Energy Estimates and Gravitational Collapse
Abstract.
In this note, we study energy estimates for Einstein vacuum equations in order to prove the formation of black holes along evolutions. The novelty of the paper is that, we completely avoid using rotation vector fields to establish the global existence theorem of the solution. More precisely, we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components. We show that, thanks to the special cancelations coming from the null structure of nonlinear interactions, desirable estimates on curvatures can be derived under the short pulse ansatz due to Klainerman and Rodnianski [4] (which is originally discovered by Christodoulou [2]).
Contents
1. Introduction
1.1. A Brief History
Penrose singularity theorem states that if in addition to the dominant energy condition, the spacetime has a trapped surface, then the spacetime contains singularities. The weak cosmic censorship conjecture asserts that under reasonable physical assumptions, singularities should be hidden from an observer at infinity by the event horizon of a black hole. Thus, by combining these two claims, to predict the existence of black holes, it suffices to exhibit one trapped surface in a spacetime,. In other words, although many supplementary conditions are required, we regard the existence of a trapped surface as the presence of a black hole.
A major challenge in general relativity is to understand how trapped surfaces actually form due to the focusing of gravitational waves. In a recent breakthrough [2], Demetrios Christodoulou gave an answer to this long standing problem. He discovered a mechanism which is responsible for the dynamical formation of trapped surfaces in vacuum spacetimes. In the monograph [2], in addition to the Minkowskian flat data on a incoming null hypersurface, Christodoulou identified an open set of initial data (this is the short pulse ansatz) on a outgoing null hypersurfaces. Based on the techniques developed by himself and Klainerman in the proof of the global stability of the Minkowski spacetimes [3], he managed to understand the whole picture of how various geometric quantities interact along the evolution. Once the estimates on curvatures are established in a large region of the spacetime, the actual formation of trapped surfaces is easy to demonstrate. Christodoulou also proved a version of the same result for the short pulse data prescribed on past null infinity. This miraculous work provides the first global large data result in general relativity (without symmetry assumptions) and opens the gate for many new developments on dynamical problems related to black holes.
In [4], Klainerman and Rodnianski extended aforementioned result of Christodoulou. They significantly simplified the proof of Christodoulou (from about six hundred pages to one hundred and twenty). They also enlarged the admissible set of initial conditions and show that the corresponding propagation estimates of connection coefficients and curvatures are much easier to derive. The relaxation of the propagation estimates are just enough to guarantee that a trapped surface still forms. Based on the trace estimates developed in a sequence of work of the authors towards the critical local wellposedness for Einstein vacuum equations, they reduced the number of derivatives needed of Christodoulou in the argument from two derivatives of the curvature (in Christodoulou’s proof) to just one. More importantly, Klainerman and Rodnianski introduced a parabolic scaling in [4] which is incorporated into Lebesgue norms and Sobolev norms. These new techniques allow them to capture the hidden smallness of the nonlinear interactions among different small or large components of various geometric objects. The result of Klainerman and Rodnianski can be easily localized with respect to angular sectors, has the potential for further developments, see [5]. We remark that Klainerman and Rodnianski only concentrated on the problem on a finite region. The question from past null infinity can be solved in a similar manner as in [2] once one understand the picture on a finite region. The problem from past null infinity has been studied in a recent work by Reiterer and Trubowitz, [6].
1.2. Novelty of the Paper
One common feature of the proofs in [2] and [4] is that, in order to derive energy estimates on one or higher derivatives of curvature components, they all constructed three angular momentum vector fields , and which essentially captured almost rotational symmetry of the spacetime. As far as the author aware, the main reason of using ’s is that the energy estimates on behave well because one can take advantage of cancelations from the pseudosymmetry of ’s. Here, the is the modified Lie derivative defined in [3] and is the curvature of the spacetime. We can observe this advantage in the proof of stability of Minknowski spacetime [3] where yields better decay estimates.
There is one obvious defect of the modified Lie derivative. As usual Lie derivatives, the is not tensorial in . In fact, it evolves one derivative of . In other words, if we use modified Lie derivatives, we may lose immediately one derivative. We have two remedies to this loss of derivatives: in [2], one relies on higher order derivative estimates; in [4], one makes use of more subtle trace estimates.
