A REGIME OF LINEAR STABILITY FOR THE EINSTEIN-SCALAR FIELD SYSTEM
WITH APPLICATIONS TO NONLINEAR BIG BANG FORMATION
IGOR RODNIANSKI∗ AND JARED SPECK∗∗
A BSTRACT. We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature
(CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions
on (0, ∞) × T3 . The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially
anisotropic spacetime that expands towards the future and that has a “Big Bang” singularity at {t = 0}.
We place initial data for the linearized system along {t = 1} ≃ T3 and study the linear solution’s behavior
in the collapsing direction t ↓ 0. Our main result is the identification of a new form of approximate L2
monotonicity for the linear solutions that holds whenever the background Kasner solution is sufficiently close
to the Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) solution. Using the approximate monotonicity, we
derive sharp information about the asymptotic behavior of the linear solution as t ↓ 0. In particular, we
show that some of its time-rescaled components converge to regular functions defined along {t = 0}. In
addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMCtransported spatial coordinates gauge can be realized as a limiting version of a family of parabolic gauges for
the lapse variable. An approximate L2 monotonicity inequality also holds in the parabolic gauges, but the
corresponding parabolic PDEs are locally well-posed only in the direction t ↓ 0.
In a companion article, we use the linear stability results to prove a stable singularity formation result for
the nonlinear equations. Specifically, we show that the FLRW solution is globally nonlinearly stable in the
collapsing direction t ↓ 0 under small perturbations of its data at {t = 1}.
Keywords: constant mean curvature, parabolic gauge, spatial harmonic coordinates, stable blow-up, transported spatial coordinates
Mathematics Subject Classification (2010) Primary: 35A01; Secondary: 35Q76, 83C05, 83C75, 83F05
June 9, 2015
TABLE OF C ONTENTS
1. Introduction
1.1. Previous work on singularities
1.2. Initial value problem formulation of the Einstein equations and a rough statement of the
main results
1.3. CMC-transported spatial coordinates gauge and the Kasner family
1.4. L2 monotonicity in a simplified model problem
∗
1
3
6
8
9
Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA.
irod@math.princeton.edu.
∗∗
Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Ave, Room E18-328, Cambridge,
MA 02139-4307, USA. jspeck@math.mit.edu.
∗
IR gratefully acknowledges support from NSF grant # DMS-1001500.
∗∗
JS gratefully acknowledges support from NSF grants # DMS-1454419 and # DMS-1162211, from a Sloan Research Fellowship provided by the Alfred P. Sloan Foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts
Institute of Technology.
i
A Regime of Linear Stability for the Einstein-Scalar Field System
ii
1.5. Comments on other matter models, higher dimensions, and the far-from-trace-free case
1.6. A related instance in which monotonicity led to global results
1.7. Paper outline
2. Notation and Conventions
2.1. Indices
2.2. Spacetime tensorfields and Σt −tangent tensorfields
2.3. Coordinate systems and differential operators
2.4. Integrals and L2 norms
2.5. Constants
3. The Einstein-Scalar Field Equations in CMC-Transported Spatial Coordinates and the
Linearized Equations
3.1. Preliminary discussion
3.2. The Einstein-scalar field equations in CMC-transported spatial coordinates
3.3. Linearized quantities
4. Norms and Energies
4.1. Pointwise norms
4.2. Sobolev norms
4.3. Energies
5. The Two Linearized Stability Theorems
5.1. Statement of the two theorems
5.2. Energy identities verified by solutions to the linearized equations
5.3. Proof of Theorems 5.1 and 5.2
5.4. Comments on realizing “end states”
6. Comments on the nonlinear problem
7. A Second Proof of Linearized Stability via Parabolic Lapse Gauges
7.1. Choice of a gauge and the corresponding formulation of the Einstein-scalar field equations
7.2. Linearizing around Kasner
7.3. Statement of the main monotonicity theorem
7.4. Preliminary estimates and identities towards the proof of Theorem 7.1
7.5. Proof of Theorem 7.1
Acknowledgments
References
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I. Rodnianski and J. Speck
1
1. I NTRODUCTION
The Einstein-scalar field system models the evolution of a dynamic spacetime1 (M, g) containing matter
known as a scalar field (denoted by φ). The scalar field is a simple matter model that has been well-studied
in mathematical general relativity in the context of asymptotically flat spacetimes; see [20–26]. Relative
to an arbitrary coordinate system, the equations are (for µ, ν = 0, 1, 2, 3):
(1.1a)
(1.1b)
1
Ricµν − Rgµν = Tµν ,
2
−1 αβ
(g ) Dα Dβ φ = 0,
where Ric denotes the Ricci tensor of g, R = (g−1 )αβ Ricαβ denotes the scalar curvature of g, D denotes
the Levi-Civita connection of g, and T denotes the energy-momentum tensor of φ ∶
(1.2)
1
Tµν = Dµ φDν φ − gµν (g−1 )αβ Dα φDβ φ.
