The Near-Boundary Geometry of Einstein-Vacuum Asymptotically Anti-de Sitter Spacetimes

We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman--Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity--boundary correspondence theorems for vacuum asymptotically Anti-de Sitter spacetimes.


Introduction
The main objective of this article is to study the geometry, near the timelike conformal boundary, of a wide class of asymptotically Anti-de Sitter spacetimes that also satisfy the Einstein-vacuum equations. More specfically, assuming that such a spacetime, which can have arbitrary boundary topology and geometry, satisfies generic and finite regularity assumptions near the boundary: (1) We derive limits of various geometric quantities at the conformal boundary.
(2) We then apply these limits in order to construct partial Fefferman-Graham expansions from the conformal boundary for these same geometric quantities. Moreover, for each such partial expansion, we obtain enough terms so that both "free" coefficientsthose undetermined by the Einstein-vacuum equations-are captured.
In addition, this article supplements the recent results obtained in [14,15] on unique continuation of waves from the conformal boundary of asymptotically Anti-de Sitter spacetimes, as well as their upcoming applications. The estimates from [15] will be applied toward proving: (1) Symmetry extension results from the conformal boundary for Einstein-vacuum spacetimes.
(2) Correspondence results between vacuum spacetimes and data at the conformal boundary. However, for these results, one needs a detailed understanding of the boundary asymptotics implied by the Einstein-vacuum equations. This task is fulfilled by the present paper.
One fundamental feature of AdS spacetime is the presence of a timelike conformal boundary. For instance, if the radial coordinate r is replaced by ρ := r −1 , then g 0 can be written as Thus, if we disregard the factor ρ −2 in (1.3), then we can formally attach a boundary (1.4) (I , g 0 ) := (R × S n−1 , −dt 2 +γ) at ρ = 0 (or equivalently, at r = ∞). In particular, (I , g 0 ) is an n-dimensional Lorentzian manifold, and g represents the limit of ρ 2 g 0 as ρ ց 0 (or r ր ∞).
In general, we loosely refer to spacetimes (not necessarily vacuum) that have a similar conformal boundary (I , g) as asympotically Anti-de Sitter, which we will also abbreviate as aAdS. (A much more precise definition of aAdS spacetimes for this paper will be given in Section 2.) In the past two decades, aAdS spacetimes have been a topic of significant interest in theoretical physics due to the AdS/CFT conjecture [18], which roughly posits a holographic correspondence between gravitational theory in the ((n + 1)-dimensional) spacetime and a conformal field theory on its (n-dimensional) boundary. Despite its substantial influence in physics, there are scant rigorously formulated mathematical results that support or refute this conjecture; see, e.g., [2,3,5,7,22].
In the context of classical relativity, one may attempt to approach this conjecture by studying the relationship between vacuum aAdS spacetimes-representing the interior gravitational theory-and appropriate data posed at the conformal boundary I . In this direction, an essential task would be to gain a precise understanding of the geometries of such spacetimes and their asymptotics near I .

Fefferman-Graham Expansions.
Much of the understanding around the above question has arisen from the works of Fefferman and Graham [11,12], which were concerned with constructing Einstein-vacuum spacetimes given conformal data on a null hypersurface. From this data and the Einstein-vacuum equations, one then derives a formal series expansion; when the data is sufficiently "analytic", this series then converges to a genuine Einstein-vacuum spacetime.
These ideas have been widely adapted by the physics community to the present aAdS setting. Here, one first simplifies the expression for the metric by adopting special coordinates, so that the aAdS spacetime metric g has a form that trivializes the (inverse) radial component: (1.5) g := ρ −2 (dρ 2 + g ab dx a dx b ).
(Here, the x a 's denote non-radial coordinates that are tangent to the boundary manifold I .) Using the Einstein-vacuum equations, one then derives a formal series expansion for g from I : (1.6) g = n−1 2 k=0 g (2k) ρ 2k + g (n) ρ n + . . . n odd, n−2 2 k=0 g (2k) ρ 2k + g (⋆) ρ n log ρ + g (n) ρ n + . . . n even, When n is odd, the above continues as a power series in ρ. However, when n is even, the series becomes polyhomogeneous beyond n-th order, containing terms of the form h · ρ k (log ρ) l .
In addition, one can observe the following features of the expansion (1.6): • The zero-order coefficient g (0) and the n-th order coefficient g (n) represent the "free" data that are not determined by the Einstein-vacuum equations. 1 • The coefficients g (1) , g (2) , . . . , g (⋆) that arise before g (n) are formally determined only by g (0) . • All the coefficients arising after g (n) are formally determined by both g (0) and g (n) . 1 More accurately, only the trace-free and divergence-free parts of g (n) are "free" in this sense.
We remark that g (0) is precisely the conformal boundary metric g from the preceding discussions. Furthermore, g (n) is closely related to the boundary stress-energy tensor that is often studied in the physics literature [9,20] and is expected to describe the boundary conformal field theory.
While Fefferman-Graham expansions have proven quite fruitful for inferring the near-boundary heuristics of vacuum aAdS geometries, they do not provide a robust-or even generic-means of constructing vacuum aAdS spacetimes. In particular, given "free" boundary data (g (0) , g (n) ), one can solve the Einstein-vacuum equations for series solutions of the form (1.6) only within the excessively restrictive class of sufficiently "analytic" metrics. 2 Therefore, if our aim is to obtain a general understanding of aAdS spacetimes near the conformal boundary, then we would need to dramatically broaden the class of metrics under consideration.
Unfortunately, the expectation is that the Einstein-vacuum equations are ill-posed given non-"analytic" data (g (0) , g (n) ) at I . This arises from the observation that one generally cannot solve hyperbolic PDEs from non-analytic Cauchy data imposed on a timelike hypersurface. Continuing this analogy with hyperbolic equations, we then expect that the appropriate venue for solving the Einstein-vacuum equations in the aAdS context is as an initial-boundary value problem: • One imposes the usual initial data on a spacelike hypersurface.
• One also imposes one piece of boundary data (say, either g (0) or g (n) ) on I . These principles were affirmed in [10,13], where the authors solved, under various assumptions, the Einstein-vacuum equations in aAdS settings via this initial-boundary value problem.
1.1.2. Partial Expansions. While we cannot rely on Fefferman-Graham expansions as a means for solving the Einstein-vacuum equations in the smooth (or finitely regular) class, the ideas could still provide powerful insights for characterizing the boundary asymptotics of vacuum aAdS spacetimes. Even in these more general situations, one might expect that the Einstein-vacuum equations could still be exploited as before to obtain partial expansions, up to a finite order, for the metric from the conformal boundary. In terms of elementary analysis, this is comparable to the distinction between Taylor series (which are applicable only to analytic functions) and the finite expansions from Taylor's theorem (which apply to finitely differentiable functions).
One objective of the present article is to give a rigorous affirmation of the above expectations. To be more specific, we will address the following question: spacetimes (such as AdS spacetime itself and the Kerr-AdS families). Moreover, the aAdS spacetimes obtained by solving the initial-boundary value problem for the Einstein-vacuum equations (for instance, in [10,13]) also satisfy these regularity criteria.
Given these very weak assumptions, the main results of this paper achieve the following: • We derive limits at the conformal boundary for the metric g and its ρ-derivatives, up to order n. We also show that these limits depend only on the boundary metric g. • We prove the existence of the boundary limit at the n-th order (which corresponds to g (n) ), which is not determined by the Einstein-vacuum equations. • We also obtain analogous limits for other geometric quantities, such as the curvature. Using these boundary limits, we then construct the expected partial Fefferman-Graham expansions, up to and including the n-th order, for the above geometric quantities. In particular, these capture both branches of "free" data at the conformal boundary.
