3.3 Initial value problems3 The Regular Conformal Field 3.1 General properties of the

3.2 The reduction process for the conformal field equations 

We have set up the conformal field equations as a system of equations for the conformal geometry of a conformal Lorentz manifold. As such they are invariant under general diffeomorphisms and, as we have seen, under conformal rescalings of the metric. In this form they are not yet very useful for treating questions of existence of solutions or even for numerical purposes. For existence results and numerical evolution the geometric equations have to be transformed into partial differential equations for tensor components which can then be used to set up well-posed initial value problems for hyperbolic systems of evolution equations.

This process, sometimes referred to as ``hyperbolic reduction'' consists of several steps. First, one needs to break the invariance of the equations. By imposing suitable gauge conditions one can specify a coordinate system, a linear reference frame and a conformal factor. Then the equations can be written as equations for the components of the geometric quantities with respect to the chosen frame in the chosen conformal gauge and as functions of the chosen coordinates. In the next step, one needs to extract from the equations a subsystem of propagation equations which is hyperbolic so that it has a well-posed initial value problem. It is often referred to as the ``reduced equations''. Finally, one has to make sure that solutions of the reduced system give rise to solutions of the full system. This step may involve the verification that the gauge conditions imposed are compatible with the propagation equations, or that other equations (constraints) not included in the reduced system are preserved under the propagation. The first two steps, choice of gauge and extraction of the reduced system, are very much related. Gauge conditions should be imposed such that they lead to a hyperbolic reduced system. Furthermore, the gauge conditions should be such that they can be imposed locally without loss of generality.

The gauge freedom present in the conformal field equations can easily be determined. The freedom to choose the coordinates amounts to four scalar functions while the linear reference frame, which we take to be orthogonal, can be specified by a Lorentz rotation, which amounts to six free functions. Finally, the choice of a conformal factor contributes another free function. Altogether, there are eleven functions which can be chosen at will.

Once the geometric equations have been transformed into equations for components, the next step is to extract the reduced system. These are equations for the components of the geometric quantities defined above as well as for gauge-dependent quantities: the components of the frame with respect to the coordinate basis, the components of the connection with respect to the given frame and the conformal factor.

There are several well-known choices for coordinates (harmonic, Gauß, Bondi, etc.), as well as for frames (Fermi-Walker transport, Newman-Penrose, etc.). These are usually ``hard-wired'' into the equations and one has no further control on the properties of the gauge. Gauß coordinates for instance have the tendency to become singular when the geodesic congruence which is used for their definition starts to self-intersect. Similarly, Bondi coordinates are attached to null-hypersurfaces which have the tendency to self-intersect thus destroying the coordinate system. In the context of existence proofs and the numerical evolution of the equations it is of considerable interest to have additional flexibility in order to prevent the coordinates or the frame from becoming singular. The goal is to ``fix the gauge'' in as flexible a manner as possible and to obtain reduced equations which still have useful properties.

A scheme to obtain the reduced equations in symmetric hyperbolic form while still allowing for arbitrary gauges has been devised by Friedrich [48Jump To The Next Citation Point In The Article] (see also [55Jump To The Next Citation Point In The Article] for various examples). The idea is based on the following observation. Cartan's structure equations which express the torsion and curvature tensors in terms of tetrad and connection coefficients are two-form equations: They are skew on two indices and the information contained in the equations is not enough to fix the tetrad and the connection by specifying the torsion and the curvature. The additional information is provided by fixing a gauge. Normally, this is achieved by reducing the number of variables, in this case the number of tetrad components and connection coefficients. However, one can just as well add appropriate further equations to have enough equations for all unknowns. The additional equations should be chosen so that the ensuing system has ``nice'' properties.

We illustrate this procedure by a somewhat trivial example. Consider, in flat space with coordinates tex2html_wrap_inline4478, a one-form tex2html_wrap_inline4480 which we require to be closed:


From this equation we can extract three evolution equations, namely




Obviously, these three equations are not sufficient to reconstruct tex2html_wrap_inline4480 from appropriate initial data. One possibility to proceed from here is to specify one component of tex2html_wrap_inline4480 freely and then obtain equations for the other three. However, it is easily seen that only by specifying tex2html_wrap_inline4494 we can achieve a pure evolution system. Otherwise, we get mixtures of evolution and constraint equations. So we may note that proceeding in this way leads to a restriction of possibilities as to which components should be specified freely and, in general, it also entails that derivatives of the specified component appear.

Another possible procedure is to enlarge the system by adding an equation for the time derivative of tex2html_wrap_inline4494 . Doing this covariantly implies that we should add an equation in the form of a divergence


where F is an arbitrary function. This results in the system





which is symmetric hyperbolic for any choice of F . Note also that F appears as a source term and only in undifferentiated form. Clearly, our influence on the component tex2html_wrap_inline4494 is now very indirect via the solution of the system, while before we could specify it directly.

