Conformal geodesics and the evolution of spacetimes with positive Cosmological constant

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild–de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.


Introduction
One of the main open problems in mathematical relativity is that of the non-linear stability of spacetimes.In 1986, Friedrich provided the first result concerning the global stability of the de Sitter spacetime and a semi-global stability result for the Minkowski spacetime [3,4].These results are obtained by using the conformal Einstein field equations to reformulate Cauchy problems which are global or semi-global in time into problems which are local in time.This strategy allows to use results obtained for quasi-linear symmetric hyperbolic systems [16,17] to prove the existence of solutions which are suitably close to known reference spacetimes.More results [1,5,19,20,21,15,25] using the conformal Einstein field equations show that these equations are a powerful tool for the analysis of the stability of spacetimes.They provide a system of field equations for geometric objects defined on a four-dimensional Lorentzian manifold (M, g), the so-called unphysical spacetime, which is conformally related to a spacetime ( M, g), the so-called physical spacetime, satisfying the Einstein field equations.The conformal Einstein field equations constitute a system of differential conditions on the curvature tensors with respect to the Levi-Civita connection of g and the conformal factor Ξ.
A problem one encounters when discussing the conformal structure of spacetimes by means of these equations is that of the gauge freedom.In the original formulation of the conformal Einstein field equations [2] the gauge is fixed by means of gauge source functions.An alternative approach of gauge fixing is by exploiting the properties of a congruence of curves which are invariants of the conformal structure.These curves are known as conformal geodesics and they have been originally introduced as a tool for the local analysis of the structure of conformally rescaled spacetimes [9].Using this gauge allows to define a conformal Gaussian gauge system in which coordinates are propagated along conformal geodesics.To combine this gauge choice with the conformal Einstein field equations it is necessary to make use of a more general version of the latter, the extended conformal Einstein field equations.These equations contain a bigger gauge freedom being expressed using a Weyl connection.This is a torsion-free connection which provides a transport equation along the conformal geodesics preserving conformally orthonormal frames and the causal nature of their vectors.
One of the advantages of the conformal Gaussian gauge system is that it gives an a priori knowledge of the structure of the conformal boundary of the spacetime.This aspect is used to obtain an alternative proof of the semi-global non-linear stability of the Minkowski spacetime and of the global non-linear stability of the de Sitter spacetime by Lübbe and Valiente Kroon [19].In [23] the results obtained in [3,19] are generalised to de Sitter-like spacetimes with compact spatial sections of negative scalar curvature.The existence and stability result follows from explicit calculations and the requirement that the data are close to de Sitter-like data.The success of this approach in the analysis of the global properties of asymptotically simple spacetimes leads to the question of whether a similar strategy can be used to study the evolution of black hole spacetimes.A first step in this direction is made in [24] where certain aspects of the conformal structure of the sub-extremal Schwarzschild-de Sitter spacetime are analysed in order to adapt techniques from the asymptotically simple setting to the black hole case.More precisely, since this solution can be studied by means of the extended conformal Einstein field equations -see [11].These equations are used to obtain a result concerning the evolution of the region of this spacetime which is bounded by the Cosmological horizon known as the Cosmological region.In particular, in analogy to the de Sitter-like case, the Cosmological region has an asymptotic region admitting a smooth conformal extension with a spacelike conformal boundary and there exists a conformal representation in which the induced 3-metric on the conformal boundary I is homogeneous.Thus, it is possible to integrate the extended conformal field equations along single conformal geodesics -see [8,10].
In this review article, the discussion of the construction of a conformal Gaussian gauge system leading to a hyperbolic reduction of the conformal Einstein field equations in the de Sitter-like case [23] and the sub-extremal Schwarzschild-de Sitter case [24] is revisited and presented in a coherent and contiguous way.

Notations and conventions
The signature convention for Lorentzian spacetime metrics will be (−, +, +, +).In this article, the abstract index notation is used.Accordingly, the lowercase Latin indices { a , b , c , ...} will denote spacetime abstract tensor indices and { a , b , c , ...} will be used as spacetime frame indices taking the values 0, ..., 3.In this way, given a basis {e a } a generic tensor is denoted by T ab while its components in the given basis are denoted by T ab ≡ T ab e a a e b b .The Greek indices µ , ν , . . .denote spacetime coordinate indices while the indices α , β , . . .denote spatial coordinate indices.An index-free notation is used where convenient.
The conventions for the curvature tensors are fixed by the relation

Tools of conformal geometry
The purpose of this section is to provide a brief summary of the technical tools of conformal geometry that will be used in the analysis of the evolution of the spacetimes under consideration.
Let ( M, g) be a vacuum spacetime satisfying the Einstein field equations with positive Cosmological constant Rab = λg ab (1) and let g denote an unphysical Lorentzian metric conformally related to g via the relation with Ξ a suitable conformal factor.The Levi-Civita connections of the metrics g and g are denoted by ∇ and ∇, respectively.The set of points for which Ξ = 0 is called the conformal boundary.

