Well-Posedness of the Ambient Metric Equations and Stability of Even Dimensional Asymptotically de Sitter Spacetimes

The vanishing of the Fefferman–Graham obstruction tensor was used by Anderson and Chruściel to show stability of the asymptotically de Sitter spaces in even dimensions. However, the existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions of the Graham–Jenne–Mason–Sparling operators. In particular, we use one of these operators to propagate Gover’s condition of being almost Einstein (basically conformal to Einsteinian metric). This allows us to study initial data also for Cauchy surfaces which cross the conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose a conformal factor such that Branson Q-curvature vanishes.


Anderson-Fefferman-Graham equation
An important issue in General Relativity is the long time, asymptotic behaviour of solutions to Einstein's equations.In the case of positive cosmological constant the problem was solved by Friedrich [1].He showed that there exists in 4 dimensions a hyperbolic system of equations for a metric and some derived variables which is satisfied if a metric is conformal to a solution of Einstein equation with a cosmological constant.This allows to study compactified versions of the solutions via conformal Penrose compactification and replace difficult long time analysis by a simpler finite time problem.Future asymptotically simple solutions are those, which conformal compactification extends smoothly to the future boundary Cauchy surface Σ + .An important example of such a spacetime is de Sitter universe and such spacetimes are often called asymptotically de Sitter.In fact, we need to assume positive cosmological constant in order the conformal boundary surface to be spacelike.From hyperbolicity of the new system one obtain immediately stability in this class of spacetimes.Moreover, the method gives explicite description of the initial data on the conformal boundary Σ + .However the Friedrich's method does not extend easily to higher dimension.Other important drawback is that no Lagrangean formulation for it exists.Such formulations are important for analysis of the initial data and conserved charges.
The alternative method proposed by Anderson in [2] and futher developed by Anderson and Chruściel in [3] is using the Fefferman-Graham obstruction tensor H µν which is defined for even dimensions d ≥ 4 [4].We will describe the original definition of [4] (see [5]) in section 4.This tensor can be defined as the variation of the Lagrangean over the metric h µν , where Q is the Branson curvature [6], a covariant object with nice conformal transformations, and c d is a constant depending on the dimension.This functional is invariant (up to boundary terms) under both diffeomorphisms and conformal transformations.Consequently, the H µν tensor has interesting properties 1.It is a covariant object built out of the metric and its derivative, 2. It is conformally covariant, namely for h 1 µν = e 2σ h 2 µν , for σ a smooth function 3. It is divergenceless ∇ µ H µν = 0 and traceless H µ µ = 0.Even more remarkably, 4 if h µν satisfies vacuum Einstein equations with a cosmological constant Λ: G µν (h) = Λh µν , then H µν = 0 [5].
The method proposed in [2,3] is to consider conformally invariant Anderson-Fefferman-Graham (AFG) equations instead of the Einstein equations and impose the later as a constraint at the initial surface Σ + .If we want to use this method, it is necessary to prove that the system (3) is well-posed after fixing diffeomorphism and conformal gauge.However, this is a tricky problem.It is shown in [3] that one can impose gauge h (x µ ) = 0 and R h = 0 (where h is a scalar d'Alembert operator for the metric h, x µ are coordinates and R h is the Ricci scalar).In this specific gauge the equations take the form where D m h µν denote m-th jets of the metric i.e. all derivatives ∂ k h µν for k ≤ m.The principal symbol is hyperbolic, but the roots have multiplicities, thus the system is not strictly hyperbolic.Such systems are complicated as we can see in the following example: Example 1 (Not well-posed).Consider an equation on R × S1 with x 1 being time coordinate The principal symbol is hyperbolic (it is a power of d'Alembert operator for a flat metric), but it has multiple characteristics.Functions φ k (x 1 , x 2 ) = e i(ω(k)x 1 +kx 2 ) are solutions for ω(k) 2 x 1 .The smooth initial data on Σ = {x 1 = 0} i n ω(k) n e −k 1/4 e ikx 2 , n = 0 . . .5, (6) does not admit a Cauchy development because for every k ≥ 0 the mode function should behave like e −k 1/4 φ k (x 1 , •) for x 1 > 0 but the series k≥0 e −k 1/4 φ k (x 1 , x 2 )e ikx 2 does not converge even in L 2 (S 1 ).We see that in order to establish hyperbolicity of the equations with a non-strictly hyperbolic principal part, one needs to control few lower order derivatives of the equation (by so called Levi conditions).Unfortunately, the Fefferman-Graham obstruction tensor is quite complicated and we need to control more and more of these terms the higher dimension we consider.This is the reason, why proofs of wellposedness of the Anderson-Fefferman-Graham equation in [2] and [3] are not correct.In particular, it is assumed in [2], that d/2 h is strongly hyperbolic, but our example (for a flat metric) shows that it is not the case.The conditions on the lower order symbols are necessary for the case of multiple characteristics.We will provide in this paper a proof for smooth data.Our method is in fact a modification of approach from [3], but the special form of the system needs to be taken into account.
The problem is less complicated in lower dimensions.It is worth to mention that the well-posedness in dimension 4 was proven in [7]. 1 Our approach can be regarded as a generalization, which put also [7] in a proper context.
where h µν is a Lorentzian metric.We introduce a normal N to Σ with respect to this metric and we assume that it is a timelike vector.Assume that (7) satisfy constraints (well-defined because we know sufficiently many derivatives) We will consider a specific (local) coordinate system given by conditions introduced in [3] where R is the Ricci scalar.As it is shown in [3] we can always locally transform the metric by diffeomorphism and rescaling, such that these conditions are satisfied.We then show that equations (3) are hyperbolic in this gauge.The standard analysis [8] thus shows: Theorem 1.The AFG equation (3) with initial data (7) and subject to constraints (8) forms a C ∞ well-posed system.Every two local solutions differ by diffeomorphisms and conformal transformations.
We will now describe our approach to the problem.Fefferman-Graham obstruction tensor is obtained by ambient metric construction.We consider expansion coefficient of the ambient metric g [k] µν for k = 0, . . .d  2 − 1 and then the equation of vanishing of the Fefferman-Graham tensor are equivalent to where µν is a part of the expansion of the Ricci tensor for the ambient metric.We will briefly describe the ambient construction in Section 4. The AFG equation is obtained by recursive determination of g µν = h µν .Instead of solving recursively, we will consider these equations as a dynamical system for g Every equation in (10) as coming from Ricci tensor is of second order, but the system is not of hyperbolic type.It is not surprising because we need gauge fixing.Following standard Choquet-Bruhat method we write where µν is of hyperbolic type and G

