Space-like infinity

3.5 Space-like infinity

As indicated above, the problem at $i0$ is one of the most urgent and important ones related to the conformal properties of isolated systems. However, it is also one of the most complicated, and a thorough discussion of all its aspects is not possible here. We will try to explain in rough terms what the new developments are, but we have to refer to [67

] for the rigorous statements and all the details.

There have been several approaches over the years towards a treatment of space-like infinity. Geroch [79 ] gave a geometric characterization along the same lines as for null-infinity based on the conformal structure of Cauchy surfaces. He used his construction to define multipole moments for static space-times [76 , 77 ], later to be generalized to stationary space-times by Hansen [87 ]. It was shown by Beig and Simon [20 , 152 ] that the multipole moments uniquely determine a stationary space-time and vice versa.

Different geometric characterizations of spatial infinity in terms of the four-dimensional geometry were given by Sommers [153 ], Ashtekar, and Hansen [11 , 7 ], and by Ashtekar and Romano [12 ]. The difficulties in all approaches that try to characterize the structure of gravitational fields at space-like infinity in terms of the four-dimensional geometry arise from the lack of general results about the evolution of data near spatial infinity. Since there are no radiating solutions that are general enough at spatial infinity to provide hints, one is limited more or less to one’s intuition. So all these constructions essentially impose “reasonable” asymptotic conditions on the gravitational field at $i0$ , and from them derive certain nice properties of space-times which satisfy these conditions. But there is no guarantee that there are indeed solutions of the Einstein equations that exhibit the claimed asymptotic behaviour. In a sense, all these characterisations are implicit definitions of certain classes of space-times (namely those that satisfy the imposed asymptotic conditions). What is needed is an analysis at space-like infinity which is not only guided by the geometry but which also takes the field equations into account (see e.g. [18 , 19 ] for such attempts using formal power series).

Recently, Friedrich [67 ] has given such an analysis of space-like infinity, which is based exclusively on the initial data, the field equations and the conformal structure of the space-time. In this representation several new aspects come together. First, in order to simplify the analysis, an assumption on the initial data (metric and extrinsic curvature) on an asymptotically Euclidean hypersurface $&tidle;Σ$ is made. Since the focus is on the behaviour of the fields near space-like infinity, the topology of $&tidle; Σ$ is taken to be $3 ℝ$ . It is assumed that the data are time-symmetric ( $χ&tidle;ab = 0$ ) and that on $&tidle; Σ$ a (negative definite) metric $&tidle; hab$ with vanishing scalar curvature is given. Let $Σ$ be the conformal completion of $(&tidle;Σ, &tidle;hab)$ which is topologically $S3$ , obtained by attaching a point $i$ to $&tidle;Σ$ , and assume furthermore that there exists a smooth positive function $Ω$ on $Σ$ with $Ω (i) = 0$ , $dΩ (i) = 0$ , and $Hess Ω (i)$ negative definite. Furthermore, the metric $2&tidle; hab = Ω hab$ extends to a smooth metric on $S$ . Thus, the three-dimensional conformal structure defined by $&tidle;hab$ is required to be smoothly extensible to the point $i$ .

From these assumptions follows that the conformal factor near $i$ is determined by two smooth functions $U$ and $W$ , where $U$ is characterized by the geometry near $i$ , while $W$ collects global information because $W (i) = 12mADM$ , while $U (i) = 1$ . With this information the rescaled Weyl tensor, the most important piece of initial data for the conformal field equations, near $i$ is found to consist of two parts, a “massive” and a “mass-less” part. Under suitable conditions, the mass-less part, determined entirely by the local geometry near $i$ , can be extended in a regular way to $i$ , while the massive part always diverges at $i$ as $1∕|x|3$ in a normal coordinate system $(x1,x2,x3)$ at $i$ unless the ADM mass vanishes.

In order to analyze the singular behaviour of the initial data in more detail, the point $i$ is blown up to a spherical set $0 I$ essentially by replacing it with the sphere of unit vectors at $i$ . Roughly speaking, this process yields a covering space $𝒞$ of (a suitable neighbourhood of $i$ in) $Σ$ projecting down to $Σ$ which has the following properties: The pre-image $I0$ of $i$ is an entire sphere while any other point on $Σ$ has exactly two pre-image points. There exists a coordinate $r$ on $𝒞$ which vanishes on $I0$ , and which is such that on each pair of pre-image points it takes values $r$ and $− r$ , respectively. The actual blow-up procedure involves a rather complicated bundle construction that also takes into account the tensorial (respectively spinorial) nature of the quantities in question. The reader is referred to [67 ] for details.

