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Conformal geodesics on vacuum space-times

We discuss properties of conformal geodesics on general, vacuum, and warped product space-times and derive a system of conformal deviation equations. The results are used to show how to construct on the Schwarzschild-Kruskal space-time global conformal Gauss coordinates which extends smoothly and without degeneracy to future and past null infinity.


Introduction
Geometrically defined coordinate systems are convenient in the local analysis of fields but they often degenerate on larger domains. It is well known, for instance, that Gauss coordinates are of limited use because the underlying geodesics tend to develop caustics. In [7] conformal Gauss coordinates have been introduced and successfully employed in local studies of conformally rescaled space-times.
Here the time-like coordinate lines are generated by conformal geodesics, curves which are associated with conformal structures in a similar way as geodesics are associated with metrics. In this case the danger of a coordinate break-down is even more severe, because the conformal geodesics generating the system can not only develop envelopes or intersections, as occur in caustics of metric geodesics, but they may even become tangent to each other. However, it will be shown in this article that the structure responsible for this difficulty also has useful aspects.
Subsequent analyses of conformal Gauss coordinates revealed that conformal geodesics possess various remarkable properties. In [4] they have been studied systematically in the context of the conformal field equations. Though this at first introduces additional complications into the equations because it requires the use of Weyl connections, it leads to unexpected simplifications. While the Bianchi equation satisfied by the rescaled conformal Weyl tensor, which is given by d i jkl = Θ −1 C i jkl where Θ denotes the conformal factor, always implies hyperbolic reduced systems of partial differential equations, the generalized conformal field equations (in vacuum, possibly with a cosmological constant) admit in the new gauge hyperbolic reductions in which the frame coefficients, the connection coefficients and the components of the conformal Ricci tensor are governed by equations of the form ∂ τ e µ k = . . . , ∂ τΓi j k = . . . , ∂ τRjk = . . . , where τ denotes the time variable in the given gauge and the right hand sides are given by algebraic functions of the unknowns e µ k ,Γ i j k ,R jk , d i jkl and known functions on the solution manifold.
Moreover, associated with a conformal geodesic is a function along that curve, the conformal factor, which is determined up to a constant factor that can be fixed on a given initial hypersurface. This function necessarily has zeros at points where the conformal geodesic crosses null infinity. It turns out that for given data on the initial hypersurface this function can be determined explicitly and acquires a form which allows us to prescribe to null infinity by a suitable choice of initial data a finite coordinate location.
In [4] these facts are used to provide a rather complete discussion of Anti-de Sitter-type solutions, including a characterization of these solutions in terms of initial and boundary data. In [5] these properties form a basic ingredient of a detailed investigation of the behaviour of asymptotically flat solutions near space-like and null infinity under certain assumptions on the Cauchy data for the solutions. In particular, the cylinder at space-like infinity introduced in [5] to remove the conformal singularity at space-like infinity is obtained as a limit set of conformal geodesics. The results obtained there strongly suggest that even under the most stringent smoothness assumptions on the conformal data near space-like infinity the solutions will in general develop logarithmic singularities at null infinity. However, they also suggest that these singularities can be avoided if the data satisfy certain regularity conditions at space-like infinity.
The conversion of these conjectures into facts requires the proof of a certain existence result for the regular finite initial value problem at space-like infinity formulated in [5]. While the basic difficulty of this proof has nothing to do with the specific features of the underlying conformal Gauss system, only the proof will show that the latter is doing what we expect it to do: to extend near space-like infinity smoothly and without degeneracy to null infinity if null infinity admits a smooth differentiable structure at all.
An independent proof that conformal Gauss systems behave this way on general space-times is difficult, because it requires the information on the asymptotic structure of the solution which we first hope to obtain by completing the analysis of [5]. The question whether conformal Gauss systems can be used globally is even more difficult.
In the context of the hyperboloidal initial value problem with smooth initial data it can be shown that these coordinates remain good for a while and that they can in fact be globally defined for data sufficiently close to Minkowskian hyperboloidal data. However, nothing is known for data which deviate strongly from Minkowskian once. There remains the possibility to study the behaviour of conformal Gauss systems on specific solutions. It has been shown in [5] that there exist conformal Gauss systems on the Schwarzschild space-time which smoothly cover neighbourhoods of space-like infinity including parts of null infinity.
One may be tempted to think that this is only possible due to the weakness of the field in that domain (although it doesn't look weak in the conformal picture) while in regimes of strong curvature a break-down of the coordinates will occur. It turns out that this is not true. It is the main results of this article that there exist conformal Gauss coordinates on the Schwarzschild-Kruskal spacetime which smoothly cover the whole space-time and which extend smoothly (in fact analytically) and without degeneracy through null infinity.
This raises hopes that conformal Gauss systems stay regular under much wider circumstances than expected so far and that we may be able to exploit the simplicity of the reduced equations and the fact that conformal Gauss systems provide by themselves the conformal compactification in time directions under quite general assumptions. How robust these systems really are, whether they will stay regular under non-linear perturbations of the Schwarschild-Kruskal space, remains to be seen. Though a general analytical investigation of these equations appears difficult at present, important information about them may be obtained by numerical calculations. Conversely, there are good reasons to believe that it should be possible to give strong analytical support to numerical work based on the use of conformal Gauss systems because the latter are amenable to a geometric analysis.
We begin the article by discussing properties of conformal geodesics on general space-times and then specialize to vacuum space-times, using also some of the results which have been obtained already in the articles referred to above. For later application we work out the conformal geodesic equations on warped product space-times. Various features of conformal Gauss systems are then illustrated by explicit examples on Minkowski space-time.
After specializing further to warped product vacuum space-times a conformal Gauss system on the Schwarzschild-Kruskal space-time will be analysed. Expressions for the conformal geodesics determined by suitably chosen initial data are derived in terms of elliptic and theta functions. It is shown that the curves extend smoothly through null infinity and that the associated conformal factor defines a smooth conformal extension if the congruence is regular. We do not attempt to derive the explicit expression of the rescaled metric in terms of the new coordinates, because this cannot be done anyway in more general situations. Instead, the regularity of the coordinate system and the conformal extension is established by analysing a system of equations which is the analogue for conformal geodesics of the Jacobi equation for metric geodesics. While specific features of the Schwarzschild solution are used here, it is clear that similar techniques apply in more general cases. We end the article by indicating as a possible application the numerical calculation of entire asymptotically flat vacuum solutions.

