Peeling or not peeling—is that the question?^*

The direct measurement of gravitational radiation [1] must be seen as a triumph for experimental, as well as for theoretical physics. In view of the graphs showing the impressive coincidence of the measured and the calculated radiation signals, one might think that everything comes to a conclusion and is understood now. But new questions will come up (see [26] for an example), and may require more precise statements. It will thus still be worthwhile to reconsider questions that have been left open in the theory of gravitational radiation. After giving an outline of the basic ideas and results concerning the underlying mathematical structure, I shall discuss some of the remaining unresolved problems.

Following a gestation period of several years, involving many workers, the basic setting for the analysis of gravitational waves was proposed in the early 1960s by Bondi et al [11], Sachs [67], Newman and Penrose [61]. It requires: the idealization of an isolated self-gravitating system; the analysis of solutions to Einstein's field equations which are asymptotically flat in null directions; the control of the evolution by Einstein's field equations on large scales; the control of the geometry on large scales; precise asymptotics at space-like and null infinity; and the definition of physical concepts related to physical observations 'far away from the system'.

Carving out the role of null and conformal geometry in the analysis of space-time structures over large scales, Penrose [62] combined the various ideas in the elegant geometric concept of asymptotic simplicity, which characterizes the expected asymptotic behavior by the requirement that the conformal structure be smoothly extendable across null infinity.

The basic model is provided by Minkowski space $\hat{M} = \mathbb{R}^4$, $\hat{g} = -{\rm d}t^2 + {\rm d}r^2 + r^2\, {\rm d}\sigma^2$, given here in spatial polar coordinates with ${\rm d}\sigma^2$ denoting the standard line element on $\mathbb{S}^2$ and coordinates $t \in \mathbb{R}$ and $r \geqslant 0$. Performing the coordinate transformation

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle t(\tau, \chi) = \frac{\sin \tau}{\cos \tau + \cos \chi}, \quad r(\tau, \chi) = \frac{\sin \chi}{\cos \tau + \cos \chi}, \nonumber \end{align*}$$

and rescaling with the conformal factor $\Omega = \cos \tau + \cos \chi = \frac{2}{\sqrt{1 + (t-r){\hspace{0pt}}^2} \sqrt{1 + (t + r){\hspace{0pt}}^2}}$, the conformal metric and its domain of definition are obtained in the form

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle g = \Omega^{2}\,\hat{g} = - {\rm d}\tau^2 + {\rm d}\chi^2 + \sin^2\chi\,{\rm d}\sigma^2, \quad \hat{M} = \{\chi \geqslant 0, \,\, \vert \tau \pm \chi\vert < \pi\}. \nonumber \end{align*}$$

This metric, the conformal factor, and the underlying manifold smoothly extend to yield conformally compactified Minkowski space with manifold

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle M = \{\chi \geqslant 0, \,\, \vert \tau \pm \chi\vert \leqslant \pi\} = \hat{M} \cup {{\mathcal J}}^{\pm} \cup i^0 \cup i^{\pm}. \nonumber \end{align*}$$

The two components ${{\mathcal J}}^{\pm} = \{\vert \tau \pm \chi\vert = \pi\}$ of the conformal boundary, on which $\Omega = 0$, ${\rm d}\Omega \neq 0$, represent future and past null infinity, they are generated by the future and past endpoints respectively acquired by the null geodesics. They are null hypersurfaces with respect to the conformal metric g.

The two points $i^{\pm} = \{\tau = \pm \pi, \, \chi = 0\}$, where $\Omega = 0$, ${\rm d}\Omega = 0$, and ${\rm Hess}_g\Omega = - g$, represent the future and past endpoints of the time-like geodesics and thus future and past time-like infinity. The space-like geodesics run in both directions towards space-like infinity, represented by the point $i^0 = \{\tau = 0, \, \chi = \pi\}$, where $\Omega = 0$, ${\rm d}\Omega = 0$, and ${\rm Hess}_g\Omega = g$. By adding this point, the Cauchy hypersurface $\{t = 0\} = \{\tau = 0, \, \, 0 \leqslant \chi < 0\}$ of Minkowski space with the metric induced by g extends to the sphere $\mathbb{S}^3$ endowed with its standard metric.

The process of extending the differential structure and the conformal structure of a given space-time $(\hat{M}, \hat{g})$ to obtain a smooth (resp. C^k with k sufficiently large) conformal extension $(M, g, \Omega)$ with boundary ${{\mathcal J}}$ so that $M = \hat{M} \cup {{\mathcal J}}$, $g = \Omega^2\hat{g}$ on $\hat{M}$ and $\Omega = 0, {\rm d}\Omega \neq 0$ on ${{\mathcal J}}$ as observed above was largely generalized in [62]. It was suggested that it applies to many solutions of Einstein's field equations. In the case of solutions which satisfy Einstein's vacuum field equations near ${{\mathcal J}}$, it turns out that ${{\mathcal J}}$ (consisting then in general of two components ${{\mathcal J}}^{\pm}$) is in fact a null hypersurface for the conformal metric g that represents (future and past) null infinity. In the situations considered by the authors mentioned above, it gives the precise fall-off behaviour required in the asymptotic analysis—and it largely simplifies the latter by offering the possibility of using, if ${{\mathcal J}}^{\pm}$ is sufficiently smooth, local differential geometry instead of taking complicated limits.

In the 1960s and 70s, a large number of articles analyzed the geometrical and physical implications of the new picture and various concepts related to ${{\mathcal J}}^{\pm}$ were discussed: the Bondi mass, the radiation field, the BMS group, ...; see e.g. [52, 54, 63, 64] and the literature cited there. However, while various concepts seemed to find a natural home in the new picture, it was not universally accepted by all workers in the field. Many competing and conflicting aspects are to be considered:

• questions of mathematical generality;
• the definability and properties of physical concepts;
• the capacity to model the various physical situations of interest;
• sharpness of results and avoidance of redundancies;
• existence of solutions to the field equations with the desired asymptotics;
• numerical or analytical calculability of observation related quantities.