Consider now a four-dimensional neighbourhood of space-like infinity. The next important step is the realization that, in order to take full advantage of the conformal structure of space-time, it is not enough to simply allow for metrics that are conformally equivalent to the physical metric, but that one should also allow for more general connections. Instead of using a connection that is compatible with a metric in the conformal class, one may use a connection $∇a$ compatible with the conformal structure, i.e. that satisfies the condition

∇c &tidle;gab = 2λc&tidle;gab

for some one-form $λa$ . If $λa$ is exact, then one can find a metric in the conformal class for which $∇a$ is the Levi–Civita connection. Generally, however, this will not be the case. This generalization is motivated by the use of conformal geodesics as indicated below, and its effect is to free up the conformal factor, which we call $Θ$ to distinguish it from the conformal factor $Ω$ given on the initial surface $Σ$ , from the connection. (Recall that two connections, that are compatible with metrics in the same conformal class, differ only by terms which are linear in the first derivative of the conformal factor relating the metrics.) As a consequence, the conformal field equations, when expressed in terms of a generalized connection, do not any longer contain an equation for the conformal factor. It appears, instead, as a gauge source function for the choice of conformal metric. Additionally, a free one-form appears which characterizes the freedom in the choice of the conformal connection.

To fix this freedom, Friedrich uses conformal geodesics [73 ]. These curves generalize the concept of auto-parallel curves. They are given in terms of a system of ordinary differential equations (ODE’s) for their tangent vector together with a one-form along them. In coordinates this corresponds to a third-order ODE for the parameterization of the curve. Their crucial property is that they are defined entirely by the four-dimensional conformal structure with no relation to any specific metric in that conformal structure.

A time-like congruence of such curves is used to set up a “Gauß” coordinate system in a neighbourhood of $i0$ and to define a conformal frame, a set of four vector fields that are orthonormal for some metric in the conformal class. This metric in turn defines a conformal factor $Θ$ that rescales it to the physical metric. The one-form determined by the conformal geodesics defines a conformal connection $∇a$ , thus fixing the freedom in the connection. In this way, the gauge is fixed entirely in terms of the conformal structure. One may wonder whether the choice of conformal “Gauß” coordinates is a practical one, since it is well known that the usual Gauß coordinates are prone to develop singularities very quickly. However, conformal geodesics are in general not geodesics in the physical space-time, and parallel transport with a Weyl connection is not the same as parallel transport in the physical space-time. If one thinks of the usual Gauß coordinates as being constructed from the world-lines of freely falling particles, then one should think of the conformal geodesics as the world-lines of some fiducial “particles” that experience a force determined by the conformal structure of the space-time. This “conformal force” could counteract the attractive nature of the gravitational field under certain conditions, thereby delaying the intersection of the world-lines and the formation of coordinate singularities. In fact, Friedrich [69 ] has shown that there exist global conformal Gauß coordinates on the Kruskal extension of the Schwarzschild space-time that extend smoothly beyond null-infinity. Thus, one would hope that the conformal Gauß systems can be used to globally cover more general space-times as well. This would make them an ideal tool for use in numerical simulations because they are intrinsically defined so that the numerical results would also have an intrinsic meaning. Furthermore, as outlined below, the use of a conformal Gauß system simplifies the field equations considerably.

If the physical space-time is a vacuum solution of the Einstein equations, then one can say more about the behaviour of the conformal factor $Θ$ along the conformal geodesics: It is a quadratic function of the natural parameter $τ$ along the curves, vanishing at exactly two points, if the initial conditions for the curves are chosen appropriately. The vanishing of $Θ$ indicates the intersection of the curves with $ℐ ±$ . The intersection points are separated by a finite distance in the parameter $τ$ .

Figure 8: The geometry near space-like infinity: The “point” $i0$ has been blown up to a cylinder that is attached to $− ℐ$ and $+ ℐ$ . The physical space-time is the exterior part of this “stovepipe”. The “spheres” $± I$ and $0 I$ are shown in blue and light green, respectively. The brown struts symbolize the conformal geodesics used to set up the construction. Note that they intersect $ℐ$ and continue into the unphysical part.

Now, one fixes an initial surface $Σ$ with data as described above, and the conformal geodesics are used to set up the coordinate system and the gauge as above. When the blow-up procedure is performed for $Σ$ , a new finite representation of space-like infinity is obtained, which is sketched in Figure 8 .

The point $i$ on the initial surface has been replaced by a sphere $0 I$ , which is carried along the conformal geodesics to form a finite cylinder $I$ . The surfaces $± ℐ$ are the surfaces on which the conformal factor $Θ$ vanishes. They touch the cylinder in the two spheres $I±$ , respectively. The conformal factor $Θ$ vanishes with non-vanishing gradient on $I$ and on $ℐ ±$ , while on the spheres $I±$ its gradient also vanishes.