Weyl connections
A Weyl connection is a torsion-free connection ∇ such that ∇a g bc = −2f a g bc .
It follows from the above that the connections ∇ a and ∇a are related to each other by where f a is a fixed smooth covector and v a is an arbitrary vector.Given that In the following, it will be convenient to define

The frame version of the extended conformal Einstein field equations
The extended conformal Einstein field equations constitute a conformal representation of the vacuum Einstein field equations written in terms of Weyl connections -see [7].These equations are formally regular at the conformal boundary.Moreover, a solution to the extended conformal equations implies, in turn, a solution to the vacuum Einstein field equations away from the conformal boundary.
The frame formulation of the extended conformal Einstein field equations is obtained by definining the following zero-quantities: where the components of the geometric curvature Rc dab and the algebraic curvature ρc dab are given, respectively, by Lb]e .
In terms of the zero-quantities (4a)-(4d), the extended conformal Einstein field equations are given by the conditions In the above equations the fields Ξ and d a are regarded as conformal gauge fields which are determined by gauge conditions.These conditions will be determined through conformal geodesics -see Subsection 2.3 below.In order to account for this it is convenient to define The conditions δ a = 0, will be called the supplementary conditions.They play a role in relating the Einstein field equations to the extended conformal Einstein field equations and also in the propagation of the constraints.
The correspondence between the Einstein field equations and the extended conformal Einstein field equations is given by the following -see Proposition 8.3 in [25]: ) denote a solution to the extended conformal Einstein field equations (5) for some choice of the conformal gauge fields (Ξ, d a ) satisfying the supplementary conditions (7).Furthermore, suppose that on an open subset U. Then the metric is a solution to the Einstein field equations (1) on U.

Conformal geodesics
The gauge used to analyse the evolution of the spacetimes under consideration is based on the properties of the conformal geodesics.Conformal geodesics allow the use of conformal Gaussian systems in which a certain canonical conformal factor gives an a priori knowledge of the location of the conformal boundary.This is in contrast with other conformal gauges in which the conformal factor is an unknown.

Basic definitions
A conformal geodesic on a spacetime ( M, g) is a pair (x(τ ), β(τ )) consisting of a curve x(τ ) and a covector β(τ ) along x(τ ) satisfying the equations where L denotes the Schouten tensor of the Levi-Civita connection ∇.A vector v is said to be Weyl propagated if along x(τ ) it satisfies the equation A congruence of conformal geodesics can be used to single out a metric g ∈ [g].This is due to the following property: Proposition 2. Let ( M, g) denote a vacuum spacetime with positive Cosmological constant.Suppose that (x(τ ), β(τ )) is a solution to the conformal geodesic equations (8a)-(8b) and that {e a } is a g-orthonormal frame propagated along the curve according to equation (9).
then one has that where the coefficients are constant along the conformal geodesic and are subject to the constraints A proof of this result can be found in [25].Thus, if a spacetime can be covered by a non-intersecting congruence of conformal geodesics, then the location of the conformal boundary is known a priori in terms of data at a fiduciary initial hypersurface S ⋆ .

The g-adapted conformal geodesic equations
As a consequence of the normalisation condition (10), the parameter τ is the g-proper time of the curve x(τ ).In some computations it is more convenient to consider a parametrisation in terms of a g-proper time τ of the curve x(τ ) as it allows to work directly with the physical metric.To this end, consider the parameter transformation τ = τ (τ ) given by dτ dτ = Θ, so that τ = τ⋆ + with inverse τ = τ (τ ).Now, consider in equations (8a)-(8b) so that one obtains the following g-adapted equations for the conformal geodesics: with β2 ≡ g♯ ( β, β).For an vacuum spacetime with Cosmological constant one has that L = 1 6 λg.

A Conformal Gaussian gauge system
One considers a region U of the spacetime ( M, g) which is covered by a non-intersecting congruence of conformal geodesics (x(τ ), β(τ )).The property of these curves stated by Proposition 2 allows to single out a canonical representative g of the conformal class [g] with an explicitly known conformal factor as given by the formula (11).Now, let {e a } denote a g-orthonormal frame which is Weyl propagated along the conformal geodesics.To every congruence of conformal geodesics one can associate a Weyl connection ∇a by setting f a = β a .It follows that for this connection one has This gauge choice is supplemented by choosing the parameter τ of the conformal geodesics as the time coordinate so that e 0 = ∂ τ .
Since the initial data for the congruence of conformal geodesics is prescribed on a fiduciary spacelike hypersurface S ⋆ .On S ⋆ one can choose some local coordinates x = (x α ).These coordinates can be extended off S ⋆ by requiring them to remain constant along the conformal geodesic which intersects S ⋆ at the point p with coordinates x.The spacetime coordinates x = (τ, x α ) obtained in this way are known as conformal Gaussian coordinates.
The collection of conformal factor Θ, Weyl propagated frame {e a } and coordinates (τ, x α ) is known as a conformal Gaussian gauge system.