[k]
µ are gauge fixing functions.The addition . . .comes from an additional conformal gauge fixing term that is basically of the form g µν γ [k] , for some additional (scale) gauge fixing functions γ [k] .As in the standard method we will solve system E [k] µν = 0.However, the system is still not strictly hyperbolic.Fortunately, it is some generalized type of hyperbolic equation for which we provide a proof of well-posedness in Section 3. The standard method now use Bianchi identity to show that gauge fixing functions G [k] µ and γ [k] propagate by a linear hyperbolic system.Here the next problem appears.We use only part of the Ricci tensor from the ambient metric.The Bianchi identities are already used to deduce that the remaining parts S ∞∞ vanish.We denote by ∞ the ambient direction.In some way, the ambient metric is already in the partially gauge fixed form and additional gauge fixing is excessive.We circumvent this problem by building G [k] µ and γ [k] from these remaining parts of the ambient Ricci tensor in such a way that Bianchi identities provide generalized hyperbolic system for the gauge fixing functions (see Section 4.1.1).
Our method is more general and allows to prove well-posedness of various equations constructed with aid of the ambient construction.In particular, it is true for Graham-Jenne Mason-Sparling (GJMS) [9] equation and its various generalizations.It is a linear system of the similar type as gauge fixed Anderson-Fefferman-Graham equation thus it has the unique development with a given initial data, which is global on globally hyperbolic spacetimes.Using this result, we show that there always exists a scale with vanishing Branson Q-curvature [10] for a given globally hyperbolic spacetime.Another application is propagation of covariantly constant tractor [11,12] from the initial Cauchy surface.Existence of the covariantly constant tractor is equivalent for metric to be conformal to Einsteinian metric, except when certain nondegeneracy condition is not satisfied what corresponds to conformal boundary (see [13]).In this way one can show that the condition of being Einstein propagates to the whole development of AFG equation in a uniform way.The initial Cauchy surface can now also cross the conformal boundary (see Proposition 18).

Generalized hyperbolic systems
We use abstract index notation and Einstein summation convention in the paper.We denote indices in M by Greek letters.We use symbol D m u to denote m jets on M of the field u on M .
We will consider a bit more general situation then a standard second order hyperbolic system.We consider a system involving family of multifields u (k) for k = 0, . . .N , where each multifield can be in other space.Consider a system of equations on M for multifields u (k) where functions µν ) depends smoothly on coordinates x and u (l) for l ≤ max(k + 1, N ), v ) is a lorentzian metric smoothly depending on x and u (0) . 2 We will call such system K n = 0 generalized hyperbolic for u (k) , k = 0, . . .N .
We assume that Σ ⊂ M is spacelike and compact.This is a condition for u| Σ if the metric is u dependent.We consider smooth initial data We are interested in well-posedness in smooth category.It means a series of important properties (see [14]): 1. Existence of a unique local solution: There exists I = (−T − , T + ), T ± > 0 such that on M = I × Σ we have a unique and maximal smooth solution with the given initial data at {0} × Σ.Moreover, every surface {s} × Σ is a Cauchy surface with respect to the metric g µν (x, u (0) ), thus M is globally hyperbolic.
2. The speed of propagation is equal to the speed of light: Namely, if two initial data u, u ′ are equal on some opens set From this we can deduce some version of well-posedness also for arbitrary non-compact Cauchy surfaces.
3. Smooth dependence on the initial data: For arbitrary T ′ ± < T ± the solution is defined for an open neighbourhood of a given initial data.The solution depends smoothly on the initial data (as a map from a Fréchet space of smooth sections on Σ to a Fréchet space of smooth sections on M ).In particular, the derivative of the family of solutions satisfies a linearized equation.
Let us first notice that the system is non-characteristic on every Cauchy surface.Suppose that Σ = {x 1 = 0}, so x 1 is a time function.