The results here can also be viewed as complementary to the well-posedness results for the initial-boundary value problem. In [10], the authors assume initial data that solve the constraint equations and have the boundary asymptotics dictated by a partial Fefferman-Graham expansion. They then solve the Einstein-vacuum equations, in turn showing that these boundary asymptotics are propagated to all times at which the solution exists. In contrast, here we assume a priori (rather than solve for) a vacuum aAdS spacetime, but we impose far weaker regularity assumptions around the conformal boundary. We then proceed to derive the boundary asymptotics (up to and including the n-order) that are suggested by the Fefferman-Graham expansions.
A Riemannian analogue of our results were established by Chruściel, Delay, Lee, and Skinner in [8]. Roughly, they showed that a conformally compact Einstein manifold has boundary regularity consistent with a partial Fefferman-Graham expansion from the conformal boundary.
We also mention here the recent results of [19], which is in some ways similar in nature to [10], but lies closer to the original context of Fefferman-Graham expansions. Here, the authors solve, from characteristic initial data, for asymptotically self-similar solutions of the Einstein-vacuum equations (with zero cosmological constant). In particular, the authors impose, on the initial outgoing null hypersurface, geometric data that is consistent with a partial Fefferman-Graham expansion, again up to the n-th order. They then proceed to rigorously prove that this form of the partial expansion is propagated along the incoming null direction.
1.1.3. Connections to Unique Continuation. Though we cannot generally solve the Einstein-vacuum equations given boundary data (g (0) , g (n) ), we could instead formulate a slightly different problemthat of unique continuation for the Einstein-vacuum equations from I : Question 1.2. Suppose we are given a vacuum aAdS spacetime (M , g). Does its "free" boundary values (g (0) , g (n) ) uniquely determine g? In other words, is there a one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and some space of boundary data?
A related, though generally simpler, problem is whether a suitably defined symmetry of the conformal boundary must necessarily be inherited by the spacetime: is a vacuum aAdS spacetime, and suppose its boundary (I , g) has a symmetry (in an appropriate sense). Must this symmetry necessarily extend into (M , g)?
A crucial step toward resolving both questions has been given in [14,15], which establish a unique continuation property from I for wave equations on a fixed aAdS spacetime. However, in order to apply the results of [14,15] toward Questions 1.2 and 1.3, we must first have a detailed understanding of the boundary asymptotics of vacuum aAdS spacetimes. In particular: • For Question 1.2, we must establish the coefficients (g (0) , g (n) ) as the appropriate boundary data, and we must understand how they relate to the spacetime geometry. We must also connect them with applications of the unique continuation results of [14,15]. • Similarly, for Question 1.3, the appropriate notion of symmetry of (I , g) is to posit that both g (0) and g (n) are invariant with respect to the corresponding transformation. Once again, we must then connect this condition with the relevant unique continuation results. The results of the present paper provide the foundations for the above tasks. Therefore, a second objective-and a key application-of this paper is as an important step toward answering Questions 1.2 and 1.3. These tasks will be addressed in future papers.
1.2. The Main Results. To conclude our introductory discussions, we give informal statements of the main results of this paper, as well as some of the main ideas of their proofs.
1.2.1. The Vertical Formulation. We begin with a few key points on the formalisms that we will use for our results and their proofs. The main objects in our analysis are vertical tensor fields, which can be viewed as one-parameter families, indexed by the radial coordinate ρ, of tensor fields on I ; see Definition 2.4 and the subsequent discussion for further details.
Throughout, we assume the spacetime metric g is expressed in a Fefferman-Graham gauge-that is, of the form (1.5). In particular, the object g in (1.5) is a vertical tensor field-which we call the vertical metric-that is also a Lorentzian metric for each fixed ρ-value. One advantage of adopting a Fefferman-Graham gauge is that the entire geometric content of g lies in the vertical metric g.
From g, we proceed to define other geometric objects associated with it, such as the Levi-Civita connection and the Riemann, Ricci, and scalar curvatures; for details, see Definition 2.13.
Next, we can make sense of boundary limits (as ρ ց 0) of vertical tensor fields as tensor fields on I ; a formal description of this is given in Definition 2.12. Therefore, as long as the appropriate limits exist, we can express vertical tensor fields in terms of (finite) expansions from the boundary in powers of ρ, with each coefficient given by a tensor field on I .
With the above in mind, we can now state a rough version of the main result: Theorem 1.4. Let (M , g) be an (n + 1)-dimensional vacuum aAdS spacetime, and suppose that g is expressed in terms of a Fefferman-Graham gauge (1.5). In addition, assume: • The vertical metric g has a boundary limit g that is a Lorentzian metric on I .
• A sufficient number of non-radial derivatives g remain locally bounded on M .
• ∂ ρ g is locally bounded on M . Then, the following statements hold: • g and its (radial and non-radial) derivatives up to and including order n satisfy boundary limits at I . Similar boundary limits also hold for the curvature R associated to g. • g can be expressed as the following partial Fefferman-Graham expansion: k=0 g (2k) ρ 2k + g (⋆) ρ n log ρ + g (n) ρ n + r g ρ n n even, where the coefficients g (0) , g (2) , . . . , g (⋆) , g (n) are tensor fields on I , and where the remainder r g is a vertical tensor field that has vanishing boundary limit. Similar expansions, also up to n-th order, hold for R and other associated geometric objects.
• The following coefficients depend only on g (0) = g: the coefficients g (k) for all 0 ≤ k < n, the anomalous coefficient g (⋆) , and both the (g-)trace and the (g-divergence) of g (n) .
In short, we assume in Theorem 1.4 that sufficiently many non-radial derivatives of g are wellbehaved up to I . We then use the Einstein-vacuum equations to deduce the expected asymptotics for ρ-derivatives of g, as dictated by the Fefferman-Graham expansion, up to order n.
As previously mentioned, the fact that the partial expansions (1.7) hold for "analytic" (in a polyhomogeneous sense) solutions of the Einstein-vacuum equations is already well-known. The main novel contribution of present article is to show that even in finite regularity, the form (1.7) is, in fact, still forced by the Einstein-vacuum equations. One particuilar point of emphasis is that the presence of the undetermined g (n) -term also needs not be assumed a priori, as the existence of the boundary limit corresponding to g (n) is derived as part of the proof of Theorem 1.4. Remark 1.5. While we also recover the expected formulas detailing how the coefficients g (k) (with 0 < k < n) and g (⋆) depend on g (0) , this part of the argument more closely resembles how the Einstein-vacuum equations are used in the "analytic" setting.
In addition, Theorem 1.4 can be connected to the rigidity results of [1] in aAdS settings. In particular, the results of [1] assume a vacuum aAdS spacetime that also satisfies the partial expansion (1.7). As a result, in this context, Theorem 1.4 demonstrates that the assumption of (1.7) is unnecessary and can be replaced by much weaker conditions. Remark 1.6. Here, we only perform our analysis up to order n, as this is the minimal regularity required in order to capture both parts of the "free" boundary data for g. If we assume additional regularity for g, then it is also possible to deduce additional terms beyond n-th order in (1.7).
We also stress that Theorem 1.4 only represents the most notable portion of our main results. The precise and full statements of the boundary limits for g and its associated geometric objects are given in Theorem 3.3 and Corollary 3.5. Formal statements of the partial Fefferman-Graham expansions are given in Theorem 3.6. The reader is referred to these statements for additional details, such as the precise sense of convergence for the various vertical quantities.
In addition, we obtain analogous partial expansions for components of the spacetime Weyl curvature in Corollary 3.7; these will be needed in upcoming results regarding Questions 1.2 and 1.3. Finally, for completeness, we recall the standard partial expansion for Schwarzschild-AdS metrics (which is well-known in the physics literature; see, e.g., [4,21]) in Corollaries 3.15 and 3.16.

1.2.2.