In this representation there is for the first time a clean separation of the issues that determine the structure of space-like infinity: The spheres $± I$ are the places where “ $ℐ$ touches $0 i$ ” while the finite cylinder $I$ serves two purposes. On the one hand, it represents the endpoints of space-like geodesics approaching from different directions, while, on the other hand, it serves as the link between past and future null-infinity. The part “outside” the cylinder where $r$ is positive between the two null surfaces $± ℐ$ corresponds to the physical space-time, while the part with $r$ negative is not causally related to the physical space-time but constitutes a smooth extension. For easy reference, we call this entire space an extended neighbourhood of space-like infinity.

The conformal field equations, when expressed in the conformal Gauß gauge of this generalized conformal framework, yield a system of equations that has properties similar to the earlier version: It is a system of equations for a frame, the connection coefficients with respect to the frame, and the curvature, split up into the Ricci and the Weyl parts; they allow the extraction of a reduced system that is symmetric hyperbolic and propagates the constraint equations. Its solutions yield solutions of the vacuum Einstein equations whenever $Θ > 0$ . The Bianchi identities, which form the only sub-system consisting of partial differential equations, again play a key role in the system. Due to the use of the conformal Gauß gauge, all other equations are simply transport equations along the conformal geodesics.

The reduced system is written in symbolic form as

with symmetric matrices $A τ$ , $Ar$ , $A 𝜃$ , $A ϕ$ , and $B$ which depend on the unknown $u$ and the coordinates $(τ,r,𝜃,ϕ )$ . This system as well as initial data for it, first defined only on the original space-time, can be extended in a regular way to an extended neighbourhood of space-like infinity, which allows for the setup of a regular initial value problem at space-like infinity. Its properties are most interesting: When restricted to $I0$ , the initial data coincide necessarily with Minkowski data, which together with the vanishing of $Θ$ implies that on the entire cylinder $I$ the coefficient matrix $r A$ vanishes. Thus, the system (37

) degenerates into an interior symmetric hyperbolic system on $I$ . Therefore, the finite cylinder $I$ is a total characteristic of the system. The two null-infinities $ℐ ±$ are also characteristics, and at the intersections $I±$ between them and $I$ , the system degenerates: The coefficient matrix $A τ$ which is positive definite on $I$ and $ℐ ±$ looses rank on $± I$ .

The fact that $I$ is a total characteristic implies that one can determine all fields on $I$ from data given on $I0$ . $I$ is not a boundary on which one could specify in- or outgoing fields. This is no surprise, because the system (37 ) yields an entirely structural transport process that picks up data delivered from $− ℐ$ via $− I$ , and moves them to $+ ℐ$ via $+ I$ . It is also consistent with the standard Cauchy problem where it is known that one cannot specify any data “at infinity”.

The degeneracy of the equations at $± I$ means that one has to take special precautions to make sure that the transitions from and to $± ℐ$ are smooth. In fact, not all data “fit through the pipe”; Friedrich has derived restrictions on the initial data of solutions of the finite initial value problem that are necessary for regularity through $I±$ . These are necessary conditions on the conformal class of the initial data, stating that the Cotton tensor and all its symmetrized and trace-removed derivatives should vanish at the point $i$ in the initial surface. If this is not the case, then the solution of the intrinsic system will develop logarithmic singularities, which will then probably spread across null-infinity, destroying its smoothness. So here is another concrete indication that initial data have to be restricted, albeit in a rather mild way, in order for the smooth picture of asymptotic flatness to remain intact.

In order to get more insight into the character of the transport equations along the cylinder $I$ and, in particular, to see whether these conditions are also sufficient, one can take advantage of the group theoretical origin of the blow-up procedure, and write all the relevant quantities according to their tensorial character as linear combinations of a complete set of functions on the sphere. (According to the Peter–Weyl theorem, the matrix entries of irreducible representations of $SU (2)$ form a complete system of functions on $SU (2)$ , these functions being closely related to the spin-spherical harmonics [133 ].) Insertion into the transport equations and restriction to the cylinder $I$ yields a set of ODE’s for the coefficients along the conformal geodesics that generate $I$ . Taking successive $r$ -derivatives and restricting to the cylinder yields a hierarchy of such ODE’s. The system of ODE’s obtained at order $n$ can be solved provided a solution up to order $n − 1$ is known. Thus, given initial conditions, one can determine recursively the expansion coefficients and thereby obtain a Taylor approximation of the fields in a neighbourhood of $I$ . These calculations are rather cumbersome and can be performed only with the aid of computer algebra.