Lemma 2. There exist smooth functions
l≤k valued in the multifields, such that the following conditions are equivalent , where Dn denotes n jets on Σ.
Proof.We will prove by induction in k 0 the statement: The statement for k 0 = 0 is tautological.Suppose that it is true for some k 0 ≥ 0. Using g 11 = 0 we can write where . . . is a smooth function of D1 for k ≤ k 0 by L k by induction hypothesis, thus we obtain desired function finishing the induction proof.Proposition 3. The generalized hyperbolic system K n = 0 (12) for the multifields u (k) for k = 0, . . .N is well-posed in the smooth category.If the system is linear (or linear with a source term) then the solution is defined on the whole globally hyperbolic spacetime.
Proof.We will prove Proposition 3 by induction with respect to the order N .For N = 0 it is a known result (see for example [15] Chapter 16. 1-16.3 and [8] Appendix III).The following system is well-posed: where g µν (x, u) is a lorentzian metric smoothly depending on coordinates x and u and F is a smooth function of coordinates and D 1 u (first jets of u).
We assume now that we proved the statement for all 0 ≤ N < N 0 .Consider generalized hyperbolic system with N 0 multifields for some functions µν ) depending on variables described in the definition of generalized hyperbolic system.Differentiating equation for u (k) with respect to ∂ ρ we get where summation over l is implicitely assumed (as well as standard Einstein summation convention).Introducing p where where all derivatives of F k and g µν retain their original variables.We introduce new multifields and the system of equations It is of the generalized form but the order is now N 0 − 1.
From solution of original system we can form solution of this system by taking We proved uniqueness by induction hypothesis.
In order to prove existence we construct initial data where ∂ 1 ∂ µ u (k) | Σ we compute from Lemma 2 (the original system is non-characteristic with respect to a Cauchy surface, thus it is possible).Now we notice that the difference ∂ ρ (24) − (23) is equal where w µ , k = 0, . . .N 0 − 1.These equations form linear generalized hyperbolic system for w (k) µ and as the initial data we have by uniqueness of solution (due to induction hypothesis) and the solution of the lower order system gives the solution of the original one.
The solution of u ′ system depends smoothly on the initial data, so it is also true for the original system.The induction is complete.
Remark 1.In fact one can show that it is well-posed in Sobolev spaces, but of different order for every u (k) .We leave the details for further investigations.

The ambient construction
We consider an ambient space with coordinates (x µ , ρ) where x µ are coordinates on M .We will denote fields on M with ˜.We regard them as a formal series in ρ.
We denote differentiation over ρ by ∂ ∞ or ′ .We denote indices in M by greek letters.We assume that x 1 is a time coordinate and in what follows Σ = {x 1 = 0} ⊂ M .We use symbol D m u to denote m jets on M of the field u on M .
Let us consider a multifield (a collection of fields) ṽ on M , we can write an expansion where ṽ[m] are rescaled Taylor expansion coefficients and O(ρ ∞ ) means a term that vanishes to infinite order at ρ = 0 surface.Every term in the expansion is a function on M .In what follows we will be interested in such formal series.

Definition 1.
Let ũ be multifield on M .We say that a formal series is a function of x and Let Dm ũ denotes m jets on M of the field ũ.If F (x, D2 ũ) is a smooth function then it is of order 2. Let f µν (x, ũ) be a smooth tensor function then where F is of order 1.Important example of generalized hyperbolic systems can be obtained from K and F is of order 1 and F [n] for n ≤ N decouple in the sense that they does not depend on O(ρ N +1 ) part of the multifield.