The above discussion can be summarized as follows: roughly speaking, a good estimate on Lie derivatives relies on the almost symmetries ’s, but Lie derivatives causes a loss of derivative as we just explained. Hence, to avoid this loss, we shall give up the use of the pseudosymmetries.
In this paper, we propose an approach to derive energy estimates on curvatures without constructing rotational vector fields at all. In particular, in stead of using modified Lie derivatives, we work with covariant derivatives to save one derivative. Of course, we have to pay a a price of controlling much more error terms. Nevertheless, this significantly simplifies the proofs compared to either [2] or [4]. In particular, compared to [2], we use only one derivative in curvature; compared to [4], instead of using trace inequality, all the estimates are derived from the classical Sobolev inequalities.
We also want to mention that in the thesis of L. Bieri, see [1], based on a more general asymptotic assumptions, she gave a simplified proof of the stability of Minkowski spacetime. She managed to derive decay from the time vector field and the conformal scaling of the spacetime which allowed her to circumvent rotational vector fields in that situation. But the current situation is different from [1]: we do not use Lie derivatives at all and the lower regularity forces us to explore more structures from the Einstein equations.
1.3. Structure of the Proof
The main observation arises from the second Bianchi identities. Roughly speaking, they explicitly show how one expresses angular or rotational derivatives of curvature components in terms of some null derivatives of curvature components plus lower order nonlinear terms. Namely, they can be written schematically as
or
where we use lower order terms to collect all nonlinear interactions and , are two standard null directions under the framework of double null foliations. Thus, up to lower order corrections, to obtain the estimates on rotational derivatives on curvatures, it suffices to control or null derivatives of curvatures. Thus, we identify our main targets to be and and we shall derive energy estimates for them.
According the above idea, we use the modified short pulse ansatz proposed Klainerman and Rodnianski in [4]. We remark that this ansatz allows more large components than the original short pulse data discovered by Christodoulou in [2]. We shall derive the energy estimates based on BelRobinson tensors associated to and . To deal with error terms, namely terms , and in the proof in following sections, we have to take account of the special structure of those terms. In reality, some generic terms in error term may cause a loss of which prevent us from closing the bootstrap argument. To avoid this loss, there are typically three techniques to use:

We bound a product of two term in by two estimates on each term instead of one estimate on one of them and one estimate on the other. In most of the situation, this trick saves a .

When we integrate a product of terms on some null hypersurface or on a domain in the spacetime, we use integration by parts to move a bad derivative, typically or , from one term (for whom this bad derivative may cause a loss of ) to another (for whom this bad derivative is not bad at all, namely, there is no loss in ). Combined with Bianchi identities, this procedure may save a .

For some generic term in error estimates mentioned above, though it appears that it causes a loss of (which can not be retrieved by using the trick 1) and 2)), we can use in fact either signature considerations or a precise computations to show that this term does not show up at all in the error estimates. This manifests the special cancelations in the error terms.^{1}^{1}1 This cancelation can also be observed much more directly by another way of deriving energy estimates, namely, multiplying Bianchi identities and integrating directly on a give domain. The author would like to thank Igor Rodnianski for communicating this idea.
The whole proof is to combine these three tricks. The paper is organized as follows: in next section, we recall basic definitions and estimates from [4] and we state the main theorem; in following sections, we derive energy estimates for derivatives of curvature components in the following order: , and then for and for .
2. Main Result
2.1. The Double Null Foliation Framework
We briefly recall the double null foliation formalism, See [2] for more precise definitions. We use to denote the underlying spacetime and use to denote the background metric. We assume that is spanned by a double null foliation generated by two optical functions and and they increase towards the future, and . We use / to denote the outgoing / incoming null hypersurfaces generated by the level surfaces of / . We use to denote the spacelike two surface . We denote by the region of defined by ; similarly, we can define .
The shaded region on the right represents the domain with . The function is in fact defined from to . When , this part of is assumed to be a flat light cone in Minkowski space with vertex located at . We use to measure the maximal radius of the flat part of . In [4], the trapped surface forms at and .