The conformal constraint equations
The conformal constraint Einstein equations are intrinsic equations implied by the conformal Einstein field equations on a spacelike hypersurface.
Let S denote a spacelike hypersurface in an unphysical spacetime (M, g) and let {e a } denote a g-orthonormal frame adapted to S. The conformal constraint equations in the vacuum case are given by -see [25]: with the understanding that and where Moreover, Ω denotes the restriction of the spacetime conformal factor Ξ to S and Σ is the normal component of the gradient of Ξ.The field l ij denotes the components of the Schouten tensor of the induced metric h ij on S. The fields d ij and d ijk correspond, respectively, to the electric and magnetic parts of the rescaled Weyl tensor.The scalar s denotes the Friedrich scalar defined as with R the Ricci scalar of the metric g.Finally, L ij denote the spatial components of the Schouten tensor of g.

de Sitter-like spacetimes
In this section, we study the evolution of de Sitter-like spacetimes which can be conformally embedded into a portion of a cylinder whose sections have negative scalar curvature as in [23].
The conformal embedding is realised by means of a conformal factor which depends on the affine parameter of the conformal geodesics.

Basic properties
A de Sitter-like spacetime ( M, g) is a solution to the vacuum Einstein field equations with positive Cosmological constant (1) given by M = R × S and where γ is a positive definite Riemannian metric over a compact manifold S with constant negative curvature.The Riemann curvature tensor r i jkl [γ] of γ is given by In particular, by setting λ = 3, it follows from the above expressions that A spacetime of the form given by ( M, g) will be known as a background solution.
The value λ = 3 for the Cosmological constant is conventional and set for convenience.This analysis can be carried out for any other positive value of λ.

Metric geodesics as conformal geodesics
The analysis of the metric geodesics x(s) on ( M, g) with ẋ ≡ ∂x ∂s = α∂ t , where α is a proportionality function, by means of the geodesic equation ∇ ẋ ẋ = 0 and the metric (16) shows that α is constant along the integral curves of ∂ t .Hence, without loss of generality one can set α = 1 so that the curves are non-intersecting timelike g-geodesics over M.These curves can be recasted as conformal geodesics by means of a reparametrisation τ → t(τ ) and a 1-form β given by the Ansatz The resulting pair (x(τ ), β(τ )) with describes a congruence of non-intersecting timelike conformal geodesics on the background spacetime ( M, g).

The conformal factor associated to the congruence of conformal geodesics
The parameter τ introduced in the previous section is used as a new time coordinate in the metric (30) so that .4 Global existence and stability using conformal Gaussian systems 439 S ?, I Penrose diagrams of de Sitter-like spacetimes obtained from Theorems 30 and ft the spacetime obtained from a standard Cauchy initial value problem; to spacetime obtained from the asymptotic initial value problem.

Geodesic completeness and asymptotic analysis
is of the existence and stability of de Sitter-like spacetimes developed 15.3 and 15.4 can be refined to include geodesic completeness.As the ter spacetime is geodesically complete, it is to be expected that suitably rbations thereof will also share this property.More precisely: n 29 (geodesic completeness of de Sitter-like spacetimes) Suitperturbations ( M, g) of the de Sitter spacetime are null and timelike lly complete.
ular, the above proposition together with the existence and stability ined in the previous sections show that suitably small perturbations of r spacetime are asymptotically simple spacetimes.
enient to divide the analysis of Proposition 29 into two cases.

Null geodesics
servation required to prove null geodesic completeness is the following: nformal representation (R⇥S 3 , ḡE ) any null ḡE -geodesic starting within cal spacetime reaches the conformal boundary for a finite value of its eter.ollows, let ( M, ḡ) be one of the de Sitter-like spacetimes obtained from, ard Cauchy initial value problem with data prescribed on a hypersurking use of a perturbative argument similar to the ones employed in s 26 and 28 and by reducing ", if necessary, it can be shown that given S ? and a fixed > 0, for all points q 2 S ?lying in an h-metric ball of ntred at p, the future directed null ḡ-geodesics starting at q will reach ite value of their a ne parameter.As S ? is a compact hypersurface, it This metric is singular at τ = ±1.This line element suggest the introduction of a new unphysical metric g via the relation with associated Levi-Civita connection to be denoted by D, whereas D is the Levi-Civita connection of the metric γ.The integral curves of the vector field ∂ τ are geodesics of the metric g given by equation (19).Moreover, since β is a closed 1-form the Weyl connection is, in fact, a Levi-Civita connection which coincides with ∇.
A Penrose diagram of conformal representation of the background solution described by the metric ( 19) is given in Figure 1.

The background spacetime as a solution to the conformal Einstein field equations
The unphysical spacetime (M, g) is recasted as a solution to the conformal Einstein field equations.This construction is done using an adapted frame formalism.