The derived equation
The equations of interest have also another important property: Definition 2. We say that a system 2. For every n < N , K n depends linearly on u (n+1) and we can determine u (n+1) from equation K n = 0 in terms of other variables.
We will consider in this paper the generalized hyperbolic systems given by K which are recursive till order N .In order to determine this property it is enough to study F .The property (2) allows us to determine higher order variables from sufficiently high jets of the lowest order u (0) .In our application we need a local version of this procedure that is described by a following lemma: Lemma 4. Let K n be recursive in u (k) for 0 ≤ k ≤ N till order N .There exist smooth functions H n K for 0 < n ≤ N depending on x ∈ M and on variables D 2n u (0) , such that for any point x ∈ M and an integer N ′ > 0 the following conditions are equivalent where D m denotes m-th jets.In the case of linear system the functions H n K are also linear.If the system does not directly depend on x then the same is true for H n K .
Remark 2. We will use subscript for H k to indicate which system is used to determine recursive functions.
Proof.We proceed by induction in k 0 .Suppose that due to linear dependence.By inserting recursively variables from the lower orders we show the result.Induction starts with k 0 = 0 where it is a trivial statement.
This lemma allows us to determine initial data for the generalized hyperbolic system from the sufficienty high jets of the lowest order field u (0) on the Cauchy surface.Important is that the evolved generalized system will have u (0) in agreement with this data.This property is guaranteed by the following fact: Lemma 5. Let K n = 0 be the generalized hyperbolic system, recursive in the multifield u (k) , k = 0, . . .N .Then a solution with initial data on a Cauchy surface satisfies Proof.Let u (k) be a development.We denote Consider a set We should show that this set is empty.By contradition assume otherwise and define We notice that m 0 ≥ 2 because of the way where . . . is a function of the terms which do not belong to A by definition.Similarly and as . . . of the same property.From the definition of (m 0 , k 0 ) the remainders . . .are equal thus and as g 11 | Σ is nonvanishing we obtain a contradition.
If the equation system K n = 0 is recursive till order N and it decouples at this order, then the equation for n = N gives us by Lemma 4 the equation of higher order for u (0) We will called it the derived equation from the system.In case of a linear system the derived equation is also linear.
If the system K n = 0 , n = 0, . . ., N is satisfied, then the equation (49) for u (0) also holds.From solution of (49) we can obtain solution to the system.Initial conditions for this equation provide also initial conditions for the system, if we know sufficiently high jets on the initial surface.

The Fefferman-Graham ambient metric construction
We are working in even d dimensions.Moreover we assume that d ≥ 4. Let us introduce an ambient space M for the spacetime with coordinates (t, x µ , ρ) and (x µ , ρ) respectively, where x µ are coordinates on M and the metric on M takes the form In the following we will denote objects on M with ˜and objects on M bold.We denote by g IJ , ∇ I , S IJ metric covariant derivative and Ricci tensor respectively on M. Indices I = 0, ∞ or µ in the case of index on M .We use g IJ to raise or lower indices.The metric gµν , connection ∇µ and Ricci tensor Rµν are depending on ρ objects on M .We use gµν to raise and lower indices for such objects.Let h µν be a given metric on M .The ambient metric on M is a metric that satisfies g[0] µν = h µν and where S IJ and S are Ricci tensor and Ricci scalar of the metric on M. Symbol F = O(ρ n ) means that lim ρ→0 ρ −n F exists.One can show that S 0I = 0 and that S IJ is t independent.Essentially, it is a function on M (see [5]) We have (eq.3.17 in [5]) where Rµν denotes the Ricci tensor in the metric gµν depending on ρ.The equations (52) are equivalent to and then other components automatically vanish.Namely, (see [5] and compare with Proposition 19), One can check that S[n] µν is recursive till order d/2 − 1.Thus, we obtain4  The Fefferman-Graham obstruction tensor for h µν is defined by The constraints H µν = 0 are equivalent to Let us notice, that the specific combination depends only on g[k] for k ≤ d/2 − 1 and its derivatives.Importantly,

Hyperbolicity of the Anderson-Fefferman-Graham equation
The system of ( 10) is not hyperbolic for the same reason as Einstein's gravity, because of the gauge transformations.The first step is to introduce a hyperbolic system in a specific gauge.We will use a natural gauge introduced in [2].Then we show that this gauge is preserved in the evolution and as a result the obtained solution is also a solution of the Anderson-Fefferman-Graham equations.It is a standard treatment in gravity (see [8] for application to Einstein's equations).
Let us remind the following known identity (see [8]) where ... means terms of order 1 (see Definition 1) and We notice that where . . .denotes terms of order at most 2. Comparing it with we obtain the formula where . . .denotes terms of order at most 2.
In order to write a slightly modified Fµ in terms of Sµ∞ and S∞∞ we extend the notion of derivatives with respect to ρ.For n > 0 we introduce n-times integration of a multifield ũ (a collection of fields on M ) We introduce additional tensors These tensors will be used in our analysis of the AFG equations.The reason for occurence of additional term gµν γ is explained in the proof below.
Proof.We will first prove that it is recursive till order d/2 − 1 and that it decouples at this order.Functions Gµ and γ are of order 1.Hence, Ẽµν is of order µν .In fact, we can compute where . . .are terms depending on µν for k ≤ d/2 − 1 and their derivatives (see also (63)).Thus, the system is recursive and it decouples.
We need to show that Ẽµν is of the form (35).As a preliminary step we prove that Fν = Gν + . . ., where . . .denotes term of order 0. Indeed, which can be compared to where . . .denotes terms depending on x and D m g[k] µν for m + k ≤ 1.Finally, by (70) we obtain where . . .denotes term of order 2.This shows (77).We thus have ∇µ Fν = ∇µ Gν + . . ., ( where . . .denotes term of order 1 (both Fµ and Gµ depends only on up to first derivatives of the metric).
Taking this and (65) into account, the following yields where . . . is of order 1.Expanding first term the form described in (35) is obtained.Well-posedness follows from Proposition 3.