Ideas of the Proof. The proof of Theorem 1.4, given in Section 3.2, can be divided into three steps. In the first step, we derive limits for g and ∂ ρ g, as well as their non-radial derivatives. The main idea is to recast the Einstein-vacuum equations as a system of equations satisfied by the vertical metric g and its curvature R. The appropriate limits can then be obtained by combining a careful analysis of these equations with the assumptions we have imposed on g.
The second step is to obtain boundary limits for ∂ k ρ g and ∂ k ρ R for all 0 ≤ k < n. This is systematically accomplished by differentiating the Einstein-vacuum equations and then analyzing the resulting system. (In particular, this is equivalent to the observations that first led to the full Fefferman-Graham expansions.) However, this process does not continue up to n-order, as the Einstein-vacuum equations do not determine limits ∂ n ρ g and ∂ n ρ R. At its most basic level, this analysis revolves around the study of ODEs of the form where y is the unknown, x is the independent variable, and f is a fixed forcing term. Note that (1.8) becomes singular as x ց 0. The key observation here is that when c > 0, the solution y becomes determined at x = 0, via the relation −c · y(0) = f (0); see Proposition 2.40.
Since the Einstein-vacuum equations and its ρ-derivatives, formulated in terms of vertical tensor fields, have the basic form (1.8), we can use the above observation in order to derive boundary limits for g and its derivatives. Moreover, in these equations (see (2.31)), the corresponding constant c remains positive until one reaches the equation for ∂ n ρ g, for which c = 0. This accounts for the fact that we can obtain boundary limits up to, but not including, the n-th order.
The final step of the proof is to examine the n-th order behavior. The first observation is that the Einstein-vacuum equation do imply boundary limits for ρ∂ n+1 ρ g (and, by extension, ρ∂ n+1 ρ R). These limits yield the anomalous logarithmic terms in (1.6). In addition, we show that these limits vanish when n is odd, so that the logarithmic terms are not present in this case. (Again, this is analogous to the derivation of the log-term in the original Fefferman-Graham expansion.) Now, the above observations yield the coefficients g (k) and g (⋆) in (1.6), for all 0 ≤ k < n. However, this does not yet imply the "free" g (n) -term, which is not determined by the Einsteinvacuum equations. To obtain this, we require a stronger type of convergence for all the preceding limits that also establishes a minimal convergence rate; see (2.9) for details.
To be more specific, this analysis at the n-th order is now based on the ODE (1.9) xy ′ = f .
From (1.9), we see that as x ց 0, the leading order behavior of y is logarithmic: This accounts for the anomalous coefficient g (⋆) . The additional observation that is responsible for g (n) is the following: if we also assume f converges to f (0) in a slightly stronger sense, then the error term in (1.10) will have a finite limit as x ց 0. (However, this limit is no longer determined by f , which leads to the fact that g (n) is "free".) See Proposition 2.41 for details. Furthermore, the above considerations force us to modify all the preceding steps of this proof. We have to derive this stronger convergence for g and ∂ ρ g in the first step of the above. Moreover, we must then propagate this stronger convergence to higher derivatives in the second step, and then also to the logarithmic term. These considerations further complicate the technical analysis.
1.2.3. Outline. In Section 2.1, we give a precise description of the aAdS spacetimes that we will study. The goal of Section 2.2 is to reformulate the Einstein-vacuum equations in terms of vertical tensor fields. Some basic estimates involving these spacetimes are given in Section 2.3.
The main results of the paper are stated in Section 3.1, while the proof of Theorem 3.3 (where the boundary limits are obtained) are given in Section 3.2. In addition, in Section 3.3, we recall the partial Fefferman-Graham expansion for the usual Schwarzschild-AdS metrics.
Finally, Appendix A consists the proofs of a number of propositions found within the main text. These contain additional details and computations that are of benefit to especially interested readers, but would also be distractions from the main presentation.
1.3. Acknowledgments. The author thanks Gustav Holzegel and Alex McGill for numerous discussions, as well as Claude Warnick for generously providing some preliminary notes and computations. The work in this paper is supported by EPSRC grant EP/R011982/1.

Asymptotically AdS Spacetimes
The aim of this section is to precisely describe the spacetimes that we will study in this paper: near-boundary regions of asymptotically AdS spacetimes satisfying the Einstein-vacuum equations with negative cosmological constant. We will accomplish this through three steps: (1) In Section 2.1.1, we define the manifolds on which our spacetimes will lie.
(2) In Section 2.1.2, we define the geometry of the spacetime.
(3) In Section 2.2, we impose the Einstein-vacuum equations on our spacetime. Moreover, in Sections 2.2 and 2.3, we discuss some basic properties of these spacetimes.
2.1. aAdS Spacetimes. In this subsection, we give a precise geometric definition of the asymptotically AdS spacetimes that we will study, as well as the Fefferman-Graham gauge condition. In addition, we define the main tensorial objects that we will use for our analysis.
2.1.1. The Spacetime Manifold. The first step is to construct the spacetime manifold M : Definition 2.1. We define an aAdS region to be a manifold (with boundary) of the form where I is a smooth n-dimensional manifold, and where n ∈ N. For brevity, we will, in the context of the above, refer only to M as the aAdS region; the remaining quantities n, I , ρ 0 are always assumed to be implicitly associated with M .  In our analysis, we will encounter three types of tensorial objects on an aAdS region M : (1) Tensor fields on M .
(2) Vertical tensor fields on M , representing ρ-parametrized families of tensor fields on I .
(3) Tensor fields on I , representing limits of vertical tensor fields as ρ ց 0. While tensor fields on M and I are self-explanatory, the notion of vertical tensor fields was not explicitly used in [14,15]. These objects can be described more precisely as follows: Definition 2.4. Let V k l M denote the vertical bundle of rank (k, l) over M , which we define to be the manifold of all tensors of rank (k, l) on each level set of ρ in M : 3 In addition, we refer to smooth sections of V k l M as vertical tensor fields of rank (k, l). Notice that a vertical tensor field of rank (k, l) on an aAdS region M can be viewed as a oneparameter family, indexed by ρ ∈ (0, ρ 0 ], of rank (k, l) tensor fields on I . Throughout, we will make use of this interpretation of vertical tensor fields whenever convenient. Definition 2.5. We adopt the following conventions for tensor fields on an aAdS region M : • We use the normal italicized font (for example, g) to denote tensor fields on M .
• We use serif font (for example, g) to denote vertical tensor fields on M .
• We use Fraktur font (for example, g) to denote tensor fields on I . Moreover, unless otherwise stated, we always assume that a given tensor field is smooth. Definition 2.6. We also note the following natural identifications of tensor fields: • Given a vertical tensor field A and σ ∈ (0, ρ 0 ], we let A| σ denote the tensor field on I that is obtained by restricting A to the level set {ρ = σ} (and identifying {ρ = σ} with I ). • Given a tensor field A on I , we will also use A to denote the vertical tensor field on M obtained by extending A as a ρ-independent family of tensor fields on I : • Any vertical tensor field A can be uniquely extended to a tensor field on M of the same rank, via the condition that the contraction of any component of A with either ∂ ρ or dρ (whichever is appropriate) is defined to vanish identically.
More succinctly, the last part of Definition 2.5 notes that tensor fields on I can be equivalently viewed as vertical tensor fields that do not depend on ρ. Once again, we will use both interpretations of tensor fields on I interchangeably, depending on context. Remark 2.7. As a special case of Definition 2.5, any scalar function on I can, at the same time, be viewed as a ρ-independent scalar function on M .
Definition 2.8. Let M be an aAdS region, and let A be a vertical tensor field. We define the Lie derivative with respect to ρ of A, denoted L ρ A, to be the vertical tensor field given by Next, we set the coordinate conventions that we will use throughout this article: Let M be an aAdS region, and let (U, ϕ) be a coordinate system on I .
• As usual, repeated indices will indicate summations over the appropriate components.
Definition 2.10. Let M be an aAdS region. A coordinate system (U, ϕ) on I is compact iff: •Ū is a compact subset of I .