The frame
Let {c i }, i = 1, 2, 3, denote a γ-orthonormal frame over S with associated cobasis The above frame is used to introduce a g-orthonormal frame {e a } with associated cobasis {ω b } so that ⟨ω b , ea ⟩ = δ a b .This is done by setting

The connection coefficients
The connection coefficients γi k j of the Levi-Civita connection D with respect to the frame {c i } are defined through the relations Similarly, for the connection coefficients Γi Using these relations, it follows that the only non-vanishing connection coefficients are Γi where χi j denote the components of the Weingarten tensor.Thus, all the connection coefficients are smooth over [τ ⋆ , ∞) × S.

Conformal fields
The components of the conformal fields appearing in the extended conformal Einstein field equations are obtained by solving the conformal Einstein constraints discussed in Section 2.5.This is done by means of an adapted frame with e 0 = ∂ τ and by making the identification Ω → Θ in equations (15a)-(15j).The analysis of these equations gives Thus, all the fields are regular up to the conformal boundary and the metric ( 19) is conformally flat.

Evolution equations
In this section we discuss the evolution system associated to the extended conformal Einstein equations (5) written in terms of a conformal Gaussian system.In addition, we also discuss the subsidiary evolution system satisfied by the zero-quantities associated to the field equations, (4a)-(4d), and the supplementary zero-quantities (6a)-(6c).

The conformal Gaussian gauge
To obtain suitable evolution equations for the conformal fields it is used a conformal Gaussian gauge.More precisely, it is assumed that a region U ⊂ M is covered by a congruence of nonintersecting conformal geodesics.Then, by choosing for τ = τ ⋆ , τ ⋆ ∈ (0, 1), Proposition 2 gives the conformal factor along the curves of the congruences.The choice of initial data for the conformal factor is associated to a congruence that leaves orthogonally a fiduciary initial hypersurface S ⋆ with τ = τ ⋆ .
Since the conformal factor Θ given by equation (22) does not depend on the initial data for the evolution equations it can be regarded as valid not only for the background solution but also for its perturbations.
Along the congruence of conformal geodesics one considers a g-orthogonal frame {e 0 } which is Weyl-propagated and such that τ = e 0 .The Weyl connection ∇a associated to the congruence then satisfies ∇τ e a = 0, L(τ, •) = 0.
By choosing the parameter, τ , of the conformal geodesics as time coordinate one gets the additional gauge condition e 0 = ∂ τ , e 0 µ = δ 0 µ .
On S ⋆ we choose some local coordinates x = (x α ).These coordinates can be extended off the initial hypersurface so that the coordinates (τ, x) thus obtained are conformal Gaussian coordinates.

Structural properties of the evolution and subsidiary equations
In the conformal Gaussian gauge, the various fields associated to the extended vacuum conformal Einstein field equations satisfy the evolution equations Letting e, Γ, L and ϕ denote, respectively, the independent components of the coefficients of the frame, the connection coefficients, the Schouten tensor of the Weyl connection and the rescaled Weyl tensor and setting, for convenience, u ≡ (υ, ϕ) with υ ≡ (e, Γ, L) and ϕ ≡ (d, d * ) one has the following: Lemma 1.The extended conformal Einstein field equations (5) expressed in in terms of a conformal Gaussian gauge imply that the evolution equations (23a)-(23f) can be written as a symmetric hyperbolic system for the components (υ, ϕ) of the form where I is the unit matrix, K is a constant matrix Q(Γ) is a smooth matrix-valued function, L(x) is a smooth matrix-valued function of the coordinates, A µ (e) are Hermitian matrices depending smoothly on the frame coefficients and B(Γ) is a smooth matrix-valued function of the connection coefficients.
Regarding the subsidiary evolution system, it follows from the system ∇0 Σb that the zero-quantities Σa c b , Ξa bcd , ∆abc , Λabc , δ ab , γ ab and ς ab satisfy, if the conformal evolution equations (23a)-(23e) hold, a symmetric hyperbolic system which is homogeneous in the zero-quantities.More precisely, upon defining X ≡ ( Σa c b , Ξc dab , ∆abc , Λabc , δ a , γ ab , ς ab ), these equations can be recasted as a symmetric hyperbolic system of the form where H(0) = 0.The particular situation in which all the zero-quantities vanish identically gives rise to the subsidiary evolution system.

A perturbative argument
In the following, we look for solutions to the system (24a)-(24b) of the form û = ů + ȗ where ů is the solution to the conformal evolution equations (23a)-(23f) implied by a background solution, while ȗ denotes a small perturbation.Accordingly, one can set it is possible to write the evolution equations for ȗ = ( υ, φ) as Since this is a symmetric hyperbolic system, existence and stability results are obtained by using known results for symmetric hyperbolic systems with compact spatial sections -see e.g.[25], Section 12.3 which, in turn, follow from Kato's theory for symmetric hyperbolic systems over R n [17].The existence and Cauchy stability of the solution to the initial value problem for the original conformal evolution problem follows from the fact that û satisfies the same properties as ȗ and then it exists in the same solution manifold and with the same regularity properties, existence and uniqueness.

