Decay of the Weyl curvature in expanding black hole cosmologies

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein’s equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null infinity. More precisely we establish decay estimates for Weyl fields which are (i) uniform (with respect to a global time function) (ii) optimal (with respect to the rate) and (iii) consistent with a global existence proof (in terms of regularity). The proof relies on a geometric positivity property of compatible currents which is a manifestation of the global redshift effect capturing the expansion of the spacetime.


Introduction
In this paper we are interested in solutions to Einstein's field equations with a positive cosmological constant Λ > 0, Ric(g) = Λg . (1.1) These equations -in fact a more general form including sources of matter -were proposed by Einstein to model the universe in the large [Ein17], (M, g) being an unknown 3 + 1 dimensional Lorentzian manifold which represents the geometry of spacetime. 1 The simplest solutions to (1.1) have non-trivial topology, and are not asymptotically flat: The Einstein universe (which contains a homogeneous fluid) is topologically a cylinder S 3 × R, and thus represents a closed universe. While Einstein's solution is static (in time), de Sitter found a solution to the vacuum equations (1.1) shortly after the cosmological constant was introduced [dS17], which is expanding: 2 The de Sitter spacetime can be embedded as a (time-like) hyperboloid H in 5-dimensional Minkowski space (R 4+1 , m), with metric h simply induced by the ambient metric m; see Fig. 7 below. Its geometric properties are central to this paper, and will be discussed in Section 2.
A model of a black hole in an expanding universe is provided by the Schwarzschild de Sitter geometry [Kot18,Wey19] discussed in Section 4. It is a solution to (1.1) with less symmetries than the de Sitter solution but still spherically symmetric, which means that the metric takes the form: where Q g is a Lorentzian metric on a 1 + 1-dimensional manifold Q, r : Q → (0, ∞), q → r(q) is the radius of a sphere q ∈ Q, and • γ the standard metric on S 2 . Its causal geometry is best understood if we depict the level sets of r in Q while keeping the null lines of Q g at 45 • , namely in the form of the Penrose diagram of Fig. 1.
On Schwarzschild de Sitter we distinguish between the black hole region B, the stationary black hole exterior S, and the cosmological region R. The stationary region S has a time-like Killing vectorfield T , and is bounded by an event horizon H towards the interior at r = r H , and a cosmological horizon C towards the exterior at r = r C . Beyond C lies the cosmological region, whose future component R + we depict separately in Fig. 2. R + is bounded to the past by the null hypersurfaces C + ∪C + , and foliated by the level sets Σ r of r: The equations (1.1) in its more general form including sources of matter can be thought of as a general relativistic version of the classical analogue ψ − Λψ = 4πρ, namely a modification of Newton's law which Einstein considered to model homogeneous mass densities ρ in space [Ein17].
2 Neither was this solution discovered in the form presented in Section 2, nor was it immediately understood that this "universe" is expanding. This was realized only later by Lemaître [Lem27]. For a historical discussion of the confusions and controversies surrounding its discovery see e.g. [NB09]. Each leaf Σ r is topologically a cylinder S 2 × R, and a spacelike hypersurface for r > r C . Since r is increasing along any future-directed causal curve in R + we also call this region expanding. It is future geodesically complete, yet has the property that any two observers are eventually causally disconnected. 3 This has the consequence that the "ideal boundary at infinity" I + is a spacelike surface. 4 (I + is not part of the spacetime, but it is intrinsically a cylinder R × S 2 and can be thought of as attached to the spacetime in the topology of the Penrose diagram.) Finally B is referred to as the black hole region, because it lies in the complement of the past of I + .
The maximal extension of the Schwarzschild de Sitter spacetime consists of an infinite chain of black hole regions B, separated by exteriors S to the future and past of which lie the cosmological regions R. We shall restrict attention to a given cosmological region, and its adjacent black hole exteriors, up to the event horizons; in particular the interior of the black hole is not considered here. While all classical solutions to (1.1) referred to here were found explicitly, a natural question to ask from the evolutionary point of view is the following: Is the picture of Fig. 3 dynamically stable? In other words, does a perturbation of Schwarzschild de Sitter data on a Cauchy hypersurface Σ give rise to a maximal development D with similar features? In particular does D contain a future geodesically complete region R with spacelike boundary I + at infinity, relative to which D contains a black hole regions B. Moreover, is the black hole exterior S = I − (C) ∩ I − (H) -where C and H are defined to be the future boundary of the past of B and I + , respectively -asymptotically stationary?
In view of the domain of dependence property of solutions to (1.1) the stability of the black hole exterior S can be treated independently of the cosmological region R (and the black hole interior B). Indeed, since S is contained in the domain of dependence of a compact subset Σ c ⊂ Σ, the behavior of the solution in S is not influenced by data in the complement of Σ c . In a remarkable series of papers [HV16b, Hin16, HV16a, HV14, Vas13] Hintz and Vasy have recently proven that solutions to the Cauchy problem for (1.1) arising from a perturbation of Schwarzschild de Sitter data on Σ c converge exponentially fast to a member of the Kerr de Sitter family on S ∪ H ∪ C, (and in particular become stationary). 5 The Kerr de Sitter geometry -given by an explicit 2-parameter family of axisymmetric solutions to (1.1), containing Schwarzschild de Sitter as a subfamily -plays a central role for the understanding of solutions in S. Indeed, the result of Hintz and Vasy shows that they parametrize all possible final states for the evolution of perturbations of Schwarzschild de Sitter data, in the domain enclosed by the event horizon H and the cosmological horizon C. We will see that for the evolution beyond the cosmological horizon this explicit family of solutions does not play an equally prominent role.
The problem approached in this paper is then the following: Consider the characteristic initial value problem (or Goursat problem) for (1.1) with data on the cosmological horizons C ∪C, c.f. Fig. 4. Suppose the characteristic data converges exponentially fast to the geometry induced by a Kerr de Sitter horizon, then is the maximal development future geodesically complete, and can future null infinity I + be attached at infinity as a spacelike surface in a suitably regular manner?
Note that the assumption -that the data be exponentially decaying to a Kerr de Sitter geometry along the cosmological horizons -is justified by virtue of the result of Hintz and Vasy [HV16b]. However, also note that in this formulation no reference is made to the Kerr de Sitter solution in the discussion of our expectation for the asymptotics: In [Sch15] I have considered a linear model problem, namely the corresponding Cauchy problem for the linear wave equation (1.4) on a fixed Kerr de Sitter background (M, g). It was shown that any solution to (1.4) arising from finite energy data on Σ, remains globally bounded yet has a limit ψ + on I + which as a function on the standard cylinder R × S 2 has finite energy. Moreover, if exponential decay is assumed along the cosmological horizons (which in this setting is justified by the results of Dyatlov [Dya11a,Dya11b]), then this "rescaled" energy of ψ + on I + decays towards time-like infinity ι + , but still need not vanish globally on I + . This means that even in the context of the linear theory, there is a non-trivial degree of freedom at infinity. Finally, the results in [Sch15] depend by no means on the symmetries of Kerr de Sitter, and have been proven therein for a large class of spacetimes without any symmetries near the Schwarzschild de Sitter cosmology. The intuition gained in the linear problem tells us that in the context of the fully non-linear problem we cannot expect convergence to a member of the Kerr de Sitter family, but merely a "nearby" geometry, which is however a priori unknown. In fact, the setting to be presented in this paper is consistent with the asymptotic geometry to differ from Kerr de Sitter -indeed de Sitter -even at the "leading order". 6 This view is echoed in an instructive series of papers by Ashtekar, Bonga and Kesavon [ABK16, ABK15a,ABK15b]. In [ABK15a] it is argued that future null infinity I + cannot be conformally flat in the presence of gravitational waves. They show in particular that the condition that I + be intrinsically conformally flat (as it is for Schwarzschild de Sitter) would suppress "half" of the gravitational degrees of freedom. This is consistent with the setting in this paper, where will will allow the spheres foliating I + to be not perfectly round. Now we will treat Einstein's equations not as a system of wave equations for the metric, but rather using the electromagnetic analogy, which has been employed so successfully in the seminal work of Christodoulou and Klainerman [CK93]. In other words, we use that (1.1) imply the homogeneous contracted Binachi equations for the Riemann curvature tensor R: div R = curl Ric = 0 (1.5) The Riemann curvature however -in the role of the Faraday tensor F -is not a suitable quantity to consider in this setting, and cannot be expected to decay; indeed de Sitter is a constant curvature space. A central idea in this work is to pass from the Riemannian curvature R to the conformal Weyl curvature W , which for any solution to (1.1) is related to R by 7 W (X, Y, U, V ) = R(X, Y, U, V ) + Λ 3 g(X, V )g(Y, U ) − g(X, U )g(Y, V ) (1.6) and thus also satisfies div W = 0 . (1.7) The conformal Weyl curvature is the prototypical Weyl field W in the sense of [CK93] and its algebraic properties allow us to construct energies using the Bel-Robinson tensor Q(W ), which can be viewed as a generalisation of the energy-momentum tensor of electromagnetic theory. A key advantage of this approach is then that certain methods 6 As we shall see the geometry is expected to be asymptotically de Sitter locally in the past of any time-like geodesic, however it is not expected to agree globally. Indeed the results in [HV16b] could be applied to the past of any point p on the future boundary at infinity I + , to infer -see in particular Theorem C.4 therein -that up to a gauge choice that depends on the solution in I − (p), the spacetime converges to de Sitter as the tip p of the backwards cone is approached. However the gauge modification depends on p ∈ I + , and with a given gauge choice for in I − (p), the spacetime does not converge to de Sitter at any point q ∈ I + in a neighborhood of p.
7 In Section 2 we will recall that de Sitter is in fact conformally flat. The passing from R to W can thus be thought of as a renormalisation of the curvature by its de Sitter values. Note also that for Λ = 0 the Weyl and Riemann curvature coincide, and thus play a notably different role only in the cosmological setting.
developed for the treatment of the linear equation (1.4) -in particular our understanding of the decay mechanism for solution to (1.4) in the cosmological region -carry over to the study of solutions to (1.7). We will elaborate on this in more detail in Section 1.4.
In this paper we will accomplish the first part of the global non-linear stability problem, as formulated above as a characteristic initial value problem for the cosmological region. Following the strategy laid out in [CK93], we will make certain assumptions -as part of an overarching bootstrap argument -on the metric g, and the connection coefficients, 8 and then prove a non-trivial statement for the Weyl curvature, which will allow us -and this is the subject of the next paper -to recover all assumptions initially made, thus yielding a full existence result.
A significant challenge of this part lies in identifying a set of assumptions which are on one hand sufficiently general to encompass the actual dynamics of the metric under the evolution of (1.1) (too restrictive assumptions would be inconsistent, and have no chance of being recovered), while on the other hand sufficiently restrictive for the decay mechanism to come into play. The latter is predominantly the expansion, which can be captured adequately on the level of mean curvatures.
Informally speaking, we establish the following: The conformal Weyl curvature decays uniformly in the cosmological region R, provided the metric and connection coefficients satisfy a set of assumptions which capture in particular the expansion of the spacetime.
In Schwarzschild de Sitter the Weyl curvature has only one non-vanishing component where m is a constant, the mass of the black hole; c.f. Section 3, 4. "Decay", and its "uniformity" refer to a parameter like r in the Schwarzschild de Sitter example, but as we shall see even the definition of a suitable time function is non-trivial, because depening on the choice of a function r : R → (0, ∞) , the set of points at "infinity" where r = ∞ is located on a different "surface", which may or may not be partly contained in the spacetime manifold. 9 With a "good choice" 8 Connection coefficients are on the level of one derivative of the metric. Alternatively these assumptions can be thought of as conditions on the geometric properties of a chosen foliation, or as conditions on the deformation tensors of a number of relevant vectorfields. 9 We will discuss this issue further in Section 1.1, and at length with an example in de Sitter in Section 2.2. Interestingly, in a different context, namely in the study of relativistic compressible fluids, a similar issue appears for the characterisation of the "singular hypersurface" in the formation of shocks; see Chapter 15 in [Chr07]. of a time function r, we will then establish that under our assumptions all components of the Weyl curvature decay at "almost the rate" indicated in (1.8).
In Section 1.1 we discuss the basic difficulties related to a suitable choice of coordinates which covers the domain of development, and correctly parametrizes future null infinity. In Section 1.2 we highlight some of the assumptions made in this paper, in particular we identify a suitable notion of expansion at the level of mean curvatures. Then we proceed to a more precise statement of the result in Section 1.3, and discuss some of the ideas and difficulties of the proof in Section 1.4. In Section 1.5 we briefly sketch some of the ideas relevant for the recovery of the assumptions, which is the topic of a subsequent paper. (Nonetheless, it is important to illustrate the consistency of the assumptions already at this stage.) Finally, we discuss the relation of this result to earlier work, in particular Friedrich's proof of the stability of de Sitter, in Section 1.6.

