The main theorems

We give now our first rough statements of the main theorems of this paper. These will correspond to the more precise statements given in the body of the paper in§10.

2.2.1. Boundedness and decay of gauge invariant quantities: the Regge–Wheeler and Teukolsky equations

The first two main theorems correspond to boundedness and decay statements for the gauge invariant quantities α, ψ and P in §2.1.5, and their corresponding underlined quantities. We will state these as independent statements for general solutions of the Regge–Wheeler and Teukolsky equations.

The first statement concerns Regge–Wheeler. The precise statement is Theorem1 in§10.1. A rough formulation is as follows.

Theorem 1. (Rough version) Let P be a solution of the Regge–Wheeler equation (37) arising from regular data described in §2.1.5, and define the rescaled quantity Ψ=

r⁵P. Then, the following statements hold:

(a) The quantity P remains uniformly bounded with respect to an r-weighted energy norm in terms of its initial flux

F[Ψ].F0[Ψ]. (43)

(b) The quantity P decays to zero in the following quantitative senses: An r-weighted integrated decay statement

I[Ψ].F0[Ψ] (44)

holds, as well as polynomial decay of energy fluxes and pointwise polynomial decay.

The flux quantities F referred to in the above theorem are suprema over integrals on constantuandv null cones (Cu andCv, respectively). The integralIis a spacetime intergal over the shaded region of Figure 2. Both will be explained in §10.1.1. The statements (a) and (b) above are in fact just special cases of an r^p hierarchy of flux bounds and integrated decay statements (cf.§2.3.3below). Although the precise form of the statements obtained in Theorem1is new, we again note previous decay-type results for the Regge–Wheeler equation in [32], [7], [29].

Using the above theorem, we can now obtain a result for general solutions of the Teukolsky equation. The precise statement is Theorem2in §10.2. A rough formulation is as follows.

Theorem2. (Rough version) Let αbe a solution of the spin+2Teukolsky equation (32) arising from regular data described in §2.1.5,and let ψ and Ψ=r⁵P be the derived quantities defined by (33)and (35). Then the following statements hold:

(a) The triple (Ψ, ψ, α) remains uniformly bounded with respect to an r-weighted energy norm in terms of its initial flux

F[Ψ, ψ, α].F0[Ψ, ψ, α].

(b) The triple (Ψ, ψ, α) decays to zero in the following quantitative senses: An r-weighted integrated decay statement

I[Ψ, ψ, α].F0[Ψ, ψ, α]

holds, as well as polynomial decay of suitable fluxes and pointwise polynomial decay.

A similar statement holds for solutions αof the spin −2Teukolsky equation and its derived quantities ψand P.

Again, the precise versions of the quantitiesFandIreferred to in the above theorem will be explained in §10.2.1. As noted above, even a boundedness statement for the Teukolsky equation on Schwarzschild was not previously known.

Applied to the full system of linearised gravity, the above yields the following state-ment.

Corollary. Let S be a solution of the full system of linearised gravity arising from regular, asymptotically flat initial data described in §2.1.6. Then,Theorem 2 applies to yield boundedness and decay for the gauge invariant hierarchy (

(1)

Ψ,

(1)

ψ,⁽¹⁾α)and (

(1)

Ψ,

(1)

ψ,⁽¹⁾α).

See Corollary10.1for the precise statement.

We next turn to the problem of boundedness for all quantities (30) associated withS, not just the gauge-invariant ones.

2.2.2. Boundedness of the full system of linearised gravity

The third main theorem is a (quantitative) boundedness statement for the full system of linearised gravity, embodying (a) of the main theorem of the introduction.

The most fundamental statement is again at the level of an energy flux, now aug-mented byL² estimates on spheres, and must be expressed with the help of the initial-data normalised solutionS_∨ discussed already in§2.1.7. The precise statement is formu-lated as Theorem3in §10.3. A rough statement is as follows.

