On emerging scarred surfaces for the Einstein vacuum equations

This is a follow up on our previous work in which we have presented a modified, simpler version of the remarkable recent result of Christodoulou on the formation of trapped surfaces. In this paper we prove two related results. First we extend the semi-global existence result, which was at the heart of our previous work, to an optimal range. We then use it to establish the formation of surfaces with multiple pre-scarred angular components.

in fact a stronger, indeed optimal result. We are happy to acknowledge that a related result is stated in theorem 8.1. of [R-T], in a different setting. We would like to thank Reiterer and Trubowitz for drawing our attention and making an effort to explain its formulation to us.
(2) We state, see theorem 2.4, an angular localized version of the global energy estimates for the null curvature components of theorem 1.14. The proof relies on a natural modification of the proof in theorem 1.14 and is discussed in section 5. (3) We give a large class of critical, sufficient conditions on the initial data, which lead to the formation of pre-scarred surfaces. The main result is stated in theorem 2.8. The proof rests on theorem 1.14 as well as on a localized version of the Ricci coefficient estimates in [K-R:trapped]. As mentioned above, the importance of this result is due to the fact that once a significant part of a surface is pre-scarred, it can be deformed to a real trapped surface.
Concerning the new propagation result stated in theorem 1.14, we note that the main new idea is to use, in addition to the small parameter δ > 0, originating in the short pulse method of [Chr:book], a new small parameter ǫ with δ 1/2 ǫ −1 sufficiently small. The parameter δ is used to define scale invariant norms, similar to those we have introduced in [K-R:trapped] but with one important modification. In the main result of [K-R:trapped], for example, the scaling was such that all null curvature components, except the component denoted by α, were bounded (in its scale invariant norms) . The behavior (in the scale invariant norm) of the anomalous component α, on the other hand, was δ −1/2 . Here we choose the scaling with respect to δ such that the scale invariant norm of α is bounded, independent of the second parameter ǫ, and the scale invariant norms of all other curvature components are proportional to ǫ, i.e. small. All results in [K-R:trapped] correspond precisely to the case when ǫ is chosen to be proportional to δ 1/2 . It is quite remarkable that the proof of the stronger propagation result in theorem 1.14 is exactly the same as in [K-R:trapped]. This is surprising, especially considering that the initial data in theorem 1.14 is allowed to be δ − 1 2 ǫ times bigger 3 than that in [K-R:trapped] (as measured in absolute, unscaled norms). In [K-R:trapped] nonlinear non-anomalous interactions were controlled by the scale invariant Hölder estimates In this work the new critical scaling does not generate a small factor of δ 1 2 in such interactions. Instead we have For non-anomalous ψ and φ the scale invariant norms on the right hand side are both of size ǫ and so is the expected value of the left hand side norm. This analysis indicates that with the new scaling the factor δ 1 2 of quadratic interactions is effectively replaced by the independent small parameter ǫ.
3 More precisely all components of the curvature tensor , except α, are δ − 1 2 ǫ times bigger. The α component behaves exactly the same as in [K-R:trapped].
In the result on the formation of a pre-scarred surface we describe a set of initial data which lead to a space-time with a surface containing approximately δ − 1 2 q angular regions of size δ 1 2 q −1 , each of which is pre-trapped for some sufficiently small parameter q.
We start by recalling the framework of double null foliations in which the results of both [Chr:book] and [K-R:trapped] are formulated.
1.1. Double null foliations. We consider a region D = D(u * , u * ) of a vacuum spacetime (M, g) spanned by a double null foliation generated by the optical functions (u, u) increasing towards the future, 0 ≤ u ≤ u * and 0 ≤ u ≤ u * . We denote by H u the outgoing null hypersurfaces generated by the level surfaces of u and by H u the incoming null hypersurfaces generated level hypersurfaces of u.
We write S u,u = H u ∩ H u and denote by H (u 1 ,u 2 ) u , and H (u 1 ,u 2 ) u the regions of these null hypersurfaces defined by u 1 ≤ u ≤ u 2 and respectively u 1 ≤ u ≤ u 2 . Let L, L be the geodesic vectorfields associated to the two foliations and define the null lapse Ω and connection, or Ricci, coefficients, χ, ω, η, η, χ, ω, (1) where e 3 = ΩL, e 4 = ΩL and D a = D e (a) . As usual we decompose the null second fundamental forms χ, χ into their traceless partsχ,χ and traceless parts, or expansions, trχ, trχ. We also introduce the null curvature components, α ab = R(e a , e 4 , e b , e 4 ), α ab = R(e a , e 3 , e b , e 3 ), β a = 1 2 R(e a , e 4 , e 3 , e 4 ), β a = 1 2 R(e a , e 3 , e 3 , e 4 ), Here * R denotes the Hodge dual of R. We denote by ∇ the induced covariant derivative operator on S(u, u) and by ∇ 3 , ∇ 4 the projections to S(u, u) of the covariant derivatives D 3 , D 4 . We note the formulas, We recall also the formula for the Gauss curvature K of S(u, u), As well known, our space-time slab D(u * , u * ) is completely determined (for small values of u * , u * ) by specifying, freely, the traceless parts of the null second fundamental formsχ, respectivelyχ , along the null, characteristic, hypersurfaces H 0 , respectively H 0 , corresponding to u = 0, respectively u = 0, and prescribing trχ together with trχ on S(0, 0). Following [Chr:book] we assume that our data is trivial along H 0 , i.e. assume that H 0 extends for u < 0 and the spacetime (M, g) is Minkowskian for u < 0 and all values of u ≥ 0. Moreover we can construct our double null foliation such that Ω = 1 along H 0 , i.e., We also introduce the notation, where trχ 0 is the flat value of trχ along the initial hypersurface H 0 . We denote by γ the induced metric on the surfaces S(u, u) of intersection between H u and H u . A space-time tensor tangent to S(u, u) is called an S− tensor, or horizontal tensor.