Propagation of the gauge
In this section we will explain that Gµ = O(ρ d/2 ) and γ = O(ρ d/2−1 ) provided that these functions vanish on the initial surface together with their time derivatives and secondly Ẽ = O(ρ d/2 ).As usual this is achieved by showing that these variables obey a system of linear hyperbolic equations.

We will prove in Proposition 23 in Section
) and B2 = O(ρ d/2−1 ).The detailed property is presented below: Proof.Inspection of the equations shows that where F 1 µ and F 2 are of order 1.Generalized hyperbolicity follows from two facts which ensure that the system decouples: and it does not depend on D m γ[k] for k ≥ n.

The dependence of [ B2 ]
and it does not depend on Let us now prove the second statement of the lemma by induction on k 0 .Suppose that for all 0 ≤ k < k 0 then taking up to d−2k 0 −2 derivatives of [ B1 µ ] [k0−1] = 0 and up to d−2k 0 −3 derivatives of [ B2 ] [k0−1] = 0 we get due to (91) which shows the induction together with a trivial statement for k 0 = 1.
As a result, we obtain: ) and on the initial surface

Gauge fixing conditions
We assume that on the initial surface and prove that in such a case We show it by noticing that the equations B1 µ = O(ρ d/2 ), B2 = O(ρ d/2−1 ) hold.We can invoke Lemma 7 to show that Comparing with (96) we see that the missing condition is ) and on the Cauchy surface Σ Proof.We will assume for simplicity that N = ∂ 1 .The modification of the proof for the general case is minor.We have the identity for jets From Lemma 4 the jets of the expansion of the ambient metric by using (64) and showing the statement of the lemma.
We can summarize the results obtained so far in the following proposition: The conditions are well-defined because x µ depends on on up to first jets of the metric, R depends on up to the second jets and H( N , •) depends on d − 1 jets of the metric on Σ.Then there exists a unique solution to AFG system H µν = 0 with the given initial data and which satisfies x ν = 0, R = 0.
Proof.As Ẽµν is recursive till order d/2 − 1 we compute by Lemma 4 the initial value data for the system by h This allows us to determine the initial data for the equation Ẽ = O(ρ d/2 ).We consider a unique solution gµν of this equation.Since the system is generalized hyperbolic, we get by Lemma 5. Thus, the solution has prescribed initial data.
In particular, it is true that and We take the trace of this equality to derive This means that the condition (95) is satisfied.From Lemma 8 and Lemma 9 we conclude We can always assume γ[d/2−1] = 0. Taking this into account, we obtain for n ≤ d/2 − 1.We have a solution with Sµν = O(ρ d/2 ).

The AFG equation in the Anderson-Chruściel gauge
In this section, the correspondence of our gauge functions to R = 0 and x µ = 0 gauge will be investigated.This gauge was proposed in [2] and [3].

Uniqueness of the solution of the AFG equation
Assume that we have a solution H µν (h) = 0 with the given initial conditions at Σ.

Lemma 11 ([2], [3]
). Suppose that we have a local coordinate system y 2 , . . .y d on Σ. Locally there exists a coordinate system x µ and the choice of the conformal factor f such that for h Here ′ is a scalar d'Alembert operator with respect to the metric h ′ .
Proof.First we find f as a solution of Yamabe problem, which is a nonlinear hyperbolic system for f (see [3] for discussion).Define x ξ for ξ ≥ 2 as a unique solution to ′ φ = 0 with initial conditions (local development) where N µ is a unit normal to Σ. Finally, we define x 1 as a unique solution with initial value One can check that at U ′ ⊂ U this new coordinates are independent, so it is also true in the small neighborhood of the Cauchy surface.
We can thus work in this gauge.From the solution to H µν = 0 we now construct iteratively by ( 60).Let us notice that g[0] µν = h µν and g [1] µν = P µν where Rh µν is the Schouten tensor [5].Due to the gauge condition, one obtains From the uniqueness of the solution of (117) we obtain the uniqueness of the solution of AFG equation (in the given gauge).