Conformal geodesics on pseudo-Riemannian manifolds
Before we introduce conformal geodesics we recall a few concepts and formulae of conformal geometry. On a pseudo-Riemannian manifold (M,g) of dimension n ≥ 3 we consider two operations preserving the conformal structure defined bỹ g: (i) conformal rescalings of the metric with smooth conformal factor Ω > 0, (ii) transitions ∇ →∇ of the Levi-Civita connection ∇ of a metric g in the conformal class ofg into (torsion free) Weyl connections∇ with respect tog. In terms of the Christoffel symbols Γ and the connection coefficientsΓ, defined by ∇ ∂µ ∂ ν = Γ µ ρ ν ∂ ρ and∇ ∂µ ∂ ν =Γ µ ρ ν ∂ ρ respectively, the transition is described by with a smooth 1-form f . For given f the difference tensor S(f ) depends only on the conformal structure of g. The transformation of the Levi-Civita connection Γ ofg under 1 is a special case of (2) in which the 1-form is exact Conversely, if the 1-form f is closed (2) arises locally from a rescaling of g with a suitable conformal factor. By (2) we havê It follows from this that∇ preserves the conformal structure ofg in the sense that with a function θ > 0 satisfying on a given curve the metric θ 2 g µν is parallely transported along x(τ ) with respect to∇. The curvature tensors of the connections∇ and ∇, defined by (∇ λ∇ρ − ∇ ρ∇λ ) X µ =R µ νλρ X ν etc., are related bŷ were indices are raised or lowered with respect to g. The tensor C µ νλρ of the curvature tensor of∇ into its trace parts and the trace-free, conformally invariant conformal Weyl tensor C µ νλρ , is related to the tensor by the equation
Thus, the changes of the initial data (10) which locally preserve the point set spread out by the curve x(τ ) are given bẏ The 1-form remains unchanged and an affine parameter transformation results if α * = 0. With suitable choices of the free constants, however, any fractional linear transformations of the parameter can be obtained and it can be arranged that τ → ∞ at any prescribed value ofτ . This fact is related to an important difference between conformal geodesics and metric geodesics. It may happen that the parameter τ on a conformal geodesic takes any value in R while the curve still acquires endpoints in M as τ → ±∞. If the transformation in (14) has a pole it may further happen that the point sets spread out by x(τ ),x(τ ) coincide only partly, namely at the points where the transformation is defined, but each of the curves extends into a region not entered by the other one. In fact, there may occur an arbitrary number of such overlaps (we will encounter and make use of this phenomenon below). In certain contexts it may be preferable to call the union of the corresponding point sets a conformal geodesic. However, it will be more convenient for us to preserve this name for a solution of the conformal geodesic equation with its distinguished parameter.
If the 1-form b is used to define a connection∇ along x(τ ) by requiring that its difference tensor with respect to∇ is given bŷ equation (8) can be written in the form∇ẋẋ = 0. Thus x(τ ) is an autoparallel with respect to∇. Equation (9), which determines the connection∇ along that autoparallel, acquires some meaning by comparing it with (7). A congruence of conformal geodesics, covering smoothly and without caustics an open set U , defines a smooth 1-form field b and thus a connection∇ on U . If ∇, f , L are replaced in (7) by∇, b,L respectively and transvect withẋ, we find that equation (9) takes the form of a restriction on the connection∇ in the direction of the congruence, given byL µνẋ µ = 0. Here it turns out important that (7) does not contain the contraction of ∇ ν f µ .
Let f be an arbitrary 1-form and∇ =∇ + S(f ) the associated Weyl connection. Observing (7) with ∇, L replaced by∇,L, we find for any curve x(τ ) in M and 1-form b(τ ) along x(τ ) the identities It follows that conformal geodesics are invariant under transitions to general Weyl connections and, in particular, under conformal rescalings ofg in the following sense: if x(τ ), b(τ ) is a solution of these equations with respect to∇, then is a solution of the conformal geodesic equation with respect to∇ =∇ + S(f ). Note that the parameter τ is an invariant of the conformal class ofg. It is determined by the initial conditions (10) but it does not depend on the Weyl connection and the metric in the conformal class chosen to write the conformal geodesics equations. Let e k , k = 0, 1, . . . n − 1, be a frame field which is parallel along x(τ ) for the connection∇ of (16) and satisfiesg(e i , e k ) = Θ −2 diag(1, . . . 1, −1, . . . , −1) at x(τ * ) with some Θ = Θ * > 0. It follows then from (4), (5), that this relation is preserved along x(τ ) with the function Θ = Θ(τ ) satisfying Consider again a congruence of conformal geodesics as above and initial data Θ * chosen such that the function Θ satisfying (19) is smooth and positive on U . Then the associated 1-form is given in terms of the Levi-Civita connection ∇ of the metric g = Θ 2g , for which the frame is orthonormal, by along x(τ ). Equations (19) and (11) imply x(τ ) We assume the congruence, the function Θ, and the frame e k to be constructed by a suitable choice of the initial data such that From (21) and where the index of h is raised with g. This implies that Thus the frame e k , which is parallely propagated with respect to∇, is in general not parallely but always Fermi-propagated with respect to ∇.
The following general observation, which follows by a direct calculation, describes the sense in which Fermi-transport is conformally invariant. Let θ > 0 be some conformal factor andg, g metrics on M such that g = θ 2g . Denote bỹ F , F the respective Fermi-transports. Let x(τ ) be any curve in M such that g(ẋ,ẋ) = 1. For any vector field X along the curve we then haveF θẋ (θ X) = θ 2 FẋX, where θẋ is the tangent vector of the curve parametrized in terms of g-arc length.
Applying this to the situation considered above, we see that theg-orthonormal frame Θe k is Fermi-transported along the conformal geodesics, if the latter are parametrized in terms ofg-arc length.
Let f : M → M be a conformal diffeomorphism ofg such that f * g = Ω 2g with some function Ω on M . Since conformal geodesics are invariants of the conformal structure, it is not surprising that f maps conformal geodesics into conformal geodesics. If K is a conformal Killing vector field its flow defines a 1-parameter family of local conformal diffeomorphism and thus maps the set of conformal geodesics into itself. Though we will occasionally make use of this fact, it will not be demonstrated it here and we refer to [10] for the formal argument.