2
In this article, in the context of cosmological spacetimes,2 we identify a new form of approximate monotonicity for solutions to linearized versions of the equations and use it to prove linear stability results. An
important aspect of our approach is that it is robust under perturbations and can be used to prove stable
blow-up for a family of cosmological solutions to the nonlinear equations; we describe the nonlinear analysis in more detail below. Our derivation of the approximate monotonicity relies on subtle combinations
of energy identities in which unexpected cancellations occur and the sign and size of various coefficients
is important. Thus, we also provide a simplified model problem in order to illustrate the main ideas; see
Sect. 1.4. Our results apply when the initial Cauchy hypersurface is T3 , and they can easily be generalized to the case of n spatial dimensions, that is, to the case of Tn for n ≥ 1. We anticipate that similar
results may hold for some other matter models with special properties and, in the case of very high spatial
dimensions, for the Einstein-vacuum equations; see the discussion in Sect. 1.5.
In addition to deriving stability results for solutions to linearized versions of (1.1a)-(1.1b), we also
outline (see Sect. 6) how to use the linear estimates to prove the nonlinear stable singularity formation
result mentioned above; we provide the detailed nonlinear analysis in the companion article [55]. More
precisely, as we describe below, in [55], we show the nonlinear stability of the well-known FriedmannLemaı̂tre-Robertson-Walker (FLRW) solution in a neighborhood of its Big Bang singularity. The approximate monotonicity driving the results of [55] is interesting in itself and moreover, some aspects of it
survive in regimes other than the near-FLRW one; see Sect. 5.2. It is for this reason that we have chosen
to isolate the main ideas behind it by studying the linearized equations of the present article.
We now describe the background solutions corresponding to our linearization. In suitable gauges that we
describe below, we linearize equations (1.1a)-(1.1b) around members of the well-known Kasner family,3
which are cosmological solutions of the nonlinear equations. Exceptional cases aside, the Kasner solutions
1
By spacetime, we mean a four-dimensional time-orientable manifold M equipped with a Lorentzian metric g of signature
(−, +, +, +).
2
By “cosmological spacetime,” we mean that M has compact Cauchy hypersurfaces and that the Ricci curvature of gµν
verifies Ricαβ Xα Xβ ≥ 0 for all timelike vectors Xµ . This Ricci curvature condition is always verified by the solutions that we
consider because Einstein’s equations imply that Ricµν = Tµν − 12 (g−1 )αβ Tαβ gµν and because the energy-momentum tensor
Tµν of a scalar field verifies the strong energy condition.
3
Many authors use the term “Kasner family” exclusively in the context of solutions to the Einstein-vacuum equations. Hence,
it would be more correct to refer to the solutions under study here as “generalized Kasner solutions.”
A Regime of Linear Stability for the Einstein-Scalar Field System
2
have Big Bang singularities along the past boundary {t = 0} where their Kretschmann scalars4 blow up
like5 t−4 . We provide their explicit form in Sect. 1.3.
Although some aspects of the approximate monotonicity are present for a large class of Kasner backgrounds, it becomes particularly strong when they are near the FLRW solution, which is spatially isotropic
(see (1.11)). Thus, under a near-FLRW Kasner background assumption, we consider “initial” data for the
linearized equations given along {t = 1} and study the behavior of the linear solution in the past direction
t ↓ 0. In addition to quantifying the approximate monotonicity (see Theorem 5.1), we also extract detailed
information about the asymptotics. In particular, we show that certain components of the linear solution
converge not to 0, but rather to a function of the spatial variable x ∈ T3 (see Theorem 5.2 and Sect. 5.4).
Our analysis also shows that the linear solution exhibits velocity term dominated (VTD) behavior at the
lower derivative levels near the background singularity {t = 0}. In general relativity, the terminology
“VTD behavior” means that the spatial derivative terms in the equations become negligible relative to the
time derivative terms near {t = 0}. This terminology is the modern descendant of the phrase “velocity
dominated,” which originated in the work [32]. The VTD nature of the linear solutions is not just a curiosity; it plays a key role in controlling error terms in the nonlinear problem, as we describe in Sect. 6.
We remark that in the case of far-from-spatially isotropic Kasner backgrounds, the methods employed in
this paper do not seem to yield stability results that are sufficient for closing the nonlinear problem; see
Sect. 1.5 for additional discussion.
It is the combined strength of the estimates provided by Theorems 5.1 and 5.2 that allows us to prove
that the FLRW solution is nonlinearly stable in the collapsing direction (towards the Big Bang). More
precisely, in the companion article [55], we prove that if we perturb6 (without symmetry assumptions)
the FLRW “data” at {t = 1}, then we can solve the Einstein-scalar field equations in the past direction
all the way down to {t = 0}, where a Big Bang singularity occurs and the Kretschmann scalar blows up
like t−4 . Actually, we prove the result for a more complicated kind of matter known as the stiff fluid,7
which reduces to the scalar field model when the vorticity vanishes. As we explain below, the proof in
[55] is based on a formulation of the equations relative to constant mean curvature (CMC from now on)transported spatial coordinates. For this reason, in the present article, we use the same coordinates in
formulating the nonlinear equations. We then study linearized versions of precisely this formulation of the
nonlinear equations.