In view of the unknown status of the above condition on the Cotton tensor, it is clear that a particularly interesting scenario is obtained for conformally flat initial data. For such data, the Cotton tensor vanishes identically, so that the conditions above are satisfied. If any logarithms appear in the solutions of the hierarchy of ODE’s, then this implies that the vanishing of the Cotton tensor and its symmetrized, trace-removed derivatives is only necessary and not sufficient.

In [162 ] Valiente Kroon has carried out these calculations up to order $n = 6$ in the general case and up to order $n = 8$ for data that were in addition assumed to be axi-symmetric. The rather surprising result is this: Up to order $n = 4$ the solutions do not contain any logarithms. This changes with $n = 5$ , where there are logarithmic terms that contribute additively to the solution with coefficients $(5) G$ proportional to the Newman–Penrose constants. This means that the solution is smooth up to $n = 5$ only if the Newman–Penrose constants vanish. Assuming this, then for $n = 6$ the solutions will again have an additive logarithmic term with some coefficients $G (6)$ (higher-order Newman–Penrose “constants”) whose vanishing is necessary for smoothness of the solution. This picture returns at the orders $n = 7$ and $n = 8$ , where again the vanishing of certain quantities $(n) G$ is necessary for the solution to be free of logarithms.

From the explicit algebraic expressions for the $G (n)$ up to order $8$ , one can guess a formula that expresses them for arbitrary orders in terms of certain expansion coefficients of the conformal factor on the initial hypersurface. Then, the vanishing of the $(n) G$ implies that the expansion coefficients of the conformal factor are correlated in a very specific way. In the special case of the Schwarzschild space-time (which is spherically symmetric, hence has conformally flat initial data), one can compute the expansion coefficients of the conformal factor explicitly, and one finds that they are correlated in exactly the way implied by the vanishing of the $G (n)$ . This observation seems to indicate that the time evolution of an asymptotically Euclidean, conformally flat, and smooth initial data set admits a smooth extension to null-infinity near space-like infinity if and only if it agrees with the Schwarzschild solution (the only static solution in the conformal class of the data) to all orders at the point $i$ . This conjecture seems to imply that the developments of the Brill–Lindquist and Misner initial data sets have non-smooth null-infinities.

While the conditions on the Cotton tensor are entirely local to $i$ , this is not so for the additional conditions discussed above. They involve the part $W$ of the conformal factor which is related to the ADM mass of the system and which is obtained by solving an elliptic equation. This part is therefore determined by properties of the initial data that are not local to $i$ . To what extent it is really the global structure of the initial data which is involved here is not so clear since the results by Corvino [35 ] and Corvino–Schoen [36 ] show that one can deform initial data in certain annular regions of the initial hypersurface while keeping the asymptotics and the inner regions untouched.

These observations also shine a new light on the static, respectively stationary, solutions. It might well be that they play a fundamental role in the construction of asymptotically flat space-times. If it turns out that such space-times have to be necessarily static (or stationary in the non-time-symmetric case) near $i0$ in order to have a smooth conformal extension, then this implies that there are no gravitational waves allowed in the neighbourhood of space-like infinity. In particular, this means that incoming radiation has to die off in the infinite future, while the system cannot have emitted gravitational waves for all times in the infinite past. Whether this is a reasonable scenario for an isolated system is a question of physics. It has to be answered by investigating whether it is possible to discuss realistic physical processes like radiation emission and scattering in a meaningful way such as whether one can uniquely define physical quantities like energy-momentum, angular momentum, radiation field, etc.

This touches upon the question as to how physically relevant the assumption of a smooth asymptotic structure will in the end turn out to be. It is after all an idealisation, which we use as a tool to describe an isolated system. If we want to find out whether this idealisation captures physically relevant scenarios, we first have to know the options, i.e. we need to have detailed knowledge about the mathematical situations that can arise. Whether these situations are general enough to admit all physical situations that one would consider as reasonable remains an open question.

The setting described in the above paragraphs certainly provides the means to analyze the consequences of the conformal Einstein evolution near space-like infinity and to understand the properties of gravitational fields in that region. The finite picture allows the discussion of the relation between various concepts that are defined independently at null and space-like infinity. As one application of this kind, Friedrich and Kánnár [71 ] have related the Newman–Penrose constants, which are defined by a surface integral over a cut of $ℐ +$ , to initial data on $Σ$ . The cut of $ℐ +$ is pushed down towards $I+$ , where it is picked up by the transport equations of system (37 ). In a similar way, one can relate the Bondi and ADM masses of a space-time.