The conformal deviation equations
Following the discussion of the Jacobi equation for metric geodesics, we derive now an analogous system of equations for conformal geodesics. Let x(τ, λ), b(τ, λ) be a family of solutions to (8), (9) depending smoothly on a parameter λ. We denote the tangent vector field of the conformal geodesics, the deviation vector field of the congruence, and the deviation 1-form by respectively. Considering x = x(τ, λ) as a map from some open subset of R 2 intoM and denoting by T x its tangent map, gives T where (S(b; X, Y )) µ = S(b) ρ µ ν X ρ Y ν for given 1-form b and vector fields X, Y . The last two terms on the right hand side of (23) indicate why conformal geodesics are potentially more useful than metric geodesics for the construction of coordinate systems. Under suitable circumstances the acceleration induced by the 1-form b may counteract curvature induced tendencies of the curves to develop caustics.
Caustics of conformal geodesics can be more complicated than caustics of metric geodesics because for a given tangent vector there exists (essentially) a 3-parameter family of conformal geodesics with the same tangent vector. Moreover, to analyse them we need to complement the conformal Jacobi equation by a equation which governs the behaviour of B.
We refer to the equations (23), (24) as to the system of conformal deviation equations. They form a linear system of ODE's for Z and B along the curves x(τ ).

Conformal geodesic equations on warped products
For later applications we work out the simplifications of the conformal geodesic equations which are obtained in the case whereg can be written as a warped product. Thus we assume that there exist coordinates x µ for which the set I = {0, 1, . . . , n − 1} in which the index µ takes its values can be decomposed into two non-empty subsets I 1 , I 2 with I = I 1 ∪ I 2 , ∅ = I 1 ∩ I 2 , such that in terms of the coordinates x A , A ∈ I 1 , and x a , a ∈ I 2 , the metric takes the form The Christoffel coefficients are then given bỹ Those which cannot be derived by symmetry considerations from the coefficients above, vanish. The curvature tensor has components where D denotes the h-Levi-Civita connection with connection coefficientsΓ A B C . Components which cannot be deduced by symmetry considerations from those above, vanish. In particular This implies L Ac [g] = 0, The conformal geodesic equations take the form

Conformal geodesics on vacuum fields
Though many of the subsequent discussions apply to more general situations, we shall assume from now on that dim M = 4, sign(g) = (1, −1, −1, −1) and that g satisfies Einstein's vacuum field equationsR µν = 0. With the resulting simplification and the notation introduced above the conformal geodesic equations and the equation for the frame e k now take the form Since these equations admit solutions with vanishing 1-form b, metric geodesics form on vacuum space-times a subclass of conformal geodesics.

