associated to that geometry, it is interesting and necessary to construct examples of constraint-satisfying initial data sets in each of the classes (CK), (B), and (A), requiring the ability to prescribe the data’s falloff rate. While the ultimate act of construction is a numerical task, it is vital to first develop a mathematical blueprint of the procedure underpinned by theorems guaranteeing its efficacy. A subset of the present authors [9] recently provided such a blueprint in the rather restrictive setting of Brill wave initial data sets, requiring both azimuthal and time symmetry. These results, however, were insufficient to guarantee the construction of the more delicate type (A) data. That work numerically constructed an initial data set of type (B) whose evolution cannot satisfy Strominger’s antipodal conjecture in [10], which is a topic of interest in the literature [11]–[13]. That conjecture posits a symmetry of a particular component of the electric part of the Weyl curvature (\(\Psi_2\) in the Newman-Penrose formalism, or \(\rho\) in Christodoulou and Klainerman’s notation) evaluated asymptotically along \(\mathscr I^+\) as compared to \(\mathscr I^-\): The conjecture says that if one evaluates this component’s limit on each of \(\mathscr I^\pm\) and proceeds along each to spatial infinity, the two resulting functions on the sphere should agree up to composition with the antipodal map on the sphere, \(p \mapsto -p\).

In this work, we significantly generalize the results of [9] both to do away with any symmetry assumptions and to handle type (A) data, establishing a sequence of results guaranteeing that one can control the falloff rates and the ADM momenta of initial data sets generated via the widely utilized conformal method. We use these results to construct an initial data set of type (A) whose evolution cannot satisfy the antipodal conjecture of [10], restricting the class of spacetimes to which this conjecture could apply. Since the conjecture holds trivially for spacetimes evolved from type (CK) data, our results indicate that the domain on which the conjecture might hold nontrivial content must be somewhat small. For simplicity of presentation, we work with \(\Sigma = \mathbb{R}^n\), but the analytical results generalize straightforwardly to all asymptotically Euclidean manifolds, even with multiple ends. Of course, one takes \(n = 3\) in standard general relativity (as above), but we have found it interesting and instructive to generalize the mathematical results presented here to arbitrary dimension \(n \geq 3\).

We work with finite regularity measured in Sobolev scales, and in particular with weighted Sobolev spaces. This common setting for building asymptotically Euclidean initial data usually focuses on decay rates within the isomorphism range of the applicable elliptic operators (e.g. [14], [15]). While this approach suffices for establishing existence and uniqueness, the slow decay rates lose asymptotic information. We work instead with faster decay rates that expose asymptotic structure, and with particular care at the critical transition rates where the Fredholm index of the operators changes (see Proposition 5 and Lemma 4). Although we work exclusively in the context of constant mean curvature (CMC) solutions of the constraints, the tools developed here are equally applicable to perturbative constructions of non-CMC initial data.

We remark that Dain and Friedrich published a work in 2001 [16] having a fair bit of conceptual overlap with (and a remarkably similar title to) the present work. They also proved a number of interesting results allowing control of asymptotic features of initial data sets constructed via the conformal method, including the prescription of a leading-order Euclidean transverse traceless tensor in \(\widetilde{K}_{ij}\) allowing control of the ADM momentum (cf. our Theorem 14). However, there is little technical overlap with the present work, either in precisely what is proved or in the analytical techniques employed. Namely, Dain and Friedrich worked (appropriately to their objectives) in the restricted setting of initial data sets admitting a conformal compactification at spatial infinity, so they were interested in obtaining asymptotics which amount to analyticity at spatial infinity, asking that \(\tilde{g}_{ij}\) and \(\widetilde{K}_{ij}\) admit a full power series expansion in integer powers of \(1/r\). The present work seeks to provide a simple procedure for constructing initial data sets in a much broader category, with emphasis on controlling the structure of the leading-order terms (especially in \(\tilde{g}_{ij}\)) and the precise (non-integral) decay rate of subleading terms, and making no demand of a full power series expansion. Broadly, we are interested in the construction of initial data sets conforming to varying notions of asymptotic flatness for the ultimate purpose of probing the extent to which physically interesting features of spacetime depend on this notion, but the results in [16] are not amenable to handling a variety of falloff behaviors. In particular, results in [16] could be described as allowing the construction of highly specialized examples of type (CK) data, but they are not relevant to the construction of type (A) data which is not (CK), which is among our chief concerns.

In a similar vein, in 1996 Beig and O’Murchadha used spherical harmonic expansions to investigate the asymptotics of solutions to the CMC vacuum momentum constraint equation 10. Working in the smooth setting in dimension \(n = 3\), they fully characterized the asymptotic forms and decay rates of solutions to the momentum constraint with seed data having given arbitrarily fast, non-exceptional falloff compatible with type (CK) data. This characterization was made up to a collection of constant “multipole moments", including the ADM momentum components, which can be prescribed via an appropriate choice of seed data (but not anticipated for an arbitrary choice of seed data). We have built upon Beig and O’Murchadha’s work by doing the following: extending some of their conclusions to lower regularity and arbitrary dimension; establishing that similar conclusions hold for solutions to the Hamiltonian constraint equation eq. ¿eq:eqn:hamiltonian? (and hence one’s entire initial data set); showing that the ADM momentum components can be computed directly from the seed data; and investigating the asymptotics of solutions if the seed data has the exceptional”threshold" falloff \(O(r^{2-n})\), allowing the construction of type (A) data which is not (CK).

The remainder of this work is organized as follows: In Section 2, we establish our notational conventions and discuss the procedure for carrying out the conformal method. To proceed with obtaining solutions of the constraint equations using the conformal method, one must solve the “conformal constraint equations" eq. ¿eq:eqn:lichnerowicz? and eq. ¿eq:eqn:conformalmomentum? for the positive scalar \(\varphi\) and the vector field \(W\). We note in that section that, so long as we presume that the mean curvature \(\text{tr} \,_{\tilde{g}}(\widetilde{K})\) for the initial data set is zero, these conformal constraint equations semi-decouple. In Section 3, we discuss the conformal momentum constraint equation eq. ¿eq:eqn:conformalmomentum? by analyzing the mapping properties of the vector Laplacian operator, and we show how to choose the”seed data" so that the extrinsic curvature of the solution has various falloff properties. We show there that the ADM momentum of the ultimate initial data set may be prescribed a priori through the choice of the seed data. In Section 4, we discuss the Hamiltonian constraint equation 11 in a manner closely analogous to the discussion of the momentum constraint, showing that the seed data can be chosen so that the conformal factor \(\varphi\) implements the desired falloff properties. In Section 5, we summarize the implications of our results for the construction of full initial data sets satisfying the desired falloff rates– those of type (CK), (A), and (B) data–, collated into Theorems 12, 13, and 14. In Section 6, we produce a numerical example of type (A) data whose evolution cannot satisfy the antipodal conjecture in [10]. We briefly summarize and conclude in Section 7. Appendix 8 presents a largely self-contained Fredholm theory of vector Laplacians, analogous to results for the scalar Laplacian-type operators in [17].

