Initial data for stationary space-times near space-like

infinity

S. Dain

Max-Planck-Institut für Gravitationsphysik

Albert-Einstein-Institut, Golm. Cargse July 29 - August 10, 2002

The quotient manifold X̃

ξ^{a}: timelike Killing vector. λ, ω : norm and twist of ξ^{a}. (M̃, g̃_{ab}): spacetime.

X̃: collection of all trajectories of ξ^{a}. X̃ = ℝ^{3} \ B_{R}: B_{R} is a ball which contains the “sources”.

g̃_{ab} = ξ_{a}ξ_{b} − \frac{γ̃_{ab}}{λ}

γ̃_{ab}: intrinsic metric on X̃. Reduction: write the field equations in terms of quantities intrinsic to X̃.

(M̃, g̃_{ab}) → (X̃, γ̃_{ab})

In the static case (ω = 0), X̃ is one of the hypersurfaces of M̃ everywhere orthogonal to ξ^{a}:

When ω ≠ 0, there is no natural way of introducing such surface on M̃.

Stationary spacetimes: initial data

The metric can be written as

g̃ = λ(dt + β_{i}dx̃^{i}) − λ^{−1} γ̃_{ij}dx̃^{i}dx̃^{j},

where λ, β_{i}, and the Riemannian metric γ̃_{ij} depend only on the spatial coordinates x̃^{k}.

The intrinsic metric of the hypersurface S̃ defined by t = constant is

h̃_{ij} = λ^{−1} γ̃_{ij} − λβ_{i}β_{j},

The vector β^{k} is the essential piece in the translation from the quotient manifold X̃ to the Cauchy slice S̃.

In the static case β^{k} = 0 and X̃ = S̃.

Vacuum equations

Hansen potentials:

ϕ̃_{M} = \frac{1}{4λ}(λ^{2} + ω^{2} − 1),
ϕ̃_{S} = \frac{1}{2λ}ω,
ϕ̃_{K} = \frac{1}{4λ}(λ^{2} + ω^{2} + 1).

These functions satisfy the relation

ϕ̃_{M}^{2} + ϕ̃_{S}^{2} − ϕ̃_{K}^{2} = −\frac{1}{4}.

Denote by ϕ̃ any of the functions ϕ̃_{M}, ϕ̃_{S}, ϕ̃_{K}. The unknowns are γ̃_{ab}, ϕ̃_{M}, ϕ̃_{S}. The vacuum field equations are:

Δ̃ ϕ̃ = 2R̃ϕ̃
R̃_{ij} = 2(D̃_{i}ϕ̃_{M} D̃_{j}ϕ̃_{M} + D̃_{i}ϕ̃_{S}D̃_{j}ϕ̃_{S} − D̃_{i}ϕ̃_{K}D̃_{j}ϕ̃_{K}),

where R̃_{ij} is the Ricci tensor of γ̃_{ij} and Δ = D̃^{i}D̃_{i}.

Conformal compactification of X̃

We assume that (X̃, γ̃_{ij}, ϕ̃_{M}, ϕ̃_{S}) is asymptotically flat in the following sense. There exists a manifold X, consisting of X̃ and an additional point i; such that:

(i) For some real constant B^{2} > 0 the conformal factor

Ω = \frac{1}{2}B^{−2}[(1 + 4(ϕ̃_{M}^{2} + ϕ̃_{S}^{2}))^{1/2} − 1]

is C^{2} on X and satisfies

Ω(i) = 0, D_{i}Ω(i) = 0

at the point i.

(ii) γ_{ij} = Ω^{2} γ̃_{ij} extends to a C^{4,α} metric on X and

D_{j}D_{k}Ω(i) = 2γ_{jk}(i).

(iii) Ω is C^{2,α} on X.

8

Define the rescaled functions ϕ = ϕ̃/√Ω .

It is possible to find an elliptic system of equations for the rescaled quantities γ_{ab}, ϕ_{M}, ϕ_{S} which is regular at i.

Theorem 1 (Beig-Simon) For any asymptotically flat solution (γ̃_{ij}, ϕ̃_{M}, ϕ̃_{S}) of the stationary, vacuum, field equations there exists a chart defined in some neighborhood of i in X such that (γ_{ij}, ϕ_{M}, ϕ_{S}, Ω) are analytic.

(γ_{ij}, ϕ_{M}, ϕ_{S}) are the “natural” fields on X. x^{j}: γ-normal coordinates x^{j} centered at i. γ_{ij}, ϕ_{M}, ϕ_{S}: are analytic with respect to x^{j}. r = (∑_{j=1}^{3} (x^{j})^{2})^{1/2} : radial coordinate. r, √Ω : are not analytic at i. λ, ω, ϕ_{K} and β^{i} are not analytic at i:

ϕ_{K} = \frac{f_{K}}{√Ω} , f_{K} = √Ω(ϕ_{M}^{2} + ϕ_{S}^{2}) + 1/4 ,

where f_{K} is analytic;

ω = \frac{ϕ_{S}}{(ϕ_{K} − ϕ_{M})} , λ = \frac{1}{2√Ω (ϕ_{K} − ϕ_{M})} .

Ω = r^{2}f_{Ω} ,

where f_{Ω} is analytic and f_{Ω} (i) = 1. The non-analytic type of behavior of these fields is given by the radial coordinate r.

Conformal compactification of the initial

data S̃

The conformal metric on S is given by:

h_{ij} = γ_{ij} − Ω^{2}λ^{2}β_{i}β_{j}.

How differentiable is h_{ab} at i?

The vector β^{i}

β^{i} satisfies the following equation

D̃_{i}ω = −λ^{2}ϵ̃_{ijk}D̃^{j}β^{k}.

We have the freedom

β_{k} → β_{k} + ∂_{k}f,

which correspond to a change in the foliation

t → t − f.

Lemma 1 There exists a solution β_{i} which, in normal coordinates x^{i}, has the following form

β_{i} = β_{i}^{1} + \frac{β_{i}^{2}}{r} ,

where β_{i}^{1} and β_{i}^{2} are analytic functions of x^{i} given by

β_{i}^{1} = ϵ_{ijk}f_{1}^{j}x^{k}, β_{i}^{2} = ϵ_{ijk}f_{2}^{j}x^{k},

where f_{1}^{i} and f_{2}^{i} are analytic. In particular, this implies that β_{i}x^{i} = 0.

Theorem 2 Let β^{i} be given by the previous lemma. In some neighborhood of i, the metric h_{ij} has the form

h_{ij} = h_{ij}^{1} + r^{3}h_{ij}^{2},

where h_{ij}^{1} and h_{ij}^{2} are analytic.

Remarks:

i) h_{ij} ∈ W^{4,p}, p < 3, because r^{3} ∈ W^{4,p}, p < 3. It implies, in particular, that the metric is in C^{2,α}.

ii) The metric is analytic in the spherical coordinates r, θ, φ.

Application: stationary spacetimes have analytic scri

g̃ = λ(dt + β_{i}dx̃^{i}) − λ^{−1} γ̃_{ij}dx̃^{i}dx̃^{j},

where λ, β_{i}, and γ̃_{ij} are analytic functions of the spherical coordinates 1/r̃, θ, φ. Using this we can prove the following:

Theorem 3 (Damour-Schmidt) Every stationary, asymptotically flat, vacuum spacetime admits an analytic conformal extension through null infinity.