Conformal geodesics on Minkowski space
Denote by x µ coordinates on Minkowski space (M, η) in which the metric takes and let ǫ k = δ µ k ∂ µ be the orthonormal standard frame. There are various possibilities to determine the solutions to (33), (34), (35). The framework of the normal conformal Cartan connection (cf. [4]), in which conformal geodesics are defined as "horizontal" curves on a certain bundle allows a purely algebraic calculation of these curves, because the bundle admits in the case of the conformally compactified Minkowski space a purely group theoretical description. Since the curves on the bundle space are known explicitly there only remains the task to calculate their projectioin onto the base space. We just give the result of the calculation. Besides the initial data (10) we prescribe the value Θ * > 0 and a Lorentz transformation A * which determines the initial value e k * = Θ −1 * A j * k ǫ j of the frame e k . The corresponding solution is given by with a Lorentz transformation A(τ ) = (A j k (τ )) given by Indices are moved in this subsection with the metric η. Conformal geodesics which satisfy η(ẋ,ẋ) = 0 at one point coincide (as point sets) with null geodesics, those for which b * = 0 are Minkowski geodesics. Ifẋ * is space-or time-like we can assume, possibly after a reparametrization, that < b * ,ẋ * > = 0. It follows that the remaining conformal geodesics fall into one of the following classes. Ifẋ * and b ♯ * generate a space-like 2-surface in the tangent space of x * , the corresponding conformal geodesics is a metric circle in the plane tangent to that 2-surface. Ifẋ * and b ♯ * generate a time-like 2-surface andẋ * is space-like, the corresponding conformal geodesics is a space-like metric hyperbola in the plane tangent to that 2-surface. Ifẋ * is space-like and b * is null, the conformal geodesics is a space-like curve in a null planed.
We shall mainly be interested in the last case in whichẋ * and b ♯ * generate a time-like 2-surface andẋ * is time-like. An example of such a conformal geodesic is given by the curve We shall use now time-like conformal geodesics to construct coordinate systems on Minkowski space where the parameter τ on the curves defines a time coordinate while the other coordinates are obtained by dragging along spatial coordinates onS. Most important for us is the fact that the construction provides a conformal rescaling and extension of Minkowski space to which the coordinates are adapted in a natural way.
We denote by r * the restriction of the standard radial coordinate r on Minkowski space to the hypersurfaceS = {x 0 = 0}. In spherical coordinates θ, φ the inner metric induced on S then takes the formh = −(dr 2 * + r 2 * dσ 2 ) with dσ 2 = dθ 2 + sin 2 θ dφ 2 the standard line element on the 2-sphere. If we write r * = tan( χ 2 ) with 0 ≤ χ < π, we find that the conformal factor Θ * = 2 (1 + r 2 * ) −1 = 1 + cos χ allows us to realize the conformal embedding of To define a congruence of conformal geodesics and a conformal factor, we choose initial data onS as follows. At The function a onS is determined by the following consideration. With the data above and τ * = 0 we get Θ Our data are then spherically symmetric and we can express the conformal factor (36) and the curves (37) in terms of the time coordinate t = x 0 and the radial coordinate r to obtain To relate these expressions to known facts, we use r * = tan χ 2 and set τ 2 = tan s 2 to obtain Θ = Ω ω, with Ω = cos s + cos χ, ω = 1 + ( t = sin s cos χ + cos s , r = sin χ cos χ + cos s .
Reading the last equation as a coordinate transformation, we get The set M E = R × S endowed with the metric g E defines the Einstein cosmos. We see that (42) realizes the well known conformal embedding of Minkowski space into the Einstein cosmos, which maps the former onto the subset |s ± χ| < π, 0 ≤ χ < π of M E . The boundary ∂M of this set in M E supplies representations of future and past null infinity, future and past time-like infinity, and space-like infinity of Minkowski space, which are given by the subsets J ± = {s ± χ = ±π}, i ± = {s = ±π, χ = 0}, i 0 = {s = 0, χ = π} of ∂M respectively.
In terms of τ and Θ we get from (40) the metric which extends smoothly to J ± but does not extend to i ± where ω → ∞. This is not due to an unfortunate choice of intial data onS. The curve τ → z(τ ) = (s = 2 arctan τ 2 , χ = 0) is a conformal geodesic on the Einstein cosmos which approaches i ± as τ → ±∞. The freedom to perform parameter transformations, characterized by (14), (15), does not allow us to find a parametrization for which curve extends simultaneously to i − and i + . Thus the separation of i − and i + realizes a conformal invariant.
If we insist on a conformal factor Θ which defines a conformal compactification of (S,h) and initial data ensuring b = Θ −1 d Θ onS, invariance under t → − t implies the existence of a point onS where b vanishes. The conformal geodesic through such a point will then be a metric geodesic. In our first example such a point is given by r * = 0 and it follows from (40) that the parameter τ takes values in R and the conformal factor is constant on the curve r = 0. Thus, in order to achieve a compactification which extends smoothly to i + for a finite value of the parameter we have to choose non-time-symmetric initial data such as (44).
Notice that s * could have been chosen to be a function onS. The possibility to change the parametrization while leaving the curves as point sets unchanged offers a large freedom to select slices of constant parameter value.

The conformal factor and the 1-form on general vacuum fields
In the following we shall derive some general, explicit information on the solutions to (33), (34), (35) (cf. the more general discussion in [4] on solutions to the vacuum field equationsR µν = λg µν with a cosmological constant λ). From (33), (34) follows where the index sharp on the symbol of a metric indicates here and in the following the contravariant version of that metric. Assume again that the congruence of conformal geodesics, Θ, and the frame have been chosen such that (22), whenceg(ẋ,ẋ) = Θ −2 holds along the congruence. Equation (11) then implies∇ẋ Θ = Θ < b,ẋ >. Taking a derivative and observing (45) yields∇ 2 Taking another derivative and observing (46) gives finally∇ In terms of the initial data (10) this yields with Θ * = (g(ẋ * ,ẋ * )) the explicit expression Similar steps lead to∇ẋ (g(ẋ,ẋ) (g ♯ (b, b)) 2 ) = 0, whence In particular, the sign ofg ♯ (b, b) is preserved as long as Θ > 0. From (34), This implies with (49) the relatioṅ While the expressions (48), (50) depend on invariants built from the initial data, they are universal in the sense that their form does not depend on the solutioñ g.
Using the representation of the 1-form with respect to the connection of g, i.e. (20), we can write by (50) on the congruence e k (Θ) = Θ < b, e k > − Θ < h, e k >. If there is a point on a given curve of the congruence where Θ vanishes and Θ b and h remain bounded on the curve, (50), (51) imply