We define systems of, local, transported coordinates along the null hypersurfaces H and H. Starting with a local coordinate system θ = (θ 1 , θ 2 ) on U ⊂ S(u, 0) ⊂ H u , we parametrize any point along the null geodesics starting in U by the the corresponding coordinate θ and affine parameter u. Similarly, starting with a local coordinate system θ = (θ 1 , θ 2 ) on V ⊂ S(0, u) ⊂ H u we parametrize any point along the null geodesics starting in V by the the corresponding coordinate θ and affine parameter u.
We also define the scale invariant norms on the 2 surfaces S = S u,u , We have, We denote the scale invariant L ∞ norm in D by ψ L ∞ (sc) . Remark 1.5. These norms correspond to a different scaling than that introduced in [K-R:trapped]. Indeed in [K-R:trapped] the scale invariant norms were based on the definition of the scale of an horizontal component of scale sc(ψ) = −sgn(ψ) + 1 2 . The norms introduced here would correspond to a new definition of scale give by sc(ψ) = −sgn(ψ). To distinguish between them we denote the old scaling byṡc. Thus, for example, Remark 1.6. With the new scale invariant norms introduced here we have, or, These differ from the situation in [K-R:trapped] where the corresponding estimates (with (sc) replaced by (ṡc)) had an additional power of δ 1/2 on the right.
Curvature norms. We introduce our main curvature norms Also, Remark 1.7. We have included the Gauss curvature K with the null components. Since K = −ρ + 1 2χ ·χ − 1 4 trχtrχ we easily deduce that, Remark 1.8. All curvature norms above have a factor of ǫ −1 in front of them except for α L 2 . These correspond exactly to the anomalous curvature norms of [K-R:trapped].
To rectify the anomaly of α we introduce, as in [K-R:trapped], an additional scale-invariant norm, obtained by evolving an angular disc S ǫ ⊂ S u,0 of radius ǫ relative to our transported coordinates. We define the initial quantity R (0) by, 1.9. Connection coefficients norms. We introduce the Ricci coefficient norms, with the supremum taken over all surfaces and, Remark 1.10. Note that the only norms which do not contain powers of ǫ −1 are the L 2 (sc) (S) norms ofχ andχ . This anomaly is also manifest in the L 4 (sc) (S) norms of the same quantities. These are precisely the same quantities which were anomalous in [K-R:trapped], with respect to theṡc scaling.
To cure the above anomaly we define the auxiliary norms, with S ǫ -an angular subset of S of size ǫ relative to our transported coordinates.
Finally we define the initial data quantity: 1.11. Initial conditions. Define the main initial data quantity, or, in the natural norms, 1.12. Main propagation result. The first result establishes the boundedness of the initial curvature and Ricci coefficent scale invariant norms Proposition 1.13. Assume that the initial data along H 0 is flat and that . Then, for δ 1/2 ǫ −1 and ǫ > 0 sufficiently small we have, with C a fixed super-linear polynomial Also, starting with R (0) < ∞ and δ 1/2 ǫ −1 , ǫ sufficiently small, we have, with C a fixed super-linear polynomial, We can now state our main propagation result.
Remark 2. The additional smallness assumption on δ 1/2 ǫ −1 is due to the lower order terms which appear in some of the calculus inequalities presented in the next section.
In the remaining part of this section we introduce norms for the deformation tensors of the geodesic null generators L, L and rotation vectorfields O and give a short sketch of the proof of theorem 1.14.
1.15. Deformation tensors norms for L, L. If π is the deformation tensor of either L or L we denote by π (s) its null component of of signature s. We now introduce the norms for (L) π and (L) π as follows, with, We introduce also the first derivative norms, We also set, were defined, see section 13 in [K-R:trapped], by parallel transport starting with the standard rotation vectorfields on S 2 = S u,0 ⊂ H u,0 along the integral curves of e 4 . Suppressing the index (i) we have, The only non-trivial components of the deformation tensor π αβ = 1 2 (∇ α O β + ∇ β O α ) are given below: The quantities, H and Z can be assigned signature and scaling, (consistent with those for the Ricci coefficients and curvature components) according to.
Similarly, assigning signatures to all other components of (O) π , we introduce the norms, 1.17. Proof of Main Theorem I. To prove the theorem we start by making a bootstrap assumption on the Ricci coefficient norm O. More precisely we assume that, Based on this assumption we state various preliminary estimates in section 3, which are simple adaptation of results proved in [K-R:trapped]. It is interesting to remark that this is the only place when we need to make a restriction for the size of δ 1/2 ǫ −1 . Using these preliminary estimates we then indicate how, by a simple adjustment of the curvature estimates in [K-R:trapped] we can prove, see section 4, the following.
Theorem 1.18 (Theorem A). There exists a positive constant a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, Next we rely on a theorem which bounds the norms Π, Π and (O) Π, for the deformation tensors of L, L and O, to the Ricci coefficients norms O.
Theorem 1.19 (Theorem B). Under the assumptions δ 1/2 ǫ −1 and ǫ sufficiently small we have, Finally we state the theorem which relates the norms O to the curvature norms R, R.
Theorem 1.20 (Theorem C). Under the assumptions δ 1/2 ǫ −1 and ǫ sufficiently small we have, with Combining theorems B and C with theorem A we deduce, under the bootstrap assumption 34, from which, for ǫ sufficiently small, Thus, back to (37) and using also proposition 1.13, which allows us to remove the bootstrap assumption and confirm the result of the main theorem I.