Consistent with our proof of nonlinear stable blow-up in [55], we could also extend our linear stability
results to apply when the background solutions are near-FLRW as measured by a Sobolev norm (and hence
are spatially dependent). We do not provide such an extension here because it would significantly lengthen
the paper without contributing substantially to the main ideas. A related feature of the results of [55] is
that we can derive them without knowing the precise “end state” (that is, the asymptotics near {t = 0})
in advance; it suffices to control the difference between the perturbed solution and the FLRW solution,
and moreover, our proof would go through if we instead controlled the difference between the perturbed
solution and any near-FLRW Kasner solution.
4
The Kretschmann scalar is Riemαβγδ Riemαβγδ , where Riem is the Riemann curvature of g.
One can compute that in terms of the Kasner exponents from (1.8), the Kretschmann scalar is equal to
4t−4 {∑3i=1 qi4 + ∑1≤i<j≤3 qi2 qj2 + ∑3i=1 qi2 − 2 ∑3i=1 qi3 } ≥ 4t−4 ∑1≤i<j≤3 qi2 qj2 .
6
The perturbations belong to a suitable Sobolev space.
7
A stiff fluid verifies
the equation of state p = ρ, where p is the fluid pressure and ρ is the proper energy density. Its speed of
√
sound, given by dp/dρ, is equal to 1.
5
I. Rodnianski and J. Speck
3
1.1. Previous work on singularities. Previous work has provided related results showing the stability
of singular solutions to the Einstein equations in various contexts, but only under under symmetry assumptions that reduce the analysis to 1 + 1 dimensional PDEs8 [28, 43, 53, 54]. There also is a body of
work that provides the construction of (but not the stability of) singularity-containing solutions to select
nonlinear Einstein-matter systems, but only under the assumption of symmetry [1, 9, 16, 17, 42, 44, 48, 59]
and/or spatial analyticity [7, 30]. Readers may also consult [2] for a more general well-posedness result
for singular initial value problems that applies to a class of symmetric hyperbolic quasilinear systems in
more than one spatial dimension. More precisely, in [2], the authors prescribe Sobolev-class asymptotics
featuring singular behavior and their main result is the existence of a Sobolev-class solution that realizes the singular asymptotics. We note, however, that [2] does not treat Einstein’s equations. A related
approach to studying Big Bang singularities involves devising a formulation of Einstein’s equations that
allows one to solve a Cauchy problem with initial data given on the singular hypersurface {t = 0} itself;9
see, for example, [8, 29, 46, 47, 61–63]. In some cases, these works included a proof that the singular
solutions exhibit VTD behavior. Readers may consult [50] for a precise comparison of these results as
well as an extension of them to prove the existence of singular solutions to the Einstein-vacuum equations
with Gowdy symmetry.10
In contrast to the regular Cauchy problem studied here and in the companion article [55], the above
works are based on prescribing the asymptotics as t ↓ 0 and then constructing a solution that achieves those
asymptotics. Most of those works are based on solving a Fuchsian PDE system that is singular at {t = 0}.
We now describe some aspects of the Fuchsian approach. A representative work is [1], in which the authors
construct singular solutions to the Einstein-vacuum equations11 with T2 symmetry under the polarized or
half-polarized condition. In Sect. 5.4, we provide a simple model problem suggesting that results similar
to those of [1] might also hold for the Einstein-scalar field system without symmetry assumptions. The
Fuchsian PDEs12 treated in [1] are of the form
(1.3)
A0 (t, x, u)t∂t u + A1 (t, x, u)t∂x u + B(t, x, u)u = f (t, x, u),
where u is the array of unknowns, the Aα and B are symmetric matrices (the energy estimates rely on
the symmetric hyperbolic framework), and f is an array, all of which verify a collection of technical
assumptions. The analysis in [1] is based on splitting the solution as u = u0 + w, where u0 is the “leading
order” part and w is an error term that one would like to show is small compared to u0 as t ↓ 0. An
important technical assumption made in [1], which is used for deriving energy estimates, is that for small
w, one can split A0 (t, x, u0 + w) = A00 (x, u0 ) + A01 (t, x, u0 + w), where A00 (x, u0 ) is symmetric positive
definite, and the map w → A01 (t, x, u0 + w) maps certain time-weighted Sobolev spaces into other timeweighted Sobolev spaces. There are various ways of constructing u0 . The most relevant way in the context
of the present article is to choose u0 to be a prescribed solution to a VTD version of (1.3) in which spatial
8
There also are stable singularity formation results in the class of spatially homogeneous solutions (in which case the equations reduce to ODEs); see [51] or [64] for an overview.
9
This method is based on formulating the equations in terms of a rescaled metric, conformal to the physical spacetime metric,
in such a way that the rescaled metric remains regular throughout the entire evolution. As such, this method can be viewed as
an extension of Friedrich’s conformal method [39, 40].
10
Gowdy solutions are a subset of the T2 −symmetric solutions characterized by the vanishing of the twist constants
′
−1 µµ′
(g ) αβµν Xα Yβ Dµ′ Xν and (g−1 )µµ αβµν Xα Yβ Dµ′ Yν , where is the volume form of g and X and Y are the Killing
fields corresponding to the two symmetries.