Without stating it explicitly in each case, the following discussion will touch, in one way or other, on most of these points.

Penrose [63] analyses the behaviour of the conformal Weyl curvature $C^{\mu}\, _{\nu \lambda \rho}[g]$ of the conformal extension of an asymptotically simple vacuum solution $(\hat{M}, \hat{g})$ in coordinates $x^{\mu}$ adapted to $g_{\mu \nu}$. He gives an argument that $C^{\mu}\, _{\nu \lambda \rho}[g]$ vanishes at ${{\mathcal J}}^+$, assuming the conformal extension is sufficiently smooth and ${{\mathcal J}}^+$ is diffeomorphic to $\mathbb{R} \times \mathbb{S}^2$. Because it involves implicit assumptions on the smoothness of the conformal extension, it is difficult to assess the precise range of validity of this argument. It certainly works if $(M, g, \Omega)$ is of class C⁴, but weakening this assumption is a very delicate matter (the argument as reconsidered in [45] starts from smoothness assumptions stronger than those in [63]). It will be seen below that the vanishing of $C^{\mu}\, _{\nu \lambda \rho}[g]$ at null infinity is in fact necessary for the smoothness of the conformal extension.

The vanishing of the Weyl tensor of the conformal metric at ${{\mathcal J}}^{\pm}$ is thus related to the smoothness of the conformal fields. But what does that mean in terms of the physical fields? It turns out that it is directly related to the very specific fall-off behaviour of the Weyl tensor $C^{\mu}\, _{\nu \lambda \rho}[\hat{g}]$ at null infinity known as Sachs peeling. This has been suggested by Sachs [66] to be characteristic of the asymptotic behaviour of self-gravitating isolated systems.

Assume that the conformal extension $(M, g, \Omega)$ with $M = \hat{M} \cup {{\mathcal J}}^+$ and $g = \Omega^2\, \hat{g}$ on $\hat{M}$ is of class C^k, $k \geqslant 3$. Let the function u on $\hat{M}$ with ${\rm d}u \neq 0$ define a family of null hypersurfaces $\{u = {\rm const}.\}$ in $\hat{M}$ that approach ${{\mathcal J}}^+$ at space-like surfaces $\sim \mathbb{S}^2$ and let $\hat{r}$ be an affine parameter along the future directed null geodesics generating the hypersurfaces $\{u = {\rm const}.\}$ so that $\hat{r} \rightarrow \infty$ at ${{\mathcal J}}^+$. Denote by $\{\hat{\kappa}_A\}_{A = 0, 1}$ a spin frame so that $\hat{e}_{AA'} = \hat{\kappa}_A\, \bar{\hat{\kappa}}_{A'}$ is a $\hat{g}$-pseudo-orthonormal frame with $\hat{e}_{00'} = {\rm grad}_{\hat{g}}u$ and assume that $<\hat{e}_{00'}, {\rm d}\hat{r}> \, = 1$.

It can then be shown that the conformal Weyl tensor $C^{\mu}\, _{\nu \lambda \rho}[g]$ (in coordinates adapted to g) vanishes on ${{\mathcal J}}^+$ if and only if the components $\hat{\Psi}_k$ of the conformal Weyl spinor corresponding to $\hat{g}$ in the spin-frame $\hat{\kappa}_A$ have the Sachs peeling property, i.e. they satisfy, with integer powers of $\hat{r}$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\Psi}_k = \hat{\psi}_k\,\hat{r}^{k - 5} + o(\hat{r}^{k - 5}), \quad \quad k = 0, 1, \ldots 4, \quad \quad {\rm{as}} \quad \hat{r} \rightarrow \infty, \nonumber \end{align*}$$

where the $\hat{\psi}_k$ can be regarded as functions of class C^k−3 on ${{\mathcal J}}^+$ (see [64] for details). We note that the function $\hat{\psi}_4$ is interpreted as the radiation field on ${{\mathcal J}}^+$.

Not everybody shared the opinion that asymptotic simplicity encodes the fall-off behaviour of self-gravitating isolated systems in an appropriate way. Workers who studied equations of motion and tried to calculate the radiation escaping from the system found it difficult to verify Sachs peeling in their settings. This provoked doubts, questions, and gave rise to heated discussions (see [32] and [21]). In the following years many colleagues who follow the subject only by hearsay still seem to consider peeling as a dubious concept; Christodoulou and Klainerman [16] write in 1993: '... it remains questionable whether there exists any non-trivial solution of the field equations that satisfies the Penrose requirements. Indeed, his regularity assumptions translate into fall-off conditions of the curvature that may be too stringent and thus may fail to be satisfied by any solution that would allow gravitational waves'.

Regardless of its geometric elegance, at the time the concept of asymptotic simplicity was just a proposal based on (profound) guess work. It then appeared natural to demand that the fall-off behaviour of gravitational fields at null infinity should not be a matter of guesses but should be derived by achieving precise control on the evolution process. But this leaves the question: Which are appropriate situations from which the fields should evolve, and what precisely is to be achieved? Moreover, far into the 1970s only local-in-time results had been obtained in the general analysis of the Cauchy problem for Einstein's field equations [12, 57].

The first results aiming at the long term evolution of gravitational fields appeared in the early 1980s:

Following Choquet-Bruhat's [34] treatment of the Einstein's vacuum equations as a non-linear system of wave equations for the coefficients of the 'physical' metric, Christodoulou and O'Murchadha [15] analysed the boost problem. They showed that any asymptotically flat initial data for Einstein's vacuum field equations have a development which includes complete space-like surfaces boosted relative to the initial surface. Future or past complete null geodesics, however, were not under control yet.

Friedrich [37, 38] studied representations of the Einstein equations in terms of the conformal fields g, Ω and derived fields, referred to as conformal Einstein equations, and introduced new hyperbolic reductions, aiming at precise and general existence results on solutions admitting smooth conformal boundaries.

To avoid permanently switching from one setting to the other, I shall in the following give up chronological orders (which can roughly be reconstructed from the references) and sketch in outline the second and then the first line of this research and the results obtained. I shall try to put them into perspective and consider questions which I think are more relevant than the one in the title.