A solution to the Einstein field equations
In this section, we discuss the connection between the solution to the conformal evolution systems and the actual solution to the Einstein field equations.
From the discussion in Section 3.5.2 it follows that the independent components of the zeroquantities X ≡ ( Σa c b , Ξc dab , ∆abc , Λabc , δ a , γ ab , ς ab ) satisfy the symmetric hyperbolic system (26).Then, a solution to the initial value problem is given by X = 0.Moreover, from Kato's theorem [17] follows that this is the unique solution.Thus, the zero-quantities must vanish on τ ⋆ , 1 × S ⋆ .This result is summarised by the following Now, given the propagation of the constraints, Proposition 3, and Proposition 1 it follows that the metric g = Θ −2 g obtained from the solution to the conformal evolution equations implies a solution to the vacuum Einstein field equations with λ = 3.
The main result of this discussion is contained in the following theorem Theorem 1.Let û⋆ = ů⋆ + ȗ⋆ denote smooth initial data for the conformal evolution equations satisfying the conformal constraint equations on a hypersurface S ⋆ .Then, there exists ε > 0 such that if then there exists a unique C m−2 solution g to the vacuum Einstein field equation with positive Cosmological constant over [τ ⋆ , ∞) × S ⋆ for τ⋆ > 0 whose restriction to S ⋆ implies the initial data û⋆ .Moreover, the solution û remains suitably close to the background solution ů.

Schwarzschild-de Sitter spacetimes
In this section, it is discussed the behaviour of the conformal geodesics in the Cosmological region of the sub-extremal Schwarzschild-de Sitter spacetime.The aim of this analysis is to adapt the technique described in the de Sitter-like setting and valid, in general, for asymptotically simple spacetimes to the black hole case as presented in [24].

Basic properties
The Schwarzschild-de Sitter spacetime ( M, g) is a spherically symmetric solution to the vacuum Einstein field equations with positive Cosmological constant (1) with M = R × R + × S 2 and line element given in standard coordinates (t, r, θ, φ) by where σ ≡ dθ ⊗ dθ + sin 2 θdφ ⊗ dφ, denotes the standard metric on S 2 .The coordinates (t, r, θ, φ) take the range This line element can be rescaled so to that where In our conventions M , r and λ are dimensionless quantities.

Horizons and global structure
The location of the horizons of the Schwarzschild-de Sitter spacetime follows from the analysis of the zeros of the function D(r) in the line element (30).
Since λ > 0, the function D(r) can be factorised as where r b and r c are, in general, distinct positive roots of D(r) and r − is a negative root.Moreover, one has that 0 < r b < r c , r c + r b + r − = 0.
The root r b corresponds to a black hole-type of horizon and r c to a Cosmological de Sitter-like type of horizon.Using Cardano's formula for cubic equations, we have where the parameter ϕ is defined through the relation The sub-extremal case is characterised by 0 < M < 2/3 √ 3 and ϕ ∈ (0, π/2) and describes a black hole in a Cosmological setting.The Penrose diagram of the sub-extremal Schwarzschild-de Sitter is well known -see Figure 2.
where ∇ denotes the Levi-Civita connection of the physical metric g and ∇ẋ denotes a derivative in the direction of ẋ.Notice that in the last expression the indices of the vectors and covectors are raised or lowered using g-unless otherwise stated, we follow this convention in the rest of this article.The symbol L denotes the Schouten tensor of g defined by:    As described in the main text, these horizons are located at r = r b and r = r c .The excluded points Q and Q ′ where the singularity seems to meet the conformal boundary correspond to asymptotic regions of the spacetime that does not belong to the singularity nor the conformal boundary.

Construction of a conformal Gaussian gauge in the Cosmological region
This study begins with the qualitative analysis of the behaviour of the conformal geodesics of the Schwarzschild-de Sitter spacetime prescribed in terms of data on hypersurfaces of constant r in the Cosmological region.

Basic setup
It is assumed that r c < r < ∞ corresponding to the Cosmological region of the Schwarzschild-de Sitter spacetime.Given a fixed r = r ⋆ , S ⋆ denotes the spacelike hypersurface of constant r = r ⋆ in this region.Points on S ⋆ are described in terms of the coordinates (t, θ, φ).
In order to prescribe the congruence of conformal geodesics, it is provided the value of the conformal factor Θ ⋆ over S ⋆ so that The second condition implies that the resulting conformal factor will have a time reflection symmetry with respect to S ⋆ .Then it is required that These conditions give rise to a congruence of conformal geodesics which has a trivial behaviour of the angular coordinates.Accordingly, the analysis of these curves is effectively given by the Finally, in order to exclude the asymptotic points Q and Q ′ , it is defined where the constant t • is assumed large enough so that D + (R • ) ∩ I + ̸ = ∅.