11
More general Fuchsian systems in one spatial dimension are also treated in [1].
12
Specifically, the PDEs are the T2 −symmetric polarized or half-polarized Einstein-vacuum equations in areal coordinates
with the singularity at {t = 0}.
A Regime of Linear Stability for the Einstein-Scalar Field System
4
derivatives have been discarded. This approach is complementary to the one taken in the present article
and [55], in which we show that VTD behavior emerges in solutions to the true equations. From the VTD
system and (1.3), one computes that the error term w solves an “error equation” depending on u0 . The main
result of [1] is that under suitable additional assumptions, there exists a solution w to the error equation
that becomes small relative to u0 as t ↓ 0 and that w is unique within well-chosen time-weighted Sobolev
spaces. The main idea of the proof is to derive uniform a priori symmetric hyperbolic energy estimates
for a sequence {wn }∞
n=1 of error equation solutions on intervals of the form [tn , δ]. More precisely, the wn
solve a standard symmetric hyperbolic Cauchy problem (to the future) with 0 initial data at time tn . Here,
δ > 0 is a small constant and {tn }∞
n=1 is a sequence of times decreasing to 0. A key point of the analysis is
that time weights are inserted by hand into the energies. More precisely, one derives energy estimates for
t−P wn , where t−P is a diagonal matrix whose non-zero entries are well-chosen negative powers of t that
are allowed to depend on x (that is, P = P (x)). The energies are also weighted by an additional overall
γ
scalar factor of e−κt , where κ and γ are positive constants. The time weights must be compatible with
the nonlinearities in the sense that the nonlinear error integrals arising in the energy estimates must be
controllable, which leads to controlled energy growth towards the future (away from the singularity) in a
neighborhood of the singularity. In particular, for well-chosen t−weights (as we illustrate in Remark 5.5,
there is some freedom in choosing them), one can derive uniform estimates for the {wn } showing that the
weighted energies cannot grow too fast towards the future; see Sect. 5.4 for a very simple linear model
problem. Then through a standard limiting procedure, one can produce a solution w to the error equation
that exists on the interval (0, δ], and it is unique within suitable time-weighted Sobolev spaces.
Although the Fuchsian approach furnishes a large class of solutions with singularities, it is inadequate
for treating the true stability problem of solving down towards {t = 0} starting from Cauchy data for u
given along a hypersurface {t = c} with c > 0. In our work here and in [55], we encounter new difficulties
that do not fall under the scope of the standard Fuchsian framework. An important difficulty is that in order
to synchronize the singularity across space, one cannot expect to be able work with a purely hyperbolic
formulation of the Einstein equations such as the one afforded by wave coordinates. This suggests that
gauges involving an infinite speed of propagation, such as the elliptic and parabolic ones for the lapse
employed in the present article and in [55], are essential. Hence, the required analysis goes beyond the
standard Fuchsian framework, which applies only to hyperbolic equations. In particular, the Fuchsian
strategy of inserting suitable time weights into the energies is not sufficient for deriving our stability
results; some of the terms in the equations are too singular to be treated in this fashion. Our proofs are
viable only because we are able to find approximate L2 monotonicity generated by subtle cancellations
that occur when the lapse estimate is coupled to the scalar field energy estimate. These cancellations are
a surprising consequence of the special structure of the equations in our gauges; see Sect. 1.4 for our
analysis of a simplified model problem.
The scalar field matter model has some special properties that allow us to derive our stability results.
We describe some of these properties in more detail in Sect. 1.5. In particular, we expect that our stability/approximate monotonicity results do not hold for general matter models. Actually, as we now explain,
for certain fluid models, Ringström obtained rigorous results showing that solutions behave in a drastically
non-monotonic fashion. In [52], Ringström studied fluids verifying the equation of state p = c2s ρ, where
the constant cs verifies the sub-stiff condition 0 < cs < 1. For the Euler-Einstein equations with a sub-stiff
I. Rodnianski and J. Speck
5
equation of state, he showed that spatially homogeneous solutions with Bianchi IX symmetry13 generically (that is, for non-Taub solutions) have limit points in the approach towards the singularity that must
be either vacuum Bianchi type I (that is, vacuum Kasner), vacuum Bianchi type VII0 , or vacuum Bianchi
type II. In particular, Ringström’s work showed that a sub-stiff fluid has a negligible effect on Bianchi IX
solutions near the singularity. Furthermore, he showed that almost all such solutions are oscillatory in the
sense that there are at least three distinct limit points, which stands in stark contrast to the approximately
monotonic behavior of our linear solutions and the nonlinear solutions in [55].