Theg-adapted form of the conformal geodesic equation
We write b =b + ζẋ ♭ (where here and below indices of 1-forms and vectors are moved withg) withb such that We introduce the parameter transformation with inverse τ = τ (τ ) and setx(τ ) = x(τ (τ )). Thenx ′ ≡ ∂τx = Θẋ satisfies g(x ′ ,x ′ ) = 1 and we obtain theg-adapted conformal geodesic equations where, by (50), along the conformal geodesic. These equations bring out the important role of b. Ifb vanishes at a point it vanishes alongx(τ ) and the curve is ag-geodesics. We determine the transformation (54) under the assumption that It follows then from (48) that Θ(τ ) vanishes at with τ + = τ − , such that and the integration of (54) gives In terms of the parameterτ we also get Suppose that the solution admits a smooth conformal extension for which Θ = 0 on J + , that the extension admits a point i + such that J + coincides with the past light cone of i + , and that one of our conformal geodesics, x(τ ) say, passes through i + for a finite value τ i of τ . Then condition (57) precludes a dicussion of the zero of Θ on x(τ ). If (58) describes the zeros of Θ on the conformal geodesics covering a neighbourhood of x(τ ), then, approaching x(τ ), we should find τ ± → τ i , and consequently β = 0, < b * ,ẋ * > = 0 on x(τ ) and τ i = τ * − 2 <b * ,ẋ * > .

Conformal geodesics and theg-adapted conformal deviation equations on warped product vacuum fields
It will be shown now that on warped product vacuum fields the discussion of the conformal geodesic equations reduces under suitable assumptions to the analysis of one equation and that a similar statement is true for theg-adapted conformal deviation equations. Ifg can be written in the form (25), the vacuum field equations take the form and the dependence of the various fields in the second equation on x a implies We shall assume from now on that p = 2 and that the indices A, a take values 0, 1 and 2, 3 respectively. Then These equations imply with (26), (27) It follows that R µ νλρ = 0 unless c = 0. If Einstein's vacuum field equationL µν = 0 holds, equations (29), (30), (31), (32) admit solutions satisfyingẋ a = 0, b c = 0, and these solutions obey equations which can be written in the form where all quantities are derived from h. We shall discuss now theg-adapted version of (70), assuming that ∆ ≡ det(h AB ) < 0. With it follows that ǫ h (ẋ, .) = |∆|(ẋ 0 d x 1 −ẋ 0 d x 0 ) and h ♯ (ǫ h (ẋ, .), ǫ h (ẋ, .)) = −h(ẋ,ẋ). Since the vectorsẋ,b ♯ are contained in the 2-dimensional space spanned by ∂ A , A = 0, 1, and <b,ẋ > = 0, the equations above and (56) imply the representation where the sign is determined by the initial conditions and the convention for β. Using this expression in the second of equations 55, gives β 2x′ = Dx′b = ± β ǫ h (Dx′x ′ , .), which can be rewritten in the form Thus, theg-adapted forms of the two equations (70) are equivalent to each other and we only have to solve (73) withb as in (72) to obtain a solution to (70). Observing the formulae for the connection coefficients, the results (28) for the curvature tensor of warped products, and equations (64), (65) are seen to be equivalent to each other and to the equation The sign here is chosen as in (72) and use has been made of the identities

Conformal geodesics on the Schwarzschild space-time
For our discussions various forms of the Schwarzschild line element will be needed. Its standard form with mass m is given forr > 2 m bỹ In terms of the retarded and advanced null coordinates the line element (76) is obtained in the forms respectively. These extend analytically into regions wherer ≤ 2 m. The retarded null coordinate w extends smoothly through J + and through the past horizon to the past singularity, while the advanced null coordinate v extends smoothly through J − and the future horizon to the future singularity. The transformation applied to (76) with m > 0 produces the Schwarzschild-Kruskal line element The coordinate r, given forr > 2 m bȳ yields the isotropic Schwarzschild line element On the hypersurface {t = 0} it induces initial data which can be extended analytically to an initial data set whereχ denotes the second fundamental form. These data may be identified isometrically with the initial data induced by the Schwarzschild-Kruskal metric 79 on the hypersurface {s = 0} by the transformation In the following discussions one may think of the conformal geodesics as being constructed on the Schwarzschild-Kerr solution (79) with initial data being prescribed on the hypersurfaceS = {s = 0}. It will be convenient, however, to specify them in terms of the coordinate r in (82). To discuss the different aspects of these curves we will use those coordinates which will appear to give the simplest formulae. Because of the symmetries s → −s and ρ → −ρ, it will be sufficient to consider the region {s ≥ 0, ρ ≥ 0} if we prescibe data for the conformal geodesics which respect these symmetries. The data will also be spherically symmetric. The functionr extends by the first of equations (80) to an analytic function onS which takes its minimumr min = 2 m at the "throat" {r = r h ≡ m 2 }. The space (S,h) has two asymptotically flat ends. In terms of (82) the end i 1 at r = ∞ has the "usual" representation, while the end i 2 at r = 0 has a coordinate representation adapted to the metric (1 + m 2 r ) −4h which may be thought of as a conformal compactification ofh at the end i 2 . The map which corresponds to the isometry ρ → −ρ allows us to show the equivalence of the spatial infinities i 1 and i 2 . It leavesr invariant and has {r = r h } as its fixed point set.