Basic Difficulties
In view of the expectation that the geometry of a dynamical solution to (1.1) in the cosmological region does not globally converge to a member of the Kerr de Sitter family, but merely to a "nearby geometry" which is a priori unknown, the choice of suitable coordinate system -which covers the entire region of existence -is non-trivial.
Consider for example any given coordinate system (x a ; y A ) on R ⊂ Q for the Schwarzschild de Sitter solution. A naïve approach would be to formulate the assumptions on the metric in this coordinate system, and try to establish the decay with respect to a parameter r = r(x a ) formally defined as in Schwarzschild de Sitter. However, such an approach turns out to be inconsistent, the reason being that the surface r = ∞ thus defined does not coincide with the true future boundary found in evolution.
To overcome this problem we work in a double null gauge, 10 namely coordinates (u, v; ϑ 1 , ϑ 1 ) such that the level sets of u, v are null hypersurfaces whose intersections S u,v are diffeomorphic to S 2 . This choice is natural for the treatment of a characteristic initial value problem, and it allows us in particular to introduce the function r as the area radius of the spheres of intersection (S u,v , g /), (1.9) It is then tempting to define future null infinity I + as the collection of spheres with infinite area radius. However, a general double null foliation still allows considerable freedom in the choice of the spheres of intersection, and it can happen that a sphere with infinite area radius is in fact partly contained in the spacetime. In such a scenario future null infinity is not correctly identified by the set of points where r = ∞. Since this is a subtle yet imporant point, we have included in Section 2 an explicit construction of a double null foliation of de Sitter which illustrates this phenomenon: There exist double null foliations of de Sitter such that the union of all spheres with area radius r ∈ (r C , ∞) is contained, but does not exhaust the cosmological region. (1.10) with the property that the level sets C u intersect the cosmological horizon C in a small ellipsoidal deformation C u ∩ C of the round sphere, yet the "corresponding sphere near infinity" -namely the intersection S u,0 = C u ∩ C 0 with a fixed "incoming" null hypersurface C 0 -is only partially contained in H. In these examples, S u,0 first "touches infinity at a point" and then gradually "disappears on annular regions" as u varies, see Fig. 5. We expect this to be the generic behavior, and it is thus important that our assumptions on the foliations to be discussed in Section 1.2 rule out such behavior of the spheres near infinity. In Section 2 we study also the transformation of the structure coefficients under the gauge transformations provided by these examples, and demonstrate explicitly that some of the assumptions we make in Section 1.2 are not satisfied. 11 The assumptions of Section 1.2 then ensure that the spheres of the foliation indeed exhaust the expanding region.
11 Essentially, any assumption that requires the smallness of the deviation of a quantity q(u, v, ϑ 1 , ϑ 2 ) from its average q(u, v) on the sphere S u,v cannot be satisfied if the area radius of S u,v diverges while a subset of points q ∈ S u,v remain in the spacetime.
Even with a "good" choice of a foliaton with the property that all spheres with infinite area radius are indeed "at infinity", it would be too restrictive to assume that the spheres foliating null infinity are intrinsically round. Our assumptions are certainly consistent with such a non-trivial asymptotic geometry, and we will already see in Section 1.5 an obstruction to the roundness of the spheres, but the precise geometric characterisation of future null infinity is the subject of a subsequent paper. 12

Assumptions on the foliation
Consider a 3 + 1-dimensional Lorentzian manifold (M, g) with past boundary C ∪ C, two null hypersurfaces intersecting in a sphere S; we think of initial data prescribed along C and C, and of R = J + (C ∪ C) -the "cosmological region" -as its future development.
Consider further a double null foliation of R by null hypersurfaces C u , and C v , namely the level sets of functions satisfying the eikonal equations g(∇u, ∇u) = 0 g(∇v, ∇v) = 0 (1.12) which are increasing towards the future, such that C = C 0 , C = C 0 , and is diffeomorphic to S 2 . Following the conventions in [Chr09] we define to be the null geodesic normals, and Ω to be the null lapse: Then normalised null normals are given bŷ L = ΩL L = ΩL (1.16) and used to define the null second fundamental forms of the spheres S u,v as surfaces embedded in C u , and C v respectively: The null expansions, namely the traces tr χ, and tr χ (with respect to g /), measure pointwise the change of the area element dµ g / , in the null directionsL, andL, respectively. Our first main assumption is that "the cosmological region is expanding": The positivity of the null expansions alone is not enough, and we will assume that they are always close to the "geodesic accelerations" 2ω, and 2ω defined by Equivalently, they are given bŷ and our second assumption is that for some constant C 0 > 0: Our third assumption is crucially related to the discussion in Section 1.1, and amounts to the condition that the null expansions are pointwise close to their spherical averages (1.20) We require that for some 0 > 0 |Ω tr χ − Ω tr χ| ≤ 0 Ω tr χ |Ω tr χ − Ω tr χ| ≤ 0 Ω tr χ (BA:I.iii ) Finally we assume for the remaing connection coefficients, namely the trace-free parts of the null second fundamental forms above, and the torsion The assumptions (BA:I ) are L ∞ -bounds on S u,v . They already allow us to prove that the L 2 (Σ r )-norm of the Weyl curvature decays; see Section 5.3. However, to recover the assumptions made here we will need the Weyl curvature to decay in L 4 (S u,v ); see discussion below. This requires us to obtain information on one derivative of the Weyl curvature as well, which can only be proven under additional assumptions on the derivatives of the connection coefficients. These assumptions, schematically bounds on :II ) are too numerous and technical to state conveniently here and will instead be collected under the label (BA:II ) below. They will allow us to prove that all tangential derivatives of W to Σ r decay in L 2 (Σ r ). Finally, several assumptions will be needed to deduce the decay rates of the energy and for the application of the elliptic theory in Sections 7, 8. Most importantly, and the remaining assumptions will be collected under the label (BA:III ).
We emphasize that none of the assumptions make explicit reference to the Schwarzschild de Sitter geometry, and capture only some of the features that the relevant "nearby" geometries have in common.
In the coordinates (u, v; ϑ 1 , ϑ 2 ) thus introduced 13 the spacetime metric takes the form In the special case that b A = 0, g / = r 2 • γ , and Ω is independent of ϑ A , the metric reduces to the spherically symmetric form (1.2). In Section 4 we will discuss various specific choices of the functions u, v on Q for the Schwarzschild de Sitter metric, and the associated values of the connection coefficients above.
In Section 3.5 we will discuss yet another form of the metric adapted to the decomposition relative to the level sets of the area radius Σ r = q ∈ R : r(q) = r .
By (BA:I.i ) these hypersurfaces are always spacelike, and the area radius plays the role of a time-function: g = −φ 2 dr 2 + g r (1.23) where g r denotes the induced Riemannian metric on Σ r . 14 While this form of the metric is advantageous in some parts of this paper, 15 and of some importance for the discussion of the asymptotics, we have chosen the double null gauge as the underlying differential structure, because it will allow us to formulate all assumptions that fix the gauge and specify the initial data only on C ∪ C. Moreover, unlike in "non-local" gauges such as "maximal" gauges which fix the mean curvature of the surfaces Σ r , the double null gauge allows us to localise any argument to the domain of dependence of any subset of Σ r ; this will be of importance for the recovery of the assumptions made in this Section. 13 The coordinates (ϑ 1 , ϑ 2 ) are chosen arbitrarily on a domain of S = C ∩ C, and then transported first along the null geodesics generating C, and then those of C v .
14 Here we choose for simplicity coordinates on Σ r which are transported along the normal geodesics, which justifies the absence of a shift term.
15 In Section 5 we will consider the "energy flux" of the Weyl curvature through Σ r , and the form (1.23) is particularly well adapted to the application of the co-area formula for integration on domains foliated by Σ r . Also the Sobolev inequality of Section 8 is applied on the manifold Σ r .