To keep this article at a reasonable length, I clearly have to ignore many contributions, for which I apologize. For the many omitted details of this highly technical subject I refer to the original articles, in particular to [8, 16, 60] with regard to the first approach and to [50] and the references given there in the case of the second.

The first results which show that the smoothness of a conformal boundary can be preserved as a consequence of the field equations was obtained by solving the hyperboloidal Cauchy problem [39].

A space-like hypersurface S in the conformal extension of an asymptotically simple space-time $(\hat{M}, \hat{g})$ is referred to as a hyperboloidal hypersurface if it extends smoothly to ${{\mathcal J}}^+$ and is also space-like there. The set $\partial S = S \cap {{\mathcal J}}^+ \sim \mathbb{S}^2$ then defines a boundary of S at which $\Omega = 0$ and $\, {\rm d}\Omega \neq 0$ (there could be inner boundaries as well, but we will not be interested in those here). If $\hat{h}_{\alpha \beta}$ and $\hat{\chi}_{\alpha \beta}$ denote the first and the second fundamental form induced by $\hat{g}$ on $\hat{S} = S \cap \hat{M}$, then—in marked contrast to the behavior of the mean intrinsic curvature of asymptotically flat standard Cauchy data, which must approach zero at space-like infinity—it holds that $\vert \hat{h}^{\alpha \beta}\, \hat{\chi}_{\alpha \beta}\vert \geqslant c = {\rm const}. > 0$ near the end at infinity represented by $S \cap {{\mathcal J}}^+$.

Examples of such hypersurfaces in the conformally compactified Minkowski space considered above are given by the sets $\{\tau = {\rm const}. \neq 0\}$—which comprise, in particular, the extension of the unit hyperbola $\newcommand{\e}{{\rm e}} \{\eta_{\mu\nu}\, x^{\mu}\, x^{\nu} = -1, x^0 > 0\} \subset \{\tau = \frac{\pi}{2}\}$ that motivated the name. There are, of course, many more general examples. For later use, we discuss a particular class of hyperboloidal hypersurfaces in the Schwarzschild space-time with metric

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{g} = - \left(1 - \frac{2\,m}{r}\right)\,{\rm d}t^2 + \left(1 - \frac{2\,m}{r}\right)^{-1}\,{\rm d}r^2 + r^2\,{\rm d}\sigma^2, \nonumber \end{align*}$$

where ${\rm d}\sigma^2$ denotes the standard metric on $\mathbb{S}^2$ and we assume $r > 2\, m \geqslant 0$. Since all structures will be spherically symmetric, angular coordinates are suppressed. In terms of the coordinates $w = t - r - 2\, m\, \log(r - 2\, m)$ and $\rho = 1/r$, it follows that $\Omega^2\, \hat{g} = g$, with

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Omega = \rho, \quad g = - (1 - 2\,m\,\rho)\,\rho^2\,{\rm d}w^2 + 2\,{\rm d}w\,{\rm d}\rho + {\rm d}\sigma^2. \nonumber \end{align*}$$

These fields extend smoothly to the set $\{\rho = 0, \, w \in \mathbb{R}\}$ which describes the future conformal boundary ${{\mathcal J}}^+$ of the Schwarzschild solution. The Cauchy hypersurface $\{t = 0, r > 2\, m\}$ is given in this representation by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle w(\rho) = - \frac{1}{\rho} - 2\,m\,\log\left(\frac{1}{\rho} - 2\,m\right), \quad 0 < \rho < \frac{1}{2\,m}. \nonumber \end{align*}$$

Choosing $\rho_0$ with $0 < \rho_0 < \min (\frac{1}{3\, m}, \frac{1}{1 + 2\, m})$, and replacing $w(\rho)$ by the C¹ function $w_*(\rho)$ with $w_*(\rho) = w(\rho)$ for $\rho_0 \leqslant \rho < \frac{1}{2\, m}$ and $w_*(\rho) = \frac{\partial w}{\partial \rho}\vert _{\rho_0}\, (\rho - \rho_0) + w(\rho_0)$ for $0 \leqslant \rho < \rho_0$, one obtains a spherically symmetric C¹ hypersurface $S'$ which is hyperboloidal. It intersects ${{\mathcal J}}^+$ in the same sphere as the outgoing null hypersurface $\{w = - \frac{\partial w}{\partial \rho}\vert _{\rho_0}\, \rho_0 + w(\rho_0)\}$, and approaches a Minkowskian hyperboloidal hypersurface as $m \rightarrow 0$. Choosing $\rho_0$ small enough and smoothing $S'$ near the sphere $\{w = w(\rho_0), \, \rho = \rho_0\}$ while preserving its space-like nature, we find:

For given $R > 2\, m$, there exist smooth hyperboloidal hypersurfaces in the Schwarzschild solution which have intersections with the Cauchy hypersurface $\{t = 0, \, 2\, m < r\}$ that comprise the set $\{t = 0, \, 2\, m < r \leqslant R\}$ and approach Minkowskian hyperboloidal hypersurfaces as $m \rightarrow 0$.

If the asymptotically simple space-time $(\hat{M}, \hat{g})$ considered above solves Einstein's vacuum field equations, the data $\hat{h}_{\alpha \beta}$ and $\hat{\chi}_{\alpha \beta}$ induced on the hyperboloidal hypersurface S satisfy the vacuum constraints on space-like hypersurfaces, and have a specific fall-off behaviour at the boundary $S \cap {{\mathcal J}}^+$ which allows them to be conformally transformed and smoothly extended to $S \cap {{\mathcal J}}^+$ so as to yield the 1st and 2nd fundamental forms induced by the smooth conformal metric g on S. We refer to data with these properties as smooth hyperboloidal Cauchy data. For the associated hyperboloidal Cauchy problem, the following holds:

Friedrich [39]: Smooth hyperboloidal Cauchy data develop into a solution to the vacuum equations which admits a smooth conformal extension ${{\mathcal J}}'^+$ in the future of S whose null generators have past end points on the boundary of S.