• ϕ extends smoothly to (an open neighborhood of )Ū .
We use compact coordinates to define local notions of convergence for vertical tensor fields: Definition 2.11. Let M be an aAdS region, and let M ≥ 0. Moreover, let (U, ϕ) be a compact coordinate system on I , and let A be a vertical tensor field of rank (k, l).
• We define the following local pointwise norm for A (with respect to ϕ-coordinates): • We also define the following local uniform norm of A: Definition 2.12. Let M , M , and A be as in Definition 2.11. Moreover, let A be a tensor field on I having the same rank as A. We then define the following: • A is locally bounded in C M iff for any compact coordinate system (U, ϕ) on I , • A locally converges in C M to A, which we abbreviate as A → M A, iff given any compact coordinate system (U, ϕ) on I , we have that Observe that (2.8) is a locally uniform coordinate limit in C M (involving only derivatives along I ) as ρ ց 0. Moreover, the ⇒-convergence defined in (2.9) describes a slightly stronger notion of limits that can be roughly interpreted as converging at a minimal rate.

2.1.2.
The Spacetime and Boundary Geometry. Having described the background manifold, as well as the types of objects we will study on them, the next step is to describe the geometry of our spacetime. For our purposes, these will be quantified using vertical tensor fields: Furthermore, given a vertical metric h on M : • The metric dual of h, denoted h −1 , is the rank (2, 0) vertical tensor field on M such that h −1 | σ is the metric dual (h| σ ) −1 of h| σ for each 0 < σ ≤ ρ 0 . • Similarly, the Levi-Civita connection associated with h is the operator on vertical tensor fields such that its restriction to any level set {ρ = σ}, 0 < σ ≤ ρ 0 , coincides with the usual Levi-Civita connection of the Lorentzian metric h| σ . • The Riemann curvature associated with h is the rank (1, 3) vertical tensor field whose restriction to any {ρ = σ}, 0 < σ ≤ ρ 0 , is the Riemann curvature associated with h| σ . Analogous definitions also hold for the Ricci and scalar curvatures of h.
The spacetimes that we will consider in this article can now be defined as follows: Definition 2.14. (M , g) is called an FG-aAdS segment iff the following hold: • M is an aAdS region, and g is a Lorentzian metric on M .
• There exists a vertical metric g on M such that • There is a Lorentzian metric g on I such that Moreover, in the above context, while we refer only to (M , g) as the FG-aAdS segment, we always implicitly assume the associated metrics g and g described in (2.10) and (2.11). Definition 2.14 captures the minimal assumptions required for a (segment of a) spacetime to be considered "asymptotically AdS". We refer to the form (2.10) for the spacetime metric g as the Fefferman-Graham (abbreviated FG) gauge condition, which is characterized by the ρ-component being (up to the conformal factor ρ −2 ) trivial and fully decoupled from the remaining components.
Definition 2.15. Given an FG-aAdS segment (M , g), we refer to the n-dimensional (Lorentz) manifold (I , g) as the conformal boundary associated with (M , g, ρ).
Remark 2.16. Imposing the FG gauge condition in Definition 2.14 does not result in any loss of generality. For more general spacetime metrics g having the same asymptotics as ρ ց 0, one can apply an appropriate change of coordinates to transform g into the form (2.10).
Remark 2.17. The conformal boundary (I , g) in Definition 2.15 is gauge-dependent, in that it depends on the choice of ρ. More specifically, through coordinate transformations, one can find other boundary-defining functionsρ with respect to which (M , g) satisfies all the conditions in Definition 2.14. Such transformations result in g being multiplied by conformal factors; see [9,16].
Definition 2.18. Given an FG-aAdS segment (M , g): • Let g −1 and ∇ denote the metric dual and the Levi-Civita connection, respectively, associated with g. Moreover, let R, Rc, Rs and W denote the Riemann curvature, Ricci curvature, scalar curvature, and Weyl curvature, respectively, associated to g. • Let g −1 and D denote the metric dual and the Levi-Civita connection, respectively, associated with the vertical metric g. Moreover, let R, Rc, and Rs denote the Riemann curvature, Ricci curvature, and scalar curvature, respectively, associated to g. • Let g −1 and D denote the metric dual and the Levi-Civita connection, respectively, associated with g. In addition, let R, Rc, and Rs denote the Riemann curvature, Ricci curvature, and scalar curvature, respectively, that are associated to g. Also, as is standard, we omit the superscript " −1 " when expressing a metric dual in index notion.
Remark 2.19. To maintain consistency, we will, by default, always treat the Riemann curvaturespacetime, vertical, or boundary-as a rank (1, 3) tensor field.
Below, we collect some basic observations on the geometry of FG-aAdS segments: Proposition 2.20. Let (M , g) be an FG-aAdS segment. Moreover, fix a coordinate system (U, ϕ) on I , and assume all indices below are with respect to ϕ and ϕ ρ -coordinates. Then: • The components of g and g −1 satisfy • The vector field N := ρ∂ ρ satisfies • The following identity holds: In particular, the left-hand side of (2.14) is precisely the second fundamental form of the level sets of ρ, as submanifolds of (M , g), with respect to the unit normal N .
Finally, we will also require the following operators later in our analysis: Definition 2.21. Let (M , g) be an FG-aAdS segment.
• Given a symmetric vertical tensor field h of rank (0, 2), we define its g-trace, denoted tr g h, and its g-divergence, denoted D · h, to be the vertical tensor fields given by • Given a symmetric tensor field h on I of rank (0, 2), we define its g-trace, denoted tr g h, and its g-divergence, denoted D · h, to be the tensor fields on I given by

2.2.
Einstein-Vacuum Spacetimes. The final assumption we will pose on our asymptotically AdS setting is that it satisfies the Einstein-vacuum equations, with negative cosmological constant: Remark 2.23. Definition 2.22 normalizes the cosmological constant to a specific negative value. This is done for convenience, in order to simplify constants from various computations.
Proposition 2.24. Suppose (M , g) is a vacuum FG-aAdS segment, and let (U, ϕ) be a coordinate system on I . Then, the following hold with respect to ϕ ρ -coordinates: Proof. See Appendix A.2.

The Vertical Formulation.
We now reformulate the Einstein-vacuum equations (2.17) in terms of vertical tensor fields, in particular the vertical metric g, its L ρ -derivatives, and its associated curvature R. We begin by converting some standard differential geometric identities, such as the Gauss-Codazzi equations, into equations for vertical tensor fields.
Proposition 2.25. Let (M , g) be a vacuum FG-aAdS segment, and let (U, ϕ) be a coordinate system on I . Then, the following identities hold with respect to ϕ and ϕ ρ -coordinates: Proof. We begin by defining the vertical tensor fields Notice that (2.10) and (2.14) imply that h and k are the (vertical) metrics and second fundamental forms on the level sets of ρ that are induced by the spacetime metric g.
Observe that with respect to ϕ-coordinates, we have that is, g and h have the same Christoffel symbols Γ a bc . This implies D is the Levi-Civita connection for both g and h. Thus, the Codazzi equations on the level sets of ρ are given by Observe that by (2.12) and (2.18), the left-hand side of (2.22) is equal to ρW ρcab . Thus, replacing k in (2.22) using (2.20) and noting that Dg = 0 yields the first identity in (2.19). Next, let S denote the Riemann curvature associated with h. Using the (ϕ-)coordinate representation of the Riemann curvature along with (2.21), we have that Lowering the first index on both sides of the above (with respect to h on the left-hand side, and with respect to g on the right-hand side) and recalling (2.20), we obtain Now, the Gauss equations on the level sets of ρ are given by Replacing k and S in the above using (2.20) and (2.23), we see that Replacing R with W using (2.18) and recalling (2.10) results in the second part of (2.19).