Ringström’s work [52] also applied to the Einstein-vacuum equations in Bianchi IX symmetry and thus
yielded the first examples of the oscillatory behavior conjectured in the work [14] of BKL (Belinsky, Khalatnikov, and Lifschitz). Specifically, in [14], the authors gave heuristic arguments suggesting that general
solutions to the Einstein-vacuum equations containing incomplete timelike geodesics should exhibit highly
oscillatory behavior near the boundary where the geodesics terminate, which should be a spacelike singularity. The so-called “BKL conjectures”14 have been seminal in stimulating the investigation of solutions
to Einstein’s equations near singularities. However, as we now explain, despite Ringström’s work, there is
immense controversy surrounding the conjectures. First, they are false as stated because of, for example,
the existence of Taub solutions, which develop a Cauchy horizon15 rather than a true singularity. One
might be tempted to weaken the conjectures by replacing the phrase “general solutions” with “generic solutions.” However, Luk has constructed [45] a class of solutions to the Einstein-vacuum equations without
symmetry assumptions such that the boundary of the maximal development contains a null portion along
which the metric remains C 0 but its Christoffel symbols blow-up in L2 . His examples, which are stable in
a certain sense, contradict the BKL vision of spacelike singularities. Moreover, as of the present, there are
no examples of Einstein-vacuum solutions away from spatial homogeneity that are rigorously known to
exhibit the kind of oscillatory behavior near a singularity conjectured in [14]. In total, given the presentday state of knowledge, it is not clear to what extent the vision of BKL is realized in Einstein-vacuum
solutions.
In the opposite direction, Belinsky and Khalatnikov [13] were the first to suggest the existence of nonspatially homogeneous approximately monotonic singular solutions to the Einstein-scalar field system. In
a later article [11], Barrow argued that fluids verifying the equation of state p = c2s ρ (where cs is a nonnegative constant representing the speed of sound) should induce a similar effect if and only if cs = 1; he
referred to the mollifying effect of a stiff fluid as quiescent cosmology. The first rigorous construction of
such solutions without symmetry was provided by the aforementioned work of Andersson and Rendall
[7]. They constructed a family of spatially analytic solutions to the Einstein-scalar field and Einstein-stiff
fluid systems that have Big Bang singularities and that exhibit approximately monotonic behavior near
them. Their proof was based on a two-step process. In the first step, they constructed a family of spatially
analytic solutions to VTD equations, which were obtained by throwing away the spatial derivative terms
from the Einstein-matter equations.16 In the second step, they constructed spatially analytic solutions to
the Einstein-matter equations by writing the true solution as the VTD solution plus error terms that were
13
Members of the Bianchi symmetry classes are spatially homogeneous and hence the corresponding solutions depend on
only a time variable. For a precise definition of these symmetry classes and the others that we mention, readers may consult
[27].
14
The statements in [14] are somewhat vague and thus it is a bit of an overstatement to refer to them as “conjectures.”
15
Roughly, a Cauchy horizon is a boundary along which the solution remains regular but beyond which it cannot be continued
uniquely as a solution due to lack of information for how to continue.
16
In [7], the Einstein equations were formulated relative to a Gaussian coordinate system in which the spacetime metric
takes the form g = −dt2 + gab dxa dxb .
A Regime of Linear Stability for the Einstein-Scalar Field System
6
shown, by Fuchsian analysis, to go to 0 as t ↓ 0. The results of [7] were extended to higher dimensions
and other matter models in [30]. The family of solutions constructed in this fashion is large in the sense
that its number of degrees of freedom coincides with the number of free functions in the Einstein initial
data. However, since the results are based on prescribing the asymptotics near the Big Bang within the
class of spatially analytic solutions, they are not true stable singularity formation results. In particular, their
work left open the possibility the map from the set of spatially analytic asymptotic states of [7] to the set of
Cauchy data (say at t = 1) might be highly degenerate in the sense that it does not extend to more physically
relevant function spaces such as Sobolev spaces; see, however, the discussion in Sect. 5.4. The primary
ingredient needed to upgrade the work of Andersson and Rendall to a true stable singularity formation
result corresponding to solving a regular Cauchy problem is a suitable statement of linear stability, strong
enough to control the nonlinear terms. This is what our linear stability theorems provide in the near-FLRW
case.
1.2. Initial value problem formulation of the Einstein equations and a rough statement of the main
results. The fundamental results [19] and [18], which are respectively by Choquet-Bruhat and ChoquetBruhat + Geroch, showed that the (nonlinear) Einstein-scalar field system (1.1a)-(1.1b) has an initial value
problem formulation in which sufficiently regular data give rise to a unique maximal globally hyperbolic
development.17 We now state some basic facts about the initial value problem. We specialize to the
context of the present article, where the initial Cauchy hypersurface is T3 . The data (for the nonlinear
equations) consist of the following fields on T3 ∶ (0gij , 0kij , 0φ, 0ψ). Here, 0gij is a Riemannian metric, 0kij
is a symmetric two-tensor , and 0φ and 0ψ are two functions. A solution launched by the data consists of
ι
a four-dimensional time-oriented spacetime (M, gµν ), a scalar field φ on M, and an embedding T3 ↪ M
such that ι(T3 ) is a Cauchy hypersurface in (M, gµν ). The spacetime fields must verify the equations
(1.1a)-(1.1b) and be such that ι∗ g = 0g, ι∗ k = 0k, ι∗ φ = 0φ, ι∗ N̂φ = 0ψ, where k is the second fundamental
form of ι(T3 ) (our sign convention is given in (3.1)), N̂φ is derivative of φ in the direction of the futuredirected normal N̂ to ι(T3 ), and ι∗ denotes pullback by ι. Throughout the article, we will often suppress
the embedding and identify T3 with ι(T3 ).