Analysis of the behaviour of the conformal geodesics
The congruence of conformal geodesics prescribed by the initial data (33) is such that β 2 = 0, so that after reparametrisation reduces to a congruence of metric geodesics.Thus, the geodesic equations imply that where γ is a constant.Evaluating at S ⋆ one readily finds that with D ⋆ < 0.Moreover, since the unit normal to S ⋆ and x′ ⋆ are parallel to each other then γ = 0.In order to study the behaviour of these curves and obtain simpler expressions, it is set λ = 3 and τ ⋆ = 0.It follows then from Proposition (2) that the conformal factor Now, since the relation between the physical proper time τ and the unphysical proper time τ is obtained from equation ( 12) so that then τ → ±2, as τ → ±∞ and since this congruence of conformal geodesics is reparametised as metric geodesics, it will reach the conformal boundary orthogonally -see [9].Now, since the dependence of the physical proper time τ on r is given by which can be written in terms of elliptic functions -see e.g.[18], it follows from the general theory of elliptic functions that τ (r, r ⋆ ) is an analytic function of its arguments.Moreover, one has that τ → ∞ as r → ∞.
Accordingly, the curves escape to infinity in an infinite amount of physical proper time.Using the reparametrisation formulae (37) the latter corresponds to a finite amount of unphysical proper time.

Analysis of the behaviour of the conformal deviation equation
In [8] (see also [10]) it has been shown that for congruences of conformal geodesics in spherically symmetric spacetimes the behaviour of the deviation vector of the congruence can be understood by considering the evolution of a scalar ω satisfying the equation where ̸ D denotes the Levi-Civita covariant derivative of l and R[ l] denotes the Ricci scalar of l.If ω does not vanish, then the congruence is non-intersecting.Since this setting r ≥ r ⋆ > r c and ω ≡ Θω, it follows that

Q r
By solving this last differential equation and reverting to ω, one has that which is non-vanishing in the limit τ → ±2.Thus, we have the following Proposition Proposition 4. The congruence of conformal geodesics given by the initial conditions (33) leaving the initial hypersurface S ⋆ reach the conformal boundary I + without developing caustics.
This behaviour of the conformal geodesics is shown in Figure [3].

Conformal Gaussian coordinates in the sub-extremal Schwarzschild-de Sitter spacetime
The congruence of conformal geodesics defined by the initial conditions (33) is used to construct a conformal Gaussian coordinate system in a domain in the chronological future of R • containing a portion of the conformal boundary I + .This analysis is carried out by considering the coordinate z ≡ 1/r in terms of which the line element (30) takes the form The above expression suggest defining an unphysical metric ḡ via ḡ = Ξ 2 g, Ξ ≡ z.
More precisely, one has Now, let SdS I denote the Cosmological region of the Schwarzschild-de Sitter spacetime -that is Moreover, denote by SdS I the conformal representation of SdS I defined by the conformal factor Θ defined by the non-singular congruence of conformal geodesics.Let z ≡ 1/r, for z < z c one has that in terms of these coordinates where z ⋆ ≡ 1/r ⋆ with r ⋆ > r c .
The conformal geodesics defined by the initial conditions (33) define a map ψ which is analytic in the parameters (τ, t ⋆ ).This map is invertible since the Jacobian of the transformation is nonzero for the given value of the parameters.The inverse map gives the transformation from the standard Schwarzschild coordinates (t, z, θ, φ) into the conformal Gaussian coordinates (τ, t ⋆ , θ, φ).This result is summarised by the following Proposition 5.The congruence of conformal geodesics on SdS I defined by the initial conditions on S ⋆ given by (33) induce a conformal Gaussian coordinate system over D + (R • ) which is related to the standard coordinates (t, r) via a map which is analytic.

The background spacetime as a solution to the conformal Einstein field equations
The Schwarzschild-de Sitter spacetime in the region is casted as a solution to the extended conformal Einstein field equations by means of a Weyl propagated frame.

The frame
Since the congruence of conformal geodesics implied by the initial data (33) satisfies β = 0, the Weyl propagation equation ( 9) reduces to the usual parallel propagation equation.Given the spherical symmetry of the Schwarzschild-de Sitter spacetime, the discussion of a frame adapted to the symmetry of the spacetime can be carried out by considering the 2-dimensional Lorentzian metric (34).The time leg of the frame is set as e 0 = ẋ so that where x′ = t′ ∂ t + r′ ∂ r .Now, upon defining with ⟨ω, x′ ⟩ = 0, one has that the radial leg of the frame is given by The Weyl propagated frame {e a } is completed by choosing two arbitrary orthonormal vectors ẽ2⋆ and ẽ3⋆ spanning the tangent space of S 2 and defining the vectors {e 2 , e 3 } on M • by constantly extending the value of the associated coefficients along the conformal geodesics.This analysis leads to the following result Proposition 6.Let x′ denote the vector tangent to the conformal geodesics defined by the initial data (33) and let {e 2⋆ , e 3⋆ } be an arbitrary orthonormal pair of vectors spanning the tangent bundle of S 2 .Then the frame {e 0 , e 1 , e 2 , e 3 } obtained by the procedure described in the previous paragraph is a g-orthonormal Weyl propagated frame.The frame depends analytically on the unphysical proper time τ and the initial position t ⋆ of the curve.