Formation of pre-scars
Relying on the results of theorem 1.14 we prove a new result concerning the formation of pre-scars. Throughout this section we assume that the assumptions and conclusions of theorem 1.14 hold true.
2.1. Local scale invariant norms. Consider a partition of S 0 = S(0, 0) into angular sectors Λ of a given size |Λ|. Let ( Λ) f (0) be a partition of unity associated to this partition, They can be extend trivially, first along H 0 and then along each H u , to be constant along the corresponding null generators. In particular we have, Then, under the assumptions and conclusions of theorem 1.14 we can easily deduce, Also, or, in scale invariant norms (assigning to f signature 0), We now introduce the localized curvature norms, and, with the supremum taken with respect to all elements of the partition. and, 2.3. Angular localized curvature estimates. Using a variation of our main energy estimates, with an additional angular localization, we can prove the following.
Theorem 2.4. Under the assumptions and conclusions of theorem 1.14, if in addition δ 1 2 |Λ| −1 is sufficiently small, then, for 0 ≤ u ≤ 1, 0 ≤ u ≤ δ, Moreover, Remark 2.5. By the standard domain of dependence argument the energy estimate can not fully localized to individual sectors (Λ) H u and (Λ) H u contained in the support of the function (Λ) f . This explains the need for the supremum in Λ in the definition of the [Λ] R, [Λ] R norms for the first part of the theorem. The second part of the theorem gives a bound for each sector individual Λ with the second term on the right hand side of (45) accounting for the defect of localization.
A proof of the theorem is sketched in section 5.
Definition 2.7. We say that the data R (0) is uniformly distributed on the scale δ Our second main result of this paper is the following.
Theorem 2.8 (Main theorem II). Assume that, in additions to the conditions of validity of theorem 1.14, the data R (0) is uniformly distributed on the scale δ 1 2 ̟ −1 for some constant ̟ << 1 and ǫ̟ −1 sufficiently small. Let Λ be a fixed angular sector of size |Λ| = q −1 δ the Λ-angular section (Λ) S u,δ of the surface S u,δ must be trapped, i.e. trχ < 0 there.
We postpone a discussion of the proof of this theorem to the last section of the paper.