Existence of the solutions of the AFG equation
Let us now assume that the initial data is given by which satisfies the constraints H( N , •)| Σ = 0 and the gauge is satisfied: The conditions are well-defined because x µ depends on up to first jets of the metric and R depends on up to the second jet.By a change of coordinates in jets of Σ we can always assume these conditions together with N = ∂ 1 where N is a normal vector.In this way, the constraints take the form We also compute by ( 119) and (120).The existence of the solution of AFG equation would follow from Proposition 10 if we were able to use this gauge globally on a compact Σ.However, it is not possible, so we need to apply some version of a gluing argument.

Proof of the Theorem 1
The harmonic gauge is well-suited for Σ = R d−1 .If we want to apply our result, we need to extend the notion of this gauge to compact Cauchy surfaces.This can be done in the case of Σ being a torus, where x µ for µ = 2, . . .d are defined modulo 2π.Due to the finite speed of propagation, the method provides an existence and uniqueness result also for open subsets of the torus.Uniqueness of the development allows to apply the standard gluing argument [8] to obtain Theorem 1.

Infinite order extension of the ambient metric
Suppose that the obstruction tensor vanishes.The results of [5] show that Taylor expansions of the metrics, which are Ricci flat of the order O(ρ ∞ ), are in one-to-one correspondence with the traceless symmetric tensors k µν satisfying where D ν is a certain 1-form (defined in eq.3.36 in [5]).The tensors k µν define trace-free part of g[d/2] µν since the trace is already determined.In the case of Euclidean manifolds, it is not obvious that such a tensor exists.We will prove that this is the case for any globally hyperbolic spacetime.
Proof.We will look for the tensor given in a special form for some covector field u µ .It is already symmetric, traceless and the equation ( 123) takes a form Taking the divergence, we obtain an additional equation where we used . We now introduce a new variable A = 1 d ∇ µ u µ and a system of equations (equivalent to (126) and (125)) It is indeed a hyperbolic second-order linear system.Thus, with the given initial data on a Cauchy surface, it has a solution.We now notice that the divergence of the left hand side of (127) minus left hand side of ( 128) is equal to zero: the whole spacetime and ∇ µ k µν = D ν .We can thus always assume that the metric is Ricci flat to an infinite order, but it is not uniquely defined except terms g[n] µν for n ≤ d/2 − 1 and tr g[d/2] .

GJMS type operators for tractor bundles
We will now concentrate on various linear systems, which arise by the ambient metric construction.They share the common property with Ẽµν , that the principal symbol is a power of the d'Alembert operator.In this part of the paper, we will also shortly describe Graham-Jenne-Mason-Sparling (GJMS) type systems.
A quite general method of introducing this type of operators is by tractor calculus.We will only focus on the essential parts of this theory in terms of the ambient metric construction (see [16]).For the short review of the tractor calculus in application to general relativity, we refer reader to [13].
We work on manifold M = R × M , T = t∂ t is a conformal Killing vector with property that for every vector field F I it satisfies We also introduce Ω = 1 2 T I T I = ρt 2 with the properties: Simplifying the notation, we will often skip tractor indices.We consider n-covector X I1...In with the property L T X = wX, ( for w ∈ R. We call w a weight of the field X.The field X is determined by its restriction to t = 1 According to [16], X[0] is a section of a tractor bundle ǫ I1...In [w − n].The ambient space (at least the uniquely determined jets of the metric) is a natural construction for the conformal structure on the manifold.The natural bundles inherit the laws of transformations under diffeomorphisms and conformal transformations of the original metric.We will not describe here the original formulation (see for example, [16,13]), but we will use this description as a definition.We will need the following results: where n is a valency of the field.
Proof.We use induction on n.For n = 0 (that means X is a functions) Lie derivative and covariant derivative agrees.Suppose that the result is true for n < n 0 .Consider a vector field F I with weight 0.
We have For any X I1...In 0 , n 0 covector with weight w, we denote by F X X I1...In 0 F I1 .( It is a n 0 − 1 covector with weight w, thus So F [(w − n 0 )X − ∇ T X] = 0.As the restriction to t = 1, F I is arbitrary, we show the induction.
The GJMS operators are constructed with the help of the d'Alembert operator in the ambient space M. We consider an operator ⊡ ⊡ ⊡ defined on n-covectors: The result has the weight w − 2. We can thus define Proposition 14.For any n, the operator ⊡w on a ρ dependent n-tractors of weight w X has the property where In other words Proof.Clearly, we can write ⊡w X = gµν where This can be done by considering fields, which depend only on the Taylor expansion coefficients for k > m.The most convenient way is to use Ω = ρt 2 in the expansion instead of ρ, because Ω is covariantly defined.We consider X = Ω m+1 F = O(ρ m+1 ) where L T F = (w − 2(m + 1))F (in order that X has the proper weight) and thus We substitute the special form of X in the following formula We use the know form of derivatives of Ω to get the nice expression: This shows that ⊡w X = (d where H[m] depends on D 2 X[k] for k ≤ m.Together with the previous expansion (141) it proves the lemma.
Together with Proposition 3 this result leads immediately to a corollary: for X[k] (n-tractors of weight w) is generalized hyperbolic, recursive and linear.Here D is linear transformation on the space of n-tractors.In particular, the Cauchy problem with smooth initial data is well-posed.
Proof.The assumption about the metric is sufficient to satisfy the requirements for the linear generalized hyperbolic system to be well-posed in Proposition 3.