It is well-known (see also Prop. 3.1) that the data are constrained by the Gauss and Codazzi equations,
which take the following form for the Einstein-scalar field system:
(1.4a)
(1.4b)
R − 0k ab 0k ba + (0k aa )2 = 2T(N̂, N̂)∣T3 = 0ψ 2 + ∇a0φ∇a 0φ,
∂
∇a 0k aj − 0∇j 0k aa = −T(N̂, j )∣T3 = −0ψ∇j 0φ.
∂x
0
Above, T(N̂, N̂) ∶= Tαβ N̂α N̂β , ∇ denotes the Levi-Civita connection of 0g, 0R denotes the scalar curvature of 0g, and indices are lowered and raised with 0g and its inverse. Equations (1.4a)-(1.4b) are sometimes
referred to as the Hamiltonian and momentum constraints.
As is well known, to obtain a hyperbolic formulation, or more generally, elliptic-hyperbolic or parabolichyperbolic formulation, of equations (1.1a)-(1.1b), one must impose gauge choices. We find that there are
two gauges that allow us to detect the aforementioned monotonicity, which plays a central role in our
analysis. The first is the CMC-transported spatial coordinates gauge, which we mentioned above and recall in detail in Sect. 3. The second is a one-parameter family of gauges that is in many ways like the
CMC-transported spatial coordinates gauge, except that the elliptic CMC lapse equation is replaced with
a parabolic evolution equation for the lapse variable that is well-posed in the past direction; see Sect. 7
17
Roughly, this is the largest possible solution to the Einstein-scalar field equations that is uniquely determined by the data.
I. Rodnianski and J. Speck
7
for the details. Both of these gauges involve an infinite speed of propagation,18 which is essential for
synchronizing the singularity across space in the nonlinear problem. To the best of our knowledge, the
earliest instance of using a parabolic gauge in general relativity is found in [10], where the authors suggested that parabolic gauges might be useful for studying the long-time behavior of solutions. Readers
may also consult [41] for a discussion of local well-posedness for the Einstein equations under various
gauge conditions involving a parabolic equation for the lapse. We now summarize our main results. We
provide the detailed statements in Theorems 5.1 and 5.2.
Main Result 1: Approximate L2 monotonicity for the linearized equations. Consider a solution
to the linearized Einstein-scalar field equations in CMC-transported spatial coordinates gauge with
data given on Σ1 ∶= {1} × T3 . Specifically, the equations are linearized around a Kasner solution
(1.8). There exists an energy E(T otal) ≥ 0 (see Def. 4.4) that controls t−weighted L2 norms of
the linearized solution variables and whose square verifies the following estimate for t ∈ (0, 1]
whenever κ ≥ 0 is sufficiently small:
(1.5)
2
2
E(T
otal) (t) ≤ CE(T otal) (1) + cκ ∫
1
s=t
2
s−1 E(T
otal) (s) ds − Coercive potential terms.
Above, C and c are positive constants, and the constant κ ≥ 0 is a measure of the size of the tracefree part of the second fundamental form of Σ1 relative to the Kasner solution metric (see (1.12b)).
Furthermore, the “Coercive potential terms” on the right-hand side of (1.5) are t−weighted quadratic spacetime integrals that are positive definite in the “potential” components (that is, components not involving time derivatives) of the linear solution. In addition, we have the following
estimate for t ∈ (0, 1] ∶
(1.6)
E(T otal) (t) ≤ CE(T otal) (1)t−cκ .
The proof of Main Result 1 is based on combining a collection of integration by parts identities in
suitable proportions and judiciously using the constraint and lapse equations, which in total yields the
cancellation of dangerous terms and the emergence of favorable ones. As we noted earlier, in Sect. 1.4,
we analyze a simplified model problem that illustrates some of the special structure that allows for an
estimates of the form (1.5)-(1.6).
Main Result 2: Improved behavior for the lower-order derivatives and convergence. Under
the above smallness assumption on κ, whenever the data for the linearized equations have higher
Sobolev regularity, the below-top-order derivatives of the linear solution enjoy less-singular-in
t behavior as t ↓ 0 compared to the behavior allowed by (1.6). Furthermore, for sufficiently
regular data, certain time-rescaled components of the linear solution have a regular, finite, spatially
dependent limit as t ↓ 0. Some of these convergent components do not generally decay to 0.
The proof of Main Result 2 is relatively standard and is based on treating the evolution equations as
transport equations with small, derivative-losing error terms that are controlled by the estimates of Main
result 1.
Remark 1.1. Similar results hold in the parabolic lapse gauge mentioned above; see Theorem 7.1 and
Remark 7.5.
18
The fundamental (gauge-independent) dynamic variables in the Einstein-scalar field equations of course propagate at a
finite speed. It is only our description of them that involves an infinite speed.
A Regime of Linear Stability for the Einstein-Scalar Field System
8
1.3. CMC-transported spatial coordinates gauge and the Kasner family. As we mentioned above,
the monotonicity is visible upon reformulating equations (1.1a)-(1.1a) relative to CMC-transported spatial
coordinates. In this well-known gauge, the spacetime metric g is decomposed into the lapse n and the
Riemannian 3−metric g on Σt ∶= {(s, x) ∈ (0, 1] × T3 ∣ s = t} as follows:
g = −n2 dt2 + gab dxa dxb .