Preliminary estimates
3.1. Transported coordinates. As mentioned in the previous section we define systems of, local, transported coordinates along the null hypersurfaces H and H. Staring with a local coordinate system θ = (θ 1 , θ 2 ) on U ⊂ S(u, 0) ⊂ H u we parametrize any point along the null geodesics starting in U by the the corresponding coordinate θ and affine parameter u. Similarly, starting with a local coordinate system θ = (θ 1 , θ 2 ) on V ⊂ S(0, u) ⊂ H u we parametrize any point along the null geodesics starting in V by the the corresponding coordinate θ and affine parameter u. We denote the respective metric components by γ ab and γ ab .
Proposition 3.2. Let γ 0 ab denote the standard metric on S 2 . Then, for any 0 ≤ u ≤ 1 and 0 ≤ u ≤ δ and sufficiently small δ The Christoffel symbols Γ abc and Γ ab , obey the scale invariant estimates 4 The proof is a trivial adaptation of proposition 4.6 in [K-R:trapped].

Calculus inequalities.
We simply adapt here the results of section 4.9 in [K-R:trapped].
Proposition 3.4. Let S = S u,u and let S ǫ ⊂ S denote a disk of radius ǫ relative to either θ or θ coordinate system. Then for any horizontal tensor φ and any p > 2 and As a consequence of the proposition we derive.  Proposition 3.9. Let ψ verify the Hodge system with D one of the Hodge operators defined in section 3.5 of [K-R:trapped]. Then, Also, Proposition 3.10. Let ψ verify the Hodge system Then, 3.11. Trace theorems. We state the straightforward adaptations of the results of section 11 in [K-R:trapped] concerning sharp trace theorems.
We introduce the following trace norms for an S tangent tensor φ, with signature sgn(φ), along H = H (0,u) u , relative to the transported coordinates (u, θ) of proposition 3.2: Also, along H = H (0,u) u relative to the transported coordinates (u, θ) of proposition 3.2 Proposition 3.12. For any horizontal tensor φ along H = H where C is a constant which depends on O (0) , R, R.
Also, for any horizontal tensor φ along H = H (u,0) u , and a similar constant C,

Global Curvature Estimates
In this section we discuss the proof of theorem A, 1.18, which is a straightforward modification of the curvature estimates of sections 14 and 15 in [K-R:trapped].
4.1. Zero order estimates. As in [K-R:trapped] all curvature estimates are based on the energy identities for the Bel-Robinson tensor Q[W ] of a Weyl field W which we take here to be either the Riemann curvature tensor R or its modified Lie derivatives L U R = L U R − 1 8 tr U πR − 1 2 Uπ · R, relative to well chosen vectorfields U. Recall There exists a positive constant a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, We sketch the proof in the particular case when X = Y = Z = L in proposition 5.2. We obtain, schematically, by signature considerations, Passing to the scale invariant norms we have, The worst term occur when s 2 = s 3 = 2 and s 1 = 0. Observe also that, since the signature of a Ricci coefficient (L) π (s 1 ) may not exceed s 1 = 1, neither s 2 or s 3 can be zero, i.e. α cannot occur among the curvature terms on the right. We use the estimate (L) π (s 1 ) There other estimates are derived in the same manner, see [K-R:trapped] 4.4. First derivative estimates. As in [K-R:trapped] the first derivative curvature estimates are based on the following.
with D(U, R) the 3-tensor of the form, schematically.
We apply these estimate for the following the choice of vectorfields, Using our scale invariant norms, and proceeding exactly as in section 15 of [K-R:trapped] we can easily derive the estimate, for some a > 1 8 , Similarly, Combining, we derive the desired first derivative estimates Proposition 4.6. There exists a positive constant a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, Combining this with proposition 4.3 we derive, which ends the proof of theorem 1.18.