Main result
We are interested in the "cosmological region" R to the future of the cosmological horizons C ∪ C; see Fig. 4. Above we have introduced the 2-dimensional closed Riemannian manifolds (S u,v , g /) ⊂ R as the intersections of "ingoing" and "outgoing" null hypersurfaces C v , and C u , the leaves of a double null foliation of R discussed in Section 1.2. Each sphere S u,v has area 4πr 2 (u, v) as discussed in Section 1.1, and we shall now introduce a dimensionless L p -norm for tensors θ on the spheres: Theorem. Let (M, g) be a 3 + 1-dimensional Lorentzian manifold, R ⊂ M a domain with past boundary C ∪ C, where C and C are null hypersurfaces intersecting in a sphere S diffeomorphic to S 2 . Assume that g can be expressed globally on R in the form (1.22) and that the double null foliation of R satisfies the assumptions (BA:I-BA:III). Suppose the Weyl field W is a solution to the Bianchi equations (1.7) and W ∈ H 1 (Σ) where Σ ⊂ R is an arbitrary but fixed level set of r. Then W obeys where 0 < < 1, and C > 0 only depend on the constants in (BA:I-BA:III).
Recall that for the example of Schwarzschild de Sitter the Weyl curvature satisfies (1.8). In terms of the decay rate the Theorem thus states almost optimal decay of all components of the conformal curvature. 16 In terms of regularity, we do not estimate the curvature in L ∞ (S u,v ) (which would require assumptions on the connection coefficients at second order of differentiablity). However, it suffices to control the curvature merely in L 4 (S u,v ) to obtain a full existence result for the Einstein equations (1.1). 17 In this paper we do not yet give an existence proof of solutions to (1.1) in R. The theorem establishes what one expects to be the first part of a larger bootstrap, or continuous induction argument. The task of the second part is to establish the converse, namely that assuming a L 4 -bound on the Weyl curvature, we can prove the L ∞ , and L 4 -estimates (BA) on the connection coefficients, under suitable assumptions on the characterisitc initial data. This, and the resulting asymptotics, are the subject of a forthcoming paper; see also Section 1.5. 16 In Schwarzschild de Sitter there is really only one non-vanishing component of the Weyl curvature. In this result all components of the Weyl curvature are on an equal footing, and at present there is no indication of any "peeling" behavior. This is consistent with the expectations formulated in [ABK16], and we will revisit this point in Section 7 in the context of the electro-magnetic decomposition of the Weyl curvature.
17 This is an insight gained in [BZ09], which has provided a significant simplification of the original proof of the non-linear stability of Minkowski space in [CK93]. It eliminates in particular the need to work with second derivatives of the curvature, but comes of course at the cost of less refined asymptotics; see [BZ09] and discussion therein.

Comments on the proof
The proof is largely a treatment of the Bianchi equations (1.7) for the Weyl curvature: In analogy to Maxwell's theory, one can construct an energy-momentum tensor Q(W ), the Bel-Robinson tensor which has the well-known properties -see for example Chapter 7.1 in [CK93] -that it is positive when evaluated on any causal future-directed vectors at a point, and divergence-free if W is a solution to (1.26): This allows us define energy currents with the help of "multiplier vectorfields" X, Y, Z, which give rise to energy identities; see derivations in Section 5.1. An important example in the context of this paper are energy identities on space-time domains bounded by the level sets Σ r : The usefulness of the resulting identity, where n denotes the unit normal to Σ r , Q(n, X, Y, Z)dµ g r 1 (1.31) depends crucially on the properties of the "divergence term", or "bulk term" on D. By (1.28) the integrand is given purely by a contraction of Q(W ) with the "deformation tensor" of the multiplier vectorfields: In fact, since Q(W ) is trace-free (and symmetric) in all indices, only the "trace-free part" of these tensors appear: In Section 5.2 we construct a multiplier vectorfield M and prove that the associated divergence terms have a sign which yields by (1.31) a monotone energy. In fact, we will use M = 1 2 has the following positivity property: Suppose the connection coefficients satisfy the assumptions (BA:I). Then there exists an > 0, such that for any solution W to (1.26): The existence of such a positive current is obviously intimately related to the assumption (BA:I.i ), 18 and the numerical prefactor in the above inequality is important, because it will directly translate into the decay rate of the energy associated to M . This part of the argument is in close analogy to my treatment of linear waves on Kerr de Sitter cosmologies in [Sch15]; c.f. discussion in Section 5.2. The inequality (1.36) lends itself to the interpretation that M captures the classical redshift effect in the cosmological region, 19 in the language of "compatible currents"; we will thus often refer to M as the "global redshift vectorfield".
In the context of the linear wave equation on Schwarzschild de Sitter spacetimes, the treatment of higher order energies, and pointwise estimates is a trivial extension of the "global redshift estimate" because the tangent space to Σ r is in this case spanned by Killing vectorfields. Indeed the commutation of (1.4) with the generators of the spherical isometries of S u,v , and of the translational isometry T along Σ r immediately gives the desired higher order energy estimates in [Sch15]. In the present context, however, this approach is not very fruitful, because even if we were to construct generators Ω (i) and T of "spherical" and "translational" actions on Σ r , these actions cannot be expected to generate asymptotic symmetries (as in [CK93]; because unlike in the asymptotically flat case future null infinity may not possess any symmetries).
In our approach then we use the global redshift vectorfield also as a commutator. 20 18 The assumption that both null expansions are positive fails in S, where tr χ < 0. Similarly in the study of asymptotically flat black hole exteriors, with Λ = 0, one has tr χ < 0. In these settings the construction of positive currents is much more subtle, and in particular sensitive to the presence of "trapped" null geodesics; see [DHR16] and references therein. 19 Much like the redshift observed in black hole spacetimes, the effect is due to the presence of horizons: An observer A who stays away from a black hole perceives a signal sent from an observer B who crosses the event horizon as redshifted, because B leaves the past of A in finite proper time; see e.g. [DR13]. In the cosmological region, where all observers are drifting away from each other (due to the expansion of space), each observer has its own cosmological horizon, which all other observers (in his past) cross in finite proper time. This effect is already present in the de Sitter cosmology, where each time-like geodesic has a cosmological horizon with positive surface gravity; c.f. Section 2 below and [Sch15]. 20 This is an idea due to Dafermos and Rodnianski that already appeared in the study of the linear wave equation on black hole spacetimes, related to the redshift effect on the event horizon; see [DR13].
In general, the conformal properties of the Bianchi equations -see e.g. Chapter 12.1 in [Chr09] -allow us to define a "modified Lie derivative"L X W with respect to any vectorfield X of a solution W to (1.26) which satisfies the inhomogeneous Bianchi equations where (X) J(W ) is a "Weyl current" which can be expressed in terms of contractions of (X)π with ∇W , and contractions of ∇ (X)π with W ; see e.g. Proposition 12.1 in [Chr09]. In the presence of an inhomogeneity in the Bianchi equations, the divergencefree property of the associated Bel-Robinson tensor fails, and (1.28) is replaced by (1.38) Consequently the energy identity derived from the current contains an additional "divergence term" of the form (1.40) A major difficulty in the proof lies in showing that this contribution to the "bulk term" can also be arranged to have a sign. As already indicated above, one could choose X = M , and it is possible to show that (under suitable assumptions) "at the highest order of derivatives" the current (1.39) has the following positivity property: However, the "lower order terms" contained in the "error" E -which are on the level of W and can in principle be controlled by the energy associated to P M [W ] -do not decay fast enough towards I + , for this energy identity to give the rate of decay of the energy associated to P M [L M W ] that would be required (in the application of the Sobolev inequality) to prove the L 4 -bound stated in the theorem. The treatment of the "commutation", or "first order energy" thus becomes the most complex part of this paper, because it requires us not only to find a sign in the divergence, but also to exhibit various cancellations in the lower order terms. This is the reason why we are forced to compute very carefully most terms contained in (1.40), including the signs and prefactors. 21 It turns out that a suitable choice of a commutation vectorfield is given by (1.43) One can think of the map M → M q as induced by a Lorentz transformation with the effect of aligning M q with the normal n to Σ r . The final commutation vectorfield X, subsequently denoted by N , is then obtained from M q by scaling with the weight Ω 2 . In Section 6 we shall prove the following: Suppose the connection coefficients satisfy the assumptions (BA:I) and (BA:II).
Then there exists an > 0, and a constant C > 0, such that for any solution W to (1.26): In this estimate we have achieved that the "error" is -small and controlled by the energy associated to P [L N W ] Mq and P [W ] Mq (first and second term on the r.h.s), up to terms which only involve tangential derivatives to Σ r (third term). Next we prove that the latter can be controlled by the first two terms, but it is again a non-trivial statement that in this estimate no "lower order terms" appear, which would obstruct the "redshift" gained with the positivity of the first term on the r.h.s. of (1.44). The "electromagnetic decomposition" of a Weyl field W relative to Σ r -much like the decomposition of the Faraday tensor F in electric and magnetic fields E, and H 21 In problems where this circle of ideas -the treatment of Bianchi equations using energies constructed from the Bel-Robinson tensor -has been applied, these terms are usually treated as "error terms". Notably for the "stability of Minkowski space" they have only been written out schematically in Chapter 8 in [Chr09]. Although they are also treated as error terms in proof of the "formation of black holes" in [Chr09], their precise nature, and "scaling", is much more important therein, and we take great advantage in this paper of the fact that at least in special cases precise algebraic expressions for the "divergence terms" have already been provided in Chapter 12-14 in [Chr09]. relative to a given frame of reference -recast the Bianchi equations in a system akin to Maxwell's equations: In Section 7 we derive an elliptic estimate for this Hodge system on Σ r 22 that allows us to control all tangential derivatives to Σ r by the energy associated to P [L N W ] (Mq) .
Here it is essential that the commutator vectorfield N has been aligned with the normal n to Σ r , and carries a weight that leads again to exact cancellations with the "lower order terms" (on the level of the second fundamental form k of Σ r ) present in (1.45).
Suppose the assumptions (BA:I) and (BA:III) hold. Then there exists a constant C > 0 such that for all solutions to (1.45), In conclusion, we obtain under the assumptions (BA:I-III ), stated with a slight abuse of notation and freely using (BA:III.i ), that Σr |W | 2 + r 2 |∇W | 2 dµ g r 1 r 3− (1.47) which then easily implies the statement of the theorem by a Sobolev trace inequality on Σ r which we discuss in Section 8.