There is no 'smallness condition' required here, and no restriction on the topology of S besides orientability and the existence of a boundary representing the asymptotic end. The 'life time' of the solution depends, of course, on the nature of the data. In general, there may be no conformal gauge in which the null generators of ${{\mathcal J}}'^+$ are future complete (see Geroch and Horowitz [53] for a notion of completeness of null infinity).

There is an important difference here from the formal expansion type analyses considered in previous studies of asymptotically simple solutions. In that case, asymptotic smoothness and thus peeling is put in by hand all along ${{\mathcal J}}^+$. In the present case, it is imposed on the initial slice but is then seen to be preserved along ${{\mathcal J}}'^+$ as a consequence of the field equations.

3.1. Strong non-linear future stability

3.2. Existence of smooth hyperboloidal data

Andersson et al [5] studied the construction of hyperboloidal data $(S, \hat{h}_{ab}, \hat{\chi}_{ab})$ on a 3-manifold with boundary with second fundamental forms $\hat{\chi}_{\alpha \beta}$ satisfying $\hat{\chi}_{\alpha \beta} = \frac{\hat{\chi}}{3}\, \hat{h}_{\alpha \beta}$ on $\hat{S} = S \setminus \partial S$, the analogue of assuming time reflection symmetric data in the standard Cauchy problem. The momentum constraint and the assumed geometry then require $\hat{\chi} = {\rm const}. \neq 0$, so that the free datum is given by the conformal class of the physical 3-metric $\hat{h}_{\alpha \beta}$.

Let $\omega \in C^{\infty}(S)$ be a boundary defining function so that $\omega > 0$ on $\hat{S}$ and $\omega = 0$, ${\rm d}\omega \neq 0$ on $\partial S$ and let $h_{\alpha \beta}$ be a smooth Riemannian metric on S. The ansatz $\hat{h}_{\alpha \beta} = \phi^4\, \omega^{-2}\, h_{\alpha \beta}$ with an unknown scalar function ϕ reduces the Hamiltonian constraint to a singular elliptic problem for ϕ:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle R[\phi^4\,\omega^{-2}\,h] = - \frac{2}{3}\,\hat{\chi}^2, \quad \quad \phi \geqslant \phi_0 = {\rm const}. > 0 \quad {\rm{on}} \quad S. \nonumber \end{align*}$$

There exists a unique solution ϕ to this problem. It is smooth on $\hat{S}$, but admits in general only a polyhomogeneous expansion at $\partial S$—i.e. an asymptotic expansion in terms of the functions $\omega^i\, (\log \omega){\hspace{0pt}}^{\,j}$. The logarithmic terms vanish, and the hyperboloidal data obtained are smooth if and only if one of the following equivalent conditions is satisfied:

• (i)  
  The trace free part of the second fundamental form induced by $h_{\alpha \beta}$ on $\partial S$ vanishes.
• (ii)  
  The conformal Weyl tensor $C^{\mu}\, _{\nu \lambda \rho}$ calculated from the data on S vanishes on $\partial S$.
• (iii)  
  The 'asymptotic shear' of the null geodesic congruence approaching $\partial S$ which defines the Cauchy horizon of the past Cauchy development of the solution determined by the data vanishes at $\partial S$.

If the fields $\hat{h}_{ab}$ and $\hat{\chi}_{ab}$ satisfy the constraints, they allow us to calculate the conformal Weyl tensor $C^{\mu}\, _{\nu \lambda \rho}[\hat{g}]$ on $\hat{S}$, where $\hat{g}$ denotes the physical solution metric determined from these data. On $\hat{S}$, it is equal to the Weyl tensor $C^{\mu}\, _{\nu \lambda \rho}[g]$, where $g = \Omega^2\, \hat{g}$ with a suitable conformal factor Ω. If g extended smoothly to the set $\{\Omega = 0\}$, the tensor $C^{\mu}\, _{\nu \lambda \rho}[g]$ would also extend smoothly. That it should in fact satisfy $C^{\mu}\, _{\nu \lambda \rho}[\hat{g}] = C^{\mu}\, _{\nu \lambda \rho}[g] \rightarrow 0$ at $\partial S$ is a non-trivial condition.

That the asymptotic shear vanishes on the conformal boundary of asymptotically simple vacuum solutions has already been observed by Penrose [63]. That this condition is decisive in the smoothness discussion for hyperboloidal data is again non-trivial.

Most important is the first condition, which is given directly in terms of the free data. It shows that the latter only need to satisfy asymptotic conditions at the boundary $\partial S$ for the data to evolve into a space-time that admits a smooth conformal boundary in its future. These conditions are easily satisfied.

This result opened the way to the construction of more general smooth hyperboloidal data. Andersson and Chruściel generalized the result in two ways [4]: the second fundamental form was only subject to the requirement $\hat{\chi} = \hat{\chi}_{ab}\, \hat{h}^{ab} = {\rm const}. \neq 0$; and free data were admitted that have polyhomogeneous expansions at $\partial S$. With these conditions they could show:

The solutions to the constraints again admit polyhomogeneous expansion at $\partial S$. The non-vanishing of the conformal Weyl tensor at $\partial S$ again contributes to the occurrence of logarithmic terms in the solutions to the elliptically reduced constraints. Conditions on the free data can be given under which the hyperboloidal data extend smoothly to $\partial S$.

While the solutions arising from smooth data admit a smooth conformal extension across null infinity, the much more complicated behaviour near null infinity of solution space-times arising from general Andersson–Chrúsciel hyperboloidal data has not been analysed yet.

Chruściel et al [21] studied general formal Bondi expansions admitting, again, asymptotic polyhomogeneous expansions. While some Bondi expansions admitting some logarithmic terms had been discussed previously (see [21]) they had not been analysed before in such a systematic way¹.