Localized Energy estimates
5.1. Localized zero order estimates. We start by modifying slightly proposition 5.2, Proposition 5.2. The following identity holds on our fundamental domain D(u, u), where π(X, Y, Z) is a linear combination of the deformation tensors of the vectorfields X, Y, Z.
As in the derivation of the global estimates we make all the choices, In each case the only new term that needs to be estimated is due to D(u,u) ( Clearly, recalling (41), Recalling also, Λ ( Λ) f = 1, and ( Λ) f · ( Λ) f = 0 except for a a few neighboringΛ, Therefore, passing to scale invariant norms, and treating the term in (L) π exactly as before, Therefore, taking the supremum over Λ on both sides, we derive Proceeding in the same manner with all other curvature components we derive, Proposition 5.3. Consider a partition of unity ( Λ) f of of length |Λ| such that δ 1/2 |Λ| −1 is sufficiently small. There exists a positive constant a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, 5.4. Localized derivative estimates. We start with a localized version of proposition 4.5.
Proposition 5.5. Let U be a vectorfield defined in our fundamental domain D(u, u), tangent to H 0 .
We apply these estimate, as before, for same choice of vectorfields, The only new terms which need to be treated are due to D(u,u) ( . For all choices of vectorfields U, , X, Y, Z we can proceed precisely as in the proof of proposition 4.5 and thus derive. Proposition 5.6. Given a partition of unity ( Λ) f of length |Λ|, such that δ 1/2 |Λ| −1 is sufficiently small, we can find a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, Combining propositions 4.6 and 5.6 we derive, Theorem 5.7. Given a partition of unity ( Λ) f of length |Λ|, such that δ 1/2 |Λ| −1 is sufficiently small, we can find a > 1 8 such that, for δ 1/2 ǫ −1 and ǫ sufficiently small, we have,

Deformation tensor estimates
In this section we sketch the proof of the estimates which relate the norms Π of the deformation tensors for L, L and O to the Ricci coefficient norms O, stated in theorem 1.19. Throughout the section we assume that δ 1/2 ǫ −1 and ǫ are sufficiently small.
6.1. Estimates for Π and Π. The null components of (L) π and (L) π are simply expressed in terms of null Ricci coefficients according to the lemma.
which establishes half of theorem B (1.19).
6.4. Estimates for (O) Π. Recall that the only non-vanishing components of (O) π are given by The quantities Z and H verify the following transport equations, written schematically, In view of equations (68) we derive, by integration, Using the trace estimates for (η, η) and (ρ, σ) we derive, (sc) ), Following precisely the same steps as section 13 of in [K-R:trapped] we derive, Also all null components of the derivatives D (O) π , with the exception of (D 3 (O) π ) 3a , verify the estimates, Recalling the definition of the norms (O) Π we deduce, Proposition 6.5. The following estimates hold true with a constant C = C(O, R, R), This establishes the remaining part of theorem B(1.19).

Estimates for the Ricci coefficients
In this section we discuss the proof of theorem C(1.20). We make the point that the proof can be derived by a straightforward modification of the arguments in sections 5-10 of [K-R:trapped].
Relying on the bootstrap assumption the boot-strap assumption (34) we first derive, see section 4.1.
Using this fact we can deduce, as in section 4.1. of [K-R:trapped], Proposition 7.1.
5 With the small quantity ǫ replaced by q here.