Theorem 3. (Rough version) Let S be a solution of the full system of linearised gravity arising from regular, asymptotically flat initial data described in §2.1.6, and let

∨G be the pure gauge solution such that

S∨ =S+

∨G

defined by (38) is normalised to initial data. Then, the solution

S∨ remains uniformly bounded with respect to a weighted energy norm F, augmented by the supremum of a weighted L²-norm on spheres (denoted D),in terms of its initial norm:

D[S_∨ ]+F[S_∨ ].D0[S_∨ ]+F0[S_∨ ]. (45) All quantities (30) of

S∨ are controlled in L² on suitable null cones or spheres by the above norms up to their projections to the `=0 and `=1 modes.

Moreover, there is a unique linearised Kerr solution Km,si, computable explicitly from initial data,such that

S∨ ⁰=

S∨ −Km,si has vanishing `=0and`=1modes,and thus the above L² control is coercive for all quantities (30)associated with

S∨ ⁰.

Again, the precise form of the fluxFand theL^∞(L²) normDis contained in§10.3.1.

We emphasise that the bound (45) and resultant control of the restriction of the solution to angular frequencies`>2 is obtained independently of identifying the correct linearised Kerr solutionKm,s_i.

From the above flux bounds we immediately obtain, by standard Sobolev inequali-ties, pointwise bounds on all quantities (30) associated withS_∨.

Corollary. For sufficiently regular, asymptotically flat initial data forS as above, all quantities (30) associated with

S∨ are uniformly bounded pointwise in terms of an initial energy as on the right-hand side of (45), and the parameters m and si of Km,s_i

(explicitly computable from—and thus also bounded by—initial data).

See Corollary 10.2 for a precise statement. Note that these pointwise bounds are againr-weighted bounds.

2.2.3. Decay of the full system of linearised gravity in the future-normalised gauge

The final part of our results is the statement of decay for the full system, embodying (b) of the main theorem of the introduction. For this, we have already discussed in§2.1.7 the necessity of adding a pure gauge solutionG^∧normalised to the event horizon.

The precise decay theorem is formulated as Theorem4in§10.4. A rough statement takes the following form.

Theorem 4. (Rough version) Let S be a solution of the full system of linearised gravity arising from regular,asymptotically flat initial data described in §2.1.6,let

S∨ be as in Theorem 3,and let

S∧= S∨ +G^∧,

defined by (40), be the solution normalised to the event horizon. Then, the pure gauge solution G^∧,and thus, in view of (45),also S^∧,satisfy boundedness statements

D[ G∧]+F[

G∧].D0[S_∨ ]+F0[S_∨ ] and D[ S∧]+F[

S∧].D0[S_∨ ]+F0[S_∨ ], (46)

from which, as in Theorem 3, it follows that all quantities (30)of

G∧(and thus

S∧) are controlled inL²on suitable null cones or spheres,while

S∧moreover satisfies “integrated local energy decay”,schematically

IS^∧ .D0

S∨

+F0

S∨

. (47)

From (47), a hierarchy of inverse-polynomial decay estimates follows for all quan-tities (30) associated with

S∧. As in Theorem 3, these estimates control the quantities (30)of

S∧up to their projections to the `=0and `=1modes.

If Km,si is the linearised Kerr solution of Theorem 3, then S^∧0=S^∧−Km,si has vanishing `=0and `=1modes and the above control is indeed coercive for all quantities associated with S^∧0.

Again, for the precise form of the estimates, see the propositions referred to in the full statement of the theorem in §10.4. In analogy with the corollary of Theorem 3, from the aboveL²bounds we immediately obtain pointwise decay estimates by standard Sobolev inequalities.

Corollary. For sufficiently regular asymptotically flat data for S as above, we have pointwise inverse polynomial decay of all quantities of

S∧given in (30)to those of Km,s_i, in particular quantitative inverse polynomial decay rates for the linearised metric quantities (20)themselves.

See Corollary10.3for the precise statement.