Preview
An important aspect of this paper is the requirement that the assumptions we make are consistent with the evolution of solutions to (1.1). This is necessary to be able to "upgrade" the results in this paper to a full existence result, which is the subject of a subsequent paper. We illustrate this informally for the assumption (BA:I.ii ), which is central to the redshift estimate of Section 5.3. In general,ω, and tr χ -the connection coefficients of a metric g expressed in the form (1.22), solving (1.1) -satisfy "propagation equations" 22 The analysis of these systems has been developed in some generality in Chapter 2-4 of [CK93]. along the null geodesics generating C u , of the form where terms associated to the cosmological constant contribute "dominantly" to the source terms. However, by virtue of the Gauss equation of the embedding of the spheres in the spacetime manifold, these leading order terms cancel if we consider specifically the propagation equation for the difference: (1.50) A major undertaking of our forthcoming paper -involving an analysis of the null structure equations -is to show that the result (1.25) implies the following bounds on the "null shears": Then (1.50) is integrable, using a suitable lower bound on ∂ v r, and we obtain |ω − Ω tr χ| ≤ C , (1.52) in agreement with (BA:I.ii ). These statements are of course sensitive both to initial gauge choices, and to the initial data, which we will discuss in a subsequent paper. Note moreover that (1.51) states as expected that the "energy densities"χ, andχ of the "gravitational waves" decay at equal rates. This is in contrast to the asymptotically flat case, where the outgoing null direction is preferred, and the associated energy flux χ decays faster in r, thanχ. Consequently, taking (1.50) and (BA:III.i ) at face value, one has in this context (χ,χ) 1 r 2 (1.53) and in view of the Gauss equation we have to expect that lim r→∞ r 2 K = 1 ; (1.54) in contrast to the asymptotically flat case where (χ,χ) r −3 and in a suitable gauge it can be arranged that r 2 K → 1; c.f. Chapter 17 in [CK93]. This raises a number of interesting questions related to the definition of physically relevant asymptotic quantities; see also [ABK16, ST15, CI16, Pen11].

Relation to earlier work
We have already mentioned the work of Hintz and Vasy on the non-linear stability of the Kerr-de Sitter family in the black hole exterior on the domain bounded by the cosmological horizon [HV16b]. 23 Another important result to be mentioned in this context is the work of Friedrich on the stability of the de Sitter spacetime [Fri86]. We will discuss briefly its relevance to the stability problem for Schwarzschild-de Sitter cosmologies outlined on page 5 above. Finally we will mention the work of Ringström [Rin08], and Rodnianski and Speck [Spe13, Spe12,RS13].
In [Fri86] Friedrich proved that the future development of Cauchy data on S 3 is geodesically complete, provided the initial data is "a small perturbation" of the datum induced by the de Sitter solution; see Fig. 7, and Section 2 for a discussion of the hyperboloidal model. Now with regard to the initial data induced by a Schwarzschild de Sitter solution, a Cauchy hypersurface Σ as in Fig. 3 cannot be expressed as a "perturbation" of de Sitter data. However, a truncation [Σ] of Σ away from the event horizons could be viewed as a perturbation of a suitable "segment" of de Sitter data, at least for small mass 0 < m 1/(3 √ Λ); see Fig. 6 (right). It is plausible that the resulting data on [Σ] [δ, π − δ] × S 2 can be glued to the "spherical caps" [0, δ] × S 2 of an S 3 , to obtain an admissible initial data set for [Fri86]; see Notably, such an argument cannot achieve a stability statement "in a neighborhood of time-like infinity ι + ;" see Fig. 6. While the results in [Fri86] are closely related to the conformal properties of (1.1), and achieve a global existence result by a reduction to a "local in time" problem, Ringström provided a treatment of the "Einstein-non-linear scalar field system" -which includes the Einstein vacuum equations with positive cosmological constant as a special case -that reproves the results in [Fri86], without resorting to a "conformal compactification", and without specific reference to the topology of the initial data [Rin08]. In fact, the set-up in [Rin08] exploits a causal feature of "accelerated expansion" already evident from the Penrose diagram of de Sitter, c.f. Fig. 6   is contained in the domain of dependence of B 4R (p), I + (B R (p)) ⊂ R \ I + (Σ \ B 4R (p)), provided Σ is "at sufficiently late time", e.g. if min [Σ] r r C . This allows Ringström to prove "global in time" results, from "local in space" assumptions on the inital data, which cover in particular perturbations of de Sitter, but are not restricted to the S 3 topology. Notably, the "asymptotic expansions" of Theorem 2 in [Rin08] show the existence of "asymptotic functional degrees of freedom", namely that the solution converges to a metric which after rescaling by the expected behavior in time differs from the rescaled de Sitter metric, even at the leading order parametrized by a free "profile" function. 25 This paper does not yet give a full global existence theorem for solutions to (1.1) on the level of [Fri86,Rin08,HV16b]. It does however accomplish what one expects to be an essential step towards the stability of the expanding region of Schwarzschild-de Sitter cosmologies: We show that the Weyl curvature decays under sufficiently general assumptions -roughly corresponding to Part II of the original proof of the non-linear stability of Minkowski space [CK93].
The underlying decay mechanism -namely the expansion of spacetime -has also played a prominent role in the work of Speck on Friedman-Lemaître-Robertson-Walker cosmologies: They were shown to be future stable in [RS13,Spe12,HS15] as solutions to the Euler-Einstein system, and it was observed in particular that the "de Sitter"like expansion prevents the formation of shocks in relativistic fluids [Spe13]. For stiff fluids Rodnianski and Speck also showed stable "big bang" singularity formulation in the past [RS14b, RS14a]. Some elements of their proof -in particular the existence of a monotone energy at the level of the commuted equations, the resulting smallness of the Weyl curvature, and the functional degrees of freedoms associated to all possible "end states" -bear some resemblance to the approach pursued in this paper. 26 Acknowledgements. I would like to thank Mihalis Dafermos for drawing my attention to this problem already when I was a student, and for his continued encouragement and support. I would also like to thank Abhay Ashtekar, and Lydia Bieri for many stimulating discussions at a conference this January in Sanya, China, and Gustav Holzegel for several useful comments following a talk in London this May.

Geometry of the de Sitter solution
In the introduction we have already introduced the de Sitter spacetime as the onesheeted hyperboloid in R 5 with the induced metric of the ambient Minkowski space.
the time-like hyperboloid in the ambient Minkowski spacetime R 1+4 with metric The manifold has topology R × S 3 , and is called de Sitter space; c.f. Fig. 7. In Section 2.1 we will briefly discuss other representations of the de Sitter geometry, derived from the above, which firstly make the conformal flatness of de Sitter manifest, and secondly relate this geometry to the Penrose diagram of Schwarzschild de Sitter.
In Section 2.2 we will construct examples of non-trivial double null foliations that fail to capture correctly the geometry of future null infinity. This illustrates the subtle obstacles that have to be overcome for the definition of foliations that correctly parametrize null infinity, and further motivates our choices in Section 1.2.

Representations of the de Sitter geometry
We will first introduce stereographic coordinates in which the spacetime is manifestly conformally flat. 28 Then we will introduce a spherical coordinates system in which the metric takes a similar form as in Schwarzschild de Sitter. In these coordinates it is evident that de Sitter contains a static patch.

Stereographic coordinates
Let us choose coordinates (t, x, x ) ∈ R 5 , with t, x ∈ R, and x ∈ R 3 , and fix (0, −1, 0) as the base point of the projection. We then project the hyperboloid H on the plane x = 1, which is 3 + 1-dimensional Minkowski space with the induced metric of the ambient space. In other words, we assign to every point (t, x, x ) ∈ H coordinates (u, y), such that (u, 1, y) is on the line from (0, −1, 0) to (t, x, x ), c.f. Fig 8. This implies that for some λ ∈ R: and conclude that (2.7) Note that the map thus defined only maps a subset of x = 1 onto a subset of H, c.f. Fig. 9. More precisely, the domain is and the image is (2.10) Lemma 2.1. In (u, y) coordinates the de Sitter metric takes the form Given that the map φ : D → R ⊂ H is explicit, and h = m| H in induced by the Minkowski metric, the proof of the Lemma is an elementary calculation of the components of In view of the conformal property of the Weyl curvature W , it follows in particular that namely that h is conformally flat.

Static coordinates
The idea is to introduce a spherical coordinate system relative to a fixed observer. We choose this observer, or the origin of this coordinate system to be the curve in the coordinates of the ambient space R 5 . Each level set of t in H is an S 3 of radius 1 + t 2 : We can think of that S 3 as (0, π) × S 2 , where then |x | is the radius of the S 2 , c.f. Fig. 10. Thus we introduce as one coordinate In addition we introduce a new time coordinate t which is constant on the level sets of t/x. An appropriate choice is Lemma 2.2. The de Sitter metric in (t , r) coordinates reads We have derived this form of the metric in the domain None of the metric coefficients depend on t , and ∂ t is orthogonal to the level sets of t , thus the spacetime metric is indeed static on S. The metric in this form is spherically symmetric, the center r = 0 being a timelike curve that lies opposite to the base point of the stereographic projection; since the latter was chosen arbitrarily, static coordinate patches can be introduced for any given timelike geodesic in H. The static region S is the intersection of the past and future of the central line r = 0. (This is particularly easy to see in the stereographic coordinates.) The boundary of S is a bifurcate null hypersurface C: the cosmological horizon. Indeed, r = 1 implies t = |x| which is a null line, namely the straight lines in R 5 that rules the hyperboloid H. In the stereographic picture this is the set u = ±|2 − |y||, a null hypersurface in R 1+3 .
We summarize the causal geometry thus described in the Penrose diagram; see Fig. 6 (top right).