(1.7)
The spatial coordinates19 {xa }a=1,2,3 are called “transported” because they are constant along the integral
curves of the vectorfield N̂ = n−1 ∂t , which is the future-directed unit normal to Σt . The basic variables to
be solved for in the nonlinear equations are gij , kij ∶= − 21 n−1 ∂t gij , n, and φ. The hypersurfaces Σt have
constant mean curvature 13 k aa . Here and throughout, k ij = g ia kaj denotes the (mixed) second fundamental
form of the constant-time hypersurface Σt . We normalize the time coordinate so that k aa (t, x) ≡ −t−1 . See
Sect. 3 for a more detailed discussion of this gauge. In particular, we provide the corresponding constraint
and evolution equations in Prop. 3.1.
The aforementioned Kasner solutions (see Footnote 3 on pg. 1) can be expressed as
(1.8)
g̊ = −dt2 + g̊,
3
g̊ = ∑ t2qi (dxi )2 ,
φ̊ = A ln t,
(t, x) ∈ (0, ∞) × T3 ,
i=1
where the constants qi are called the Kasner exponents and A ≥ 0 is a constant denoting the value of ∂t φ
at t = 1. These quantities are constrained by the equations
3
∑ qi = 1,
(1.9a)
i=1
3
∑ qi2 = 1 − A2 .
(1.9b)
i=1
(1.9a) corresponds to our normalization condition k aa (t, x) ≡ −t−1 , while (1.9b) is a consequence of
k aa (t, x) = −t−1 plus the Hamiltonian constraint equation (1.4a). In this article, we only consider Kasner background solutions in which
(1.10)
qi > 0,
(i = 1, 2, 3).
It is important to note that it is not possible to have all three qi > 0 in the absence of matter. The FLRW
solution is a special case of (1.8) in which all three qi are equal to 1/3 ∶
(1.11)
√
3
2
gF LRW = −dt2 + gF LRW ,
gF LRW = t2/3 ∑(dxi )2 ,
φF LRW =
ln t,
(t, x) ∈ (0, ∞) × T3 .
3
i=1
It is the only spatially isotropic member of the Kasner family (1.8).
In our study of solutions to the linearized equations, an important role is played by the constants qM ax >
0 and κ ≥ 0 defined by
(1.12a)
qM ax ∶= max{q1 , q2 , q3 },
3
(1.12b)
κ2 ∶= ∑ qi2 −
i=1
19
1 3
1 2 2
= ∑ (qi − ) = − A2 .
3 i=1
3
3
Technically, the spatial coordinates only locally defined on T3 , even though the coordinate vectorfields ∂i can be globally
defined so as to be smooth.
I. Rodnianski and J. Speck
9
As we have mentioned, we prove the strongest version of our linearized stability results whenever κ is
sufficiently small (and thus all qi are near 1/3). The relevance of κ is: for Kasner metrics (1.8), the traceˆ
free part of the second fundamental form k̊ ij of Σt (see (3.1)), defined by k̊ ij ∶= k̊ ij − 13 k̊ aa I ij = k̊ ij + 13 t−1 I ij
ˆ ˆ
ˆ
(where I ij = diag(1, 1, 1) denotes the identity transformation), verifies (with ∣k̊∣2g̊ ∶= g̊ab (g̊ −1 )ij k̊ ai k̊ bj )
(1.13)
ˆ
∣k̊∣g̊ = κt−1 .
Note that the parameter κ drives the possible blow-up rate our L2 −based energies for linear solutions near
t = 0; see (1.6), Theorem 5.1, and the analysis of the model problem located in Sect. 1.4.
1.4. L2 monotonicity in a simplified model problem. In this section, we analyze a model linear elliptichyperbolic system. Our primary goal is to illustrate the main ideas behind the approximate L2 monotonicity for the Einstein-scalar field system in a simplified context. Our analysis also illustrates one of the
important differences between our work and previous work by authors who constructed singular solutions
via the Fuchsian approach (see Sect. 1.1): unlike in those works, in order to obtain sufficient approximate monotonicity, we must find subtle combinations of terms that, in the energy estimates, lead to the
cancellation of the worst terms and the unexpected emergence of favorable ones.
Our model system does not have quite enough structure to yield the full strength of the approximate
monotonicity that we are able to derive for the Einstein-scalar field system. Thus, at the relevant point in
the exposition, we highlight the additional structure found in the Einstein-scalar field system (specifically,
the availability of the momentum constraint equation (1.4b)) and explain how it contributes to the proof
of monotonicity. Moreover, we present a caricatured version of the momentum constraint equation and
explain how it can be used to help derive an L2 approximate monotonicity result for the model system.
The main estimate of interest, which relies on the caricature constraint, is inequality (1.17).