Cutler and Wald [27] managed to construct a parameter dependent family of smooth asymptotically flat standard Cauchy data for the Einstein–Maxwell equations on $\mathbb{R}^3$ that are isometric to Schwarzschild data in a neighbourhood of space-like infinity. As we have seen above, the developments in time of such data contain smooth hyperboloidal hypersurfaces which carry smooth hyperboloidal initial data. Since the standard Cauchy data constructed by the authors approach Minkowskian standard Cauchy data for suitable values of the parameter, the hyperboloidal initial data approach Minkowskian hyperboloidal initial data. Invoking the strong stability result discussed above, they were able to conclude:

There exist non-trivial solutions to the Einstein–Maxwell equations whose causal geodesics are complete and which admit smooth conformal extensions with complete null infinity ${{\mathcal J}}^{\pm}$ and regular points $i^{\pm}$ that represent past and future time-like infinity.

This was the first demonstration of the existence of non-trivial solutions to Einstein's field equations with smooth and complete asymptotics. At the time the data used here looked rather contrived but ten years later they turned out to be special examples of a much larger class of similar data.

4.1. The Corvino gluing construction

The hyperboloidal Cauchy problem makes a clear distinction between asymptotically smooth and non-smooth data, and the smooth data develop into solutions that admit smooth conformal extensions at their future null infinity. In the standard Cauchy problem, the situation at the asymptotic end at space-like infinity is not so clear. Compactified Minkowski space is smoothly foliated by the slices $\{\tau = \tau_* = {\rm const}.\}$. These are hyperboloidal if $\tau_* \neq 0$, while the slice $\{\tau = 0\}$ is asymptotically Euclidean, extending to the regular point i⁰ that represents space-like infinity. When $m_{\rm ADM} > 0$, conformal extensions in which space-like infinity is represented by a regular point do not exist, and the transition from an asymptotically Euclidean slice to hyperboloidal slices is in general more complicated.

To understand possible obstructions to asymptotic smoothness arising in standard Cauchy problems, we need to analyse in detail the structure of solutions in a domain where space-like and null infinity come close to each other. In the physical standard representation of the metric, in terms of which the structures referred to are at infinity, it is not clear what should be meant by this, and the analysis requires complicated limits. On the other hand, conformally compactified Minkowski space, in which space-like infinity is represented by the one point i⁰ is not a good guide if $m_{\rm ADM} \neq 0$. In such a picture, the rich structure discussed below would be compressed into one point, and it would be impossible to analyse the field equations.

The requirement above acquires a concrete meaning in a setting introduced by Friedrich [43], where space-like infinity is represented by a cylinder $I = ] -1, 1[ \times \mathbb{S}^2$ which should be thought of as a further piece of boundary of the physical space-time. It intersects an extended Cauchy hypersurface in the sphere $I^0 = \{0\} \times \mathbb{S}^2$ and touches the sets ${{\mathcal J}}^{\pm} = \{\Omega = 0, {\rm d}\Omega \neq 0\}_{\pm} \sim \mathbb{R} \times \mathbb{S}^2$ at the critical sets $I^{\pm} = \{\pm 1\} \times \mathbb{S}^2$. All these sets, which define boundaries and edges of the physical space-time manifold $\hat{M}$, are given at a finite location in a certain type of coordinate system. The setting and the gauge, including the coordinates, a g-orthonormal frame, and the conformal factor, are determined, apart from some conditions on the initial slice, by the field equations and the conformal structure of its solutions. The conformal factor and thus the location of the prospective hypersurfaces ${{\mathcal J}}^{\pm}$ are known explicitly (see [44] for illustrations and [46] for explicit formulas in the case of Minkowski space). Because the gauge is based on conformal geodesics, the analysis should generalize to more general initial data than those considered below. This requires a careful analysis, however, because the cylinder I, which is not a part of the physical manifold, is generated by limits of these curves.

No smallness conditions are needed here, but to analyse the resulting—somewhat special—initial boundary value problem, it is convenient (though most likely not necessary) to require the initial data to be asymptotically clean in the sense that they are smooth and admit expansions in terms of powers of a radial coordinate $\hat{r}$ with $\hat{r} \rightarrow \infty$ at space-like infinity. Prescribing 'free data' which satisfy this condition, Dain and Friedrich [30] analyse the contraints by standard methods. Besides the solutions of the desired form, there are also some with terms $\hat{r}^k\, \log \hat{r}$, $k \in \mathbb{Z}$ that are related to a non-vanishing linear ADM momentum. As in [16], these are omitted in the following discussion. In a suitable conformal scaling, and in suitably adapted coordinates, the boundary I⁰ of the initial slice at space-like infinity is a sphere at a finite location, and the data for the conformal field equations extend smoothly to I⁰.

In this setting, the reduced conformal field equations are hyperbolic on $\hat{M} \cup I$ and, if the frame admits a continuous extension, also at null infinity ${{\mathcal J}}^{\pm}$. The hyperbolicity is lost, however, at the critical sets $I^{\pm}$. That a standard Cauchy problem underlies the construction is reflected by the fact that the boundary I is a total characteristic: the system of reduced equations reduces on I to a system of interior equations on I. As a consequence, it allows us to calculate a formal expansion of the space-time along I in terms of a coordinate $\rho \geqslant 0$ with $\rho = 0$ on I by integrating interior equations on I. The initial data on I⁰ for this procedure are provided by the data for the conformal field equations on the initial slice and their derivatives of all orders at I⁰ with respect to the radial coordinate ρ. The main observations are:

Even when the data on the initial slice are smooth near I⁰, the solutions on I will in general not extend smoothly to $I^{\pm}$ but develop a polyhomogeneous behaviour.

If the setting is linearized at Minkowski space, so that the equations reduce essentially to the spin-2 equation, the polyhomogeneous behaviour at $I^{\pm}$ spreads along the characteristics represented by ${{\mathcal J}}^{\pm}$ [46].

The occurrence of these logarithmic terms is not a problem of the setting, but a consequence of the evolution equations and the structure of the data. We cannot expect the situation to be any better in the non-linear case. That it will not be worse has been confirmed recently by the results of [55]. The situation can be improved:

The logarithmic terms do vanish after suitable changes of the Cauchy data near I⁰. In particular, if the data are static or stationary near space-like infinity, the whole setting is as smooth near $I \cup I^{\pm} \cup {{\mathcal J}}^{\pm}$ as one could wish [2, 47].