The Weyl connection
The connection coefficients associated to a conformal Gaussian gauge are made up of two pieces: the 1-form defining the Weyl connection and the Levi-Civita connection of the metric ḡ.The congruence of conformal geodesics discussed in Section 4.3 arises from initial data chosen so that the curves with tangent given by x′ satisfy the standard (affine) geodesic equation.Consequently, the (spatial) 1-form β vanishes.Now, since x′ = r ′ ∂ r , by observing equation (35) for r ′ with γ = 0 and by introducing z = 1/r, it follows that Then, from the conformal transformation rule and by recalling that Ξ = z, it follows that β vanishes at I + .However, β ̸ = 0 away from the conformal boundary.

The connection coefficients
Since the coordinates and connection coefficients associated to the physical connection ∇ are not well adapted to a discussion near the conformal boundary we resort to the unphysical Levi-Civita connection ∇ to compute ∇.

The connection coefficients Γa
These coefficients are analytic at z = 0. Since a contraction with the coefficients of the frame does not change this, it follows that the Weyl connection coefficients Γa b c are smooth functions of the coordinates used in the conformal Gaussian gauge on the future of the fiduciary initial hypersurface S ⋆ up to and beyond the conformal boundary.

The rescaled Weyl tensor
Given a timelike vector, the components of the rescaled Weyl tensor d abcd can be encoded in the electric and magnetic parts relative to the given vector.For the vector ē0 these are given by where d * abcd denotes the Hodge dual of d abcd .A computation using the package xAct for Mathematica readily gives that the only non-zero components of the electric part are given by not determined by either the evolution equations or the gauge conditions satisfy a symmetric hyperbolic system which is homogeneous in the zero-quantities.As a result, if the zero-quantities vanish on a fiduciary spacelike hypersurface S⋆ , then they also vanish on the domain of dependence -see [16].

The perturbative existence argument
Let û ≡ (v, φ) and ů denotes the background solution being a solution to the evolution equations arising from the initial data ů⋆ prescribed on S⋆ .Solutions to the evolution equations which can be regarded as a perturbation of the background solution are constructed by introducing a perturbative argument û = ů + ȗ with ȗ being a small perturbation.This means, in particular, that one can write ê = e + ȇ, Γ = Γ + Γ, φ = φ + φ.
The components of ȇ, Γ and φ are our unknowns.Making use of the decomposition (41) and exploiting that ů is a solution to the conformal evolution equations one obtains the equations denote, respectively, expressions which are quadratic, linear and constant terms in the unknowns.
In terms of the above expressions it is possible to rewrite the system (42a)-(42b) in the more concise form Ā0 (τ, x, ȗ)∂ τ ȗ + Āα (τ, x, ȗ)∂ α ȗ = B(τ, x, ȗ).(43) These equations are in a form where the theory of first-order symmetric hyperbolic systems [17] can be applied to obtain a existence and stability result for small perturbations of the initial data ů⋆ .

A solution to the Einstein field equations
The evolution equations (42a)-(42b) imply the same subsidiary system as for de Sitter-like spacetimes.Thus, the propagation of the constraints follows from the same argument -see Proposition 3.1 Section 3.7.Given the propagation of the constraints and Proposition 1, one has the metric g = Θ −2 g obtained from the solution to the conformal evolution equations implies a solution to the vacuum Einstein field equations with positive Cosmological constant on M ≡ D + (R • ).
The main result of this discussion is contained in the following theorem In particular, the resulting spacetime ( M, g) is a non-linear perturbation of the sub-extremal Schwarzschild-de Sitter spacetime on a portion of the Cosmological region of the background solution which contains a portion of the asymptotic region.

