Asymptotically Flat Initial Data with Prescribed Regularity at Infinity

S. Dain

Max-Planck-Institut für Gravitationsphysik

Albert-Einstein-Institut, Golm. Cargse July 29 - August 10, 2002 Initial data set for the Einstein equation An initial data set for the Einstein vacuum equations is a triplet (S̃, h̃_{ab}, Ψ̃_{ab} ) where – S̃ : 3-dimensional manifold.

– h̃_{ab} : Riemannian metric.

– Ψ̃_{ab} : symmetric 2-tensor. Which satisfy the constraint equations on S̃

D̃^{a}Ψ̃_{ab} − D̃_{a}Ψ̃ = 0,

R̃ + Ψ̃^{2} − Ψ̃_{ab} Ψ̃^{ab} = 0,

where

– D̃_{a} : covariant derivative of h̃_{ab}.

– R̃ : Ricci scalar of h̃_{ab}.

– Ψ̃ = h̃^{ab}Ψ̃_{ab} .

Asymptotically flat initial data set

We say that an initial data set is asymptotically flat if the complement of a compact set in S̃ is diffeomorphic to the complement of a ball in ℝ^{3}, and there exists a coordinate system x̃^{j} in a neighborhood of infinity such that, in this coordinates,

h̃_{ij} = (1 + \frac{2m}{r̃})δ_{ij} + O(r̃^{−2}),

Ψ̃_{ij} = O(r̃^{−2}),

as

r̃ = ( 3 ; ∑ ; j=1 (x̃^{j})^{2})^{1/2} → ∞.

Where the constant m is the mass of the data, and δ_{ij} is the flat metric.

The Problem we want to solve

We want to consider the problem of the existence of a class of asymptotically flat initial data for which the higher order terms of h̃_{ab} and Ψ̃_{ab} have an asymptotic expansion in powers of r̃ of the form

h̃_{ij} ∼ (1 + \frac{2m}{r̃})δ_{ij} + _{k≥2}^{∑} \frac{h̃_{ij}^{k}}{r̃^{k}} ,

Ψ̃_{ij} ∼ _{k≥2}^{∑} \frac{Ψ̃_{ij}^{k}}{r̃^{k}} ,

where h̃_{ij}^{k} and Ψ̃_{ij}^{k} are smooth functions of x̃^{j}/r.

Maximal initial data

If we assume that

Ψ̃ = 0,

then the constraint equations reduce to

D̃^{a}Ψ̃_{ab} = 0,

R̃ − Ψ̃_{ab} Ψ̃^{ab} = 0.

After a conformal rescaling

h̃_{ab} = θ^{4}h_{ab}, Ψ̃_{ab} = θ^{−2}Ψ_{ab}, we have

D^{a}Ψ_{ab} = 0 on S̃,  (1)
(D_{b}D^{b} − \frac{1}{8}R)θ = −\frac{1}{8}Ψ_{ab}Ψ^{ab}θ^{−7}  on S̃. (2)

Conformal compactification of the initial data

Let S be the compact manifold defined by S = S̃ ∪ {i}, and define coordinates near the point i by the inversion

x^{i} = \begin{matrix} & x̃^{i} \\ & r̃^{2} \end{matrix}, r = ( 3 ; ∑ ; j=1 (x^{j})^{2})^{1/2} = \frac{1}{r̃} .

Then the asymptotically flat condition for the initial data are

Ψ_{ab} = O(r^{−4})  as r → 0,  (3)
lim_{r→0} rθ = 1 .  (4)

Our result

B_{a} : small ball center at i

Definition 1

E^{∞}(B_{a}) = {f = f_{1} + rf_{2} : f_{1}, f_{2} ∈ C^{∞}(B_{a})}

Theorem 1 Let h_{ab} be a smooth metric on S with positive Ricci scalar R. Assume that Ψ_{ab} is smooth in S̃ and satisfies on B_{a}

r^{8}Ψ_{ab}Ψ^{ab} ∈ E^{∞}(B_{a}).  (5)

Then there exists on S̃ a unique solution θ of equation (?? ), which is positive, satisfies (?? ), and has in B_{a} the form

θ = \frac{θ̂}{r}, θ̂ ∈ E^{∞}(B_{a}), θ̂(i) = 1.

General solutions of ∂_{a}Ψ^{ab} = 0

(n^{a}, m^{a},m̄^{a}): orthonormal, complex, tetrad.

ξ = \frac{1}{2}r^{3}Ψ_{ab}n^{a}n^{b}, η_{1} = √2r^{3}Ψ_{ab}n^{a}m^{b},

μ_{2} = r^{3}Ψ_{ab}m^{a}m^{b}.

λ arbitrary complex function and λ_{2} = ð^{2}λ. Then, the general solution of ∂_{a}Ψ^{ab} = 0 is given by

ξ = ð̄^{2}λ_{2}^{R} + A + r Q + \frac{1}{r} P,
η_{1} = −2 r ∂_{r} ð̄λ_{2}^{R} + ð̄λ_{2}^{I} + r ðQ − \frac{1}{r} ðP + i ðJ,
μ_{2} = 2 r ∂_{r}(r ∂_{r} λ_{2}^{R}) − 2 λ_{2}^{R} + ðð̄λ_{2}^{R} − r ∂_{r} λ_{2}^{I}.

where

P = \frac{3}{2}P^{a}n_{a}, Q = \frac{3}{2}Q^{a}n_{a}, J = 3J^{a}n_{a},

and A, P^{a}, Q^{a}, J^{a} are arbitrary constants.

Ψ_{P}^{ab} = \frac{3}{2r^{4}} (−P^{a}n^{b} − P^{b}n^{a} − (δ^{ab} − 5n^{a}n^{b}) P^{c}n_{c}) ,
Ψ_{J}^{ab} = \frac{3}{r^{3}}(n^{a}ϵ^{bcd}J_{c}n_{d} + n^{b}ϵ^{acd}J_{c}n_{d}),
Ψ_{A}^{ab} = \frac{A}{r^{3}} (3n^{a}n^{b} − δ^{ab}),
Ψ_{Q}^{ab} = \frac{3}{2r^{2}} (Q^{a}n^{b} + Q^{b}n^{a} − (δ^{ab} − n^{a}n^{b}) Q^{c}n_{c}) .

λ gives the “quadrupolar” of Ψ^{ab}, it doesn’t contribute to the linear or angular momentum of the data.

Theorem 2 If rλ ∈ E^{∞}(B_{a}) and P^{a} = 0, then r^{8}Ψ_{ab}Ψ^{ab} ∈ E^{∞}(B_{a}).