Proof of the first part: Constructing the data

In this section we shall prove the first part of TheoremA.1, i.e. the statement that the solution is determined from seed data with the bounds (439) holding onCu0∪Cv0. We will focus on establishing the latter bounds fork=0. The statement for arbitrary angular commutations is then easily obtained from the fact that r /∇ commutes with Ω∇/₃ and Ω∇/₄ and has good commutation properties with angular derivatives in the sense that

|[r /∇_A, r /∇_B]ξ|6C|ξ|.

To obtain the remaining tangential derivatives and the transversal derivatives, we will use the null structure and Bianchi equations directly in conjunction with an inductive procedure which is outlined below.

For the remainder of the proof, we will allow ourselves to drop thesubscript from all quantities as well as the “in” and “out” from

(1)

/g, as it will be clear from the context which cone we are on.

Elliptic equations on the horizon sphereS_∞,v²

0

We first note that the seed data determines on the horizon sphere:

• ⁽¹⁾σ uniquely from (146);

•

(1)

Kuniquely from the fact that

(1)

ˆ/ g and

(1)

p/g/p

/gare part of the seed data and (221);

• ⁽¹⁾% uniquely from (147);

•

(1)

β uniquely from (145);

• ⁽¹⁾η uniquely from⁽¹⁾η=−⁽¹⁾η+2∇/_AΩ⁻¹

(1)

Ω (equation (134)).

Transport equations along Cv₀: Part I We now integrate our seed data fromS_∞,v²

0 along the coneC_v0.

AlongCv₀, the tensor

(1)

ˆ/

g is part of the seed data. Note that this determines uniquely, alongC_v0, the quantities Ω⁻¹

(1)

χbvia (131) and Ω⁻²⁽¹⁾α via (139).

Next, from (137), we have

∂_u

0, the above ODE determines

(1)

(Ω trχ) uniquely alongC_v0. Note that now

(1)

p/g/p /g is now uniquely determined alongC_v0 by (131).

Since

(1)

β was already uniquely determined on S_∞,v²

0 above, we can integrate the Bianchi equation (156) written as

Ω∇/₃[r⁴Ω⁻¹

(1)

β] =−r⁴div/ ⁽¹⁾α

alongC_v0, which, since the right-hand side is uniquely determined from seed data, de-termines

(1)

β uniquely onCv₀. The Bianchi equations (151) and (154) read as ODEs along Cv₀ now uniquely determine ⁽¹⁾% and⁽¹⁾σ, since the initial value of these quantities onS²_∞,v

0

were already determined by seed data above. Similarly,⁽¹⁾η is determined uniquely from (142). Using (134) we conclude that⁽¹⁾η is also uniquely determined alongCv₀. We finally use (136) written as

∂u(r

(1)

(Ω trχ)) =Q,

withQ uniquely determined onCv₀ to determine uniquelyr

(1)

(Ω trχ) along Cv₀. Noting that

(1)

Kis uniquely determined by (147) alongC_v0, we conclude that we have determined all geometric quantities of a solution S uniquely along Cv₀ except

(1)

Estimates on the sphereS_u²

0,v0

We note that, since

(1)

• the quantity⁽¹⁾α is determined uniquely on S_u²

0,v₀ by (139);

• the quantity⁽¹⁾ω is determined uniquely onS_u²_0,v0 by (134).

Hence, by (145) and the fact that we already determined⁽¹⁾η and

(1)

(Ω trχ) uniquely on S_u²

0,v₀ above, the quantity

(1)

β is also uniquely determined onS²_u

0,v₀.

Transport equations along Cv₀: Part II

We can now determine the missing quantities

(1) that they have been determined uniquely on S_u²

0,v₀: Use (133) to determine

(1)

b, (140) to determine

(1)

χ, (150) to determineb

(1)

β, (148) to determine ⁽¹⁾α and finally ⁽¹⁾ω from (144) all uniquely alongCv₀.

We have determined all geometric quantities in terms of seed data and uniform bounds for all quantities which extend smoothly to the horizonH⁺alongCv₀.