Conclusion
This review article provides a discussion based on [23] and [24] describing how the extended conformal Einstein field equations and a gauge adapted to the conformal geodesics can be used to study the evolution of vacuum spacetimes with positive Cosmological constant.In the Sitterlike case, this analysis identifies a class of spacetimes for which it is possible to prove non-linear stability and the existence of a regular conformal representation.More precisely, it is identified a class of de Sitter-like spacetimes which can be conformally embedded into a portion of a cylinder whose sections have negative scalar curvature.The conformal embedding is realised by means of a conformal factor Θ which depends quadratically on the affine parameter τ of the conformal geodesics and this parameter is also used as a time coordinate for the physical metric.This result led to wonder whether this technique can be adapted to black hole type of spacetimes.The analysis of the conformal geodesics in the Cosmological region of the Schwarzschild-de Sitter spacetime shows that it is possible to construct a conformal Gaussian gauge system.In particular, it shows that it is possible to construct solutions to the vacuum Einstein field equations in this region containing a portion of the asymptotic region and which are non-linear perturbations of the exact Schwarzschild-de Sitter spacetime.Crucially, although the spacetimes constructed have an infinite extent to the future, they exclude the asymptotic points Q and Q ′ .From the analysis of the asymptotic initial value problem in [11] it is known that these points contain singularities of the conformal structure.Thus, they cannot be dealt by the approach used in the article.In order to have a complete statement on the non-linear stability of the Cosmological region it is necessary to address the asymptotic points.Moreover, since the initial hypersurfaces S ⋆ considered in the article are spacelike and the evolution doesn't include the Cosmological horizon r c .A complete statement should also include the case in which r = r c .This suggests reformulating the existence and stability results in [24] in terms of a characteristic initial value problem with data prescribed on Cosmological horizons.Again, to avoid the singularities of the conformal structure, the characteristic data has to be prescribed away from the asymptotic points.Alternatively, one could consider data sets which become exactly Schwarzschild-de Sitter near the asymptotic points.The associated evolution problem by means of a generalisation of the methods used in [12] should allow to reach any suitable hypersurface with constant r.

Figure 1 :
Figure 1: Penrose diagram of the background solution.The conformal representation discussed in the main text has compact sections of negative scalar curvature.The vertical lines Γ 1 and Γ 2 correspond to axes of symmetry.The solution has a singularity in the past and a spacelike future conformal boundary.Hence, in our discussion we only consider future evolution of the initial hypersurface S ⋆ .

kj
of the Levi-Civita connection ∇ with respect to the frame {e a } one has that ∇a eb = Γa c b ec , Γa c b ≡ ⟨ω c , ∇a eb ⟩.

Figure 1 .
Figure 1.Penrose diagram for the subextremal Schwarzschild-de Sitter spacetime.The serrated line denotes the location of the singularity; the continuous black line denotes the conformal boundary; the dashed line shows the location of the black hole and cosmological horizons which are located at r = r b and r = r c respectively.The excluded points Q and Q ′ where the singularity seems to meet the conformal boundary correspond to asymptotic regions of the spacetime that does not belong to the singularity nor the conformal boundary.

Figure 2 .
Figure 2. Penrose diagrams for the extremal Schwarzschild-de Sitter spacetime.Figure (a) corresponds to a white hole which evolves towards a de Sitter final state while figure (b) is a model of a black hole with a future singularity.The Killing horizon is located at r = r H as described in the main text.Similar to the subextremal case, the excluded points denoted by P , Q represent asymptotic regions of the spacetime that do not belong to the singularity nor the conformal boundary.
Figure 2. Penrose diagrams for the extremal Schwarzschild-de Sitter spacetime.Figure (a) corresponds to a white hole which evolves towards a de Sitter final state while figure (b) is a model of a black hole with a future singularity.The Killing horizon is located at r = r H as described in the main text.Similar to the subextremal case, the excluded points denoted by P , Q represent asymptotic regions of the spacetime that do not belong to the singularity nor the conformal boundary.

Figure 2 :
Figure 2: Penrose diagram of the sub-extremal Schwarzschild-de Sitter spacetime.The serrated line denotes the location of the singularity; the continuous black line denotes the conformal boundary; the dashed line shows the location of the black hole and Cosmological horizons denoted by H b and H c respectively.As described in the main text, these horizons are located at r = r b and r = r c .The excluded points Q and Q ′ where the singularity seems to meet the conformal boundary correspond to asymptotic regions of the spacetime that does not belong to the singularity nor the conformal boundary.

Figure 3 :
Figure 3: The conformal geodesics are plotted on the Penrose diagram of the Cosmological region of the sub-extremal Schwarzschild-de Sitter spacetime.The purple line represents the initial hypersurface S ⋆ corresponding to r = r ⋆ .The red lines represent conformal geodesics with constant time leaving this initial hypersurface.

bc
are defined through the relation ∇a e c = Γa b c e b .The only non-vanishing Christoffel symbols Γµ ν λ are given by Γt t z = − Γz z

Theorem 2 .
Let û⋆ = ů⋆ + ȗ⋆ denote smooth initial data for the conformal evolution equations satisfying the conformal constraint equations on a hypersurface S ⋆ .Then, there exists ε > 0 such that if||ȗ ⋆ || S⋆,m < ε, m ≥ 4then there exists a unique C m−2 solution g to the vacuum Einstein field equation with positive Cosmological constant over [τ ⋆ , ∞) × S ⋆ for τ⋆ > 0 whose restriction to S ⋆ implies the initial data û⋆ .Moreover, the solution û remains suitably close to the background solution ů.