2 Preliminaries and The Conformal Method↩︎

The problem of constructing initial data sets \((\Sigma,\tilde{g}, \widetilde{K})\) is more analytically challenging than one might like primarily due to the constraint equations, which are geometrically imposed relations between the geometry of \((\widehat M, \widehat g)\) and \((\Sigma, \tilde{g}, \widetilde{K})\) which read (in vacuum) \[\begin{align} R(\tilde{g}) - |\widetilde{K}|_{\tilde{g}}^2 + \text{tr}_{\tilde{g}}(\widetilde{K})^2 & = 0, \tag{7} \\ \text{div}_{\tilde{g}}(\widetilde{K}) - d(\text{tr}_{\tilde{g}}(\widetilde{K})) & = 0. \tag{8} \end{align}\] The first of these is the Hamiltonian constraint, and the second is the momentum constraint. There is a broad literature on the study of the constraint equations, spanning from methods of constructing solutions, either from scratch or from existing solutions, to properties that solutions enjoy, such as the positive mass theorem and the Riemannian Penrose inequality. We are interested in the former, and we seek to construct initial data sets corresponding to a maximal time slice in \(\widehat M\), wherein \(\text{tr} \,_{\tilde{g}}(\widetilde{K}) = 0\). The vacuum constraint equations then take the following form: \[\begin{align} R(\tilde{g}) & = |K|_{\tilde{g}}^2, \tag{9} \\ \text{div}_{\tilde{g}}(\widetilde{K}) & = 0 \tag{10}. \end{align}\] We pose these equations on \(\mathbb{R}^n\), to be solved for \(\tilde{g}\) and \(\widetilde{K}\). These are underdetermined, as they impose only \(n+2\) conditions (including \(\text{tr} \,_{\tilde{g}}(\widetilde{K}) = 0\)) on the two symmetric (0,2)-tensors \(\tilde{g}\) and \(\widetilde{K}\). To determine a parameterized set of solutions, one must supplement these conditions, effectively specifying part of \(\tilde{g}\) and \(\widetilde{K}\) and leaving the rest of their content to be determined by the constraints.

A historically powerful approach in the analysis of these equations, known as the conformal method [18], [19], is to specify the conformal class of \(\tilde{g}\), amounting to stipulating an arbitrary “unphysical" metric \(g\), as well as a traceless and symmetric \((0,2)\)-tensor \(A_{ab}\) (the conformal velocity) and a scalar function \(N = 1+\delta N : \mathbb{R}^n \to \mathbb{R}_+\) (the lapse function). Once this seed data comprised of \(g\), \(A_{ab}\), and \(\delta N\) has been specified on \(\mathbb{R}^n\), one sets \[K_{ab} := \frac{1}{2N}\left( A_{ab} - (\mathbb{L}_g W)_{ab} \right)\] and \(q_n := \frac{2n}{n-2}\) and proceeds to solve the PDE system \[\begin{align} -(q_n+2)\Delta_g \varphi + R(g) \varphi & = \varphi^{-q_n-1} |K|_g^2, \tag{11}\\ \text{div}_g\left( \frac{1}{2N} (\mathbb{L}_g W)_{ab} \right) & = \text{div}_g\left( \frac{1}{2N} A_{ab} \right) \tag{12} \end{align}\] for the vector field \(W^a\) and the positive scalar field \(\varphi\). Here, \(\mathbb{L}_g\) is the conformal Killing operator, which is defined to be the following symmetrized and traceless covariant derivative: \[\mathbb{(}\mathbb{L}_g W)_{ab} := \nabla_a W_b + \nabla_b W_a - \frac{2}{n} g_{ab} \nabla_c W^c.\] Once \(W^a\) and \(\varphi\) have been obtained by solving equations 11 and 12 , one sets \[\begin{align} \tilde{g} & := \varphi^{q_n-2} g, \notag \\ \widetilde{K}_{ab} & := \varphi^{-2} K_{ab}. \notag \end{align}\] Using the conformal divergence identity \(\text{div}_{\tilde{g}}(\varphi^{-2} K) = \varphi^{-q_n} \text{div}_{g}(K)\) for a traceless and symmetric \((0,2)\)-tensor \(K_{ab}\) as well as the well-known conformal transformation of scalar curvature, one readily confirms that this choice of the physical metric \(\tilde{g}\) and the second fundamental form \(\widetilde{K}_{ab}\) solves the constraint equations 9 10 .

A key feature of the conformal method for maximal data is that the conformal constraint equations 11 12 are semi-decoupled in the sense that 12 can be solved for \(W^a\) independently of \(\varphi\), and then equation 11 – the Lichnerowicz equation– can be solved for \(\varphi\). We note that in the Einstein vacuum case discussed here, these equations admit unique solutions for each choice of the seed data in appropriate function spaces (discussed below) [20].

We are interested in prescribing the falloff rates of \(\tilde{g}\) and \(\widetilde{K}\) by means of building them into the seed data \(g\), \(A_{ab}\), and \(\delta N\). To leverage analytical control over the decay rates of these quantities, we operate in weighted Sobolev spaces, which we now define:

Intuitively (and literally, provided \(k > n/p\)), the role of the weight \(\delta\) is to impose that these spaces are comprised of functions that decay more quickly than \(|x|^\delta\) at infinity, being \(o(|x|^\delta)\), with control of (distributional) derivatives up to order \(k\). The “plus" space \(W^{k,p}_{\delta^+}\) further allows terms decaying at exactly the rate \(|x|^\delta\), as well as for log terms such as \(\left<x\right>^\delta \log(\left<x\right>)\), and the”minus" space \(W^{k,p}_{\delta^-}\) ensures one has some power law decay strictly faster than \(|x|^\delta\). A weight \(\delta\) is called exceptional if it is an integer and either \(\delta\le 2-n\) or \(\delta\ge0\), and non-exceptional otherwise. Exceptional weights are precisely the possible growth rates of homogeneous harmonic functions on flat space– these act as barriers at which the analytical properties of our elliptic operators acting on these spaces, such as their kernel and cokernel dimensions, may change. Extension of Definition 1 to negative values of \(k \in \mathbb{Z}\) is necessary for complete coverage of the results obtained herein, but such technicalities are reserved for discussion in the appendix: this classical definition is sufficient to parse the main body of this work. We now define asymptotically Euclidean metrics.

We note that the condition \(k>n/p\) ensures that \(g_{ab}\) is Hölder continuous and, since \(\tau < 0\), that it converges uniformly to \(\delta_{ab}\) as \(x\to\infty\). In keeping with this definition, we work with metrics and lapses in the following category:

We remark that the core results of this work are true under the relaxed hypothesis \(k > n/p\), provided that one additionally assumes that \(g\) admits no nontrivial conformal Killing fields (vector fields in the kernel of \(\mathop{\mathrm{{\mathbb{L}}}}\)) of class \(W^{k,p}_{\delta}\) for any \(\delta < 0\), i.e. which vanish at infinity⁷. The following essential multiplication lemma is an immediate consequence of the more general Lemma 6 discussed in Appendix 8, and we note that it suffices for all of our work here.

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3 The Momentum Constraint↩︎

The asymptotics of a second fundamental form constructed using the conformal method follow from the mapping properties of the lapse-weighted vector Laplacian \(P_{g,N} := \mathop{\mathrm{\mathrm{div}}}_g (\frac{1}{2N} \mathop{\mathrm{{\mathbb{L}}}})\) appearing in equation 12 along with the asymptotic structure of its source term, the conformal velocity \(A_{ab}\). Indeed, setting \(Z_a := \nabla^b \left( \frac{1}{2N} A_{ab} \right)\), equation (12 ) becomes the inhomogenenous linear equation \[\label{eqn:pmomentum} (P_{g,N} W)_a = Z_a\tag{13}\] which is the main focus of this section.