Remark 3. The ambient metric construction determines g[n]
µν for n = 0, . . ., d/2 − 1 and tr g[d/2] .Consequently, only equations depending on this part of the metric expansion are defined uniquely.They provide conformal equations on the spacetime M .This holds if 0 > w − n > 1 − d/2.Interestingly, it is also true for ⊡0 on scalars of weight 0, where the operator depends on the aforementioned trace.If n ≥ 1 and w − n ≥ 0, then the system of equations explicitly depends on the choice of the extension of the ambient metric.
We will denote the derived equation for ˜ w by G w .The initial data for the derived equation is given by ∂ k 1 X[0] | Σ , k ≤ 2N + 1 and we get from Lemma 5 and Proposition 3 the unique global development.

Spacetimes with the vanishing Q curvature
An immediate application of Collorary 15 is uniqueness and existence of the Cauchy development for GJMS operators as they are the derived system for ⊡w acting on scalar functions of weight −k, 0 ≤ k ≤ d/2 − 1.We can apply the result to the problem of finding a conformal factor, which yields the vanishing Branson Q curvature [6].This is an important and quite mysterious object in conformal geometry (see [17,18] for an introduction to the application and meaning of the Q curvature).Proposition 16.On every globally hyperbolic spacetime, there exists a function φ such that Q(e 2φ h) = 0.
Proof.For the metric h the Branson Q curvature can be constructed as follows.We find a scalar function f such that It is always possible to find such a function and it is unique up to O(ρ d/2 ).We define where c d is a dimension dependent constant [19].Let us notice that Thus, it is a scalar function of weight 0. The conformal rescaling of the metric corresponds to a diffeomorphism, which acts by t ′ = te −φ at ρ = 0. Suppose that f satisfying then we can define t ′ = e f [0] = te F [0] .The metric rescaled by e 2F [0] has a vanishing Branson curvature.
Let us now notice that the equation is well-posed for F = F| t=1 .For linear (affine) systems, there exists a solution on the whole globally hyperbolic spacetime, if we provide arbitrary initial data on some Cauchy surface.
For future reference, we can also compute We notice that up to terms of order O(ρ d/2 ) it depends only on the part of the metric, which is determined by the ambient construction.