Conformally compactified Schwarzschild-Kruskal data and initial data for a congruence of conformal geodesics
We shall in the following prescribe initial data onS for a congruence of conformal geodesics whoseg-adapted initial vectorsx ′ coincide with the future directed unit normals ofS. Byr * will be denoted the restriction toS of the functionr on the Schwarzschild-Kruskal space-time and r will be considered as a coordinate onS. For the conformal factor we choose It follows that the transformation r = tan χ 2 , 0 < χ < π (and, to get simple equations, m 2 = tan µ 2 ) could be used to make the conformal compactification achieved by Θ * manifest in terms of coordinates, since it realizes an embedding ofS into the unit 3-sphere S 3 with the poles at χ = 0, χ = π corresponding to the ends i 2 , i 1 respectively.
The choice (84) is particularly well adapted to the Schwarzschild-Kruskal geometry and implies simple formulae. However, it is important to note that our basic results do not depend on it. For the analysis of the field near space-like infinity other choices might be preferable (cf. [5]), because the metric (85) does not extend smoothly to space-like infinity.
We set furthermore and, observing F such that β is analytic onS and β = −g ♯ (b * ,b * ) = 2 r * F (r * ) > 0 near the end i 1 in analogy to our procedure on Minkowski space. Ifh is used to raise indices,b ♯ * is outward pointing at the ends i 1 , i 2 , as it should. With the choices above the general formula (59) reduces to where functions with subscript * are assumed to be constant along the conformal geodesics. Note that our initial data and gauge conditions are preserved by the isometry (83), and are adapted to the spherical symmetry and the time reflection symmetry. We have β → −β under (83) and β(r) = 0 precisely for r = m 2 . This means that conformal geodesic satisfying the initial data above at points with r = m 2 will be metric geodesics in the hypersurface {ρ = 0} of the Schwarzschild-Kruskal space-time.

The conformal geodesic equations on the Schwarzschild space-time
To discuss conformal geodesics on the Schwarzschild space-time we write thẽ g-adapted form of the conformal geodesic equation in terms of the line element (76), which can be written in the form (25) with the metric h given by Initial data for the conformal geodesics will be given on the hypersurfaceS = {t = 0} with τ * = 0,τ * = 0 onS. Because of (72) (where due to our conventions we have to use the minus sign), equations (73) take the form with β satisfying (56). Assuming onS that the initial vector is the future directed unit normal toS, the initial data onS are given by Theg-normalization gives Solving for t ′ > 0 and inserting the result into equation (90) gives Dividing by the square root and multiplying withr ′ leads to d( F + (r ′ ) 2 − βr)/dτ = 0, whence with the constant of integration given by γ = F (r * ) − β(r * )r * . It follows that with a sign which depends on the value ofr * . Givenr(τ ), we obtain t(τ ) by integrating (t ′ ) 2 = (βr + γ) 2 F −2 for the given initial conditions. However, the integration of t is in general not particularly interesting, because t does neither extend smoothly through J + nor through the future horizon.
To study the extension of the conformal geodesics through J + it is convenient to use the first of the line elements (78). The conformal geodesic equations then read These equations imply forr the equations obtained above. The normalization of the tangent vector reads F (w ′ ) 2 + 2 w ′r′ = 1. Solving for w ′ and requiring w ′ > 0 onS gives which has to be integrated with the initial conditions w * = −r * − 2 m log(r * − 2 m). Thus w(τ ) can be obtained by a simple integration oncer(τ ) has been determined.
To study the extension through the horizon, it is convenient to use the second of the line elements (78). The conformal geodesic equations then read The normalization of the tangent vector gives now F (v ′ ) 2 − 2 v ′r′ = 1, which leads to the equation which has to be integrated with the initial conditions v * =r * +2 m log(r * −2 m).

Conformal geodesics on whichr is constant
The explicit solution of the key equation (94) requires the discussion of three different cases. We first consider the borderline solution.
A change of the sign in (94) should occur near conformal geodesics along whichr is constant. Requiringr ′ = 0 in (92) gives 1/2 F ′ = β √ F . With β given by (87) this condition has the solution m.
Insertingr ′ = 0 into (91), using (60) with τ * = 0 ,τ * = 0 , and observing (84), (87) with r = r + gives on the conformal geodesics on whichr =r and thus t → ∞ as τ → τ i ≡ 2 Θ * (r+) β(r+) . It follows now from equations (92), (94) that each of the conformal geodesics specified by our data falls into one of the following four classes: (i) the conformal geodesics passing through points with r = m 2 , which coincide with metric geodesics in the Schwarzschild-Kruskal space-time and approach the singularity, (ii) the conformal geodesics passing through points with r = r ± , which are tangent to the static Killing field ∂ t and approach time-like infinity for the finite value τ i of their parameters, (iii) the conformal geodesics passing through points with 0 < r < r − or with r + < r < ∞, for whichr is monotonically increasing, (iv) the conformal geodesics passing through points with r − < r < m 2 or with m 2 < r < r + , for whichr is monotonically decreasing. Though it will be seen below that the integration procedure for (94) depends on the classes specified above, it is clear that our data, which are smooth onS, determine a smooth congruence of conformal geodesics which is free of caustics nearS.