Transport equations along C_u0

We finally turn to the conjugate coneCu₀. Recall that all geometric quantities have been determined uniquely on the sphereS_u²_0,v0, and that moreover the seed data

(1)

We now determine all quantities alongC_u0. Starting with (137), we have

∂v

We see that the right-hand side is integrable by the asymptotic flatness condition there-fore, producing the bound

|r²

(1)

(Ω trχ)|< C,

whereC can be computed explicitly from the seed data (and depends on 0<s61). From (131) and the asymptotic flatness condition on

(1)

b, we immediately conclude

Of course, the same bounds hold for arbitrary many angular derivatives r /∇ of these quantities.

Remark A.2. One also sees that the quantities

(1) derivativesr /∇ of these quantities) have smooth limits at null infinity. By this, we mean that there exists a symmetric 2-tensor

(1) any spherical coordinate patch

∂u

and

vlim!∞

(1)

ˆ/ gAB

p/g =(

(1)

ˆ/ g∞)AB

p/g , lim

v!∞

(1)

p/g p/g=

(1)

p/g

∞

p/g , in the limit along the coneC_u0.

We now note that

(1)

β is uniquely determined from (149) producing the uniform bound

|r^3+s

(1)

β|< C.

Similarly, we can determine ⁽¹⁾η and⁽¹⁾η. For this, we note the equations (following from (142) and (134)):

∇/₄(r²⁽¹⁾η) =∇/₄(r²⁽¹⁾η)−∇/₄(r²⁽¹⁾η+r²⁽¹⁾η)

= 2Ω

r(⁽¹⁾η+⁽¹⁾η)−r²

(1)

β−2∇/₄∇/_Ar²(Ω⁻¹

(1)

Ω) =−r²

(1)

β−2r²∇/⁽¹⁾ω .

(440)

Note that the right-hand side is uniquely determined alongCu₀ and integrable, leading to⁽¹⁾η being uniquely determined alongCu₀ with the uniform bound

|r²⁽¹⁾η|< C.

We also have, by the relation (134), that ⁽¹⁾η is uniquely determined along C_u0 with the uniform bound

|r⁽¹⁾η|< C.

We turn to (151) and (153), which clearly determine ⁽¹⁾% and⁽¹⁾σ uniquely alongCu₀ with the uniform bounds

|r³⁽¹⁾%|+|r³⁽¹⁾σ|< C.

In fact, since we can repeat the above procedure commuting with angular derivatives, we also have in particular

|r /∇(r³⁽¹⁾%)|+|r /∇(r³⁽¹⁾σ)|< C.

It is now easy to see that (155) determines

(1)

β uniquely with the uniform bound

|r²

(1)

β|< C,

and that (141) determined

(1)

χbuniquely with the uniform bound

|r

(1)

χ|b < C.

Similarly, (157) determines⁽¹⁾α uniquely with the uniform bound

|r⁽¹⁾α|< C.

For the remaining components, note that (144) determines⁽¹⁾ω uniquely with the uniform bound

|⁽¹⁾ω|< C.

The quantity

(1)

(Ω trχ) is determined uniquely from (136) with the bound

|r

(1)

(Ω trχ)|< C, where we have used that we can write (136) as

∂_v(r

(1)

(Ω trχ)) = RHS

with the right-hand side uniquely determined and integrable alongC_u0. Finally, equation (147) determines

(1)

Kuniquely along C_u0, with the uniform bound

|r²

(1)

K|< C.

In fact, by the remark above and the fact that the right-hand side in ∇/₄(r²

(1)

K)=RHS is integrable, the weighted linearised Gaussian curvaturer²

(1)

K extends smoothly to null infinity.

We note once more that the same bounds hold for arbitrarily many angular deriva-tivesr /∇of the quantities estimated either by trivial commutation with angular momen-tum operatorsΩior, if the reader prefers, tensorial commutation withr /∇and inductively estimating lower-order terms.

We conclude the proof by estimating the remaining weighted tangential derivatives r /∇₄ and the transversal derivative∇/₃on the coneC_u0. The procedure onC_v0 is analo-gous (but easier, since there are no weights at null infinity) and is hence omitted.

For the remaining weighted tangential derivativer /∇₄, we use

• equation (149) pointwise forr /∇₄

(1)

β;

• equation (151) pointwise forr /∇₄⁽¹⁾%;

• equation (151) pointwise forr /∇₄⁽¹⁾σ;

• equation (155) pointwise forr /∇₄

(1)

β;

• equation (157) pointwise forr /∇₄⁽¹⁾α;

Dans le document The linear stability of the Schwarzschild solution to gravitational perturbations (Page 196-0)