For the remaining equality in (2.19), we set N := ρ∂ ρ as in Proposition 2.20, and we note that Recalling (2.14), the second part of (2.20), and the above, we then obtain Since ∇ N N vanishes by (2.13), then (2.14), (2.20), and the above imply Moreover, turning again to the second part of (2.20), we see that which, when combined with (2.24), yields The last part of (2.19) now follows from (2.10), (2.12), and (2.18), which together imply that From the identities in Proposition 2.25, we obtain our main relations for L ρ g: Proposition 2.26. Let (M , g) be a vacuum FG-aAdS segment, and let (U, ϕ) be a coordinate system on I . Then, the following identities hold with respect to ϕ-coordinates: Proof. For the first part, we take a g-contraction of the first identity of (2.19), in the components given by b and c. Since W is (g-)trace free, the left-hand side evaluates to and the first equality of (2.25) follows immediately. Next, we take a g-contraction of the second identity in (2.19), with respect to the components a and c. Using that W is (g-)trace free, along with (2.12), the left-hand side becomes Rearranging the above (that is, replacing the free indices b, d by a, b, respectively) and then replacing the left-hand side using the last identity in (2.19), we obtain Multiplying both sides by 2ρ results in the second part of (2.25).
We will also require corresponding differential equations for R, as well as general commutation formulas for L ρ -derivatives. These are somewhat standard geometric identities (for instance, in the Ricci flow literature; see [6]) adapted to our current vertical setting.
Proposition 2.27. Let (M , g) be an FG-aAdS segment, let (U, ϕ) be a coordinate system on I , and let A be a vertical tensor field of rank (k, l). Then, in terms of ϕ-coordinates, we have and where c 1dj c l denotes the sequence of indices c 1 . . . c l but with c j replaced by d.
Proposition 2.28. Let (M , g) be an FG-aAdS segment, and let (U, ϕ) be a coordinate system on I . Then, the following identity holds, with respect to ϕ-coordinates: Proof. See Appendix A.4.

2.2.2.
Higher-Order Relations. For our analysis, we also need higher-order equations satisfied by L ρ -derivatives of g and R. However, the exact forms of these identities tend to be quite lengthy.
Fortunately, the precise forms of various terms that lie below the leading order are not important. Thus, in order to describe and treat them more conveniently, we adopt the following schematic notations that highlight only the essential structures of "error" terms.
Definition 2.29. Let (M , g) be an FG-aAdS segment, and fix k 1 , l 1 , k 2 , l 2 ∈ N. Let Q be a linear operator mapping vertical tensor fields of rank (k 1 , l 1 ) to vertical tensor fields of rank (k 2 , l 2 ).
• We say Q is schematically trivial iff Q is a composition of the following operations: component permutations, (non-metric) contractions, and multiplications by a constant. • Q is said to be schematically g-trivial iff Q is a composition of the following operations: component permutations, (non-metric) contractions, multiplications by a constant, tensor products with g, and tensor products with g −1 .
Definition 2.30. Let (M , g) be an FG-aAdS segment. For any N ≥ 1 and any vertical tensor fields A 1 , . . . , A N on M , we define the following schematic notations: • We write S (A 1 , . . . , A N ) to represent any vertical tensor field of the form where T ≥ 0, and where each Q j , 1 ≤ j ≤ T , is a schematically trivial operator. • We write S (g; A 1 , . . . , A N ) to represent any vertical tensor field of the form where T ≥ 0, and where each Q j , 1 ≤ j ≤ T , is a schematically g-trivial operator.
For example, we can reformulate the equations (2.25) and (2.27) using schematic notations: Proposition 2.31. Let (M , g) be a vacuum FG-aAdS segment. Then, the following identities hold: Proof. See Appendix A.5.
To extract higher-order terms of the FG expansion, we take derivatives of the equations (2.25) and (2.27) and commute. The results of these computations are summarized below: Proposition 2.32. Let (M , g) be a vacuum FG-aAdS segment. Then, for any k ≥ 2, Proof. First, note that for any vertical tensor field A, we have Applying L k−1 ρ to the first part of (2.30) and using the above, we obtain We now apply the Leibniz rule repeatedly to the third and last terms on the right-hand side of (2.32). For the third term, if all of L k−1 ρ hits the L ρ g, we obtain the term tr g L k ρ g · g. Otherwise, if at least one L ρ -derivative hits elsewhere, then we obtain terms with schematic form Similarly, applying the Leibniz rule to the last term of (2.32) yields terms of the forms j1+···+j l =k 1≤jp<k S (g; L j1 ρ g, . . . , L j l ρ g), j1+···+j l =k+1 1≤jp≤k ρ · S (g; L j1 ρ g, . . . , L j l ρ g), depending on whether one of the L ρ 's hits the factor of ρ. This results in the first part of (2.31); the second part of (2.31) is obtained by taking a trace of the first part.
2.3. Some General Estimates. In this section, we derive some general bounds and limit properties on FG-aAdS segments that will be useful in later sections.
2.3.1. Geometric Bounds. We begin by establishing some bounds for various geometric quantities. For this, we first set some notations that will make future discussions more convenient:

Then:
• The following estimates hold: • If M ≥ 2, then the following also hold: Proof. Throughout the proof, we will always index and differentiate with respect to ϕ-coordinates. First, recall from basic linear algebra that we can write for any pair of indices b and c, where P 0 is some polynomial function of its arguments, and where V ϕ (g) and V ϕ (g) denote the determinants of the matrices formed from the ϕ-components of g and g, respectively. Note that from the assumptions (2.35), we have We now take m derivatives of (2.38), for every 0 ≤ m ≤ M , which yields where P m is a polynomial function of its arguments. Thus, applying (2.35) to (2.40) yields which imply the first two estimates of (2.36). Furthermore, from (2.39) and (2.40), we see that where 0 ≤ m ≤ M , and where we also used (2.35) in the last step. Summing the above inequality over all indices and all 0 ≤ m ≤ M results in the final estimate of (2.36). Next, from standard formulas for the coordinate components of R and R in terms of the corresponding Christoffel symbols for g and g, respectively, we see-for any indices b, c, d, e-that , whereP 0 a polynomial. If M ≥ 2, then differentiating the above yields, for each 2 ≤ m ≤ M , for any 2 ≤ m ≤ M . This implies the third estimate of (2.37). The fourth, fifth, and sixth parts of (2.37) follow trivially from the first three, since the Ricci curvature is a (non-metric) contraction of the Riemann curvature. The seventh and eight bounds of (2.37) are consequences of the definition of scalar curvature, (2.36), and previous bounds in (2.37): Finally, since Rs − Rs = (g ab − g ab )Rc ab + g ab (Rc ab − Rc ab ), then (2.36), the fourth part of (2.37), and the sixth part of (2.37) imply which is the last inequality of (2.37).
Proposition 2.37. Let (M , g) be an FG-aAdS segment, let (U, ϕ) be a compact coordinate system on I , and let M > 0. In addition, assume that the bounds (2.35) hold. Then, for any vertical tensor field A on M and any tensor field A on I of the same rank, we have the following: Proof. Let (k, l) be the rank of A and A, and fix indices a 1 , . . . , a M , b 1 . . . , b k , c 1 , . . . , c l . Then, indexing and differentiating with respect to ϕ-coordinates, we see from the coordinate formula for covariant derivatives that for each 0 < m ≤ M , there is a polynomial R m such that Consider now the difference of the two identities in (2.43). First, recalling (2.44), we see that Summing the above over all indices and over 0 < m ≤ M yields the first bound of (2.42). Similarly, shuffling the terms in (2.43) around, we can once again use (2.44) to estimate Summing the above over all indices and then adding |A − A| m−1,ϕ to both sides, we obtain Combining the above with an induction argument over m yields the second bound in (2.42).

Boundary Limits.
Next, we prove a number of general boundary limit properties for vertical tensor fields. We begin by stating some trivial but useful properties: Proof. See Appendix A.9.