These are just the simplest examples. The point here is, however, not so much the staticity or stationarity of the data in a full neighbourhood of I⁰. What matters instead is the detailed structure of the coefficients of the expansion of the data near I⁰ in terms of the coordinate ρ. It holds, in fact:

It suffices that the data be asymptotically static or stationary at space-like infinity (at all orders or up to some prescribed order) for the integration on I to be (at all orders or up to some prescribed order) free of logarithmic terms at $I^{\pm}$.

In the case of time reflection symmetric data there is a certain amount of evidence that asymptotic staticity of the Cauchy data at space-like infinity is also necessary for the non-occurrence of logarithmic terms at $I^{\pm}$ [43, 68]. Less is known about necessary conditions for the smoothness at $I^{\pm}$ in the case of Cauchy data with non-vanishing second fundamental form. It can be expected that asympotic stationarity of the data at space-like infinity is sufficient for the existence of a smooth conformal boundary at null infinity.

In cases in which sufficient smoothness of the boundary at null infinity can be established, the setting above allows us to perform detailed and explicit calculations which relate the behaviour of the Cauchy data near space-like infinity I⁰ to the behaviour of the fields on ${{\mathcal J}}^+$ near $I^{\pm}$ where the cylinder at space-like infinity meets null infinity [51].

All the results mentioned so far on the existence of 'general' solutions admitting smooth conformal extensions were obtained by using the conformal field equations for certain conformal fields Ω, g, ... $W^{\mu}\, _{\nu \lambda \rho}$ derived from the physical metric $\hat{g}$ and a conformal factor Ω subject to certain gauge conditions. An important subsystem of the equations is given by the conformally covariant Bianchi or spin-2 equation which reads in the vacuum case

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \nabla_{\mu}\,W^{\mu}\,_{\nu \lambda \rho} = 0, \nonumber \end{align*}$$

where $\nabla$ denotes the connection defined by the conformal metric g and $W^{\mu}\, _{\nu \lambda \rho}$ the rescaled version of the conformal Weyl tensor $\hat{C}^{\mu}\, _{\nu \lambda \rho} = C^{\mu}\, _{\nu \lambda \rho}[\hat{g}]$, i.e.

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle W^{\mu}\,_{\nu \lambda \rho} = \Omega^{-1}\,\hat{C}^{\mu}\,_{\nu \lambda \rho} \quad \quad {\rm{on}} \quad \hat{M}. \nonumber \end{align*}$$

This shows that trying to exploit the conformal properties of the Einstein equations in the most direct way has its advantages and its risks. The conformal field equations lead to complete, sharp results in situations in which the solutions to be constructed admit a smooth conformal extension. In cases in which peeling does not hold, however, the unknowns develop a singular behaviour as exemplified above by the situation at space-like infinity. This is reflected by the energy estimates for the symmetric hyperbolic equations of first order for $W^{\mu}\, _{\nu \lambda \rho}$, which is implied by the overdetermined system above in the gauges employed: the integrand in the energy estimates is given by components of the rescaled Bel–Robinson tensor, which become singular if peeling fails.

The results on the global non-linear stability by Christodoulou and Klainerman [16], Bieri and Zipser [8], Lindblad and Rodnianski [60], Lindblad [59], and—most recently—by Hintz and Vasy [55] are less detailed as far as the precise asymptotic behaviour is concerned, but much more complete than the results referred to above in that they start from fairly general asymptotically flat standard vacuum Cauchy data on $\mathbb{R}^3$ and control the past and future completeness of all causal geodesics. The results are obtained by working in terms of the physical metric $\hat{g}$. In [16] and [8] the Bianchi equation $\hat{\nabla}_{\mu}\, \hat{C}^{\mu}\, _{\nu \lambda \rho} = 0$ for the conformal Weyl tensor also plays an important role; its conformal properties are exploited indirectly and the properties of the Bel–Robinson tensor are used extensively to derive the relevant estimates. The work in [55, 59, 60] is based again on the representation of the Einstein equations as a system of wave equations obtained by imposing a wave gauge. Only a few observations of relevance to our discussion will be presented here.

All authors assume the first and second fundamental form $\hat{h}_{ab}$ and $\hat{\chi}_{ab}$ to be smooth, to be close to Minkowskian data in a well-defined sense, and to satisfy certain fall-off conditions near space-like infinity. In the case of [16], these read

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{h}_{ab} = \left(1 + 2\,m\,\vert x\vert ^{-1}\right)\,\delta_{ab} + o_4(\vert x\vert ^{-3/2}), \quad \quad \hat{\chi}_{ab} = o_3(\vert x\vert ^{-5/2}), \nonumber \end{align*}$$

which implies the vanishing of the linear ADM momentum. This is not the case for the generalization given by Bieri in [8], which requires

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{h}_{ab} = \delta_{ab} + o_3(\vert x\vert ^{-1/2}), \quad \quad \hat{\chi}_{ab} = o_2(\vert x\vert ^{-3/2}) \,\, {\rm as}~ \vert x\vert \rightarrow \infty. \nonumber \end{align*}$$

In both cases, x denotes a $\mathbb{R}^3$-valued coordinate near the asymptotic end. It is shown that The causal geodesics of the maximal globally hyperbolic solutions determined by these data are complete, and their curvature tensors $\hat{C}^{\mu}\, _{\nu \lambda \rho}$ approach zero asymptotically in all directions.