Double null foliations of de Sitter
A convenient feature of the representation of de Sitter as a time-like hyperboloid embedded in Minkowski space is that the causal structure of de Sitter is then induced by the ambient Minkowski space. In particular cones in Minkowski space emanating from a point on the hyperboloid trace out null hypersurfaces in de Sitter spacetime. This yields an elegant approach to construct solutions of the eikonal equation in de Sitter, which we will employ in particular to construct non-spherical double null foliations.

Spherically symmetric foliations
We first construct spherically symmetric double null foliations by intersecting cones emanating from antipodal lines. This will lead to coordinates such that the metric takes the form h = −4 r 2 − 1 du * dv * + r 2 • γ . (2.21) Most importantly, we will find the function u * , v * : H → R explicitly it terms of the ambient coordinates, which will be used in Section 2.3. For any point (t, x, x ) ∈ R × R × R 3 let us denote by C ± (t,x,x ) the forward/backward cone emanating from (t, x, x ) in R 1+4 : Now consider the two anti-podal geodesics Γ, Γ in H: Note that for |y | = 1, and moreover with Γ(t) = (t, √ 1 + t 2 , 0), Let us now define Lemma 2.3. The intersections S t,t = C t ∩C t are round 2-spheres in H cocentric around Γ, and Γ. Moreover, "future null infinity" can be identified with 29 We omit the proof here, but let us note that for any (s, x, x ) ∈ S t,t we have which shows in particular that Recall that with t = t (x, t) defined as in (2.18) the metric in (t , r)-coordinates takes the form (2.19). Since null coordinates are given by we are lead to the following functions whose level sets are the null hypersurfaces C t : Lemma 2.4. Let u, v : H → R be the following functions: then for any t, t < 0, Proof. Let us assume that t < 0, and observe that on C t : x − t < 0. Then using (2.30) it is easy to check that for (s, x, x ) ∈ C t : and thus by direct computation: Similarly for points on C t .
Note in particular that A suitable set of null coordinates are thus obtained by setting Moreover, these are of course precisely the double null coordinates introduced above, such that

Examples of non-spherically symmetric foliations
We shall now construct a double null foliation for which the intersections are not spheres but ellipsoids. Recall that the spherically symmetric foliation was constructed by considering intersections of cones emanating from antipodal geodesics Γ, Γ. We shall now consider cones emanating from points which are slightly displaced from these geodesics.
For simplicity let us leave the null hypersurfaces C t emanating from points Γ(t) = (t, t , 0) unchanged. In fact, let us fix in particular the null hypersurface C 0 = C + Γ(0) ∩H emanating from (0, 1, 0), and observe that C 0 is contained in the hyperplane x = 1: Note that indeed on C 0 (where the subscript refers to t = 0) we have that u| C 0 = 1, and thus u * | C 0 = 0. On C 0 we can also derive a simple relation between v * and t = |x | > 1: In Section 2.2.1 we considered the cones intersected with H and vertices on Γ: Then the intersection of the null hypersurfaces C and C 0 is a sphere Also note that v * | C = + O( 2 ). In particular the "sphere at infinity" where C 0 , and C 0 "meet in the Penrose diagram" is identified with S 0 . Let us now consider a null hypersurface in H emanating from a vertex slighly displaced from Γ: We also introduce an angle in the plane spanned by x δ , and x such that It is then easy to calculate that This means in particular that all sections S ( ,δ) are ellipsoids (with eccentricity δ/ ). Moreover, for fixed < 0 the deformation of the sphere S (which lies in C 0 "away from infinity") to the ellipsoid S ,δ cannot "move any point to infinity" as long as δ < | |. However, if δ = | | then S ,δ "touches infinity" at exactly one point (the antipodal point to x δ ); meanwhile it is clear that the intersections of C δ, with the cosmological horizon C is a "small deformation" of C ∩ C (as we will show below). Finally, if δ > | | then only the "hemisphere" 0 ≤ ϑ ≤ arccos /δ of S ,δ remains in the spacetime. This case occurs also when we move the sphere S ,δ "to infinty" by taking → 0 while keeping δ > 0 fixed.

Explicit parametrizations of ellipsoidal foliations
Another way to parametrize the transformation of the foliation above is to introduce new coordinates (t,x,x ) such that the vertex of C ,δ is at ( , − , 0) in the new coordinates: this can obviously be achieved by a rotation. Then in these coordinates we can write down the level set of the cone emanating from this point, which we then can express in the original coordinates (t, x, x ) as desired.
Lemma 2.5. Let (u * , v * ; ϑ 1 , ϑ 1 ) be the (spherical) double null coordinates introduced above. Then for any < 0, and −π < ϕ < π, the level sets of In fact, Remark 2.6. For small displacement angles, 0 < ϕ Moreover let us look at the situation when v * < 0, |v * | 1, and u * → −∞: One may solve approximately for v * (ϑ 1 ) on a level set of v ϕ to find that By comparison, on the intersection with C 0 , where u * = 0, in the context of the small angle approximation: Again, an approximate solution for v * (ϑ 1 ) on a level set of v * ϕ is given by Observe that while identical in form to the formula found on C, the range sweeped out in v * is now twice as large, namely contained in [−| | − |ϕ|, −| | + |ϕ|]. In particular if we choose ϕ = , then C ,ϕ ∩ C is contained in the interval v * ∈ [−3| |/2, −| |/2], while it is possible for C ,ϕ ∩ C 0 to "touch infinity", namely at points where v * = 0.

Transformation of the optical structure coefficients
In the previous section we have constructed an explicit family gauge transformations (parametrized by |ϕ| < π) such that the new level sets C v of v are again null hypersurfaces in de Sitter. We have seen that for small v = < 0, | | 1 the intersection of the null hypersurface C v with the cosmological horizon C −∞ is a small ellipsoidal deformation of the round sphere (with eccentricity proportional to ϕ), while the intersection S 0,v of C v with a fixed incoming null hypersurface C 0 going to infinity is again an ellipsoidal deformation of a sphere near infinity, which however contains points (first a point, and then annular regions surrounding this point) which "run off to infinity" as v → 0, (while keeping ϕ fixed).
We shall now calculate explicitly 30 the transformations of all optical structure coefficients associated to the gauge transformation (2.64), at least for "small displacement angles", i.e. for |ϕ| 1; (the parameter ϕ measures the displacement of the basepoint of the cones, see Fig. 11, and corresponds to the eccentricity of the ellipsoid in the cosmological horizon). We are interested in the details of the gauge transformation on the sphere near infinity, i.e. for v = , | | 1. We begin with the calculation of the null normals; in general, given two optical functions u, v we define the corresponding null geodesic vectorfields by The null lapse is then defined by Lemma 2.7. For small displacement angles |ϕ| 1, the null vectorfields on S 0,v for v = < 0, | | 1 are given by up to terms quadratic in (ϕ, ). Moreover Ω Ω * r.
Remark 2.8. The function r that appears in the approximation for Ω is by no means constant on S 0, . We will derive below an explicit dependence of r(ϑ 1 ) for small displacement angles, using the formulas obtained in Section 2.3.

Proof. Ommited.
It is now straighforward to calculate various connection coefficients: Lemma 2.9. For small displacement angles |ϕ| 1, the gauge transformation (2.64) induces the following transformations of the null structure coefficients (2.69), on S 0,v for v = < 0, with | | 1: We emphasize that in the statements of Lemma 2.7, 2.9 , the radius r is a function on S 0,v . In fact, and moreover, for small displacement angles we have found in (2.63b) the following relation between v * and ϑ 1 on This allows us to calculate the volume element on S 0, .
Lemma 2.10. The volume element on S 0, = C 0 ∩ C ,ϕ is given by Remark 2.11. Note that as expected while keeping the displacement angle ϕ fixed.
Finally we calculate the average of the above transformed null structure coefficients: Lemma 2.12. For small displacement angles |ϕ| 1, we have at the same rate as the area, namely Proof. Omitted.

Comparison to bootstrap assumptions
The above statements are important for the interpretation of the bootstrap assumptions in Section 1.2, in particular the conditions These assumptions are all trivially satisfied in the spherically symmetric double null gauge discussed in Section 2.2.1. We shall now investigate if these assumptions are also satisfied for the above examples of non-spherically symmteric gauges.
In the following discussion we restrict ourselves to foliations with small eccentricity, or displacement angles |ϕ| 1. Moreover we discuss these assumption on a "sphere" S 0, near infinity, | | 1, and are interested in particular in the limit as a point on this sphere "touches" infinity, corresponding to, approximately | | |ϕ|. According to Lemma 2.9 we have tr χ 2 tr χ 2 + 4ϕ r cos ϑ 1 (2.75) Recall here also from (2.72) that so the assumption (2.74a) on the null expansions being positive is still satisfied, at least for |ϕ| 1. Furthermore Lemma 2.9 also tells us that ω 1ω 1 (2.77) and thus the assumption (2.74b) related to the redshift is in fact satisfied: at least for |ϕ| 1. However, the last assumption (2.74c) stipulating that the values of Ω tr χ remain close to its average on S 0, ceases to be satisfied. To see this, note that according to Lemma 2.7, Ω tr χ 2r(ϑ 1 ) + 4ϕ cos ϑ 1 (2.79) where the function r(ϑ 1 ) is given in Lemma 2.10: In particular on the equator of the sphere S 0, , for ϑ 1 = π 2 , we have and (2.74c) is not satisfied with 0 < 1. Since for any non-round spheres -not just of the ellipsoidal type discussed herewhich "touch infinity" at one point the area blows up, we can expect that the assumption (2.74c) cannot be satisfied. This means that foliations that do have the property (2.74c) this behaviour cannot occur, and the spheres in these foliations approach null infinty "uniformly", and are eventually "contained" in null infinity. Indeed, if one point did "run off" the above argument shows that (2.74c) ceases to be satisfied.