Conformal geodesics on whichr is increasing
Because of the symmetry of the data it is sufficient to discuss the case r + < r < ∞. Under this assumption the "+" sign holds in (94). The radicand in this equation factorizes as Because γ = − F (r * ), it follows that .
Since α <r * and α →r asr * →r, the polynomial on the right hand side of (97) has under our assumption three different zeros while in the limit above two zeros will coincide and the integration procedure needs to be changed. The use of the notation t ≡ (α +r)r * (α +r * )r in (94) gives, after an integration, where, by our assumptions, 1 < e < √ 2, 1 √ 2 < k < 1. It is well known how to express the elliptic integral of the third kind which occurs above in terms of theta functions (for all manipulations involving elliptic integrals, elliptic functions, theta functions, etc. we refer to [9]). The substition t = sn(u, k) with Jacobi's elliptic function sn yields .
Here Z denotes Jacobi's zeta function Thus Z is an analytic function on the real line with Z(u) > 0 for 0 < u < K and Z(0) = Z(K) = 0. The theta functions are given in their product representations by (1 − q 2 n ) (1 − 2 q 2 n cos 2 z + q 4 n ), (1 − q 2 n ) (1 + 2 q 2 n cos 2 z + q 4 n ), with q = exp(−π K ′ K ). The constant K > 0 is the complete integral of the first kind , Setting now x = K − u to exhibit the symmetries of the solution, we obtain . Since .
In the limit when r → r + we have k → 1 and thus sn(u, k) → tanh u. The formula (100) thus suggests that the solutionr(τ, r) of (100) approaches the constant solutionr =r in that limit, as we know already by general arguments. However, since K → ∞ while c → 1/2 log(( √ 2 + 1)/( √ 2 − 1)) as k → 1, the precise behaviour of the solution in that limit requires a quite careful discussion involving also the behaviour of G(x, k).
From the discussion above it is clear that the first of the functions on the right hand side is an analytic function of both variables for r ∈]r + , ∞[, τ ∈ [−τ scri (r), τ scri (r)]. We show that this holds true also for the second function. It is clear that the function w is analytic nearS. Parametrizing w in terms of z, we obtain from (94), (95) The assertion now follows because the function on the right hand side is analytic for z in an open interval of the form ] − ǫ, 1 r * [ with some ǫ > 0. It follows in particular that there exists a smooth functionŵ(r), with r ∈ ]r + , ∞[, such that w(τ, r) →ŵ(r) as τ → τ scri (or, for symmetry reasons, as τ → −τ scri ). We show that w(r) → ∞ as r → r + ,ŵ(r) → −∞ as r → ∞.
The first assertion follows from the observation that along the conformal geodesic withr =r and the fact that the solutions are jointly smooth in the initial data and the parameter. The second assertion follows from a comparison of w with the solution to which satisfies u = w at z = 1 r * . Since 0 ≥ d w d z ≥ d u d z for z ∈ [0, 1 r * ], it follows that the valueû(r) of u at z = 0 gives an upper estimate forŵ(r). Since the direct integration givesû(r) withû(r) → −∞ as r → ∞, the assertion follows.

Conformal geodesics on whichr is decreasing
It is sufficient to discuss the case m 2 < r < r − . Now (94) must hold with the "−" sign. The function α in the factorization (97) then* satisfiesr * < α and α → ∞ asr * → m 2 while α →r asr * →r. If we set such that e > 1, 0 < k < 1, the integration of (94) yields The substitution k sn(u, k) = 1/t and c ∈]0, K[ such that sn(c, k) = 1/e give . Setting x = K − u and observing (99), we finally obtain The function F (x) is real analytic on R with F ′ (x) ≥ 0 and F ′ (x) = 0 iff x = (2 m + 1) K with m ∈ Z. Thus (104) can be solved to obtain an analytic function x(τ ) which maps the interval ] − τ s , τ s [, with 2 Θ * /β < τ s ≡ G(K, k), diffeomorphically onto ] − K, K[. The solution can then be written From the discusssion above it follows thatr(τ ) → 0 and dr/d τ → ∓∞ as τ → ±τ s . Thus in this case there does not exists a smooth (though a continuous) extension ofr(τ ) beyond its physical domain. The limit r → r + implies k → 1. In particular, it follows from (104), (105) that the right hand side of (105) goes in this limit tor for constant τ and τ s → τ i .
To follow the conformal geodesics through the horizon, one has to integrate for givenr(τ ) the equation whose right hand side defines a positive analytic function ofr forr > 0. The conformal geodesics are then obtained in the form where the function on the right hand side are analytic for (τ, r) with r ∈] m 2 , r + [, τ ∈ [0, τ s (r)[.