Next, we derive boundary limits for vertical tensor fields satisfying differential equations.
Proposition 2.40. Let (M , g) be an FG-aAdS segment, and let M ≥ 0. Furthermore, let A, G be vertical tensor fields on M , and suppose they satisfy the following relation: • For any ρ * ∈ (0, ρ 0 ] and any compact coordinate system (U, ϕ) on I , we have that where G is a tensor field on I , then where G is a tensor field on I , then Proof. Let (k, l) be the ranks of both A and G. Moreover, we fix an arbitrary compact coordinate system (U, ϕ) on I , and we always index with respect to ϕ-coordinates. First, observe that (2.45) can be written as Fixing 0 < ρ * ≤ ρ 0 and integrating the above from ρ = ρ * yields We now differentiate the above along I , and we apply the mean value and dominated convergence theorems to move the derivatives under the integral. This yields, for all 0 ≤ m ≤ M , Taking absolute values of both sides of (2.49) and summing appropriately, we obtain (2.46). Suppose now that G → M G. Since G converges and (U, ϕ) is compact, it follows that Applying (2.49), with ρ * = ρ 0 , results in the bound Summing the above over m and over all the indices, we obtain that Let ρ 1 > 0 be small enough so that ρ 2 1 ≤ ρ 1 ≤ ρ 0 . Applying (2.49) again, at ρ = ρ 2 1 < 0 and with ρ * := ρ 1 < ρ 0 , we then obtain, for any 0 ≤ m ≤ M , that Letting I denote the last term on the right-hand side of (2.52), we see that which, when combined with (2.52), implies Letting ρ 1 ց 0 in the above results in the first limit of (2.47); this follows immediately from (2.51), the assumption G → M G, and that (U, ϕ) is a compact coordinate system. The remaining limit in (2.47) then follows from the above and from (2.45).
Finally, assume in addition that G ⇒ M G. We rewrite the identity (2.49), with ρ * = ρ 0 , as Dividing the above by ρ and integrating its absolute value over (0, ρ 0 ], we obtain The second term on the right-hand side can be simplified using Fubini's theorem, hence we have Note the right-hand side of the above is indeed finite, since G ⇒ M G and (U, ϕ) is compact, and thus we conclude −cA ⇒ M G. The remaining limit ρL ρ A ⇒ M 0 now follows from (2.45).
Proposition 2.41. Let (M , g) be an FG-aAdS segment, and let M ≥ 0. In addition, let A be a vertical tensor field on M , and suppose that ρL ρ A ⇒ M G for some tensor field G on I . Then, then there exists another tensor field H on I such that Proof. Let (k, l) be the rank of A, and let (U, ϕ) be a compact coordinate system on I . Applying the fundamental theorem of calculus, we obtain (with respect to ϕ-coordinates) Differentiating the above yields, for any 0 ≤ m ≤ M , Rearranging the above, we then obtain The first two terms on the right-hand side of (2.54) are constant in ρ and hence do not change as ρ ց 0. Moreover, notice that for the last term in (2.54), we have Define now the tensor field H on I by We now differentiate (2.56) along I . Using the dominated convergence theorem and the assumption ρL ρ A ⇒ M G, we see these derivatives can be moved inside the integral in the last term of (2.56). Therefore, combining (2.54), (2.55), and (2.56), we conclude which completes the proof of (2.53).

Fefferman-Graham Boundary Expansions
In this section, we give (in Theorem 3. 3) the precise statement of our main result, which characterizes the near-boundary geometry of vacuum FG-aAdS segments. We also describe several immediate corollaries of Theorem 3.3, including the following: • (Theorem 3.6) A partial Fefferman-Graham expansion of the spacetime metric, up to n-th order (which, in particular, includes both undetermined parts of the metric boundary data). • (Theorem 3.7) Similar partial expansions for the spacetime Weyl curvature. In Section 3.2, we give the detailed proof of Theorem 3.3. Finally, in Section 3.3, we derive this partial expansion for a familiar family of manifolds: Schwarzschild-AdS spacetimes.
3.1. Statements of Results. The main objective of this subsection is to state the main result of this paper. In addition, we establish a number of its immediate consequences; in Section 3.1.2, we derive partial Fefferman-Graham expansions for various geometric quantities.
3.1.1. The Main Result. One major feature of our main result is that the boundary limits of lower (ρ-)derivatives of the vertical metric depend only on the boundary metric and some finite number of its derivatives. The following definition gives a precise characterization of this dependence: Definition 3.1. Let (M , g) be an FG-aAdS segment, let M ≥ 0, and let A be a tensor field on I . We say A depends only on g to order M , abbreviated A = D M (g), iff A can be written as where for each 1 ≤ j ≤ N , we have that N j ≥ 1, and that Furthermore, another feature of the boundary limits in our main result is that they vanish for all odd orders (at least, up to order n). Therefore, in order to simplify the technical writing, we define the following notational shorthand to capture this property: Definition 3.2. Let (M , g), M , A be as in Definition 3.1. For k ∈ Z, we write A = D M (g; k) iff: • A = D M (g) whenever k is even.
• A = 0 whenever k is odd.
With the above, we can now give a precise statement of the main result: Theorem 3.3. Let (M , g) be a vacuum FG-aAdS segment, and fix M 0 ≥ n + 2. Furthermore, assume n = dim I > 1, and suppose that the following conditions hold: • g is locally bounded in C M0+2 -that is, for any compact coordinate system (U, ϕ) on I , • L ρ g is "weakly locally bounded"-for any compact coordinate system (U, ϕ) on I , Then, the following statements hold: • g and g −1 satisfy the following boundary limits: Furthermore, R, Rc, and Rs satisfy the following boundary limits: • For any 0 ≤ k < n and 1 ≤ l < n, there exist tensor fields on I , such that the following limits hold for each 0 ≤ k < n and 1 ≤ l < n: on I , such that the following boundary limits hold: . In general, g (⋆) satisfies the following constraints: (3.11) tr g g (⋆) = 0, D · g (⋆) = 0.
Remark 3.4. In particular, the first part of (3.12) implies that when n > 2, the limit g (2) is precisely the Schouten tensor associated with the boundary metric g.
We defer the proof of Theorem 3.3 until Section 3.2. In the remainder of the present subsection, we discuss some variations and consequences of Theorem 3.3.
First, combining Proposition 2.37 with the conclusions of Theorem 3.3, we see that the boundary limits (3.6), (3.8), (3.10), and (3.13) can also be expressed covariantly: Corollary 3.5. Assume the hypotheses of Theorem 3.3, and let be as in the conclusions of Theorem 3.3. Then: • The following boundary limits hold for any 0 ≤ v ≤ M 0 − 2: • For any 0 ≤ k < n and 1 ≤ l < n, we have the following boundary limits: • The following boundary limits hold: • The following boundary limits hold: for m, u, v as in (3.18). Thus, it suffices to show-for the same m, u, v-that For this, we let (U, ϕ) be any compact coordinate system on I . Observe that (2.42), the fundamental theorem of calculus, (3.5), and (3.8) imply, for any 0 ≤ m ≤ M 0 − n, that M0−n−1,ϕ |g − g| M0−n,ϕ |ρ log ρ| · L ρ g M0−1,ϕ |ρ log ρ|, from which the first limit in (3.19) follows. By similar methods, we also obtain for all 0 ≤ u ≤ M 0 − n − 2 and 0 ≤ v ≤ M 0 − n − 1, which yield the remaining parts of (3.19).
3.1.2. Partial Expansions. We now use Theorem 3.3 to recover the partial Fefferman-Graham expansions for g, R, and DL ρ g, up to and including the n-th order term. All the information that is required for the expansion is already present in the boundary limits (3.5), (3.6), (3.8), (3.10), and (3.13). The remaining step is to systematically construct the expansion via Taylor's theorem (and to account for the anomalous logarithmic terms when n is even).
