The Einstein metrics with smooth scri

We consider solutions of the Einstein equations with cosmological constant $\Lambda\neq 0$ admitting conformal compactification with smooth scri $\mathscr{I^+}$. Metrics are written in the Bondi-Sachs coordinates and expanded into inverse powers of the affine distance $r$. Unlike in the case $\Lambda=0$ all free data are located on the scri. There are linear differential constraints on the Bondi mass and angular momentum aspects. All other components of metrics are defined in a recursive way.


Introduction
This paper is a continuation of [1] where the case Λ = 0 is considered. We assume that metricg satisfying vacuum equations with Λ = 0 is conformally equivalent to metric g which is smooth in a neighbourhood of the scri I + . We consider its Bondi-Sachs form with the affine distance r. Because of this choice our approach slightly differs from the standard one using the luminosity distance (see [2,3] and references). If metric is expanded into inverse powers of r almost all Einstein equations can be solved in an algebraic way with respect to metric coefficients. Only the mass aspect M and the angular momentum aspect L A undergo some linear differential constraints on I + . All free data are defined on the scri. The constraints on M and L A reduce to a single Laplace-Beltrami equation in special coordinates. If metric tends asymptotically to the (anti) de Sitter solution this equation implies preservation of the generalized Bondi energy and the oscilatory (for Λ < 0) or exponential (for Λ > 0) behaviour of the linear momentum.

Recursive solvability of the Einstein equations
Let (M, g) be a partial conformal compactification of spacetime (M ,g) with a boundary I + = R × Σ foliated by 2-dimensional spacelike surfaces. In the standard way [1] in this neighbourhood we introduce a system of the Bondi-Sachs coordinates u, Ω = 1/r, x A such that whereĝ (the hat denotes value on I + ). We assume that components g µν admit the Taylor expansion in Ω of any order. The physical metric is given bỹ whereg 00 ,g AB are of the order r 2 andg 0A = O(r). We impose on (3) the Einstein equations with cosmological constant Λ = 0 In terms of the compactified metric they are where and |µ denotes the covariant derivative defined by g. Assume thatR µν + Λg µν is finite on I + . Then Y µν must be also finite. In order to avoid a second order pole at Ω = 0 one has to assume that Thus, metric induced by g on the scri readŝ and I + is spacelike if Λ > 0 and timelike if Λ < 0. For the (A)dS metric there isĝ AB = −s AB . Still we are free to change coordinates u, x A provided thatĝ preserves form (8) modulo a conformal factor. It means that we can impose one condition onĝ AB , e. g. we can try to obtain with a prescribed function f . For instance, if I + = R × S 2 , where S 2 is the 2-dimensional sphere, a natural condition would be Equation (10) is a reminiscent of the Bondi luminosity gauge det g AB = det s AB .
Regularity of Y 1A yields A lack of singularities in Y AB is equivalent to the relation between expansion coefficients of g AB All other components of Y µν are nonsingular at Ω = 0 under conditions (7)-(13).
Unlike in the case Λ = 0, equation (13) defines n AB in terms ofĝ AB and a. This relation can be further simplified by means of a shift of r leading to We will not assume (15) at this stage since another condition where n =ĝ AB n AB , may be more convenient. Note that (16) is equivalent to thanks to (13). Note that under (9) conditions (15) and (16) are equivalent. We will test usefulness of the above conditions after obtaining field equations.
Folowing [1] we can reduce a number of the Einstein equations (4) by means of the Bianchi identity. To this end it is sufficent to replaceR µν byR µν + Λg µν andG µν byG µν − 1 2 Λg µν in equation (24) in [1]. In this way one obtains a minimal system of equations given bỹ and where the subscript (k) denotes coefficient of order k in an expansion with respect to Ω. EquationR 11 = 0 andR 1A = 0 are similar in character to those for Λ = 0.
where brackets on the r. h. s. denote expressions depending on functions within the bracket. For k = 0 equation (21) reads EquationR (1) 0A . It is equivalent to where operations on indices and covariant derivatives are defined byĝ AB . For Λ = 0 we used in [1] the topology of I + = R × S 2 to show that (23) yields For Λ = 0 equation (24) follows algebraically from (R AB + Λg AB ) (0) = 0. The latter equation also implies whereR is the Ricci scalar ofĝ AB . From the point of view of equations (24) and (25) the most convenient gauge condition is (16). In this gauge n AB is proportional to the traceless part of the exterior curvature of the section u = const of I + and p AB is proportional to metric of this section. Equation (R AB + Λg AB ) (k) = 0 with k = 1 does not contain any new information. For k ≥ 2 it can be splitted into its trace (with respect toĝ AB ) and a traceless part. The trace part is equivalent to the following composition of equationsĝ It allows to obtain g (k+2) 00 in terms of lower order coefficients The traceless part of (R AB + Λg AB ) (k) = 0 yields the traceless part of g Note that in the case Λ = 0 instead of (28) one obtains an expression forg in terms of lower order coefficients [1]. Thus, (28) is the second equation, after (13), which makes a qualitative difference between Λ = 0 and Λ = 0. For Λ = 0 there is no need of initial data for g AB . Moreover, tensor The last equations to consider are (19). They are equivalent to equations 11,A = 0 (31) having the following structure Given a,ĝ AB and N AB equations (32) and (33) constitute a conjugate system of linear equations for M and L A . Condition (9) allows to simplify them to where f and h A are known functions. Let us introduce function M ′ such that Then (35) yields and (34) becomes where∆ is the covariant Laplace operator with respect toĝ AB . Thus, equations (32) and (33) where coefficients are defined in the following way: • u-dependent metricĝ AB and a traceless (with respect toĝ AB ) tensor N AB can be arbitrarily chosen up to a gauge condition, e.g. (9).
• Function a is arbitrary but it can be gauged away, e.g. by means of (17).
• Components n AB , p AB , N , q A and b are defined by a andĝ AB according to (13) The main difference between present situation and that for Λ = 0 (see Proposition 2.1 in [1]) is that now all free data are located at I + . It is interesting that for Λ < 0 there is no need of the Cauchy data (I + is timelike). Evolution of the Cauchy data given on I + for Λ > 0 was first obtained by Friedrich [4].