Weyl curvature
Presently we shall focus on the conformal decomposition of the curvature tensor, which plays an important role in this context. Recall the Schouten tensor where R denotes the scalar curvature. We observe that for any solution to (3.1) the Schouten tensor is simply The Weyl curvature W , in general, is defined by which for solutions to (3.1) then reduces to: Note that W has the same algebraic symmetries as the curvature tensor R, and in addition is totally trace-free. We shall thus proceed in Section 3.2 with the null decompositon of the Weyl curvature.

Null decomposition of the Weyl curvature
We have already referred to the symmetries of the Weyl curvature. We note that the Weyl curvature (3.5) is a "Weyl field" in the sense of Chapter 12 in [Chr09]: It is anti-symmetric in the first two and last two indices, and satisfies the cyclic identity: Moreover, the Weyl curvature satisfies the trace conditon: The dual of W is defined by (as we know, left and right duals coincide) There are 10 algebraically independent components of a Weyl field. Let (e A : A = 1, 2; e 3 , e 4 ) be an orthonormal null frame field. Then the 2-covariant tensorfields (3.9) account for 2 components each, because they are symmetric and trace-free: Also the 1-forms account for 2 components each, which leaves us with 2 functions Note that with (3.5) we have Here we used g 33 = g 3A = g 44 = g 4A = 0, and g 34 = −2.
Thus the only component that differs from the corresponding null decompositon of the curvature tensor R (which is only a Weyl field in the case Λ = 0) is ρ.
Note that σ[W ] can equally be defined by The remaining components of W are expressed as, c.f. (12.34) in [Chr09],

Bianchi identities
Recall that in general the curvature tensor R satisfies the Bianchi identities: which, by setting α = ν and sum, yields the contracted Bianchi identies: Schematically, these equations say div Riem = curl Ric (3.20) But here, of course, for any solution to the vacuum equations with positive cosmological constant, Ric(g) = Λg, and by metric compatibility of the connection. Thus, as in the case Λ = 0, This implies now that for a solution to (3.1) also the Weyl curvature is divergence free: With the same formula, (3.23), we see that the "Bianchi identity" (3.18) is also true for the Weyl curvature: or, for short, the homogeneous Bianchi equations hold: This is consistent with general principles, according to which the equation (3.26), for the Weyl field W , also written as by the symmetries of a Weyl field, which are precisely the equations (3.24). Now we are in the situation where we have a Weyl field W satisfying the "inhomogeneous" Bianchi equations (3.24): Therefore by Proposition 12.4 in [Chr09] the null decomposition of (3.29) takes precisely the form given therein. In other words, the Bianchi equations are verbatim those of the vacuum equations in the case Λ = 0, with the understanding that the null components refer to the null decomposition of the Weyl curvature tensor.

Double null gauge
We follow the conventions of Chapter 1 in [Chr09], for the definition of the double null foliation.
We have already introduced in Section 1.2 the optical functions u, v as solutions to the eikonal equations (1.12) such that the surfaces of intersection of the level sets of u, v, (1.13), are spheres diffeomorphic to S 2 . Null geodesic normals L , and L are introduced as in (1.14), and with their help the null lapse function Ω in (1.15). Note that with we have Moreover, local coordinates on S u,v are introduced as in Chapter 1.4 in [Chr09]: We choose coordinates (ϑ 1 , ϑ 2 ) on S 0,0 , which are then transported first along the geodesics generated by L on C, and then along the null geodesics generated by L on C u . In this "canonical coordiante system" the metric takes the form (1.22).

Area radius
Recall that we have already introduced in (1.9) the area radius r(u, v) of S u,v . Since, by definition where Φ v is the 1-parameter group generated by L, we have where by definition Df = Lf for any function f , and thus where · denotes the average of a function on the sphere

Areal time function
Besides the double null gauge, which is particularly suited for the characteristic initial value problem, other gauges typically involve the choice of a time function. This concept appears here naturally in the form of the area radius which is increasing towards to future -this is one manifestation of the expansion of the cosmological region. However, while we do use the decomposition of the Einstein equations relative to a given time-function, we do not impose an equation on its level sets, such as in [CK93], or [RS14b, RS14a]; here the time function is chosen once the double null foliation is fixed.
Given an "areal time function" r, we define and the associated lapse function by Then the unit normal to the level sets of r, Σ r is In the following it will be useful to express these in terms of quantities associated to the double null foliation: and the normal n to each leaf Σ r is given by Proof. Here we need explicit expressions for the components of the inverse: so in particular This yields 1 Ω Ω tr χL + Ω tr χL (3.45) This implies and thus the statement of the Lemma.

Induced metric
Let us discuss here the metric on Σ r , in particular as r tends to infinity. On Σ r we may use (u, ϑ 1 , ϑ 2 ) as coordinates. Recall from Lemma 3.1 the expression for the normal to Σ r , and that in general g r , the induced metric on Σ r is given by Lemma 3.2. The metric on Σ r in (u, ϑ 1 , ϑ 2 ) coordinates, takes the form g r = q −2 Ω 2 du 2 + g / AB dϑ 1 dϑ 2 (3.48) and the volume form on Σ r is and so (g r ) uu = q −2 Ω 2 . (3.52) Moreover, Remark 3.3. There appears no "shift" in the induced metric, because with our present choice the angular coordinates are Lie transported along the ingoing null geodesics.

Second fundamental forms
Following the discussion of the first fundamental form, g r , we now turn to the second fundamental form k r of Σ r . Recall the Codazzi equations: where X, Y, Z are tangent to Σ r . We use a "convenient frame": and Moreover the Gauss equations are: Ric ij + tr kk ij − k m i k mj = Ric ij +R 0i0j (3.59b) R + (tr k) 2 − |k| 2 = R + 2 Ric 00 = 2Λ (3.59c) The "acceleration of the normal lines" is given by ∇ n n = ∇ log φ. In particular if φ is constant on Σ r the normal lines are geodesics parametrized by arc length. The second variation equation reads in the above frame: Finally we note the associated connection coefficients: Note the frame is not "Fermi transported".

Schwarzschild de Sitter cosmology
In this Section we briefly discuss some aspects of the geometry of the Schwarzschild de Sitter solution [Kot18,Wey19]. Its global geometry -as depicted in the Penrose diagram of Fig. 1 -has already been discussed in Section 3 of [Sch15]; c.f. [GH77].
Here we are mainly interested in the values of the structure coefficients for different choices of double null foliations, which has partly motivated our assumptions in Section 1.2. We restrict ourselves to the cosmological region R, and spherically symmetric foliations. 32

General properties
The Schwarzschild de Sitter spacetime is a spherically symmetric solution to (1.1), and distinguishes itself from de Sitter solution by the presence of a mass m > 0. The manifold is Q × SO(3), and the metric g takes the form (1.2). Moreover -as we have seen in Section 1.2 -in double null coordinates the metric takes the form (1.22), which simply reduces to The mass m, representing the "mass energy contained in a sphere" S u,v , can be defined unambiguously in spherical symmetry as a function m : In vacuum, the Einstein equations (1.1) then imply that m is a constant, (which parametrizes this 1-parameter family of solutions.) This allows us further to pass from the unknown r : Q → (0, ∞) to the "Regge-Wheeler coordinate" which by virtue of (1.1) satisfies the simple p.d.e.
The various double null coordinates discussed below can be thought of as different choices of functions f , g appearing in the general solution r * (u, v) = f (u) + g(v) of (4.4), and constants of integration in (4.3).
Let us also note that the polynomial in r on the l.h.s. of (4.2) has three real distinct roots r C , r H , r C provided 0 < m < 1/(3 √ Λ), the two positive ones r H and r C coinciding with the event, and cosmological horizons H, and C, respectively, (where ∂ u r = 0, or ∂ v r = 0, by the equation). In the following we are only interested in charts covering the cosmological region, and horizons, namely the domain r ≥ r C .

Eddington-Finkelstein gauge
In "Eddington-Finkelstein" coordinates we choose and thus cover R + by Note that the cosmological horizons C + ∪ C + at r * = −∞, are not covered by this chart, but strictly only its future; moreover future null infinity I + can be identified with the surface u + v = 0. In these coordinates the metric takes the form and we note specifically With the definitions of the null normals (L,L) of Section 1.2 it is then straight-forward to verify that 33 ω =ω (4.9c) and χ AB = g • (∇ e AL , e B ) = Ω r g / AB (4.10a) The Gauss equation (1.49) now allows us to calculate the ρ component of the Weyl curvature: Since the spheres S u,v are round, we have K = r −2 , and we obtain with (4.10) that

Gauge transformations and "regular" coordinates
The choice of null coordinates in Section 4.2 has a shortcoming: the coordinates do not extend to the cosmological horizons. While Eddington-Finkelstein coordinates provide a natural notion of "retarded and advanced time", we will now discuss coordinates which extend beyond the cosmological horizons. The following discussion highlights in particular the gauge dependence of the structure coefficients, and is relevant for the dynamical problem.

Kruskal coordinates
Let us denote the Eddington-Finkelstein coordinates of Section 4.2 by (u * , v * ). Then "Kruskal coordinates" (u K , v K ) are obtained with the following transformation: In these coordinates r(u, v) is implicitly given by where α H , α C > 0 are positive exponents (depending on Λ, m) satisfying α H + α C = 1; c.f. (3.16) in [Sch15]. In particular, in these coordinates the cosmological horizons C + , and C + , are at u K = 0, and v K = 0, respectively, and the future boundary r = ∞ lies on the hyperbola u K v K = 1. The metric takes the form (4.1) where Ω is non-degenerate on the C ∪ C; in fact and where κ C > 0 is the surface gravity of the cosmological horizons; see Section 3 of [Sch15] for derivations. It is then straight-forward to calculate that in this gauge, We also calculatê

"Initial data" gauge
We give an example of a double null system which retains "retarded time" u of "Eddington-Finkelstein type" along C + , and "advanced time" v of "Eddington-Finkelstein type" along C + , yet is regular at the past horizons. It is trivially obtained by "patching" the above coordinate systems, but its features are worth studying, because it mimics a suitable gauge choice for the characteristic initial value problem.
Let us define where This means that in this gauge, along C + , for u * > 0, the null lapse behaves like and along the null infinity, for u * > 0, Let us calculate the structure coefficients in the region u * > 0; (the region v * > 0 is entirely analogous). In the same way as in Section 4.2, we find, relative to the normalised frame,L = 1 so that tr χ| C + = 0 (4.25c) Similarly, we find in particular It remains to calculate the values ofω,ω in this gauge. We find and in particular

Gauge invariance
In view of the assumptions on the structure coefficients outlined in Section 1.2, we discuss the gauge -dependence and -invariance of the relevant quantities for the Schwarzschild-de Sitter example.
In Table 1 we summarize the asymptotics towards null infinity of the values of the connection coefficients for the Schwarzschild de Sitter metric in the gauges discussed above.
Note that each quantity, Ω,ω,ω, tr χ, tr χ, has the same asymptotics in r (towards null infinity) in all gauges, but different behavior in u * along null infinity. In particular note that Ω differs by a prefactor even at the leading order.