• g (n) , r (n) , and b (n) are C M0−n , C M0−n−2 , and C M0−n−1 tensor fields on I , respectively. Furthermore, if n is odd, then tr g g (n) and D · g (n) vanish; whereas if n is even, then tr g g (n) and D · g depend only on g to orders n and n + 1, respectively. • The "remainders" r g , r R , and r D are vertical tensor fields satisfying Proof. First, observe that the conclusions of Theorem 3.3 hold in this setting, since the hypotheses of Theorems 3.6 and 3.3 are the same. As a result, we let the tensor fields be as in the conclusions of Theorem 3.3. Define now the vertical tensor fields Differentiating h, S, and B and recalling (3.8), we see that for each 0 ≤ k < n and 1 ≤ l < n. Moreover, taking one more ρ-derivative yields where the constants C n and C n−1 are given by C n = 1 −1 + · · · + n −1 , C n−1 = 1 −1 + · · · + (n − 1) −1 .
Finally, we derive corresponding partial expansions for the spacetime Weyl curvature: Corollary 3.7. Let (M , g) be a vacuum FG-aAdS segment, assume n = dim I > 2, and fix M 0 ≥ n + 2. In addition, assume the bounds (3.3) and (3.4) hold for any compact coordinate system (U, ϕ) on I . Then, with respect to any arbitrary compact coordinate system on I , the components of the spacetime Weyl curvature W can be expressed as partial expansions in ρ, cab ρ n−3 + r x cab ρ n−3 n even, (3.29) abcd ρ n−4 + r y abcd ρ n−4 n even, ab ρ n−4 + r z ab ρ n−4 n even, where the following properties hold: , and z (2k) (where 2 ≤ k < n 2 ) are tensor fields on I that depend on g to order 2k + 1, 2k, and 2k, respectively. • x (⋆) , y (⋆) , z (⋆) are tensor fields on I depending on g to orders n + 1, n, and n, respectively. • x (n) , y (n) , and z (n) are C M0−n−1 , C M0−n , and C M0−n tensor fields on I , respectively.
• r x , r y , and r z are vertical tensor fields that satisfy Proof. The first expansion of (3.29) follows immediately from the third part of (3.20) and the first equation in (2.19). For the remaining expansions, we adopt the following shorthand: we say (a, r) is a remainder pair iff a is a C M0−n tensor field on I and r is a vertical tensor field with r → M0−n 0. For the second part of (3.29), we begin by applying Taylor's theorem along with (3.8), (3.10), and (3.13) as before to obtain, for some remainder pair (a 1 , r 1 ), the expansion L ρ g = n−1 2 k=1 D 2k (g) · ρ 2k−1 + a 1 ρ n−1 + r 1 ρ n−1 n odd, n−2 2 k=1 D 2k (g) · ρ 2k−1 + D n (g) · ρ n−1 log ρ + a 1 ρ n−1 + r 1 ρ n−1 n even.
3.2. Proof of Theorem 3.3. This subsection is dedicated to the proof of the main result. Throughout, we will assume that the hypotheses of Theorem 3.3 hold.
3.2.1. The Initial Limits. The first step is to establish the lower-order limits (3.5) and (3.6). In proving this, we will also set up the iteration process for obtaining the higher-order limits.
Also, applying Proposition 2.40 and (3.43)-(3.45) to the first part of (2.30) yields (3.35). Now, by the fundamental theorem of calculus, we have, in any compact coordinate system (U, ϕ), Differentiating the above (in ϕ-coordinates) and recalling (3.35), we see that From the above and Lemma 3.8, we immediately obtain which yields the first limit in (3.5). This, along with the last part of (2.36) and (3.3), then yields second limit of (3.5). Similarly, the limits in (3.6) follow from (2.37), (3.3), (3.5). Finally, we establish some consequences of (3.5): Lemma 3.9. Let 0 ≤ M ≤ M 0 and N ≥ 1. Furthermore, let A 1 , . . . , A N denote vertical tensor fields, and let A 1 , . . . , A N denote tensor fields on I . Then: • If A j → M A j for every 1 ≤ j ≤ N , then there exists a tensor field G on I that satisfies S (g; A 1 , . . . , A N ) → M G. Moreover, G is tensor field defined by replacing every instance of A 1 , . . . , Proof. This is an immediate consequence of the Definitions 2.29 and 2.30, the limit properties in Proposition 2.39, and the limits (3.5) (which yield the restriction M ≤ M 0 ).
Lemma 3.10. Let 0 ≤ M ≤ M 0 , let A be a vertical tensor field, and let A be a tensor field on I of the same rank as A. Then, the following statements hold: Proof. Since both statements are trivial when M = 0, we assume henceforth that M > 0. First, if A → M A, then the first part of (2.42) implies that Lemma 3.11. The following limits hold: Proof. Since (3.35) and Lemma 3.10 imply D 2 L ρ g ⇒ M0−2 0, then Lemma 3.9 yields (3.49) S (g; D 2 L ρ g) ⇒ M0−2 0.
As a result, the third equation in (2.30) and (3.49) yield the first limit in (3.48).
By Proposition 2.38, (3.35), Lemma 3.9, and Lemma 3.10, the schematic terms satisfy The final limit of (3.48) now follows from (3.50) and the above.

3.2.2.
The Non-Anomalous Limits. In Lemmas 3.8 and 3.11, we obtained zeroth and first-order (in L ρ ) boundary limits for g and R. Next, we derive the boundary limits (3.7) and (3.8), that is, the limits up to (but not including) order n, before the anomalous logarithmic power. We begin with the limits for g and R. First, observe that the cases k = 0 and k = 1 in already follow from (3.5), (3.6), Lemma 3.8, and Lemma 3.11: For the remaining cases 2 ≤ k < n, we apply an induction argument over k: Lemma 3.12. Fix 2 ≤ k < n, and assume the following hold for all 0 ≤ p < k: Then, the following limits also hold: Proof. We begin by claiming that The first two limits in (3.54) follow from the induction hypothesis (3.52) (with p := k − 2) and from the observation that L ρ commutes with (non-metric) contractions. For Rs, we first write Applying Lemma 3.9 and (3.52) to (3.55) yields Moreover, when k is odd, Lemma 3.9 and (3.52) also imply (For the second limit, note that in each schematic term of the sum, one of j, j 1 , . . . , j l must be odd.) In particular, (3.55) and the above together imply the claim (3.54). Next, we claim the following limits: , (3.57) Similar to before, the limit for S 1 ⇒ M0−k D k−1 (g) follows immediately from Lemma 3.9 and (3.52). Moreover, when k is odd, then within each schematic term comprising S 1 , one of j 1 , . . . , j l must also be odd. Therefore, Lemma 3.9 and (3.52) imply that S 1 ⇒ M0−k D k−1 (g; k). For S 2 , we consider each term S * 2 in the corresponding summation in (3.57) separately. First, if none of j 1 , . . . , j l in S * 2 is equal to k, then Proposition 2.38 and the first part of (3.57) immediately imply S * 2 ⇒ M0−k 0. This leaves only the case j 1 = k and j 2 = 1: S * 2 = S (g; ρL k ρ g, L ρ g).
Again, Lemma 3.9 and (3.52) imply S * 2 ⇒ M0−k 0 in this case. The above results in the second limit of (3.57) and hence completes the proof of the claim (3.57).
Similarly, by Lemma 3.9, Lemma 3.10, (3.52), and the first part of (3.53), we obtain (Both k + 2 and M 0 − k − 2 arise from schematic terms of the form S (D 2 L k ρ g).) Again, when k is odd, then one of j 1 , . . . , j l in each schematic term in (3.59) must also be odd, and hence Lemma 3.10 and (3.52) imply that the right-hand side of (3.59) vanishes. Combining this observation with the first part of (2.34) and (3.59) yields the third limit in (3.53).