A routine computation using Lemma 1, under Assumption 1, yields that for a smooth vector field \(X^a\) one has \[\label{eqn:pdiff} (P_{g,N} X)_a = (\overline{P} X)_a + \sum_{|\alpha|\le 2} B_{ab}^\alpha \partial_\alpha X^b,\tag{14}\] where \(\overline{P} := P_{\overline{g}, 1} = \frac{1}{2} \overline{\mathop{\mathrm{\mathrm{div}}}}(\overline{\mathop{\mathrm{{\mathbb{L}}}}})\) is the Euclidean vector Laplacian (note that we have an extra factor of \(1/2\) relative to some other works), associated with the Euclidean metric \(\overline{g}_{ab}=\delta_{ab}\), and where each \(B_{ab}^{\alpha} \in W^{k-2+|\alpha|,p}_{\tau-2+|\alpha|}\) depends on \(g\) and \(N\). Since \(\tau-2+|\alpha|<0\) for each of these coefficents, the mapping properties of \(P_{g,N}\) can be deduced from those of \(\overline{P}\). Appendix 8 contains these details, which imply the following:

The isomorphism range \(2-n< \delta < 0\) appearing Proposition 1 is well studied in the literature and is a routine tool used to find solutions of the momentum constraint (e.g. [14], [15]). Understanding the asymptotics of these solutions, however, requires a finer analysis of the next decay range \(1-n< \delta < 2-n\). Part [part:P-solvable] relates solvability in this range to elements of the kernel of \(P_{g,N}\) that are bounded at infinity, and we have the following characterization of this kernel:

Now suppose \(Z^a \in W^{k-2,p}_{\delta-2}\) with fast decay \(1-n<\delta<2-n\). Although Proposition 1 ensures equation 13 can be solved for \(W^a\in W^{k,p}_{(2-n)^+}\), the solution does not have faster decay in general. Instead, Proposition 1([part:P-solvable]) indicates that \(W^a\in W^{k,p}_\delta\) only if each of the integrals \[\label{eqn:momentumints} \mathcal{O}_{i}(Z_a) := \int_{\mathbb{R}^n} Z_a k_{(i)}^a dV_g\tag{15}\] vanishes. In fact, these obstructions turn out to be closely related to the ADM momentum, which we now examine.

Consider a \(C^1\) symmetric, traceless tensor \(S_{ab}\) with \(O(r^{1-n})\) growth. Assuming units with \(8\pi G=1\) the ADM momentum of \(S_{ab}\) has components \[\label{eqn:admmomentum} \mathcal{P}_i(S_{ab}) := \lim_{R\to\infty} \int_{\partial B_R} S_{ab} n^a e_{(i)}^b\; d\overline{A},\tag{16}\] with \(n^a := x^a/r\) the Euclidean radial unit vector, so long as the limit exists. We can express this quantity equally well in terms of the metric \(g_{ab}\), its area element \(dA\), its normal vector \(\nu^a\) to the spheres, and the kernel basis \(k^a_{(i)}\). Because \(n^a-\nu^a\), \(dA-d\overline{A}\) and \(k^a_{(i)}-e^a_{(i)}\) each decays at the rate \(O(r^\tau)\), we find \[\label{eqn:admmomentum-2} \mathcal{P}_i(S_{ab}) := \lim_{R\to\infty} \int_{\partial B_R} S_{ab} \nu^a k_{(i)}^b\; dA.\tag{17}\] The divergence theorem then yields \[\begin{align} \int_{\partial B_R} S_{ab} \nu^a k_{(i)}^b\; dA & = \int_{B_R} \nabla^a (S_{ab} k_{(i)}^b)\; dA \nonumber \\ & = \int_{B_R} S_{ab} (\nabla^a k_{(i)}^b)\; dA + \int_{B_R} (\nabla^a S_{ab}) k_{(i)}^b\; dA. \end{align}\] If \((\nabla^a S_{ab}) k_{(i)}^b\) and \(S_{ab} (\nabla^a k_{(i)}^b)\) are \(L^1\) functions, we can take a limit in \(R\), and we define \[\begin{align} \mathcal{G}_i(S_{ab}) & := \int_{\mathbb{R}^n} S_{ab} \nabla^a k_{(i)}^b dV_g,\\ \mathcal{R}_i(S_{ab}) & := \int_{\mathbb{R}^n} (\nabla^a S_{ab}) k_{(i)}^b dV_g. \end{align}\] Note that the quantity \(\mathcal{R}_i\) defined here for symmetric, trace-free \((0,2)\)-tensors is related to the obstruction values \(\mathcal{O}_i\) for covectors by \[\mathcal{R}_i( S_{ab} ) = \mathcal{O}_i( \nabla^a S_{ab}).\] The integrability conditions for the integrands defining \(\mathcal{G}_i(S_{ab})\) and \(\mathcal{R}_i(S_{ab})\) ensure that the ADM momentum \(\mathcal{P}_i\) is well defined if both \(\mathcal{G}_i\) and \(\mathcal{R}_i\) are, and we have \[\mathcal{P}_i(S_{ab}) = \mathcal{G}_i(S_{ab}) + \mathcal{R}_i(S_{ab}).\] This decomposition depends on the specific kernel basis \(k^a_{(i)}\). In particular, since the operator \(P_{g,N}\) depends on the lapse, so does the splitting. For reasons we explain below, we call \(\mathcal{G}_i\) the gravitational momentum, whereas \(\mathcal{R}_i\) is the residual momentum. For the moment, it suffices to observe that if \(S_{ab}\) is trace-free and satisfies the vacuum momentum constraint, then \(\mathcal{R}_i(S_{ab})=0\) and hence \(\mathcal{P}_i(S_{ab})=\mathcal{G}_i(S_{ab})\).

Now recall the vector field \(W^a\in W^{k,p}_{(2-n)^+}\) introduced above solving \((P_{g,N} W)_a = Z_a\), where \(Z^a\in W^{k-2,p}_{\delta-2}\) with \(1-n < \delta < 2-n\). We claim that the ADM momentum of \(1/(2N) (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab}\) is well defined and is determined precisely in terms of the obstruction coefficients of \(Z^a\): \[\mathcal{P}_i\left( \frac{1}{2N} (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab}\right) = \mathcal{O}_i(Z_b).\] This identity is a consequence of the following lemma, which shows that tensors of the form \(1/(2N)(\mathop{\mathrm{{\mathbb{L}}}}W)_{ab}\) cannot carry gravitational momentum in this gauge: that is, \(\mathcal{G}_i(1/(2N) (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab}) =0\). Hence \[\label{eq:pi95is95oi} \mathcal{P}_i\left( \frac{1}{2N} (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab} \right) = \underbrace{\mathcal{G}_i\left( \frac{1}{2N} (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab} \right)}_{=0} + \mathcal{R}_i\left( \frac{1}{2N} (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab} \right) = \mathcal{O}_i( Z_b ),\tag{18}\] as claimed.

Equation 15 suggests a strategy for precisely describing the asymptotics of \(W^a\) solving equation 13 if \(Z_a\) is rapidly decreasing. We first identify a representative basis \(W^a_{(i)}\), \(i=1,\ldots,n\) of \(O(r^{1-n})\) vectors that carry linear momentum: \(\mathcal{P}_i(W^a_{(j)})=\delta_{ij}\). With these in hand, the obstructions \(\mathcal{O}_{i}(Z_a)\) can be removed, and Proposition 1[part:P-solvable] then permits finding the zero momentum correction to the solution. The following proposition specifies a construction of such a representative basis, and it yields explicit expressions for the leading order asymptotic expansion of the basis in terms of the Green’s function of the flat vector Laplacian \(\overline{P}\), \[\label{eqn:greens} G^{ab} = - 2C_n r^{2-n}(A_n \delta^{ab} + B_n n^a n^b),\tag{20}\] with dimensional constants \[A_n = 3n-2,\qquad B_n = (n-2)^2, \qquad C_n = \frac{1}{4(n-1)(n-2)|S^{n-1}|}.\] We remark that we have an explicit factor of \(2\) in equation 20 due to the factor of \(1/2\) in our definition of \(\overline{P}\). This Green’s function can often be most effectively leveraged in terms of the \(n\) column vector fields \(G_{(j)}^a\) with components \(G_{(j)}^i = G^{ij}\). In the statement below, and throughout the remainder of this work, we denote the region exterior to a ball by \(E_R := \{x \in \mathbb{R}^n \,:\, |x| > R\}\).