Global redshift effect
This Section contains a central part of this paper: We will prove a non-trivial bound for the Weyl curvature in spacetimes that satisfy our assumptions. This is achieved by means of energy estimates for the Bianchi equations, which are recalled in some generality in Section 5.1. In Section 5.2 we will construct a suitable "multiplier vectorfield" whose associated energy is "redshifted", or "damped" in a fashion that is related to the expansion of the spacetime. This approach will then be further developed in Section 6 to obtain also bounds on the derivatives of the Weyl curvature.

Energy identity
In view of the trace-free property of Q[W ] it is actually only the trace-free part of the deformation tensor that enters here: Using also the symmetry with respect to any index we finally obtain: We defined P as a 1-form. Let * P be the dual of the corresponding vectorfield P , which is a 3-form: * P νκλ = P µ µνκλ (5.6) Here = dµ g is the volume form of g. The exterior derivative of * P is a 4-form, and hence must be proportional to the volume form: This implies, that integrated on any spacetime region D, we have by virtue of Stokes theorem, Let the domain D be as in Figure 12, namely where superscript c denotes that these surfaces are appropriately "capped". We have where n is the unit normal to Σ r ; note that in the boundary integrals arising in Stokes theorem, the normal is always outward pointing, in particular it will have the opposite sign on the past boundary Σ r 1 . For the null boundaries we recall first from (1.204-6) in [Chr09] that = dµ g = 2Ω 2 det g / du ∧ dv ∧ dϑ 1 ∧ dϑ 2 (5.14a) dµ g / = det g /dϑ 1 ∧ dϑ 2 (5.14b) and then calculate, using that (u, ϑ 1 , ϑ 2 ) are coordinates on C, where we used that (5.17) On the null hypersurface C u we have to be more careful because and so g(P, Therefore, similarly Cu * P = * P vϑ 1 ϑ 2 dvdϑ 2 ϑ 2 = 2Ω 2 P u det g /dvdϑ 1 ϑ 2 To summarize we have proven the following: where Q[W ] denotes the Bel-Robinson tensor of W , and Proof. By (5.9),
(5.28) Note M is time-like future-directed, and the associated energy flux (5.13) is positive. Its crucial property however is that also the associated divergence (5.9) has a sign and bounds the energy flux, which lends it the name of a "redshift vectorfield".
Remark 5.2. The choice (5.28) is motivated by the form of the "global redshift vectorfield" used in our treatment of linear waves on Schwarzschild de Sitter cosmologies in [Sch15]. Therein we introduced Alternatively, using the gradient vectorfield V of r introduced in (3.36), the this vectorfield can be expressed as and in the coordinates introduced in Section 4.2, In fact, as discussed in Section 4.1 of [Sch15] it is equivalent to use the vectorfield which takes a remarkably simple form, and coincides precisely with (5.28).

Fluxes
We will derive an energy identity associated to the multiplier vectorfield M on a domain foliated by level sets of the area radius r. Let us first look at the energy flux of a current constructed from M through a surface Σ r . Recall here Lemma 3.1 concerning the normal to Σ r .

Deformation Tensor
Next we calculate the components of the trace-free part of the deformation tensor of M , which enter the expression for K (M,M,M ) in (5.24).

Lorentz Transformations
We will see in Section 5.3 that the simple choice (5.28) for M suffices to obtain the desired energy estimate for the Weyl curvature W . However, it turns out in Section 6 that a more refined choice is necessary to obtain an estimates for higher order energies.
The required adjustment amounts to co-aligning the vectorfield with the normal to Σ r . This can be achieved by formally keeping exactly the same definition of M , but changing the null frame that is used in (5.28). The fact that we can keep this simple definition in terms of another null frame will be computationally very advantageous.
A simple Lorentz transformation is given by: we can expect that for some function a χ , as r → ∞, Now we see clearly that the function a χ appearing in the asymptotics of tr χ, and tr χ, is the required rescaling of the null vectors in (5.56). More precisely, with a = a χ , we can expect that M , formally given by (5.28) relative to a frame resulting from the Lorentz transformation (5.56), satifies asymptotically M → φn . and e A : A = 1, 2 an arbitrary frame on the spheres. Note that both (L,L; e A ) and (e 3 , e 4 ; e A ) are null frames. Here the frame (L,L; e A ) is the null frame derived from the double null coordiantes, and we continue to denote byω,ω, η, etc. the associated structure coefficients. Now for the frame (e 3 , e 4 ; e A ) we find the following connection coefficients: ∇ 3 e 3 = aω +La e 3 ∇ 4 e 4 = a −1ω +La −1 e 4 (5.62a) aχ AB e 4 (5.62e) Also note: e 3 log Ω = aω e 4 log Ω = a −1ω (5.63) In other words, the Lorentz transformation (5.56) induces the following transformations of the structure coefficients: 1 Ω a tr χ + a −1 tr χ − 1 2 1 Ω L a +La −1 (5.66b) where χ, χ, η, η, ζ,ω,ω are the structure coefficients associated to the null frame (L,L; e A ).

Global redshift estimate
In this Section we will show that the energy on Σ r associated to the current P M [W ] decays uniformly in r. The decay mechanism lies in the expansion of the spacetimeas manifested in our assumptions, in particular tr χ > 0, and tr χ > 0 -and results in the positivity of the K M , which we have proven in Section 5.2. In order to exploit the positivity of K + [W ] in the bulk term -in comparison to the flux terms associated to P M [W ] -we need a version of the coarea formula: We foliate the spacetime domain D by the level sets of r(u, v) = c, and first note that we have already calculated the normal separation of the leaves in Lemma 3.1: We add up these two inequalties, and in view of the formula (5.82) for the lapse function, it then follows immediately from Lemma 5.5 and Lemma 5.3 that φK + dµ g r ≥ 6(1 − ) 2 r * P M and therefore by the co-area formula It remains to estimate the error terms occuring in Lemma 5.5.
The assumptions for Lemma 5.13 are referred to as "stronger" because the smallness parameter of (5.84) is replaced by Ω −1 , which by (BA:III.i ) will be assumed to satisfy We will state the main conclusion for the energy current associated to M q , but this result of course also holds for the energy associated to M and the same proof applies.
Proposition 5.14. Assume (BA:I) and (BA:III.i ), for some 0 > 0 and C 0 > 0. Then there exists an 0 < < 0 , and a constant C(r 0 ) such that for any solution W to (1.26), Proof. Apply Proposition 5.1 to the energy current P q [W ] to obtain the inequality for any r 2 > r 1 > r 0 . Let us assume without loss of generality that r 0 is chosen sufficiently large, so that Then by the co-area formula (5.83), and by Lemma 5.13, which implies the inequality A Gronwall-type argument then implies the statement of the Proposition, see Section 6.4.
Remark 5.15. An equivalent statement can also be derived for a weighted null flux.

First order redshift
The aim of this Section is derive an energy estimate for ∇W , similar to the redshift estimate for W in Section 5.3. This is achieved by commuting the Bianchi equations (1.26) with a vectorfield X, which yields an inhomogeneous equation of the form (6.6) for the modified Lie derivativeL X W . The strategy here is to choose X to be future-directed time-like, in fact colinear with the normal n to Σ r , and to derive a redshift estimate for solutionsL X W to (6.6), which can then be used to control all derivatives ∇W tangential to Σ r . This last step relies on an elliptic estimate in the context of the electro-magnetic decomposition of W with respect to Σ r , which we will discuss separately in Section 7. A natural choice of the commutator would be One reason to expect that this vectorfield produces the correct lower order terms can already be inferred without computing the terms in (6.3): The modified Lie derivative is an expression of the form (6.5). Given control onL N W , we only obtain control on all tangential derivatives by the elliptic estimate of Section 7, if the lower order term in the expression forL N W match precisely the lower order terms in the "Maxwell equations" (7.3) of the electro-magnetic decomposition on Σ r , c.f. Lemma 7.1. This gives a condition on tr (N ) π which can motivate the rescaling by Ω 2 in (6.4).
where the Weyl current (N ) J(W ) is detailed below in (6.8). Therefore, according to Lemma 12.3 in [Chr09] we have the important formula whereα q ,β q ,ρ q ,σ q ,β q ,α q refer to the null components ofL N W , andΞ q ,Ξ q ,Λ q ,Λ q , K q ,K q ,Θ q ,Θ q refer to the null components of (N ) J(W ). Remark 6.1. Throughout this Section we use a null decomposition with respect to the null frame (e 3 , e 4 ; e A ). However the null structure coefficents are still associated to the frame coming from the foliation (L,L; e A ). To avoid confusion we append a subscript q to any null components decomposed relative to (e 3 , e 4 ; e A ). In particular in reference to (6.7) we have and 35 p µ := (N ) p µ := ∇ α(N )π αµ (6.10a) Remark 6.2. Note that only the part J 1 (W ) contains terms ∇W , and thus only the null decomposition of J 1 may containα,α,β,β,ρ,σ. We sometimes refer to these as "principal terms". In the first place it suffices then to look at the components of J 1 to show the presence of positive quadratic terms in ∇W in (6.7). Its null components are calculated in Lemma 14.1 in [Chr09]; in fact, the formulas in Lemma 14.1 in [Chr09] are only true in the special case that two components of the deformation tensor n = n = 0 (defined below) vanish, and we will discuss the general case in Section 6.2. The discussion of the "lower order" terms, involving W , then requires the inclusion of the parts J 2 , and J 3 . We emphasize that these parts cannot be treated separately, because cancellations appear accross the expressions for J 1 , J 2 , and Proof. Recall that we have already calculated the deformation tensor of M q in Lemma. 5.8. Moreover, (N ) π 34 = −2 e 3 + e 4 Ω + Ω 2 (Mq) π 34 (N ) π 33 = −4e 3 Ω + Ω 2 (Mq) π 33 (N ) π 44 = −4e 4 Ω + Ω 2 (Mq) π 44 (N ) π AB = Ω 2 (Mq) π AB and tr (N ) π = 2 e 3 + e 4 Ω + Ω 2 tr (Mq) π Now recall that e 3 Ω = qD log Ω = qω e 4 Ω = q −1 ω tr (Mq) π = 1 Ω q tr χ + q −1 tr χ +Lq +Lq −1 to infer that tr (N ) π = Ω 2qω + 2q −1ω + q tr χ + q −1 tr χ +Lq +Lq −1 Moreover, using the results of Lemma 5.8, Given that in this Section we work mainly with the null decomposition with respect to (e 3 , e 4 ; e A ) we will state here for convenience the form of the Bianchi equations relative to this frame.
Proposition 6.4. The Bianchi equations decomposed in the frame (e 3 , e 4 ; e A ) read as in Proposition 12.4 in [Chr09] with the following replacements:  for any 1-form ξ, and any 2-form θ, (similarly for the conjugate equations) and thus the above replacements of the structure coefficients are consistent with the replacements