In a similar way, one proceeds in the present case of the conformal field equations. Note, however, that this way of fixing a gauge is not at all specific to these equations. Since it depends essentially only on the form of Cartan's structure equations it is applicable in all cases where these are part of the first order system. The Cartan equations can be regarded as exterior equations for the one-forms tex2html_wrap_inline4516 dual to a tetrad tex2html_wrap_inline4518 and the connection one-forms tex2html_wrap_inline4520 . Similar to the system above, the equations involve only the exterior derivative of the one-forms and so we expect that we should add equations in divergence form, namely


with arbitrary gauge source functions tex2html_wrap_inline4522 for fixing coordinates and tex2html_wrap_inline4524 for choosing a tetrad. Note that tex2html_wrap_inline4526 implies tex2html_wrap_inline4528 .

In a given gauge (i.e., coordinates and frame field are specified) the gauge sources can be determined from


In fact, these equations are exactly the same equations as (23Popup Equation) except that they are written in a more invariant form. Now it is obvious that the gauge sources contain information about the coordinates and the frame used. What needs to be shown is that any specification of the gauge sources fixes a gauge. In fact, suppose we are given functions tex2html_wrap_inline4530 and tex2html_wrap_inline4532 on tex2html_wrap_inline4534 then there exist (locally) coordinates tex2html_wrap_inline4536 and a frame tex2html_wrap_inline4538 so that in that coordinate system the gauge sources are just the prescribed functions tex2html_wrap_inline4530 and tex2html_wrap_inline4532 . This follows from the equations



These are semi-linear wave equations which determine a unique solution from suitably given initial data close to the initial surface. Note that on the right hand side of (26Popup Equation) there is a function of the tex2html_wrap_inline4536, and not a source term. The equations can be solved in steps. Once the coordinates tex2html_wrap_inline4536 have been determined from (26Popup Equation), the right hand side of (27Popup Equation) can be considered as a source term.

Finally, we need to discuss the gauge freedom in the choice of the conformal factor tex2html_wrap_inline3721 . In many discussions of asymptotic structure the conformal factor is chosen in such a way that null-infinity is divergence-free, in addition to the vanishing of its shear, which is a consequence of the asymptotic vacuum equations. That means that infinitesimal area elements remain unchanged in size as they are parallelly transported along the generators of tex2html_wrap_inline3905 . Since they also remain unchanged in form due to the vanishing shear of tex2html_wrap_inline3905, they remain invariant and hence they can be used to define a unique metric on the space of generators of tex2html_wrap_inline3905 . This choice simplifies many calculations on tex2html_wrap_inline3905, still leaving the conformal factor quite arbitrary away from tex2html_wrap_inline3905 . Yet, in numerical applications this choice of the conformal factor may be too rigid and so one needs a flexible method for fixing the conformal factor.

It turns out that one can introduce a gauge source function for the conformal gauge as well. Consider the change of the scalar curvature under the conformal rescaling tex2html_wrap_inline4560, tex2html_wrap_inline4562 : It transforms according to


Reading this transformation law as an equation for tex2html_wrap_inline3875 we obtain


It follows from this equation that we may regard the scalar curvature as a gauge source function for the conformal factor: For, suppose we specify the function tex2html_wrap_inline4568 arbitrarily on tex2html_wrap_inline3889, then Equation (28Popup Equation) is a non-linear wave equation for tex2html_wrap_inline3875 which can be solved given suitable initial data. This determines a unique tex2html_wrap_inline3875, hence a unique tex2html_wrap_inline4576 and tex2html_wrap_inline4578 such that the scalar curvature of the rescaled metric tex2html_wrap_inline4578 has scalar curvature tex2html_wrap_inline4568 . Note that these considerations are local. They show that locally the gauges can be fixed arbitrarily. However, the problem of identifying and fixing a gauge globally is very difficult but also very important because only when the gauges are globally known one can really compare two different space-times.

Having established that the gauge sources do in fact, locally, fix a unique gauge we can now split the system of conformal field equations into evolution equations and constraints. The resulting system of equations is exhibited below. The reduction process is rather straightforward but tedious. It is sketched in Appendix  6 . Here, we only describe it very briefly. We introduce an arbitrary time-like unit vectorfield tex2html_wrap_inline4584 which has a priori no relation to the tetrad field used for framing. We split all the tensorial quantities into the parts which are parallel and orthogonal to that vector field using the projector tex2html_wrap_inline4586 . The connection coefficients for the four-dimensional connection tex2html_wrap_inline4099 are treated differently. We introduce the covariant derivatives of the vectorfield tex2html_wrap_inline4584 by


They account for half (9+3) of the four-dimensional connection coefficients. The other half is captured by defining a covariant derivative tex2html_wrap_inline4594 which has the property that it annihilates both tex2html_wrap_inline4584 and tex2html_wrap_inline4598 and agrees with tex2html_wrap_inline4099 when acting on tensors orthogonal to tex2html_wrap_inline4584 (see Equations (40Popup Equation)). Note that we have not required that tex2html_wrap_inline4584 be the time-like member of the frame, nor have we assumed that it be hypersurface orthogonal. In the latter case, tex2html_wrap_inline4606 is the extrinsic curvature of the family of hypersurfaces orthogonal to tex2html_wrap_inline4584 and hence it is symmetric. Furthermore, the derivative tex2html_wrap_inline4594 agrees with the Levi-Civita connection of the metric tex2html_wrap_inline4612 induced on the leaves by the metric tex2html_wrap_inline3923 .

