The reduction process for the conformal field equations

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 that 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 that 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 that 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 that 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 [58 ] (see also [65 ] 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 $(xμ) = (t,x1,x2,x3 )$ , a one-form $ω$ which we require to be closed:

∂μω ν − ∂νωμ = 0.

From this equation we can extract three evolution equations, namely

∂tω1 − ∂1ω0 = 0, ∂tω2 − ∂2ω0 = 0, ∂tω3 − ∂3ω0 = 0.

Obviously, these three equations are not sufficient to reconstruct $ω$ from appropriate initial data. One possibility to proceed from here is to specify one component of $ω$ freely and then obtain equations for the other three. However, it is easily seen that only by specifying $ω 0$ can we 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 $ω0$ . Doing this covariantly implies that we should add an equation in the form of a divergence

∂ μωμ = F,

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

∂tω0 − ∂1ω1 − ∂2ω2 − ∂3ω3 = F, ∂tω1 − ∂1ω0 = 0, ∂tω2 − ∂2ω0 = 0, ∂tω3 − ∂3ω0 = 0,

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 $ω0$ 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 $ωμa$ dual to a tetrad $eaμ$ and the connection one-forms $ωμaν = ω μc∇aecν$ . 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 $μ F$ for fixing coordinates and $μ F ν$ for choosing a tetrad. Note that $ωaμ= ∇ax μ$ implies $F μ = □x μ$ .

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 (23

, 24

) 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 $ˆμ F$ and $ˆ μ F ν$ on $4 ℝ$ ; then there exist (locally) coordinates $μ xˆ$ and a frame $a ˆeμ$ so that in that coordinate system the gauge sources are just the prescribed functions $ˆF μ$ and $Fˆμν$ . 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 (27

) there is a function of the $ˆx μ$ , and not a source term. The equations can be solved in steps. Once the coordinates $ˆxμ$ have been determined from (27

), the right hand side of (28

) can be considered as a source term.

Finally, we need to discuss the gauge freedom in the choice of the conformal factor $Ω$ . 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 $ℐ$ . Since they also remain unchanged in form due to the vanishing shear of $ℐ$ , they remain invariant and hence they can be used to define a unique metric on the space of generators of $ℐ$ . This choice simplifies many calculations on $ℐ$ , still leaving the conformal factor quite arbitrary away from $ℐ$ . 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 $gab ↦→ ˆgab = 𝜃2gab$ , $Ω ↦→ Ωˆ = 𝜃Ω$ : It transforms according to

( ) −2 □𝜃 Λ ↦→ ˆΛ = 𝜃 Λ + --- . 4𝜃

Reading this transformation law as an equation for $𝜃$ we obtain

It follows from this equation that we may regard the scalar curvature as a gauge source function for the conformal factor: Suppose we specify the function $ˆ Λ$ arbitrarily on $ℳ$ , then Equation (29

) is a non-linear wave equation for $𝜃$ which can be solved given suitable initial data. This determines a unique $𝜃$ , hence a unique $ˆΩ$ and $ˆgab$ such that the scalar curvature of the rescaled metric $ˆgab$ has scalar curvature $ˆΛ$ . 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 can one really compare two different space-times.

Having established that the gauge sources do, in fact, fix a unique gauge locally, 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 vector field $ta$ , which has a priori no relation to the tetrad field used for framing. We split all the tensorial quantities into the parts that are parallel and orthogonal to that vector field using the projector $b b b ha = δa − tat$ . The connection coefficients for the four-dimensional connection $∇a$ are treated differently. We introduce the covariant derivatives of the vector field $a t$ by

χab = hcb∇cta, χa = tc∇cta.

They account for half ( $9 + 3$ ) of the four-dimensional connection coefficients. The other half is captured by defining a covariant derivative $∂a$ that has the property that it annihilates both $a t$ and $ta$ and agrees with $∇a$ when acting on tensors orthogonal to $ta$ (see Equations (42 )). Note that we have not required that $ta$ be the time-like member of the frame, nor have we assumed that it be hypersurface orthogonal. In the latter case, $χ ab$ is the extrinsic curvature of the family of hypersurfaces orthogonal to $a t$ and hence it is symmetric. Furthermore, the derivative $∂a$ agrees with the Levi–Civita connection of the metric $hab$ induced on the leaves by the metric $gab$ .

We write the equations in terms of the derivative $∂a$ and the “time derivative” $∂$ , which is defined in a way similar to $∂a$ (see Equation (42 )), because in this form it is quite easy to see the symmetric hyperbolicity of the equations.

As they stand, the Equations (93 , 94 , 95 , 96 , 97 , 98 , 99 , 100 , 101 , 102 , 103 , 104 , 105 , 106 ) form a symmetric hyperbolic system of evolution equations for the collection of 65 unknowns

{cμ,cμa,χab,χa, Γ abc,Λab,ϕab,ϕa,Eab, Bab,Ω, σa,σ,S}.

This property is present irrespective of the particular gauge. For any choice of the gauge source functions $μ F$ , $Fa$ , $Fab$ , and $Λ$ , 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 do these functions themselves appear in the equations, but so do 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 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 [46 ] 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 to write the equations. Apart from various possibilities to specify the gauges that 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. [62 ] and also [45 ]). 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 [65 , 94 ]), 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 that evolve uniquely from suitable smooth data given on an initial surface. We have the

Theorem 2 (Friedrich [58 ]): For functions $Fμ$ , $Fa$ , $Fab$ , $Λ$ on $ℝ4$ and data given on some initial surface, let $u$ be the solution of the reduced equations. If $u$ satisfies the conformal field equations (16 , 17 , 18 , 19 , 20 , 21 ) 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 (15 )), 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.