Hyperboloidal Initial Data and Constraints

Hyperboloidal initial data solve the ordinary physical Einstein constraints on a noncompact spacelike hypersurface, but are constructed from fields on a compact manifold with boundary. The boundary represents a cut of future null infinity. Regularity there is not automatic: the conformal free data must satisfy an elliptic system, their tangential shear must match the boundary embedding, and further boundary compatibility conditions must remove logarithmic obstructions.

Required background. The Riemann Curvature Tensor supplies the curvature and Weyl decompositions used in the regularity statement.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, and Causality and Global Structure.

Compact CMC conformal data

The statement on this page concerns one vacuum constant-mean-curvature conformal method. Let $\overline M$ be a smooth compact three-manifold with boundary $S=\partial M$. Choose a smooth Riemannian metric $h_{ij}$, a smooth tracefree symmetric tensor $A_{ij}$, and a boundary defining function $\rho$ satisfying

$$\rho\gt 0\text{ on }M, \qquad \rho=0,\quad d\rho\ne0\text{ on }S.$$

Use $\widetilde K_{ij}=-\tfrac12\mathcal L_{\widetilde n}\widetilde h_{ij}$ with future-directed physical normal. Fix a positive CMC length scale $a$ and measure lengths in units of $a$ for this construction. In the resulting dimensionless equations set $\Sigma=-1$, so $\widetilde K=3\Sigma=-3$; restoring dimensions gives $\widetilde K=-3/a$. Set $\widehat h_{ij}=\rho^{-2}h_{ij}$ on the interior. The normalization $\Sigma=-1$ is part of the formulas for the free data and boundary expansion below, not an additional restriction on the sign-appropriate CMC radius.

Near $S$, use geodesic boundary coordinates with $\rho=x$ and

$$h=dx^2+q_{AB}(x,y)dy^A dy^B.$$

Let $\eta=\partial_x$ point toward increasing $x$, hence into the physical interior. Define the second fundamental form of $S\subset(\overline M,h)$ by

$$\lambda_{AB}=h(D_Ae_B,\eta) =-\frac12\mathcal L_\eta q_{AB}.$$

This inward orientation is essential. For future null infinity the shear-free relation on the free data is

$$A^{\mathrm{TF}}_{AB}\big|_S =\frac{\Sigma}{|\Sigma|}\lambda^{\mathrm{TF}}_{AB}\big|_S =-\lambda^{\mathrm{TF}}_{AB}\big|_S.$$

It is the convention-translated form of Andersson 2002, § 2, equation (3), pp. 184–185. Using an outward normal without changing the right-hand sign would impose the wrong shear.

The degenerate elliptic construction

Let $L$ denote the conformal Killing operator of $h$,

$$(LY)_{ij}=D_iY_j+D_jY_i-\frac23(D_kY^k)h_{ij}.$$

First solve the vector equation that removes the longitudinal part of $A_{ij}$, and define a transverse tensor $\widehat\sigma_{ij}$. Hatted covariant derivatives, Laplacians, curvatures, and contractions use $\widehat h$; unhatted ones use $h$. In particular, $(\rho^{-1}A)^{ij}$ below means that the covariant tensor $\rho^{-1}A_{ij}$ is raised with $\widehat h^{ij}$. The steps are

$$\widehat D_i\widehat L(Y)^{ij} =\widehat D_i(\rho^{-1}A)^{ij},$$ $$\sigma_{ij}=A_{ij}-\rho^{-1}(LY)_{ij}, \qquad \widehat\sigma_{ij}=\rho^{-1}\sigma_{ij}, \qquad \widehat D_i\widehat\sigma^{ij}=0.$$

Then solve the Lichnerowicz equation

$$-8\widehat\Delta u+\widehat R\,u -\widehat\sigma_{ij}\widehat\sigma^{ij}u^{-7} +6\Sigma^2u^5=0$$

for a positive $u$. The physical data and their boundary defining conformal factor are

$$\widetilde h_{ij}=u^4\widehat h_{ij}, \qquad \widetilde K_{ij}=u^{-2}\widehat\sigma_{ij}+\Sigma\widetilde h_{ij}, \qquad \Omega=u^{-2}\rho.$$

Equivalently, the regular tracefree field is $B_{ij}=u^{-4}\sigma_{ij}$, since

$$\widetilde K_{ij}=\Omega^{-1}B_{ij}+\Sigma\widetilde h_{ij}.$$

To identify the unphysical extrinsic curvature, let $n$ be the future unit normal in the conformal spacetime and write it as $K^{\mathrm{conf}}_{ij}$. The conformal transformation gives

$$K^{\mathrm{conf}}_{ij} =B_{ij}+\Omega^{-1}\bigl[\Sigma-n(\Omega)\bigr]h_{ij}.$$

Choose the conformal normal data $n(\Omega)=\Sigma$ initially. Then $K^{\mathrm{conf}}_{ij}=B_{ij}$ and its $h$-trace vanishes. This gauge identifies the tensor in the boundary regularity conditions below; a different normal gauge would require translating those conditions as well.

At a geodesic boundary, $\widehat R\to-6$, while the leading Lichnerowicz equation gives $u|_S=|\Sigma|^{-1/2}=1$. Hence $B|_S=A|_S$ when the vector correction has the boundary behavior stated below. If $\Sigma$ is left arbitrary while keeping the same ansatz, $B|_S=\Sigma^2A|_S$ instead; one must then rescale the free-data shear condition as well. This is why the CMC normalization has been fixed explicitly.