Armed with the momentum carrier vector fields \(W_{(j)}^a\), we obtain the following precise description of the solution of equation 13 if the right-hand side is rapidly decaying:

Combining Proposition 4 and Proposition 3[part:W-decomp] we find that if \(Z_a\) has fast decay, \(o(r^{-n})\), then the solution of equation 13 has a leading order \(O(r^{2-n})\) term that is a linear combination of the columns of the Green’s function of the Euclidean vector Laplacian, with a correction decaying at a faster rate depending on both \(\tau\) and \(\delta\). We now investigate the borderline case such that \(Z_a\) decays at the threshold rate \(r^{-n}\). The situation is more delicate because the vector Laplacian is no longer Fredholm as a map on the full weighted space \(W^{k,p}_{2-n}\). To compensate for this, we confine our attention to the case such that \(Z_a\) admits the expansion \(Z_a=r^{-n} F_a + H_a\) on some exterior region \(E_R\), where \(F_a\) is homogeneous of degree zero and where \(H_a\) lies in a weighted space with decay faster than \(r^{-n}\). In this scenario, we seek a solution of the form \(W^a= r^{2-n} U^a + V^a\), where \(U^a\) is homogeneous of degree zero and where \(V^a\) decays faster than \(r^{2-n}\). The following result concerning the Euclidean vector Laplacian is the tool needed for the solvability for the leading order homogeneous terms.

With this result for the Euclidean operator \(\overline{P}\) in hand, we can now solve the broader problem for \(P_{g,N}\), supposing that our leading order source term \(F_a\) satisfies the solvability condition:

We now show that the obstruction criteria ?? of the previous proposition are automatically satisfied for our primary use case where the source term is a divergence.

Compiling the results in this section, we have the following theorem summarizing the prescription of the asymptotics of the unphysical second fundamental form \(K_{ab}\) in the conformal method:

This summary indicates the precise extent of control one has over the structure of the unphysical second fundamental form \(K_{ab}\) with regards to its ADM momentum and decay properties. In particular, if one has sufficient decay to make sense of each of \(\mathcal{G}_i(S)\) and \(\mathcal{R}_i(S)\), the ADM momentum components are \(\mathcal{P}_i(K) = \mathcal{G}_i(S) = \mathcal{P}_i(S) - \mathcal{R}_i(S)\). Given conformal method seed data \(g_{ab}\) and \(N\) satisfying Assumption 1 and \(A_{ab}\) with at least \(O(r^{1-n})\) decay, \(\mathcal{G}_i(A_{ab}/2N)\) therefore directly measures the ADM momentum of the eventual initial data set, and it follows that this momentum can be readily prescribed a priori. Moreover, the quantity \(\mathcal{R}_i(A_{ab}/2N)\) specifies the momentum which must be removed to satisfy the momentum constraint. These results motivate our use of the monikers “gravitational momentum" for \(\mathcal{G}_i\) and”residual momentum" for \(\mathcal{R}_i\).

In all cases, one can precisely control the decay rate of \(K_{ab}\) (beyond any \(r^{1-n}\) terms that arise) by appropriately choosing the decay rates of one’s seed data. To illustrate, we now discuss the precise decay structure of \(W^a\) and \(K_{ab}\) which we obtain in solving 13 in the fast-decay setting of \(A_{ab}\in W^{k-1,p}_{\delta-1}\) with \(1-n < \delta < 2-n\), part [part:K-fast-falloff] above. Abbreviating \(p_i := \mathcal{G}_i(S) = - \mathcal{O}_i(Z)\), we find that \(W^a\) contains an \(O(r^{2-n})\) piece given by \(\sum_j -p_j W_{(j)}^a\), and the remainder \(V^a\) beyond this has the desired \(o(r^\delta)\) decay inherited from \(A_{ab} \in W^{k-1,p}_{\delta-1}\). In light of Proposition 3[part:W-decomp], however, one can be more explicit in describing the asymptotic form of \(W^a\): For every \(\epsilon > 0\) and any \(\ell \in \mathbb{N}\) such that \(k-1 > \ell+n/p\), one has \[W^a = \sum_{j=1}^n -p_j G_{(j)}^a + o_{\ell+1}(r^{\max(\delta, \, 2-n+\tau+\epsilon)}),\] and the corresponding asymptotics of \(K_{ab}\) are the following (using 24 ): \[\begin{align} K_{ab} & = \frac{1}{2}\sum_{j=1}^n p_j \cdot (\overline{\mathop{\mathrm{{\mathbb{L}}}}}G_{(j)})_{ab} + o_\ell (r^{\max(\delta-1, \, 1-n+\tau+\epsilon)}) \notag \\ & = \frac{2n(n-2) C_n}{r^{n-1}} \left[ p_a n_b + p_b n_a + ((n-2) n_a n_b - \delta_{ab}) p_j n^j \right] + o_\ell(r^{\max(\delta-1, \, 1-n+\tau+\epsilon)}). \label{eqn:K-fast-decay} \end{align}\tag{29}\] In particular, for \(n=3\) one has \[K_{ab} = \frac{3}{16 \pi r^{2}} \left[ p_a n_b + p_b n_a + (n_a n_b - \delta_{ab}) p_j n^j \right] + o_\ell(r^{\max(\delta-1, \, -2+\tau+\epsilon)}).\] In the case of fast decay, we may therefore anticipate the complete structure of the leading \(O(r^{1-n})\) term, and we may dictate the decay rate of \(K_{ab}\), and hence of \(\widetilde{K}_{ab}\), beyond this term through the decay of the provided seed data \(g_{ab}\) and \(N\) (specifying \(\tau\)) and \(A_{ab}\) (specifying \(\delta\)). One obtains as many derivatives of control in the error term as desired by choosing the seed data to allow \(k\) sufficiently large. Although it is not typically of physical interest, we remark that one could obtain similar results down to \(-n < \delta < 1-n\), or for an arbitrarily negative value of \(\delta\), by expanding Proposition 2 to characterize the kernel of \(P_{g,N}\) on higher-weighted spaces and expanding Proposition 3 to introduce the faster-decaying carrier vector fields dual to these additional kernel elements.

While we can anticipate the complete structure of the leading \(O(r^{1-n})\) term of \(K_{ab}\) in the case of fast decay discussed above, there is more freedom if \(A_{ab}\) is itself given with the threshold \(O(r^{1-n})\) decay, as in part [part:K-threshold-falloff] of Theorem 7. A question then remains regarding what kinds of leading terms \(r^{1-n} D_{ab}\) (\(D_{ab}\) is analogous to the quantity in brackets in equation 29 ) are possible for \(K_{ab}\) in this case. The answer is that one can choose any Euclidean transverse traceless tensor of this form: the possible homogeneous of degree zero tensors \(D_{ab} \in W^{k-1,p}_{\mathrm{loc}}(\mathbb{R}^n \setminus \{0\})\) are precisely those which are symmetric and traceless and satisfy \(\overline{\nabla}^a(r^{1-n}D_{ab}) = 0\). This form is necessary because isolating the leading order \(r^{1-n} D_{ab}\) term in \(K_{ab} = \frac{1}{2N}(A_{ab}-(\mathop{\mathrm{{\mathbb{L}}}}W)_{ab})\), with \(A_{ab}\) and \(W^a\) as in 27 and 28 on some \(E_R\), yields that symmetry and tracelessness are immediate from these properties of \(K_{ab}\) overall, and the leading order term in \(\nabla^a K_{ab} = 0\) reads \(\overline{\nabla}^a(r^{1-n}D_{ab}) = 0\).

Conversely, beginning with a given Euclidean transverse traceless tensor \(r^{1-n} D_{ab}\), if one sets \[A_{ab} = 2N\chi r^{1-n} D_{ab} \in W^{k-1,p}_{(1-n)^+}\] with \(\chi\) a cutoff function equal to \(0\) on \(B_{R/2}\) and \(1\) on some \(E_R\), then on \(E_R\) one has \[Z_b := \nabla^a\left( \frac{1}{2N} A_{ab} \right) = \nabla^a\left(r^{1-n} D_{ab} \right) = (\nabla^a - \overline{\nabla}^a)\left(r^{1-n} D_{ab} \right) = o(r^{\tau-n}),.\] This result implies that \(Z_b \in W^{k-2,p}_{(-n)^-}\), and consequently Proposition 4 yields the decomposition ?? of the solution \(W^a \in W^{k,p}_{(2-n)^+}\). The leading order \(O(r^{1-n})\) terms in the expansion of \(K_{ab} = \frac{1}{2N}(A_{ab}-(\mathop{\mathrm{{\mathbb{L}}}}W)_{ab})\) are then \(r^{1-n}D_{ab}\) and the leading term in equation 29 , with \(p_j = -\mathcal{R}_j(r^{1-n} D_{ab})\). This extraneous term can be removed by perturbing \(A_{ab}\) at lower order (specifically, by subtracting some \(B_{ab} \in W^{k-1,p}_{(1-n)^-}\) satisfying \(\mathcal{R}_j( \frac{1}{2N}B_{ab}) = \mathcal{R}_j(r^{1-n} D_{ab})\)), resulting in \(K_{ab}\) having the leading order term \(r^{1-n} D_{ab}\), as desired.