Commutations
Lemma 6.5. The null components ofL N W are given by The stated formulas then follow with (6.12).
Proof of Lemma 6.5. Using the commutation relations obtained in Lemma 6.6 we calculate where in the last formula we also used (12.46) in [Chr09]. Adopting the notation of (12.48) in [Chr09] we can write and find, similarly to (12.49) in [Chr09] that: The formulas given in the statement of the Lemma then follow.

Weyl Currents
The components of the first order Weyl current have been calculated for a general commutation vectorfield X, and are presented in the special case n = n = 0, and relative to the frame (L,L; e A ) in Lemma 14.1 in [Chr09].
In the following Lemma we list the formulas for the components in the general case, n = n = 0, decomposed relative to the frame (e 3 , e 4 ; e A ). These formulas can be inferred from the expressions in Lemma 14.1 in [Chr09] using the replacements (5.64).
Proof. Consider for example Let us write out the terms which have j as a common factor. This can arise either from (µν) = (43), or (µν) = (AB), because tr i = j. Indeed Furthermore we have the following contribution: and again inserting the frame relations (5.62) (with a = q) gives the same result as inserting (1.175) in [Chr09] followed by the replacements (5.64). For definiteness, In conclusion, we have calculated that the terms in Λ which come with a factor j are given by This coincides precisely with the formula given in Lemma 14.1 in [Chr09] (p 448-449) modulo the replacements indicated in the statement of this Lemma. Similarly for all other components.
It might be instructive to compare this result to the conclusion that we would obtain by commuting with M q . Recall that (BA:I ) includes the assumption which is stronger than needed to prove the redshift property at the level of the energy of W as shown in Lemma 5.11. Indeed it is possible to show under similar assumptions even a positivity property at the level of ∇W , which is the content of Proposition 6.9 below. This illustrates that the stronger assumptions -where the smallness parameter 0 in the assumptions of Lemma 5.11, and Proposition 6.9 is replaced by the weight Ω −1 in (BA:I ) -are related to the lower order terms, specifically the remainders in W * L N W after cancellations occured due to the use of N , as opposed to M q , as a commutator.
37 Note specifically that this Lemma does not require the stricter version (BA:I.iv ,vi ) to hold.
Proof. We use Lemma 6.7, and Lemma 14.1 in [Chr09] as discussed above, to write out the null components of the current (N ) J 1 (W ). We begin with In the first place we are only interested in terms whose factors are either j, or n, n, and ignore all other terms with factors m, m, andî; thus in the following formula we set By Lemma 6.7, where we have used the Bianchi equations in the form of Prop. 6.4. We symmetrize the terms with coefficients in n, and n, and use commutation Lemma 6.5 to obtain: In this symmetrization, we have on one hand gained that the sum yields an additional positive term, while on the other hand by the Bianchi equations of Prop. 6.4 the differ-ence leaves us with terms only involving angular derivatives: Remark 6.11. With M q as commutator the prefactor that appears in bold in (6.27c) would be different, and this term would fail to cancel with similar terms appearing iñ Λ 2 +Λ 2 discussed below. The cancellations that do occur with N as a commutator are discussed in Section 6.3.3.
In the second instance we consider all terms with factors m, m, andî, and set j = 0 , n = 0 , n = 0 , (6.28a) in the following formula: where we used the Bianchi equations in the second step to eliminate Dβ q , Dβ q , Dβ q , and Dβ q in terms of derivatives tangential to the spheres. Let us employ here already that by Lemma 6.3, In particular we have the following contributions to the divergence (6.7): Remark 6.12. Note that for M q as a commutator we would have by Lemma 5.8, j + 1 4 (n + n) = 1 2Ω q tr χ + q −1 tr χ + 2qω + 2q −1ω ≥ 0 which explains the positive terms in Lemma 6.9.
6.3.2 J 2 , and J 3 Here we analyse the currents J 2 , and J 3 . They contain "lower order" terms at the level of W , but also factors at the level of ∇π. We are interested in their precise structure to show cancellations with "lower order" terms from J 1 , and control the remainder with assumptions on ∇Γ.

Integral inequality
We discuss briefly the implications of the integral inequality obtained with the help of Prop. 6.8. The remaining error term is the subject of Section 7, where it will be shown that after integration on Σ r it is controlled by the same error terms already introduced in (6.20), and thus does not alter the following discussion.

Gronwall argument
Consider an integral inequality for a positive function f : R + → R + of the form where C > 0 is a constant, h : R + → R an arbitrary function, and κ : R + → R + a positive function with the property that for some positive constants 0 < κ 0 < κ 1 < ∞. With we then obtain d dr F (r 2 , r)r κ 1

Conclusions
We conclude this Section with the main estimate for the energy flux associated to the solutionL N W of (6.6). The proof will also make use of the main estimate of Section 7.

Electromagnetic decomposition
The electromagnetic formalism is a decomposition of the Weyl curvature -and more generally of any "Weyl field" -into "electric" and "magnetic" parts relative to a spacelike hypersurface Σ. 39 The decomposition with respect to Σ r -which we use in addition to the null decomposition of W introduced in Section 3.4 -occurs naturally in the redshift estimate of Sections 5.3, 6, and is also necessary to control the remaining error that we have seen in Section 6.3. The electromagnetic decomposition has been used in [CK93], which can serve as a reference for some of the results that we shall discuss. In Section 7.1 we discuss the from of the Bianchi equations relative to the electromagnetic decomposition, in Section 7.2 its relation to the null decomposition, and in Section 7.3 an elliptic estimate for the resulting system on Σ r .

Bianchi equations
We consider a domain foliated by the spacelike level sets of the area radius r: where each leaf Σ r is diffeomorphic to a cylinder R × S 2 . We denote by n the time-like unit normal to Σ r , and by φ the associated lapse function of the foliation, c.f. Section 3.5.

Relation to null decomposition
In this Section we discuss the relation between the electromagnetic and null decompositions. In particular, when (7.3) is viewed as a Hodge system on Σ r , div E = ρ E curl E = σ H (7.6a) div H = ρ H curl H = σ E (7.6b) we establish that the "source terms" coincide -including the lower order terms -with components of the modified Lie derivativeL N W . This is a non-trivial consequence of our choice of N in (6.4), and is important for the application of the elliptic estimate of Section 7.3. Recall first n = 1 2 (e 3 + e 4 ) , and also define X = 1 2 e 3 − e 4 (7.7) then e 3 = n + X e 4 = n − X .
We record, c.f. (7.3.3e) in [CK93], We now compare the components of L n E, L n H to the null components of the modified Lie derivative of W with respect to N = Ωn.

Elliptic estimate for Maxwell system
We quote the following result from Section 4.4 in [CK93] for symmetric Hodge systems.

Sobolev inequalties
We will prove a Sobolev trace inequality on the spacelike hypersurfaces Σ r to relate the bounds on the energy fluxes obtained in Sections 5-6 to L p estimates on the spheres.

Preliminaries
Here (Σ r , g r ) are 3-dimensional Riemannian manifolds diffeomorphic to a cylinder R×S 2 , and there exists a differentiable function u : Σ r → R, namely the restriction of the null coordinate u to Σ r , such that the level sets of u| Σr are spheres S u,r diffeomorphic to S 2 , and u has no critical points. In fact, in Lemma 3.2 we proven that the metric on Σ r in (u, ϑ 1 , ϑ 2 ) coordinates, takes the canonical form g r = q −2 Ω 2 du 2 + g / AB dϑ 1 dϑ 2 . (8.1) We also found that the volume form is given by dµ g r = q −1 Ω du ∧ dµ g / = q −1 Ω det g / du ∧ dϑ 1 ∧ dϑ 2 (8.2) The "lapse function" of the foliation of Σ r by spheres S u := S u,r is thus given by a = Ω q (8.3) Recall that in Section 7.2 we introduced X as the unit tangent vector to Σ r , orthogonal to S u . Thus we have aXf = ∂ u f , for any function f (u, ϑ 1 , ϑ 2 ) on Σ r . We denote by θ the second fundamental form of the surfaces S u embedded in Σ r . Viewed as a tensor on Σ r we can write θ(Y, Z) = g /(∇ ΠY X, ΠZ) (8.4) where Π denotes the projection to the tangent space of S u , and thus θ ij = ∇ i X j + X i ∇ / j log a because ∇ X X = −∇ / log a. Therefore We also have the first variational formula which gives tr θ = 1 2a ∂ u log det(g / AB ) Consequently, d du A(u) = Su a tr θdµ g / (8.6) and more generally, for any function f , d du Su f dµ g / = Su a Xf + f tr θ dµ g / (8.7) Remark 8.1. In the spherically symmetric setting, det g / = r 4 is constant on Σ r , leading to the vanishing of the mean curvature tr θ = 0. Thus it cannot be expected that in the present setting tr θ has a sign.