Analytic coordinates covering the Schwarzschild-Kruskal space-time and its null infinity
If we set now y 1 = r, y 2 = φ, y 3 = θ onS, drag these coordinates along with the congruence of conformal geodesics constructed in the previous section, and set y 0 = τ , we obtain a smooth coordinate system nearS. The purpose of the following discussion is to establish the following result (we ignore here the coordinate singularity arising from the use of coordinates on S 2 which can easily be removed): The conformal Gauss system y ν defines a smooth global coordinate system on the Schwarzschild-Kruskal space-time which extends smoothly to null infinity.
It remains to be shown that the conformal geodesics of the congruence do not develop caustics. Because of the symmetry of the functions (79) and of our family of curves under s → −s, ρ → −ρ, it is sufficient, to control the behaviour of the congruence on the subset {s ≥ 0, ρ ≥ 0} of the Schwarzschild-Kruskal space-time. Since the coordinates y µ are regular nearS, they are known to be regular in particular near the set {s = 0, ρ = 0}. Therefore we can work with the coordinates w,r or z,r or v,r respectively.
Using the functions on the right hand sides of (103) and (107), we can in principle express the corresponding line elements in terms of the coordinates y µ . Though the resulting metric coefficients will be smooth where (103) and (107) are analytic, the Jacobian of the transformation may drop rank at various places and we may end up with a degenerate representation of the metric. Having available the explicit solution, we could try to check this by a direct calculation. Since the explicit calculation will be tedious and, in particular, because this method will not apply to more general cases, we prefer to employ an argument which is similar to the analysis of the behaviour of metric geodesics congruences in terms of the Jacobi equation.
Consider the transformation provided by (107). We need to show that the tangent vector fieldẋ = Θ −1x′ = Θ −1 X and the connecting vector field x ,r = x ,r −ẋ τ ,r = Z −ẋ τ ,r are linearly independent on their domain of definition. In terms of the 2-form (71) (with h = F d v 2 − 2 d v dr, x 0 = v, x 1 =r) this can be expressed as the requirement that the invariant Θ −1 ǫ h (X, Z) does not vanish. In the case of (103) the nondegeneracy up to and in fact beyond null infinity would follow if it could be shown that −(r Θ) 2 (ẇ z ,r −ż w ,r ) = 0 for r * ∈]r + , ∞[, τ ∈ [0, τ scri (r)].
If z is replaced byr again, this condition translates in terms of x(τ, r) = (v(τ, r),r(τ, r)) and (71) (with h = F d w 2 + 2 d w dr, x 0 = w, x 1 =r) into the requirement that in the domain given above. Here the factor Θ is of course most significant because it vanishes at τ scri .
The proof that these requirements are met by our transformations relies on a differential equation satisfied by ǫ h (X, Z). To make use of (75), we observe that the various representations of the Schwarzschild metric with warping function f =r and second metric k = −dσ 2 give the value c = 1 2 f 2 D A D A f = −m for the constant in (69). It follows and similarly where in the second equation D X Z = D Z X, h(X, X) = 1, and (75) have been used. The invariant ǫ h (X, Z) thus satisfies the ODE with k = 2 m r 3 , and onS the initial conditions The quantity D Z β, which is constant along the conformal geodesics, is given by It vanishes atȓ ± ≡ 2± √ 3 2 m, wherer(ȓ ± ) = 3 m >r, and satisfies D Z β > 0 for m 2 ≤ r <ȓ + , D Z β < 0 forȓ + < r.
A lower estimate for ǫ h (X, Z) can be obtained as follows. Denote by u and v the solutions to the ODE problems w ′′ − (β 2 + k) w = f, w(0) = 1, w ′ (0) = 0, where f = 0 in the case of u and f = −1 in the case of v. Then u is strictly increasing with u ≥ cosh(βτ ) forτ ≥ 0, and ǫ h (X, Z) can be given in the form Since (u − v) ′′ − (β 2 + k) (u − v) = 1 and the function u − v has vanishing initial conditions atτ = 0, it follows that u ≥ v forτ ≥ 0. Since v changes its sign, a further estimate is needed. We derive a representation of v in terms of u. Since u > 0 there exists a function f with v = f u. The ODE's satisfied by u and v imply for f the equation which has because of the initial conditions for u and v the solution Since u is strictly increasing, it follows A direct calculation gives −1 < 2 β −2 D Z β < 0 for r >ȓ + . It follows that forȓ + < r.

Global numerical evolution for a class of standard Cauchy data
to evolve such data numerically (cf. [2], [8] and the references given there) we could in principle calculate their evolution in time.
If such data, satisfying (111) for a fixed χ 0 , can be constructed for sufficiently small mass m they will be close to corresponding Minkowskian data and the result of [3] suggests that they will evolve into solutions possessing complete past and future null infinities and regular points i ± at future and past time-like infinity. If we use a gauge in which b * picks up on S \ {i π } a suitable component in the direction of the tangent vectors of the conformal geodesics, it will be possible to construct a gauge of the type considered above which smoothly covers the complete future of the initial hypersurface as well as J + ∪ {i + }. The work in [8] suggests that such solutions can be calculated numerically in their entirety.
The numerical calculation of such space-time is, of course, not an end in itself. In fact, the solution will have somewhat curious features. It follows from [6] that they will have vanishing Newman-Penrose constants. If the solution admit regular points i + , it then follows from [7] that the rescaled conformal Weyl tensor will vanish at those points. This situation, which is more special than the one considered in [8], suggests that the method of gluing a Schwarzschild end to a given solution of the constraints produces data of rather restricted radiation content. However, such calculations will allow us to perform detailed tests of the code under completely controlled assumptions and to study the robustness of the code and of the gauge conditions by choosing data with an increasing value of the mass which eventually yields the time evolution of a collapsing gravitational field.

Concluding remarks
We have described in detail the construction of a global system of conformal Gauss coordinates on the Schwarzschild-Kruskal solution which extend smoothly and without degeneracy through null infinity. Furthermore, we have shown that the conformal factor naturally associated with this system defines a smooth conformal extension of the Schwarzschild-Kruskal space-time which gives to null infinity a finite location in the new coordinates which is determined by our choice of initial data.
We did not try to work out in detail the behaviour of the fields in the conformal extension constructed here. An analysis of the fields near space-like infinity can be found in [5]. The behaviour of the fields near time-like infinity, which is of considerable interest for the numerical calculation of space-times, has still to be investigated.
There is a property of the Gauss system which we only indicated but which may turn out to be quite important. While the regularity of a conformal Gauss systems is essentially decided by its underlying conformal geodesics (considered here as point sets), there always exists a huge class of different time slicings based on one and the same underlying congruence of conformal geodesics. The consequences of this freedom still have to be explored.
Of course, there are many more such coordinate systems. It would be interesting to see whether the initial data for congruences of conformal geodesics which lead to such coordinates can be characterized in a general way.