We remark that the problem of characterizing and constructing Euclidean transverse traceless tensors is well-studied in the literature [21], [22], so numerous such objects can be readily constructed. The procedure outlined in part [part:K-threshold-falloff] of Theorem 7 and the above discussion then allows one to build any example into a working \(K_{ab}\) as the leading order \(O(r^{1-n})\) term if desired.

4 The Hamiltonian Constraint↩︎

Assured that we may construct a continuous \(K_{ab}\) with any requisite asymptotics, we suppose now that we have done this and turn to the vacuum Hamiltonian constraint, which in the conformal method takes the form of the Lichnerowicz equation (11 ), \[\label{eqn:lichnerowicz2} -(q_n+2)\Delta_g \varphi + R(g) \varphi = \varphi^{-q_n-1} |K|_g^2\tag{30}\] (we recall \(q_n := \frac{2n}{n-2}\)), to be solved for the conformal factor \(\varphi > 0\) satisfying \(\lim_{|x| \to \infty} \varphi = 1\). Despite the nonlinearity of this equation, the problem of existence and uniqueness of solutions has been thoroughly resolved in the literature [14], [20]. One first observes that, in addition to Assumption 1, it is necessary to require that \(g\) is Yamabe positive, which means that the conformal invariant \[Y_g := \inf_{\substack{f\in C^\infty_c\\ f\not\equiv 0}} \frac{\int_{\mathbb{R}^n} \left[ (q_n+2) |\nabla f|^2 + R(g) f^2 \right] \; dV_g}{||f||_{L^{q_n}}^2}\] is strictly positive. To see that this is a necessary condition, we suppose that there exists a positive solution \(\varphi\) to equation 30 with \(\varphi-1 \in W^{k,p}_\delta\) for some \(\delta < 0\). Under the conformal change specified by \(\varphi\), one has \(Y_{\tilde{g}} > 0\) because \(R(\tilde{g}) = |K|_{\tilde{g}}^2\) is positive and continuous, and because Sobolev embedding ensures that \[\|f\|_{L^{q_n}}^2 \lesssim \|\nabla f\|_{L^2}^2 + \| R(\tilde{g})^{1/2} f\|_{L^2}^2 .\] The notation “\(\lesssim\)" here indicates that one has a standard inequality, with”\(\leq\)", up to an overall positive multiplicative constant independent of the quantities of interest (in this case, of \(f \in C_c^\infty\)). Conformal invariance now yields \(Y_g > 0\).

It turns out that \(Y_g > 0\) is also a sufficient condition, as indicated by the following existence theorem for solutions of the Lichnerowicz equation for asymptotically Euclidean seed data. This theorem is implied by Theorem 4.4 of [20]; for uniqueness, see the discussion in Section VIII of [14].

0◻

It is useful to also state precisely the implication of this result in the case of faster decay:

0◻

As in Section 3, understanding the asymptotics of these solutions requires a finer analysis of the next decay range \(1-n< \delta < 2-n\). Since we are now dealing with the simpler scalar Laplacian rather than the vector Laplacian, much of the requisite analysis is a simplified version of that developed in Section 3 and is well understood in the literature. To make the relation between the analyses of these two Laplacians explicit, we present this theory in a manner closely analogous to the discussion in Section 3. To most naturally incorporate the boundary condition \(\varphi \to 1\), we write \(\varphi = 1 + u\) and recast 30 as follows: \[\label{eqn:lichnerowicz-u} -(q_n+2) \Delta_g u + (1+u)R(g) = (1+u)^{-q_n-1} |K|_g^2\tag{31}\] This is the equation we analyze. We begin our analysis by noting the close connection between \(\Delta_g\) and the Euclidean Laplacian \(\overline{\Delta}\) induced by the following equation: \[\label{eqn:lap-diff} \Delta_g u = \overline{\Delta} u + \sum_{|\alpha| = 1}^2 h^\alpha \partial_\alpha u,\tag{32}\] with each \(h^{\alpha} \in W^{k-2+|\alpha|,p}_{\tau-2+|\alpha|}\). Since the following statement is well known, and since the details are much the same as those of Proposition 1, we omit the proof.

0◻

The following proposition and corollary are a variation of Proposition 1 in [9] in that they extract the mass term from \(\varphi\) to control the remaining decay. Following the thread of Section 3, we reformulate this as relying on the existence of a “mass carrier" function– compare this to Proposition 3 (the constant function \(1\) now plays the role analogous to that of the kernel vector fields \(k_{(j)}^a\)).

As in Proposition 4 with the momentum carrier vector fields, this result now allows us to peel off a mass term and control the remainder’s decay:

0◻

While this reads as a statement about the Poisson equation, it can be applied to the Lichnerowicz equation 31 as follows: If \(1-n < \tau < 2-n\) and \(u \in W^{k,p}_{(2-n)^+}\) solves 31 with \(K_{ab} \in W^{k-1,p}_{(1-n)^+}\), then \(\Delta_g u = f\) for \(f \in W^{k-2,p}_{\tau-2}\) defined by \[\label{eqn:f-lich} f = \frac{1+u}{q_n+2}R(g) - \frac{(1+u)^{-q_n-1}}{q_n+2} |K|_g^2.\tag{33}\] Combined with Proposition 10[part:u0exp], Corollary 2 now ensures that there is a constant \(C\) and a function \(v \in W^{k,p}_\tau\) such that \[\varphi := 1 + u = 1 + \frac{C}{r^{n-2}} + v.\] This yields the expected mass term \(Cr^{2-n}\) plus a remainder with prescribed decay. We observe that unlike the ADM momentum components in Proposition 4 and Theorem 7, the analogous integral \(\int_{\mathbb{R}^n} f dV_g\) is not computable directly from the seed data here, so this result may not be leveraged to prescribe the ADM mass of the resulting initial data set a priori.

It remains to discuss the more delicate case of allowing threshold decay \(O(r^{2-n})\) terms in \(g_{ab}\), which we handle in a manner directly analogous to Section 3. Moving forward, we slightly alter Assumption 1 to assume that \(g\) is asymptotically Euclidean of class \(W^{k,p}_{(2-n)^+}\), still satisfying \(k > 1+n/p\). Given \(K_{ab} \in W^{k-1,p}_{(1-n)^+}\), Corollary 1 still guarantees the existence and uniqueness of a solution \(u \in W^{k,p}_{(2-n)^+}\) to equation 31 as long as \(g\) is Yamabe positive, but we wish to ensure that no logarithmic terms arise and that we can control the decay of any remainder beyond the \(O(r^{2-n})\) terms. We supplement Corollary 2 with the following (compare Proposition 6):

In the absence of the integral condition \(\int_{S^{n-1}} \zeta \; dV_{S^{n-1}} = 0\), logarithmic terms generally arise in this lemma’s conclusion. To combine this result with Corollary 2, we must check that the leading order \(O(r^{-n})\) term in equation 33 coming from the scalar curvature \(R(g)\) satisfies this integral condition if \(g_{ab}\) is taken to have a natural form (compare to Lemma 3):

Once again, we compile the results of the present section into the following theorem summarizing the prescription of the asymptotics of the conformal factor \(\varphi = 1+u\) in the conformal method (compare Theorem 7):

We observe that in part [part:phi-threshold-falloff], the resulting physical metric \(\tilde{g}_{ab} = \varphi^{q_n-2} g_{ab}\) has the leading order \(O(r^{2-n})\) term beyond \(\delta_{ab}\) given by \[h_{ab} = r^{2-n}\left( \beta_{ab} + (q_n-2)\omega \delta_{ab} \right),\] so that a sufficient, but almost certainly not necessary, condition guaranteeing that this is not a constant multiple of \(r^{2-n} \delta_{ab}\) is that \(\beta_{ab}\) is not a function on \(S^2\) multiplying \(\delta_{ab}\).