We write the equations in terms of the derivative tex2html_wrap_inline4594 and the ``time derivative'' tex2html_wrap_inline4618 which is defined in a way similar to tex2html_wrap_inline4594 (see Equation (40Popup Equation)), because in this form it is quite easy to see the symmetric hyperbolicity of the equations.

As they stand, the Equations (90Popup Equation, 91Popup Equation, 92Popup Equation, 93Popup Equation, 94Popup Equation, 95Popup Equation, 96Popup Equation, 97Popup Equation, 98Popup Equation, 99Popup Equation, 100Popup Equation, 101Popup Equation, 102Popup Equation, 103Popup Equation) form a symmetric hyperbolic system of evolution equations for the collection of 65 unknowns


This property is present irrespective of the particular gauge. For any choice of the gauge source functions tex2html_wrap_inline4522, tex2html_wrap_inline4626, tex2html_wrap_inline4628 and tex2html_wrap_inline4630, the system is symmetric hyperbolic. The fact that the gauge sources appear only in undifferentiated form implies that one can specify them not only as functions of the space-time coordinates but also as functions of the unknown fields. In this way, one can feed information about the current status of the evolution back into the system in order to influence the future development.

Other ways of specifying the coordinate gauge, including the familiar choice of a lapse function and a shift vector, are not as flexible because then not only these functions themselves appear in the equations, but also their derivatives. Specifying them as functions of the unknown fields alters the principal part of the system and, hence, the propagation properties of the solution. This may not only corrupt the character of the system but it may also be disastrous for the numerical applications because an uncontrolled change of the local propagation speeds implies that the stability of a numerical scheme can break down due to violation of the CFL condition (see [38Jump To The Next Citation Point In The Article] for a more detailed discussion of these issues). However, due to the intuitive meaning of lapse and shift they are used (almost exclusively) in numerical codes.

There are several other ways of writing the equations. Apart from various possibilities to specify the gauges which result in different systems with different numbers of unknowns, one can also set up the equations using spinorial methods. This was the method of choice in almost all of Friedrich's work (see e.g. [52Jump To The Next Citation Point In The Article] and also [39Jump To The Next Citation Point In The Article]). The ensuing system of equations is analogous to those obtained here using the tetrad formalism. The main advantages of using spinors is the fact that the reduction process automatically leads to a symmetric hyperbolic system, that the variables are components of irreducible spinors which allows for the elimination of redundancies, and that variables and equations become complex and hence easier to handle.

Another possibility is to ignore the tetrad formalism altogether (or, more correctly, to choose as a basis for the tangent spaces the natural coordinate frame). This also results in a symmetric hyperbolic system of equations (see [55, 81Jump To The Next Citation Point In The Article]), in which the gauge dependent variables are not the frame components with their corresponding connection coefficients but the components of the spatial metric together with the usual Christoffel symbols and the extrinsic curvature.

The fact that the reduced equations form a symmetric hyperbolic system leads, via standard theorems, to the existence of smooth solutions which evolve uniquely from suitable smooth data given on an initial surface. We have the

Theorem 2:  [Friedrich [48]]  For functions tex2html_wrap_inline4522, tex2html_wrap_inline4626, tex2html_wrap_inline4628, tex2html_wrap_inline4630 on tex2html_wrap_inline4534 and data given on some initial surface let u be the solution of the reduced equations. If u satisfies the conformal field equations (16Popup Equation, 17Popup Equation, 18Popup Equation, 19Popup Equation, 20Popup Equation, 21Popup Equation) on the initial surface then, in fact, it satisfies them on the entire domain of dependence of the initial surface in the space-time defined by u .  

The proof of this theorem relies on the existence of a ``subsidiary system'' of equations for the zero-quantity Z (see Equation (15Popup Equation)), whose vanishing indicates the validity of the conformal field equations. This system turns out to be linear, symmetric hyperbolic and homogeneous . Thus, one has uniqueness of the solutions so that Z vanishes in the domain of dependence of the initial surface if it vanishes on the surface. Hence, the conformal field equations hold. It can be shown that solutions obtained from different gauge source functions are in the same conformal class, so they lead to the same physical space-time.

3.3 Initial value problems3 The Regular Conformal Field 3.1 General properties of the

image Conformal Infinity
Jörg Frauendiener
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