Let be the null geodesic generators of the double null foliation and we define the lapse function by . The normalized null pair is defined by . On sphere we choose an arbitrary orthonormal frame . We call a null frame.^{2}^{2}2 We use Greek letters to denote an index from to and Latin letters to denote an index from to . We also to denote the null direction either or . Repeated indices are always understood as taking sums
We use to denote LeviCivita connection defined by the metric and we define the connection coefficients as follows,
where . On , is the induced connection; and are the projections to of the covariant derivatives and .
Given a Weyl field , we introduce its null decomposition with respect to the given null frame,
where is the spacetime Hodge dual of . When is the Weyl curvature tensor, we use to denote its null components.
We recall the null structure equations for the Einstein vacuum spacetimes (see [4]). The originally Einstein field equations are
where is the Ricci curvature of the underlying spacetimes. We express this tensorial equations by using the null frame, by definition, this yields the null structure equations. We only list the transport type null structure equations which are relevant to the current work.
(2.1) 
(2.2) 
(2.3) 
(2.4) 
(2.5) 
(2.6) 
(2.7) 
(2.8) 
(2.9) 
We also express the Bianchi equations relative to the null frame to derive null Bianchi equations.^{4}^{4}4 We can eliminate by .
(2.10) 
(2.11) 
(2.12) 
(2.13) 
(2.14) 
(2.15) 
(2.16) 
(2.17) 
(2.18) 
(2.19) 
2.2. Energy Estimates Scheme
We review our scheme for energy estimates on Weyl fields, see [3] for the original resource. Assume a Weyl field and its Hodge dual solve following divergence equations with source terms
(2.20) 
and are called Weyl currents.
Remark 2.1.
For vacuum, the curvature tensor is a Weyl field with zero currents
(2.21) 
The BelRobinson tensor associated to is defined as follows ^{5}^{5}5 We shall use short hand notations for , for , for , for , …, if there is no confusion in the context.
It is fully symmetric and traceless in all pair of indices. Moreover, it satisfies the dominant energy condition which allows one to recover estimates for Weyl field . In pratical terms, this condition can be expressed by formulas,
We also list other null components of for future use,
(2.22)  
In view of (2.20), enjoys the following divergence equations
(2.23) 
Given vector fields , and , we define the current associated to , , and by
Thus,
(2.24) 
where is the deformation tensor of defined by and
We integrate (2.24) on to derive the fundamental energy identity ^{6}^{6}6 and are corresponding normals of the null hypersurfaces and .
(2.25)  
We list nonzero components of deformation tensors of and as well as the nonzero component of and for future use^{7}^{7}7 We do not use and in order to avoid and which do not have certain estimates, see [4] for details.:
We consider one derivative of curvature in a null direction . One easy but important observations is that is still a Weyl field. We can commute with (2.21) to derive^{8}^{8}8 Recall that commutes with Hodge operator.
(2.26) 
where
(2.27) 
and
(2.28) 
2.3. Short Pulse Ansatz and Scale Invariant Formulation
We briefly recall the notions of signature and scale introduced by Klainerman and Rodnianski in [4]. Let be either a null component of curvature or a connection coefficient, we use , and to denote the number of times , respectively and appearing in the definition of . The signature of , , and the scale of , , are defined as
We list the signatures and scales for all connection coefficients and curvature components,
signature  scale  signature  scale  signature  scale  

1  2  0  
0  0  
0  1 
We impose following rules on signatures,
We define the scale invariant norms for . Along null hypersurfaces or ,
On a two dimensional surface ,
Those norms are obviously related by formulas,
Those scale invariant norms come up naturally with a small parameter . Roughly speaking, it captures the smallness of the nonlinear interaction. We have Hölder’s inequality in scale invariant form,
(2.29) 
Similar estimates hold along null hypersurfaces.
Remark 2.2.
The rule of thumb for treating the nonlinear terms is, whenever one has a product of two terms, (2.29) gains a . We do have cases that (2.29) does not gain any power in . In fact, if is a bounded (in usual sense) scalar function (say, bounded by a universal constant), the best we can hope is . In particular, in this paper, for can be where , we have to pay special attentions to the appearance of , see [4] for more detailed descriptions.
We introduce a family of scale invariant norms for connection coefficients where or , ^{9}^{9}9 We use shorthand notations
as well as for curvature components,
Finally, we introduce total norms. We define