5 Constructions↩︎

In Sections 3 and 4, we have established a sequence of results which guarantee that one can use the conformal method to construct general relativistic initial data sets \((\mathbb{R}^n, \tilde{g}, \widetilde{K}_{ab})\) with a number of prescribed features. These results have largely been framed as statements regarding the analytical features of the solutions to equations (9 ) and (31 ) and are not directly connected to the properties of the the ultimate initial data set. We now make formal statements collating our results into theorems, Theorems 12-14 below, summarizing precisely what can be said regarding the goal of constructing data of each of the types (CK), (B), and (A) discussed in the introduction. Each of these theorems follows readily from the results in Sections 3 and 4, so we mostly omit further argumentation. Our first summarizing theorem pertains to the construction of type (B) data. Following immediately from the well-known results stated in Theorems 7[part:K-slow-falloff] and 11[part:phi-slow-falloff], it contains no novel content:

0◻

To illustrate with \(\tau = 5/2-n\), one first chooses \(g_{ab}\) to be Yamabe positive and asymptotically Euclidean of class \(W^{k,p}_{5/2-n}\), as well as \(A_{ab} \in W^{k-1,p}_{3/2-n}\), so that \(Z_a = \mathop{\mathrm{\mathrm{div}}}_g \left( \frac{1}{2N} A_{ab} \right) \in W^{k-2,p}_{1/2-n}\). Proposition 1[part:iso] ensures that one may uniquely solve equation 12 , that is \((P_{g,N} W)_a = Z_a\), for \(W^a \in W^{k,p}_{5/2-n}\). One now has \(K_{ab} = \frac{1}{2N}(A_{ab} - (\mathop{\mathrm{{\mathbb{L}}}}W)_{ab}) \in W^{k-1,p}_{3/2-n}\), so Theorem 8 ensures that we may find a conformal factor \(\varphi\) solving equation 11 and satisfying \(\varphi-1 \in W^{k,p}_{5/2-n}\). For \(n = 3\) and for sufficiently large \(k\) (\(k > 3+n/p\)), the conformally transformed metric \(\tilde{g} = \varphi^4 g\) has the form of equation 3 , and the physical second fundamental form \(\widetilde{K}_{ij} = \varphi^{-2} K_{ij} \in W^{k,p}_{5/2-n}\) has the form 4 , which corresponds to type (B) data.

In attempting to repeat this procedure for either type (CK) or type (A) data, equations 1 2 and 5 6 , with all the falloff weights shifted down by one, one finds that \(Z_a \in W^{k-2,p}_{-1/2-n}\), so that the desired domain of \(P_{g,N}\) now has \(\delta = 3/2 - n < 2 - n\), and we cannot simply invoke Proposition 1[part:iso]. This obstruction is not surprising: Proposition 1 indicates that we should expect the solution \(W^a\) to equation (13 ) to be contained in \(W^{k,p}_{(2-n)^+}\), ultimately corresponding to \(\widetilde{K}_{ab} \in W^{k-1,p}_{(1-n)^+}\) and allowing an \(O(r^{1-n})\) term in \(\widetilde{K}_{ab}\) which gives rise to the initial data set’s linear ADM momentum. The same obstruction arises in seeking \(\varphi\): The conclusion of Theorem 8 is limited due to the generic appearance of an \(O(r^{2-n})\) term in \(\varphi\), corresponding to the mass term in the metric and screening out control of any remaining decay. While the problem of extracting the \(O(r^{2-n})\) term from \(\varphi\) to control the remaining decay has been treated in the literature (see Proposition 1 of [9] or Theorem 1.17 of [17]), previously existing results are insufficient to guarantee the emergence of type (A) data in which the mass term \(h_{ij}\) in the metric is anisotropic, or to fully understand the \(O(r^{1-n})\) term in the second fundamental form. As demonstrated in Theorems 13 and 14 below, we are nevertheless able to treat both of these concerns.

Our second summarizing theorem, relevant to the construction of type (CK) data, partially overcomes these hurdles, the remainder being left for our final theorem. Theorem 13 contains the novel result that the ADM momentum components of the constructed initial data set may always be fully anticipated, and hence prescribed as a consequence of one’s choice of the seed data, thereby allowing one to build initial data sets in the center of mass frame with nontrivial \(\widetilde{K}_{ab}\) if desired. The following is a consequence of Theorem 7[part:K-fast-falloff] (and the immediately following discussion) and Theorem 11[part:phi-fast-falloff]:

0◻

We recall that \(k_{(j)}^a\) denotes the vector fields of Proposition 2, which comprise a particular basis of the kernel of \(P_{g,N}\) on \(W^{k,p}_{0^+}\). In order to use this result to prescribe the ADM momentum components in practice, one should fix \(g\) and \(N\) according to the above hypotheses, numerically construct the kernel vector fields \(k^a_{(j)}\), and compute each of the integrals 36 for \(n\) different choices of \(A_{ab}\). This procedure generically produces \(n\) independent momentum vectors, resulting in a basis that one can use to construct any desired momentum vector by specifying an appropriate linear combination of one’s choice of the \(A_{ab}\)’s. To most simply construct an initial data set in the center of mass frame, one should choose \(A_{ab} = 0\). To construct an initial data set in the center of mass frame which is not time-symmetric, one should compute the momentum vector for one more choice of \(A_{ab}\) (for a total of \(n+1\) choices, necessarily producing a linearly dependent set of momentum vectors), thereby allowing one to build a nontrivial linear combination with zero momentum.

Our third and final summarizing theorem is relevant to the construction of type (A) data. It contains the novel result that an anisotropic mass term can be built into the metric \(\tilde{g}_{ij}\) of one’s initial data set in practice, providing an intermediate regime between type (B) and type (CK) data. It further indicates that a Euclidean transverse traceless tensor which is homogeneous of degree \(1-n\) can be built into \(\widetilde{K}_{ij}\) as its leading order term if desired– this is precisely the momentum analogue of the anisotropic mass term allowed for in \(\tilde{g}_{ij}\). The following statement is a consequence of Theorem 7[part:K-threshold-falloff] and Theorem 11[part:phi-threshold-falloff] (and the immediately following discussion):

0◻

We remark that the discussion at the end of Section 3 further implies that, with an appropriate choice of \(A_{ab} \in W^{k-1,p}_{(1-n)^+}\), one may arrange that the leading order term in \(\widetilde{K}_{ij}\) can be any given Euclidean transverse traceless tensor \(D_{ij}\) that is homogeneous of degree \(1-n\). That is, this term can be prescribed, and it could readily be made nontrivial even with zero ADM momentum.

Taken together, Theorems 12-14 answer the question of how one can practically construct initial data sets of each of the types (CK), (B), and (A) (and a bit beyond) using the formalism of the conformal method. These theorems indicate that one can precisely control the initial data set’s ADM momentum and the leading (or next-to-leading) order decay rates of the physical metric \(\tilde{g}_{ij}\) and the second fundamental form \(\widetilde{K}_{ij}\), and that one may build an anisotropic mass term into \(\tilde{g}_{ij}\), or the analogous term into \(\widetilde{K}_{ij}\), if desired. As an application, these results guarantee that a simple example numerically constructed below is of type (A), allowing us to place the strongest known conditions (cf. [9]) on the collection of spacetimes to which the antipodal matching conjecture of [10] could apply.