For our discussion, the rate at which that happens along null geodesics going out to null infinity is important. The constructions are based on level surfaces H_t and C_u of a time function t and a retarded time function u. The function $\hat{r} = \hat{r}(t, u) > 0$ is chosen to satisfy ${\rm Vol}_{\hat{g}}(H_t \cap C_u) = 4\, \pi\, \hat{r}^2$ on the spherical intersections of the level surfaces. Along the null geodesics generating the null hypersurfaces C_u it holds then that $\hat{r} \rightarrow \infty$ as they run out to future null infinity. Adapting the notation for the components of the conformal Weyl tensor used in the discussion of Sachs peeling, it follows that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \vert \hat{\Psi}_k\vert = O(\hat{r}^{-7/2}), \,\,k = 0, 1, \quad \,\, \,\, \vert \hat{\Psi}_k\vert = O(\hat{r}^{k - 5}), \,\,k = 2, 3, 4,\, \nonumber \end{align*}$$

in the case of [16], and in the case of the generalization given by Bieri in [8] that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \vert \hat{\Psi}_k\vert = o(\hat{r}^{-5/2}), \,\,k = 0, 1, 2, \quad \vert \hat{\Psi}_k\vert = O(\hat{r}^{k - 5}), \,\,k = 3, 4, \quad \nonumber \end{align*}$$

as $\hat{r} \rightarrow \infty$ along a fixed null generator of C_u.  For $k = 3, 4$ the behaviour is thus similar to Sachs peeling, while it deviates from it, in the case of solutions for which these estimates are sharp, for $k = 0, 1$. What is the origin of these deviations?

The result by Hintz and Vasy [55] is based on Cauchy data that admit a polyhomogeneous expansion at space-like infinity, and it is shown that the solutions are polyhomogeneous at null infinity. The logarithmic terms observed at the critical set in the analysis of space-like infinity outlined above should contribute to them, and the logarithmic term observed on the initial slice in the construction of clean initial data [30] should contribute to the stronger deviation in the case of [8]. It would be interesting to know which of the logarithmic terms observed in [4, 21, 55] can in fact occur in the solutions considered in [16] and [8]. Moreover, if the data are specified in terms of weighted Sobolev spaces, or if they are just required to be smooth and to satisfy the fall-off at space-like infinity indicated above, there is a large freedom to have besides logarithmic terms all kinds of other terms in the data which may spoil the smoothness of any conformal extension at null infinity at higher orders.

The question of whether the coefficients coming with the logarithmic terms or any other non-smoothness properties admitted by the Sobolev norms are of any physical significance or just represent 'noise' is left untouched.

Analytical and numerical approximations are of utmost importance, because they allow us to relate measured data to theoretical results. Nevertheless, I shall only make some sketchy remarks about certain aspects related to my topic.

The analytical approximation theory designed to produce quantitative results on the radiation generated e.g. by the merger of black holes should have a counterpart formulated in terms of the Cauchy problem. It certainly would be most useful if more were known about this. The relation between the abstract and the approximative analytical understanding is not easily extracted from the literature, however, because the latter usually immediately intertwines general considerations with the technical details of the approximation method.

Blanchet and Damour note in their extensive work on approximation methods begun in [10] the difficulty of verifying the peeling behaviour (see also the remarks by Christodoulou [13]). On the other hand, they impose near space-like infinity conditions to exclude radiation coming in from the infinite past. As stated more explicitly by Blanchet [9] and Damour and Schmidt [31], it amounts to requiring the solutions to be stationary near space-like infinity. In these articles are also given arguments that the solution will then admit a smooth conformal extension near space-like infinity (a gap in the argument in [31] has been filled in by Dain [29]).

This raises the question of why the asymptotic smoothness should be lost at a later (in retarded time) stage of the development. Is this due to the eruptive behaviour of some matter system? But why should it happen in the pure vacuum case, e.g. in the merger of black holes? Corvino's result and its generalizations were not available at the time. But even if they had been known already, it is hardly conceivable that something like the gluing procedure could be realized in the context of an approximation method. Is the loss of asymptotic smoothness possibly just an artifact of the approximation method?

To keep their numerical grids finite, most relativists who develop numerical 3  +  1 codes for the standard Cauchy problem to calculate binary black hole merger (say) wave forms use cut-off procedures, and essentially ignore space-like and null infinity—thus also the finer details of the asymptotic behaviour there. The radiation field, originally defined at null infinity, is calculated only approximately at a finite, somewhat arbitrary location (see, however, the work by F. Beyer et al [7] and Frauendiener and Hennig [36] which takes first steps towards calculating entire solutions determined by asymptotically flat Cauchy data). This cut-off deletes a neighbourhood of ${{\mathcal J}}^{\pm} \cup i^0$ which is of infinite extent as measured in terms of affine parameters on the outgoing null geodesics. Nevertheless, for the time being, the results seem to be satisfactory.

Characteristic or hyperboloidal Cauchy problems with data prescribed on null or space-like hypersurfaces that extend to null infinity have also been solved numerically. The freedom in the choice of data near ${{\mathcal J}}^+$ can be used to extend the data smoothly to ${{\mathcal J}}^+$. The difference from the standard Cauchy problem is that the wave form extraction can be done at the well and uniquely defined hypersurface ${{\mathcal J}}^+$.

We note that, in all three cases—the standard Cauchy problems, the hyperboloidal problems, and in characteristic initial value problems—there is a large arbitrariness in choosing the data near the asymptotic end of the initial slice.

The following two results are of particular interest in our discussion. Allen and Stavrov Allen [3] show:

Polyhomogeneous, asymptotically hyperbolic, constant mean curvature data of Andersson–Chruściel type for the vacuum Einstein equations can be approximated arbitrarily closely in certain Hölder norms by smooth hyperboloidal constant mean curvature vacuum data.

Corvino and Schoen [25] state another density result by which:

Asymptotically flat initial data for the vacuum Einstein equations on a three-manifold $\hat{S}$ can be approximated by data on $\hat{S}$ which agree with the original data inside a given compact domain, and are in a given end identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball.

It should be noted that these approximations are controlled in terms of Sobolev norms which are weighted so that $\hat{h}_{ab} - \delta_{ab}$ and $\hat{\chi}_{ab}$ and the corresponding approximating data are consistent with the fall-off behaviour required for these fields in [16], for example. We state the results here without further details, since different function spaces may be needed if one wants to answer the most interesting question provoked by these results:

Are the asymptotically simple vacuum solutions in some sense dense in a set of asymptotically flat vacuum solutions as considered in the non-linear stability results above?