These fields solve the vacuum Hamiltonian and momentum constraints with $\widetilde K=3\Sigma$. The construction and its precise powers are given in Andersson 2002, § 4, equations (6)–(7d), pp. 187–188. Here elliptic solvability includes compatible boundary behavior, absence or treatment of conformal-Killing kernels, and existence of a positive solution. Those are hypotheses, not consequences of the symbolic ansatz.

Boundary regularity statement and hypotheses

Under the following hypotheses, the selected construction has a sharp smoothness test:

• $h_{ij}$ and $A_{ij}$ are smooth on $\overline M$, $\rho$ is a smooth defining function, and the vacuum equations are used;
• $\Sigma=-1$ in the selected length units, the vector and Lichnerowicz problems are solvable with the stated boundary behavior, and $u\gt 0$;
• the shear-free relation holds;
• conformal gauge is chosen so the boundary mean curvature $q^{AB}\lambda_{AB}$ vanishes, and the auxiliary gauge $A_{1j}|_S=0$ is imposed.

With these assumptions, the expansions begin

$$\Omega=x+O(x^3), \qquad Y^i=O(x^3).$$

Let $\mathcal D_A$ be the connection of $q_{AB}|_S$, let $R_{AB}[h]$ be the tangential projection of the three-dimensional Ricci tensor of $h$, and take all boundary tensor contractions with $q$. In these gauges, the additional boundary conditions are

$$\begin{aligned} \partial_xB_{AB}-\frac12q^{CD}\partial_xB_{CD}\,q_{AB}&=0,\\ \mathcal D^A\mathcal D^B\lambda_{AB}+R_{AB}[h]\lambda^{AB}&=0,\\ \mathcal D^B\left(\partial_xB_{AB} +\frac32\lambda_{CD}\lambda^{CD}q_{AB}\right)&=0. \end{aligned}$$

The curvature in the second line is not the intrinsic two-dimensional Ricci tensor of the cut. Replacing it by that tensor would change the obstruction. Under the construction and simplifying gauges just stated, these conditions are the smoothness criterion for $\Omega$, $B$, and the rescaled spacetime Weyl tensor. They are equations (10a)–(10c) of Andersson 2002, § 4.3, pp. 7–8 of the preprint, PDF. The first controls the tracefree normal derivative; the other two remove coupled logarithmic obstructions. They are boundary tests, not a replacement for solving the elliptic equations in the interior.

The scope matters. Shear-free data are necessary for a $C^2$ conformal spacetime, but shear freedom alone is not sufficient for a $C^\infty$ rescaled Weyl field. Nor does this result say that arbitrary asymptotically hyperboloidal data possess smooth null infinity. Matter requires separately stated conformal weights and falloff; it is not included in this vacuum theorem.

Why logarithms appear

The elliptic system degenerates at $x=0$. The model normal operator for the Lichnerowicz equation has indicial roots $-1$ and $3$. Consequently, smooth coefficients can generate

$$u=1+u_1x+u_2x^2+u_{3,1}x^3\ln x+u_3x^3+\cdots.$$

The boundary compatibility conditions set the obstruction coefficient $u_{3,1}$, and its coupled analogues, to zero. If they fail, the data may still have a meaningful polyhomogeneous completion, but not the smooth completion assumed by the regular conformal field equations. The indicial calculation and the distinction between smooth and polylogarithmic expansions are explicit in Andersson 2002, § 4.2, pp. 188–189.

On the evolution side, smooth compactification remains a hypothesis. In a general wave reduction, the preferred conformal gauge makes the formally singular source terms finite only when the evolved completion is smooth; Zenginoğlu 2008, § 3.2, Proposition 2, pp. 6–7 states that assumption explicitly.

Independent shear check

At $S$, the future generator of fixed-coordinate $\mathscr I^+$ is proportional to $n+\eta$, where $n$ is the future unit normal to $M$ in the unphysical spacetime and $\eta$ is the inward spatial normal. With the site’s definitions, the screen-projected deformation of $n+\eta$ has tracefree part proportional to

$$-\left(B^{\mathrm{TF}}_{AB}+\lambda^{\mathrm{TF}}_{AB}\right).$$

Thus $B^{\mathrm{TF}}_{AB}=-\lambda^{\mathrm{TF}}_{AB}$ cancels the outgoing null shear for $\Sigma\lt 0$. If the mismatch is nonzero, conformal gauge rescaling can change traces but cannot erase this tracefree shear obstruction.

Exercises

Suppose $S$ is totally umbilic in $h$, so $\lambda^{\mathrm{TF}}_{AB}=0$, and take future hyperboloidal data with $A^{\mathrm{TF}}_{AB}|_S=0$. Verify the shear-free condition. Does this alone prove a smooth rescaled Weyl tensor?

Solution

For $\Sigma\lt 0$, shear freedom requires $A^{\mathrm{TF}}_{AB}=-\lambda^{\mathrm{TF}}_{AB}$. Both sides vanish, so the leading null shear cancels. It does not follow that $\Omega^{-1}C$ is smooth: elliptic solvability and the additional boundary equations (10a)–(10c) remain necessary. Failure of one of those conditions can produce an $x^3\ln x$ term even though the leading shear is zero.

References

• Andersson, Lars. “Construction of Hyperboloidal Initial Data.” In The Conformal Structure of Spacetimes: Geometry, Analysis, Numerics, edited by Jörg Frauendiener and Helmut Friedrich, Lecture Notes in Physics 604, 183–194. Springer, 2002. arXiv:gr-qc/0205083.
• Zenginoğlu, Anıl. “Hyperboloidal Evolution with the Einstein Equations.” Classical and Quantum Gravity 25 (2008): 195025. DOI. Open PDF.