For the remaining limit of (3.53), we first claim that To show (3.60), we consider each term S * of the sum separately. If none of j 1 , . . . , j l is equal to k + 1, then S * ⇒ M0−k−2 0 by Proposition 2.38 and (3.59). The only remaining possibility is which also satisfies S * ⇒ M0−k−2 0 by Lemma 3.9 and the second limit of (3.53). This completes the proof of the claim (3.60). Finally, combining (3.60) with the first equation in (2.34)-with k + 1 in the place of k-we obtain the final limit in (3.53).
Now, from the base case (3.51) and the inductive case proved in Lemma 3.12, we obtain the first two relations in (3.7) and the first four limits of (3.8).
(In particular, when l is odd, then in each schematic term on the right-hand side of (3.62), one of j 0 , . . . , j k must also be odd, hence this term converges to 0.) Next, applying (2.33) again, we have that Again, Lemma 3.9, Lemma 3.10, and the first two parts of (3.8) imply that (3.64) ρL l ρ (DL ρ g) ⇒ M0−l−1 0. (In the schematic terms, ρ can always be paired with the factor with the highest derivative of g.) In particular, the limits (3.63) and (3.64) complete the proofs of (3.7) and (3.8).
3.2.3. The Anomalous and Free Limits. Lastly, we consider the boundary limits at order n. We establish the remaining boundary limits (3.9), (3.10), and (3.13), which include both anomalous logarithmic limits (when n is even) determined entirely by g and non-logarithmic limits that need not depend on g. In this process, we will also establish the relations (3.11) and (3.14). We begin by recalling the equations (2.31) in the special case k = n: ρ · S (g; L j1 ρ g, . . . , L j l ρ g), ρ · S (g; L j1 ρ g, . . . , L j l ρ g).
(In particular, one always pairs the factor of ρ with an instance of L n ρ g.) Combining (3.74), (3.75), and the above then yields the second identity in (3.11); this completes the proof of (3.11).
Moreover, applying Proposition 2.41 to this limit yields the second limit in (3.13).
3.3. Schwarzschild-AdS Spacetimes. In this subsection, we connect the partial expansions obtained from Theorem 3.6 to the special family of Schwarzschild-AdS spacetimes. We first recall the definition of these spacetimes (near the conformal boundary) in standard coordinates: • The metric g M is given by It is well-known that (M M , g M ) is a solution of the Einstein-vacuum equations (normalized with Λ given by (2.17)). The goal of this discussion is to express (M M , g M ) as a vacuum FG-aAdS segment and thus read off the partial expansion for g M from the conformal boundary. The first of the above objectives is accomplished in the subsequent proposition: Proposition 3.14. Fix M ∈ R and n > 1, and let the spacetime (M M , g M ) be as in Definition 3.13. Then, (M M , g M ) is isometric to a spacetime (M , g) satisfying the following: • There is some ρ 0 ≪ 1 such that • The metric g is of the form where ρ and t are the projections to the first and second components of M , whereγ denotes the unit round metric on the last n − 1 components of M M , and where the remainders α 1 , α 2 : [−ρ 0 , ρ 0 ] → R are real-analytic functions.
• In particular, when n = 2 and n = 4, we have explicit expressions for g:
From the Fefferman-Graham gauge derived in Proposition 3.14, we can extract the corresponding partial expansions near the boundary for the Schwarzschild-AdS metric: Corollary 3.15. Fix M ∈ R and n > 1, let (M M , g M ) be as in Definition 3.13, and let (M , g) be as in Proposition 3.14. Then, (M , g) is a vacuum FG-aAdS segment, with vertical metric where the vertical (0, 2)-tensor field r is locally bounded in C M for any M ≥ 0. Moreover, in the special case n = 2, we have the following explicit formula for g: Proof. It is clear from (3.90) that M is an aAdS region (see Definition 2.1), with I := R × S n−1 . Moreover, from (3.91), we see that g can be written in the form (2.10), with vertical metric Expanding the above immediately yields (3.93). (In particular, r depends only on ρ, so the C Mboundedness of r for any M ≥ 0 is trivial.) A similar computation using the first part of (3.92) yields the remaining formula (3.94). Observe also that (2.11) holds, with Lorentzian limit g := −dt 2 +γ.
The above proves that (M , g) is indeed a vacuum FG-aAdS segment. Finally, since (M M , g M ) is known to satisfy the Einstein-vacuum equations, (M , g) must be vacuum as well. Note in particular that g (0) is independent of M . One can also check that g (2) satisfies the last formula of (3.12), with g = g (0) . Furthermore, g (n) is the first coefficient that depends on the mass parameter M , and each value of M corresponds to a unique value of g (n) . Finally, note that the logarithmic coefficient g (⋆) always vanishes, regardless of dimension.

Appendix A. Details and Computations
This appendix contains additional proofs, computations, and details for readers' convenience. In particular, propositions which were stated but not proved in the main text are proved here.
A.3. Proof of Proposition 2.27. Letting Γ a bc denote the Christoffel symbols for g with respect to ϕ-coordinates, then a direct computation yields the following identity: To determine ∂ ρ Γ b ac , we apply (A.1), with A := g, to obtain D a L ρ g bc = L ρ D a g bc − [L ρ , D a ]g bc = ∂ ρ Γ d ab · g dc + ∂ ρ Γ d ac · g bd . Using the above, we then conclude that g bd (D a L ρ g dc + D c L ρ g da − D d L ρ g ac ) = g bd (g ec ∂ ρ Γ e ad + g de ∂ ρ Γ e ac ) + g bd (g ea ∂ ρ Γ e cd + g de ∂ ρ Γ e ca ) − g bd (g ec ∂ ρ Γ e da + g ae ∂ ρ Γ e dc ) = 2 · ∂ ρ Γ b ac .
A.4. Proof of Proposition 2.28. Let X be a vector field on M that is both vertical and independent of ρ. By the definition of the Riemann curvature and (2.26), we have that Applying (2.26) again to the right-hand side and recalling that L ρ X = 0, we obtain L ρ (R c dab X d ) = 1 2 D a [g ce (D b L ρ g ed + D d L ρ g eb − D e L ρ g bd )X d ] − 1 2 D b [g ce (D a L ρ g ed + D d L ρ g ea − D e L ρ g ad )X d ] + 1 2 g ce (D a L ρ g ed + D d L ρ g ea − D e L ρ g ad )D b X d − 1 2 g ce (D b L ρ g ed + D d L ρ g eb − D e L ρ g bd )D a X d = 1 2 g ce (D ab L ρ g ed + D ad L ρ g eb − D ae L ρ g bd )X d − 1 2 g ce (D ba L ρ g ed + D bd L ρ g ea − D be L ρ g ad )X d .
Taking X to be the coordinate vector field ∂ x d in the above yields L ρ R c dab = 1 2 g ce (D ab L ρ g ed + D ad L ρ g eb − D ae L ρ g bd − D ba L ρ g ed − D bd L ρ g ea + D be L ρ g ad ), which, after reindexing, is precisely (2.27).
A.5. Proof of Proposition 2.31. The first equation in (2.30) follows from the second part of (2.25) and Definition 2.30, while the third equation in (2.30) is a consequence of (2.27). Taking a g-trace of the first identity of (2.30) and then commuting L ρ with tr g yields (A.2) ρL ρ (tr g L ρ g) − ρL ρ g ab L ρ g ab − (2n − 1)tr g L ρ g = 2ρ · Rs + ρ · S (g; L ρ g, L ρ g).
From standard identities regarding Lie derivatives of metric duals, we see that L ρ g ab = −g ac g bd L ρ g cd , ρL ρ g ab L ρ g ab = ρ · S (g; L ρ g, L ρ g).
The second identity of (2.34) now follows from (A.4) and the above. Q j (A 1 ⊗ · · · ⊗ A N ).