6 A Numerical Example↩︎

In this section, we present a numerically computed Brill wave solution of type (A) using the method of [9]. Brill waves are initial data sets on \(\mathbb{R}^3\) for which \(\widetilde{K}=0\) and \(\tilde{g}\) is axisymmetric. The momentum constraint is then trivially satisfied, while the Lichnerowicz equation 11 becomes \[-8 {\Delta_g}\varphi + R(g) \varphi = 0. \label{Brill1}\tag{37}\]

We find it convenient to perform the numerical computations in spherical polar coordinates \((r,\theta,\phi)\). In these coordinates, the line element of the unphysical metric \(g\) takes the form \[d {s^2} = {e^{2q}} (d {r^2} + {r^2} d {\theta ^2}) + {r^2} {\sin^2} \theta d {\phi^2}.\] Here, in keeping with axisymmetry, the function \(q\) depends only on \(r\) and \(\theta\). Equation 37 then takes the form \[- \overline{\Delta} \varphi + S \varphi = 0, \label{Brill2}\tag{38}\] where \(\overline{\Delta}\) is the flat Laplacian and \(S\) is given by \[S = - {\frac{1}{4}} \left ( {\frac{{\partial ^2}q}{\partial {r^2}}} + {\frac{1}{r}} \, {\frac{\partial q}{\partial r}} + {\frac{1}{r^2}} \, {\frac{{\partial ^2}q}{\partial {\theta ^2}}} \right ).\]

We define \(F := \ln \varphi\) and decompose \(F={F_1}+{F_2}\), with \(F_1\) a solution to \[\overline{\Delta} {F_1} = S. \label{F1eqn}\tag{39}\] Equation (38 ) now becomes \[\overline{\Delta} {F_2} = - {\vec{\nabla}}F \cdot {\vec{\nabla}}F, \label{F2eqn}\tag{40}\] which we seek to solve for \(F_2\). Given a numerical method to invert the flat Laplacian, we first solve equation (39 ) and proceed to solve equation (40 ) by iteration. That is, having found \(F_1\), we make the initial guess of zero for \(F_2\) and repeatedly solve equation (40 ) for an improved version of \(F_2\), where the right hand side of equation (40 ) is computed using the previous version of \(F_2\). At each step, the current version of \(F_2\) is stored, with the final version used to compute \(F\).

Our numerical method to invert the flat Laplacian relies on the standard Green’s function for axisymmetric functions (see e.g., [23]) where we compute all integrals numerically. Explicitly, for equation (39 ) we have \[{F_1} = {\sum _{\ell =0} ^\infty} \left [ {g_\ell}(r) {r^{-(\ell+1)}} + {h_\ell}(r) {r^\ell}\right ] {P_\ell}(\cos \theta ),\] where \({g_\ell}(r)\) and \({h_\ell}(r)\) are given by \[{g_\ell}(r) = {\int _0 ^r} {{\tilde{r}}^{\ell+2}} d {\tilde{r}} {\int _0 ^\pi} \sin \theta \, d \theta \, S({\tilde{r}},\theta){P_\ell}(\cos \theta),\] \[{h_\ell}(r) = {\int _r ^\infty} {{\tilde{r}}^{1-\ell}} d {\tilde{r}} {\int _0 ^\pi} \sin \theta \, d \theta \, S({\tilde{r}},\theta){P_\ell}(\cos \theta),\] and \(P_\ell\) are the Legendre polynomials.

To assess whether our constructed initial data set conforms to the antipodal conjecture of [10], we examine the behavior of the curvature component \(\rho\), which is the principal curvature corresponding to the plane that contains the gradient of \(r\) and the normal to the initial data surface. For Brill wave initial data, \(\rho\) is given asymptotically (as \(r \to \infty\)) by the following expression: \[\begin{align} {r^2} \rho &=& - {r^2} {\frac{{\partial ^2}q}{\partial {r^2}}} - {\frac{{\partial ^2}q}{\partial {\theta^2}}} - \cot \theta {\frac{\partial q}{\partial \theta}} + 2 {r^2} {\frac{\partial q}{\partial r}} {\frac{\partial F}{\partial r}} - 2 {\frac{\partial q}{\partial \theta}} {\frac{\partial F}{\partial \theta}} \nonumber \\ && - 4 {r^2} {\frac{{\partial ^2}F}{\partial {r^2}}} - 4 r {\frac{\partial F}{\partial r}} - 2 {\frac{{\partial ^2}F}{\partial {\theta^2}}} - 2 \cot \theta {\frac{\partial F}{\partial \theta}} - 4 {{\left ( {\frac{\partial F}{\partial \theta}} \right ) }^2}. \end{align}\]

In order to produce an example that is not antipodally symmetric, we consider \(q\) of the form \[q={a_0}{r^3}\cos (\theta) {\sin ^2}(\theta) {{({r^2}+{r_0^2})}^{-\gamma}}, \label{qformula2}\tag{41}\] where \({a_0}, \, {r_0}, \,\) and \(\gamma\) are constants. In order to obtain type (A) behavior, we choose \(\gamma = 2\)– that the resulting initial data set is of type (A) is now implied by Theorem 14. Figure 1 graphs the numerically computed \(\varphi\) for the case \({a_0}= 1, {r_0}=10, \gamma = 2\) up to the radius of \(r=100\). Figure 2 plots \({r^3}\rho\) for two different values of \(r\): \(r=10000\) and \(r=15000\). The fact that the two curves agree corroborates the claim that the asymptotic behavior of \(\rho\) is \(\rho \propto {r^{-3}}\), as expected.

We observe in Figure 2 that \(\rho\) is not antipodally symmetric. This is relevant to the conjecture of Strominger [10] that in the limit of early time at null infinity there is a symmetry involving the combination of time reflection and antipodal mapping. Since Brill wave initial data sets have zero extrinsic curvature, it follows that their time evolution has time reflection symmetry. The evolution of the Brill wave initial data set whose features are plotted in Figures 1 and 2 therefore cannot satisfy the conjecture of [10]. This result should not be regarded as a counterexample to this conjecture, but rather as a statement regarding the sort of asymptotic flatness under which this conjecture can hold: that is, the conjecture of [10] does not hold for metrics of type (A).

7 Conclusion↩︎

In this work, we have thoroughly investigated the problem of constructing maximal solutions \((\mathbb{R}^n, \tilde{g}, \widetilde{K}_{ab})\) to the vacuum constraint equations 9 10 via the conformal method with prescribed asymptotic behavior. While it has long been known that one can construct unique solutions for any choice of seed data \(g_{ab}\), \(A_{ab}\), and \(N\) taken in appropriate function spaces, we have clarified the precise extent of the technical control which can be exerted over the asymptotics of these solutions. At a broad level, we have proceeded by first characterizing the solutions to the conformal constraint equations 11 12 explicitly at leading order, using the “momentum carrier" vector fields \(W^a_{(j)}\) and the”mass carrier" function \(u_0\) in the event of rapid decay in the seed data, or using understood solutions from analogous problems involving the flat operators \(\overline{P}\) and \(\overline{\Delta}\) in the event of threshold decay corresponding to the natural forms 27 and 34 for \(A_{ab}\) and \(g_{ab}\). Generally, we have shown that peeling off this leading-order behavior of the solutions renders the equations solvable in a space with faster decay, asserting control of the decay rate of subleading corrections. Theorems 7 and 11 summarize these results at the more abstract level of the vector field \(W^a\) and conformal factor \(\varphi\) directly solved for in the conformal method, while Theorems 12, 13, and 14 summarize these results at the level of the ultimately constructed initial data set \((\mathbb{R}^n, \tilde{g}, \widetilde{K}_{ab})\). We remark that the leading-order terms may be understood well enough to prescribe the initial data set’s ADM momentum components, and (in working with prescribed threshold decay) to impose that the metric include an anisotropic mass term.