Propagation of the the Einsteinian condition
Let us introduce a tractor connection (see [13]).Let X I1•••In be an n-tractor of weight w field on the spacetime M .We define for a vector field Y where we choose X such that L T X = wX and X| t=1,ρ=0 = X and Y µ = Y µ , Y 0 = Y ∞ = 0.The definition does not depend on a particular choice of X.If w = n then the tractor derivative has particularly nice properties.Usually, one restricts the definition to this special type of tractor.We define the tractor derivative for 1-tractor X I of weight 1 (see tractor derivative, for example, in [12] with a correction for covectors) by the formula: An almost conformally Einstein metric (according to Gover [20]) is a metric for which there exists X I of weight 1 satisfying This is equivalent to for a function where tf denotes a trace-free part (see, for example [13]).It is known [20] that the metric is conformal to the Einstein metric if and only if it is almost conformally Einstein and the following non-degeneracy condition holds The conformal factor rescaling metric to Einsteinian is given by e φ = X 0 and the cosmological constant is equal Λ = c d X I X I where c d is a dimension dependent constant [21].Conformal boundary corresponds to X 0 = 0.This set is a hypersurface with vanishing extrinsic curvature and it enjoys very special properties [21].In almost conformally Einstein spaces, the Fefferman-Graham obstruction tensor vanishes.It is known [12] that one can prolong the covariant tractor from ρ = 0 surface.We will need a detailed statement of this result.We remind the following fact (proven in [12], see also [20]) Proposition 17. Suppose that ∇ T µ X I = 0 for a tractor of weight 1, then there exists X I of weight 1 on the ambient space such that X I | t=1,ρ=0 = X I and In particular, in the case of an Einstein metric h µν , we define an infinite order extension We In particular, we see that ⊡ ⊡ ⊡f = O(ρ ∞ ).
In general, we do not have a distinguished infinite order extension unless we already know the covariant tractor.We would like to determine how many of the properties of f = T I X I extend to an arbitrary Ricci flat extension and to the conformal boundary.Suppose in a first step that we work in an arbitrary Ricci flat extension of a non-degenerate almost Einstein manifold.We compute for f = t(1 − λρ) Let us introduce Ãµ ν := gµξ g′ ξν .For an arbitrary extension of the Einsteinian ambient metric, we have The by ( 56).Consequently, we obtain a property of kµ Finally, we get where g0 µν is given by (159).Independently of the Ricci flat extension, the following is true We will see that the condition (166) can be also satisfied on the conformal boundary.In general, we cannot ensure vanishing of ⊡ ⊡ ⊡f to higher order on the boundary unless the metric is even with respect to the conformal boundary (see [5] for the definition of even Poincare-Einstein metrics).Equation ( 166) is a quite important observation.As f is a scalar of weight 1, this is a generalized hyperbolic system for f = f | t=1 .We can evolve it from the Cauchy surface to the whole globally hyperbolic development.In order to define evolution we need to specify the choice of extension of the ambient metric.We choose arbitrary extension that is Ricci flat to all orders.We will now show that if (166) holds, then ∇ I X J satisfies the linear hyperbolic equation too and then what remains is to show that the initial data for this system vanish.
Let us notice an identity for an arbitrary function where we introduced an operator ⊞ ⊞ ⊞ on 2-covectors where S K L I J , S K I are the Riemann tensor and Ricci tensor on M. If S IJ = O(ρ ∞ ) and (166) holds, then from (167) it follows that This is also a hyperbolic equation for D Proposition 18. Suppose that we have initial data Then the development of the AFG equation is almost Einstein with a covariant tractor given by where f is the solution of the scalar GJMS equation of weight 1: with the given initial data.The solution is computed in an arbitrary Ricci flat extension.
Proof.We need to prove (170).As ⊞ ⊞ ⊞ is recursive till order d/2 − 2 it is enough to show (due to Lemma 5) that A symbol D n denote jets in directions of M .We compute using [∇ Let us now define The condition (171) gives by (153) The nontrivial condition for ∇ T µ X ∞ is shown by divergence of (171) (see [13]).For this reason, we only get the condition for D d−2 jets.The equation (177) means for some F I and D IJ such that L T F = −2F and L T D = −D in order that L T ∇X = ∇X.Moreover, from T I ∇ I ∇ J f = 0 we have Finally, due to SIJ = O(ρ ∞ ) we derive Substituting the form of ∇ I X J from (178) we obtain t=1,Σ = 0 which means that and the initial condition (170) are satisfied.
Remark 4. In the case f [0] | t=1 = 0 on the Cauchy surface, we can change conformally the metric such that it satisfies Einstein constraints on the initial surface.Therefore, we can evolve Einstein equations.
The result needs to agree with the metric evolved with AFG equation up to conformal rescaling and diffeomorphism.In this way we obtain propagation of the Einsteinian condition up to the conformal boundary (see [2]).Surface f [0] | t=1 = 0 is more delicate.Our method has the advantage that it allows to treat all cases simultaneously.For example, the initial Cauchy surface can cross the conformal boundary.

Summary
The Fefferman-Graham obstruction tensor and GJMS operators are very special objects.One additional nice property is related to their behavior as evolution systems.Both GJMS as well as Fefferman-Graham tensor in the suitable gauge are not strongly hyperbolic, but still they enjoy well-posed Cauchy problem.
In addition, the property of being almost Einstein propagates from the initial surface.We proved it in the smooth category, but with an arbitrary Cauchy surface.Namely, the Cauchy surface can cross or partially coincide with the conformal boundary of the spacetime.This allows to use AFG equation for proving the stability of asymptotically simple solutions (as was done in [2,3]), which was the initial motivation for studying AFG equations.We should notice that this is not the most effective proof of stability as the metric needs to be of high regularity.However, it provides some advantages: it is a Lagrangean theory, which allows to apply various techniques like Noether charge definition, Hamiltonian formulations on the level of conformally compactified spacetime.The meaning of such defined charges for Einsteinian solutions is still unclear.The relation to GR charges should be investigated in future.

A Bianchi identities
For convenience of the reader, we provide here a proof of the Bianchi identities in the ambient space.Proof.Let us notice the identity Now g 0∞ = g ∞0 = t −1 , g ∞∞ = −2ρt −2 and g µν = gµν the rest of the components vanishes.Moreover, √ g = t d+1 √ g.Thus, remembering that F 0 = 0 it follows that where we used 1 √ g ∂ µ ( √ gg µν Fν ) = ∇µ Fµ .
Lemma 21.Let D IJ be a symetric tensor and X a vector field, then where L X is a Lie derivative Proof.Follows from L X g IJ = ∇ I X J + ∇ J X I and symmetry of D IJ .