Solutions with the (A)dS boundary metric
If metricg tends asymptotically to the (anti) de Sitter solution then In this case it follows from equations (7)-(13) and (20)-(25) that in the gauge a = 0 there isg The r. h. s. of equations (34) and (35) is given by Let us introduce a tensor N ′ AB such that Equation (37) yields Using decompositionL we can incorporate f into M ′ and h into N ′ AB except the dipole component of h (see Lemma 3.1 in [1]). Thus, without loss of generality we can assume that where Y lk are spherical harmonics and α k = const. In spherical coordinates θ, ϕ related to the axis defined by α k one obtains expressioñ which is the angular momentum term known from the Kerr solution. In view of (49) and (51) equation (34) transforms into where ∆ is the standard Laplace operator on the sphere. If we integrate equation (53) over the sphere we obtain the known result The l. h. s. of (54) is, modulo 4π, rather unique candidate for the total energy in this case. One can interpret (54) as a lack of gravitational radiation. If we integrate (53) with Y 1k we get equation The Bondi linear momentum P k is proportional to the time derivatives of M ′ k , so it satisfies equation Hence, it either oscilates when u changes (for Λ <) or behaves in an exponential way (for Λ > 0). In the second case its square must exceed the total energy either for increasing or decreasing u, so a reasonable physical assumption would be M k = 0. Higher moments of M ′ satisfy nonhomogeneous equations coming from (53). Their solutions are defined in quadratures up to oscillatory or exponential functions depending on sign of Λ. An alternative approach to equation (53) is to treat it as an equation for tensor N ′ AB . We can represent this tensor by two scalar functions f and h [1] Substituting (58) into (53) yields Function f exists if conditions (54) and (56) are satisfied. Then we can write Formula (60) can be realised by decomposition of M ′ into Y lk . In this approach functions M ′ and h are arbitrary modulo conditions (54) and (56). We summarize above results in the following proposition.

Stationary merics
If metric considered in section 2 admits a timelike Killing vector K then we can transform it to the form (39) with u-independent coefficients but we cannot assume (2) (see Section 3 in [1]) The KIlling vector is given by K = ∂ 0 . Let us assume (14) and expansions with coefficients depending on coordinates x A . Tensor (6) is finite on the boundary iffĝ and where indices A, B are raised by means ofĝ AB . Sinceĝ 0Aĝ A 0 ≤ 0 and we exclude Λ = 0 it follows from (63)-(65) and timelike property of K = ∂ 0 that Thus, metrics are of the AdS type. It is convenient to shift coordinate r in order to obtain Equation (65) is then equivalent to the following relations defining n AB and a Thus, again n AB is proportional to the traceless part of the exterior curvature of the section u = const of I + . Transformation of coordinates x A allows us to obtain where γ is a positive function of x A . Still there is a freedom of supertranslation which can be used to impose a condition onĝ 0A , γ or higher order coefficients in metric g. All the Einstein equations except (19) can be solved algebraically as in the general case in Section 2. Equations (19) are now a complicated system of constraints on M , L A , γ,ĝ 0A and N AB , which is linear in M , L A and N AB . The space of stationary solutions is much bigger than in the case Λ = 0 [1]. In general, solutions do not tend asymptotically to the AdS metric. It is so if the boundary metriĉ is conformally equivalent tô Metric (73) is conformally flat (note that it coincides with the conformal compactification of the 3-dimensional Minkowski metric). Thus, metric (72) must be also conformally flat. Since K = ∂ u is the Killing field of (72) it is a conformal Killing vector of the flat metric. Such vectors compose the 10-dimensional algebra so(2, 3). Given one of them in terms of the Cartesian coordinates of flat space one can find (at least in principle) coordinates u, x A such that K = ∂ u . Writing flat metric in these coordinates shows what metrics (72) are available if the physical metricg is asymptotically AdS (in transformed coordinates).

Summary
In this paper we considered the Einstein metrics admitting a conformal compactification with smooth scri. For nonvanishing cosmological constant the scri is either spacelike (Λ > 0) or timelike (Λ < 0). Metrics can be transformed to the Bondi-Sachs form with components which can be expanded into powers of a radial distance. The Einstein equations imply that the expansion coefficients can be expressed by coefficients of lower order (see Proposition 2.1). Beside these recursive formulas there are 3 differential conditions constraining the analog of the Bondi mass and angular momentum aspect. For a special foliation of the scri they can be reduced to one second order equation for one function. All free data are located on the scri. Asymptotic form of metric simplifies considerably if it tends to the (anty) de Sitter solution (see Proposition 3.1). In this case the total energy calculated according to the Bondi prescription is constant and the total momentum oscilates or behaves exponentially.
Finally we considered solutions with the timelike Killing field. In this case, in contrary to Λ = 0, there is no big difference in solving the Einstein equations with respect to the general case. However, existence of stationary solutions with nonvanishing asymptotic twist (thenĝ 0A = 0) can be important for a notion of the total energy in general case with Λ < 0. We expect that a crucial ingredient in any definition of the energy should be the mass aspect M related to coordinates such that equations (8) and (11) are satisfied. If the timelike Killing field K is present then transformation between coordinates used in (8) and those in (72) must depend on time u. It seems unavoidable that the total energy also depend on time, what seems unacceptable in a stationary spacetime.