If a definite answer could be given to this question, the 'mathematical exercises' discussed in the previous sections could be brought to a conclusion. The proof of any such density results should completely clarify the situation. 'Peeling or not peeling' may just become a matter of deciding between technically more or less convenient representations. It may also show that the questions raised in our discussion of analytic and numerical approximations may be essentially harmless. The situation would be somewhat reminiscent of the introduction of L²-Hilbert spaces in the analysis of hyperbolic equations as an intermediary step towards obtaining existence results about smooth solutions. A positive result should provide interesting information about the precise way in which solutions that are 'rough' at null infinity are approached by solutions that have smooth ${{\mathcal J}}^+$, and in which way concepts that are easily defined on smooth hypersurfaces ${{\mathcal J}}^+$ can be transferred (if at all) to concepts on the 'rough' future null infinity.

On the other hand, showing that such a density property cannot hold should explain in which sense rough asymptotics can be superior to smooth asymptotics. It should give information about physical systems of interest which cannot be modeled in the class of asymptotically simple solutions, and tell us precisely what is lost if we restrict to asymptotically simple solutions. It should further give answers to the following questions.

The logarithmic terms at null infinity mentioned so far come with certain coefficients. Which information is encoded in these coefficients? What is the physical information in the coefficient in the 'free data' underlying the construction of hyperboloidal or Cauchy data which must be set to zero to get rid of logarithmic terms? Does doing so lead to a loss of essential physical information? What is the role of the logarithmic term on the initial slice which is related to the linear ADM momentum? What is its effect on the structure of the radiation field, or other quantities of physical interest on null infinity? If it can be shown that the logarithmic terms found at the critical sets are indeed related to the deviation of the data from being asymptotically stationary, could this be interpreted as saying that radiation coming in from past null infinity has to be excluded (up to some order) close to space-like infinity to achieve asymptotic smoothness (of a prescribed order)?

Most of the considerations above are related to the structure of the asymptotic end at space-like infinity in the standard and to the asymptotic end at null infinity in the hyperboloidal Cauchy problem. But in the 'real world' of our cosmos, a system which we would like to see as one representing a self-gravitating isolated system does not possess anything like an asymptotic end at space-like or null infinity.

The best we can do is to consider an open, relatively compact subset $S'$ of a time-slice S of our cosmos with the Cauchy data $d'$ induced on it so that its domain of dependence $D(S')$ contains the essential part of the object and any related process of interest but no further comparable objects. As a next step, we could try to attach an asymptotically flat or hyperboloidal end smoothly to $(S', d')$².

If S is assumed to be compact and all the matter fields are ignored, space-time engineering as suggested by Chruściel and Pollack [19] allows us in fact to glue an asymptotically flat or hyperboloidal end to S. The resulting standard Cauchy data will contain, however, a huge number of other systems—which we wish to exclude—and there may even be something like a minimal surface close to the location of the gluing process. What we want is closer to the results of Czimek [28], who constructs asymptotically flat extensions of solutions to the vacuum constraints on a compact manifold with boundary that have vanishing mean extrinsic curvature. Since this is done, so far, only for data close to Minkowskian data, the result—as it stands—does not cover systems containing e.g. black holes. Another possibility might be to extend the set $(S', d')$, possibly after some modification close to its boundary, along the lines of R. Bartnik's parabolic constructions of constrained data discussed in [6]. All these studies suggest that it is not too far-fetched to assume, as we shall do, that $(S', d')$ can be embedded isometrically into some asymptotically flat standard Cauchy set. Our earlier discussions then show that it can equally well be embedded into smooth hyperboloidal initial data sets.

Whatever one does, while suggested by a large class of static or stationary exact solutions, and while being consistent with the field equations under much more general assumptions, the asymptotically flat or hyperboloidal end is a just figment. One of its main virtues is to allow the field equations themselves to construct, in a marvelously effective way, a null infinity, and the radiation signal to unfold while approaching that null infinity along outgoing null rays.

As we have seen, however, there exists a huge freedom to choose or modify asymptotic ends of initial data near space-like or null infinity, leaving the data unchanged on a large interior subset. The transition from one choice to another will affect the solution space-time in a neighbourhood of null infinity, and the gluing region may give rise to some spurious radiation which we may not accept as being associated with the system we wish to study (a phenomenon well known to numerical relativists). But who says that some such radiation had not been encoded already in the first choice of data? Those of us who spent much of their time working on the Kerr family or other real analytic stationary solutions may find it strange, but if we consider ends of class $C^{\infty}$ or C^k, we have to face the fact that 'the' correct asymptotic end simply does not exist. We can only hope to optimize the situation in some sense.

This raises the question: How is the best use to be made of the freedom to choose the asymptotically Euclidean or hyperboloidal end? And more specifically: To what extent do radiation fields and other physically relevant quantities defined at null infinity depend on the precise structure of the initial data near space-like (or null) infinity?

Analysing these questions should give deeper insight into some physics, because the answers will depend very much on the nature of the system we wish to model and the conclusions we want to draw. The inspiral, merger and ring down of black holes can be expected to be accompanied by strong, well-pronounced signals in the domain of dependence of some interior domain. Reasonable changes near the ends at space-like or null infinity are likely to have little effect on these, and we may well choose the end to be stationary. In scattering problems involving weak fields, however, we can, in principle, only toy with the data on ${{\mathcal J}}^- \cup \, i^-$ (see [14] and, for a full treatment of a neighbourhood of i⁻, also [22, 49]). Then we have to watch how things develop at space-like infinity, and what comes out at ${{\mathcal J}}^+$. There are data on ${{\mathcal J}}^-$ which make the end at space-like infinity stationary, but whether they will be useful in this context depends very much on the questions to be answered.

I would like to thank Bieri, Corvino, Chruściel, Garfinkle, Isenberg, Schoen and Wald for discussions. Part of this work was supported by the program on Geometry and Relativity at the Erwin Schroedinger Institute, Vienna. I wish to thank the Institute for support, hospitality and the opportunity to talk about the subject with various colleagues.