The results established herein ensure that, with an appropriate choice of seed data, one can construct initial data sets of any of the three types discussed in the introduction. We have illustrated this in Section 6 by numerically constructing a simple example of a type (A) spacetime guaranteed to include an anisotropic mass term, which previously available results were unable to handle. This example is constructed such that its spacetime evolution cannot satisfy the antipodal matching conjecture of [10], restricting the collection of spacetimes within which one can expect this property of physical interest to be realized. It is our hope that our results furnish the ability to further construct a wealth of initial data sets with various decay features, allowing the relativity community to probe, both numerically and analytically, how physically interesting properties of spacetimes may depend on one’s choice of physically motivated asymptotic behavior.

Acknowledgements↩︎

Lydia Bieri is supported by NSF grant DMS-2204182 to The University of Michigan. James Wheeler was also partially supported by NSF grant DMS-2204182 to The University of Michigan. David Garfinkle was supported by NSF grant PHY-2102914 to Oakland University.

8 Fredholm Properties of the Vector Laplacian↩︎

This appendix contains auxiliary technical results on the Fredholm properties of vector Laplacians. Although the body of our work above is restricted to the setting of metrics with Hölder continuous first derivatives, for the sake of generality and potential other applications we now relax the condition \(k>n/p+1\) of Assumption 1 with \(k>n/p\) instead.

The vector Laplacian is then \(P_{g,N} = \mathop{\mathrm{\mathrm{div}}}_g (\frac{1}{2N} \mathop{\mathrm{{\mathbb{L}}}})\), which we abbreviate as \(P\) in this section. The model Euclidean vector Laplacian with \(N=1\) is then \(\overline{P} = \frac{1}{2} \overline{\mathop{\mathrm{\mathrm{div}}}}(\overline{\mathop{\mathrm{{\mathbb{L}}}}})\).

The natural Fredholm theory for a self-adjoint operator such as the vector Laplacian makes use of dual function spaces, and in particular distributions having a negative order of differentiability. Hence we generalize Definition 1 to allow weighted Sobolev spaces \(W^{k,p}_\delta\) with \(k \le 0\). Recall that if \(k<0\), then \(W^{k,p}(\Omega)\) for an arbitrary domain \(\Omega \subset \mathbb{R}^n\) is the topological dual space \(W^{k,p}(\Omega) := (W^{-k,p'}_0(\Omega))^*\), with the induced dual norm, where \(\frac{1}{p} + \frac{1}{p'} = 1\) and where \(W^{-k,p'}_0(\Omega)\) is the closure of \(C_c^\infty(\Omega)\) in \(W^{-k,p'}(\Omega)\).

It follows from Bartnik’s scaling technique [17] that for \(k\ge 0\) the norm above is equivalent to that of Definition 1. Although we do not require it, we remark that Definition 3 generalizes to non-integral scales of differentiabilty, and indeed elementary arguments show that \(W^{k,p}_\delta\) defined here agrees with the space \(f^s_{p,2,\mu}\) of [24] with \(s=k\) and \(\mu = - \delta p + sp - n\). See also the analogous definition for the special case \(p=2\) in [15]. General properties of these spaces in the case \(k\ge 0\) can be found in [17], and their generalizations to the broader family appear in [24] and [15]. The sole exception is the compactness of the embedding \(W^{k,p}_\delta \hookrightarrow W^{j,p}_{\delta'}\) when \(k\ge j\) and \(\delta'<\delta\). This is appears in the special case \(p=2\) in [15] and can be extended to general \(p\) using the same technique.

The following multiplication lemma provides criteria under which the product of elements of two weighted Sobolev spaces determines an element of a third weighted space. The proof follows from Bartnik’s scaling technique and the corresponding Sobolev multiplication properties on unweighted spaces (see, for example, Theorem 2.5 of [25]). See also [26] Proposition 3.11, which is the equivalent result for asymptotically hyperbolic manifolds.

0◻

A straightforward application of Lemma 6 (see Proposition 2.6 of [25] for the corresponding result in unweighed spaces) establishes the following elementary mapping property of the vector Laplacian: \[P: W^{j,q}_\delta \to W^{j-2,q}_{\delta-2}\] is continuous so long as \[\begin{align} 2-k &\le j \le k, \tag{42} \\ \frac{1}{p} - \frac{k}{n} &\le \frac{1}{q} -\frac{j}{n} \le \frac{1}{p'} - \frac{2-k}{n}. \tag{43} \end{align}\]

Using the standing assumption \(k>n/p\) it is easy to show \(\mathcal{S}^{k,p}\) is nonempty and contains \(j=1\), \(p=2\). This observation and related facts concerning \(\mathcal{S}^{k,p}\) are discussed in [25].

The dual space \((W^{j,q}_{\delta})^*\) for scalars can be identified with \(W^{-j, q'}_{-n-\delta}\); see [24] for one approach. This identification does not rely on a metric, but it is more useful for our purposes to use a metric-specific representation of the dual space that aligns with the adjoint structure of \(P\). The following result is a variation of Proposition 3.18 of [26], which proves the analogous fact in the asymptotically hyperbolic setting. The proof follows the same technique, and is omitted.

0◻

Using the specific identification of \((W^{j,q}_\delta)^*\) with \(W^{-j,q'}_{-n-\delta}\) from Lemma 7, the following lemma shows that the vector Laplacian is self-adjoint.

We have the following elementary elliptic regularity estimate, a variation of [17] Proposition 1.6.

Fredholm properties of the vector Laplacian follow from those of the Euclidean model operator \(\overline{P}\), which we establish now.

The next two results concern the asymptotics of solutions of \(\overline{P}X = Z\). The first shows that if \(Z\) has \(O(r^{\delta-2})\) decay then \(\overline{P}X = Z\) is solvable, up to an error on a compact set, with \(X\) having \(O(r^{\delta})\) decay. The second concerns the specific asymptotic structure of a solution of \(PX = Z\) when \(Z\) has compact support.

We now establish multipole expansions for solutions of \(\overline{P} X = Z\) when \(Z\) has compact support. The monopole term in this expansion is used to construct the momentum carrier vector fields \(W^a_{(j)}\) of Proposition 3.

We now return to the general case of the vector Laplacian \(P = P_{g,N}\). A standard perturbation technique (compare with [17] Theorem 1.10) provides the following coercivity estimate, generalizing equation (?? ):

It remains to prove the following technical lemma used in the proof of Proposition 16.

Combining estimate ?? from Proposition 16 with the compact embedding \(W^{j,q}_{\delta} \hookrightarrow W^{j-2,q}_{\delta'}\) for \(\delta'>\delta\), a standard argument (e.g. Theorem 1.10 of [17]) implies that \(P\) is semi-Fredholm:

0◻

As a final step before establishing the full Fredholm theory, we now show that the kernel of \(P\) depends on \(\delta\) but not on the Sobolev parameters. This fact is not immediate from Lemma [lem:basic-reg], which allows for an improvement of the number of derivatives, but only at a fixed integrability parameter \(q\).

We now obtain the primary Fredholm result.

References↩︎

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1. lbieri@umich.edu↩︎

2. garfinkl@oakland.edu↩︎

3. isenberg@uoregon.edu↩︎

4. damaxwell@alaska.edu↩︎

5. jcwheel@umich.edu↩︎

6. For \(f \in C^m(\mathbb{R}^n)\), we define \(f = o_m(r^\delta)\) provided that \(D^\alpha f = o(r^{\delta-|\alpha|})\) as \(r \to \infty\) for any multi-index \(\alpha\) of order \(|\alpha| \leq m\).↩︎

7. It is a reasonable conjecture that this additional assumption is implied by \(k > n/p\), but a proof remains elusive.↩︎

8. We use parenthetical indices to denote simple labels, distinguished from the usual tensorial indices.↩︎

9. \(\lfloor \cdot \rfloor\) denotes the floor operation, which rounds its input down to the nearest equal or smaller integer.↩︎