The spacetime domain for our model system is (t, x) ∈ (0, 1] × T, and the unknowns are the three scalar
functions H, ϕ, and ν. Roughly, H corresponds to a renormalized version of the Riemannian 3−metric perturbation, ∂t H a renormalized version of the trace-free part of the second fundamental form, ϕ the scalar
field perturbation, and ν the lapse perturbation. We assume that the data (H∣t=1 , ∂t H∣t=1 , ϕ∣t=1 , ∂t ϕ∣t=1 )
are given (ν∣t=1 is obtained by solving the elliptic PDE (1.15b)), and we are interested in the behavior of
the solution as t ↓ 0. To illustrate the dependence of our results for the Einstein-scalar field system on
the size of the trace-free part of the background Kasner second fundamental form, we incorporate three
non-negative parameters into the model system: A, κ, and q, which are constrained by
√
√
2
2
1
1/2
0≤A≤
(1.14)
,
κ ∶= (2/3 − A2 ) ,
q ∶= +
κ.
3
3
3
The above definitions of κ and q are designed to emulate the Kasner exponent restriction and trace-free
size restriction from (1.12b). In particular, κ corresponds to the size of the trace-free part of the Kasner
second fundamental form. We define our model system to be:
(1.15a)
(1.15b)
(1.15c)
−∂t (t∂t H) + t1−2q ∂x2 H = −2t1−2q ∂x2 ν + κt−1 ν,
t2−2q ∂x2 ν = (1 − 2A2 )ν + 2At∂t ϕ + κt∂t H,
−∂t (t∂t ϕ) + t1−2q ∂x2 ϕ = −A∂t ν + At−1 ν.
A Regime of Linear Stability for the Einstein-Scalar Field System
10
Equation (1.15a) is a scalar caricature of equations20 (3.15a)-(3.15b), while (1.15b) and (1.15c) are respectively caricatures of equations (3.14a) and (3.16).
We will study the behavior as t ↓ 0 of an energy of the form
(1.16)
2
2
2(1−q)
E(T
(∂x H)2 dx
otal);θ (t) ∶= θ ∫ (t∂t H) + t
Σt
1
+ ∫ (t∂t ϕ)2 + t2(1−q) (∂x ϕ)2 + ν2 + t2(1−q) (∂x ν)2 dx,
3
Σt
where θ > 0 is a constant to be chosen below. Above, dx is the standard flat volume form on Σt ≃ T. Our
aim is to show that there exist constants C > 0, c > 0, and θ∗ > 0 such that if κ is sufficiently small, then
solutions to (1.15a)-(1.15c) verify the following estimate21 for t ∈ (0, 1]:
(1.17)
2
2
E(T
otal);θ∗ (t) ≤ CE(T otal);θ∗ (1)
1
1 1
2
− θ∗ ∫ s−1 ∫ s2(1−q) (∂x H)2 dx ds − ∫ s−1 ∫ s2(1−q) (∂x ϕ)2 dx ds
3
6 s=t
s=t
Σs
Σs
1
1
1
1
− ∫ s−1 ∫ s2(1−q) (∂x ν)2 dx ds − ∫ s−1 ∫ ν2 dx ds
6 s=t
2 s=t
Σs
Σs
+ cκ ∫
1
s=t
s−1 ∫ (s∂t H)2 dx ds.
Σs
The following energy estimate is a simple consequence of (1.17) and Gronwall’s inequality:
(1.18)
E(T otal);θ∗ (t) ≤ CE(T otal);θ∗ (1)t−cκ .
The estimate (1.18) is an analog of the energy estimate (5.2) for solutions to the linearized Einstein-scalar
field equations of Prop. 3.2. We stress that the mild energy blow-up rate of (5.2) (mild because κ is small)
is the main ingredient that we use to treat the high-order derivatives in the nonlinear analysis of [55].
Remark 1.2. Our analysis can easily √
be extended to show that under the constraint (1.19), inequality
(1.18) still holds when κ is large (near 2/3). The same remark applies to our energy estimate (5.2) for
the linearized Einstein-scalar field system.
As we mentioned above, we are not able to derive (1.17) without the additional “constraint” assumption
(1.19)
t∂x ∂t H = −A∂x ϕ + P κ∂x H,
where P > 0 is a positive constant representing the combined strength of the second and third terms22 on
the right-hand side of (3.13b). The size of P affects the allowable size of κ; as we will see at the end of the
discussion, the larger P is, the smaller κ must be in order for inequality (1.17) to hold in its stated form.
The identity (1.19) is not generally satisfied by solutions to (1.15a)-(1.15c), but an analog of it is satisfied
The variable H in (1.15a) corresponds to t−2q h in (3.15a)-(3.15b), while t∂t H corresponds to −2K. The term κt−1 ν on
the right-hand side of (1.15a) is a model of the term −t−1 K̊ ij ν on the right-hand side of (3.15b), where we included the factor
κ in our model term because only the trace-free part of the term −t−1 K̊ ij ν enters into the energy estimate for equation (3.15b).
We remark that a more accurate caricature of (3.15a)-(3.15b) would be obtained if we added a term proportional to ∂t ν to the
right-hand side of (1.15a); such a term would account for the factor n in the nonlinear equation (3.6a). However, a modification
of the arguments given in this section would reveal that this term would have a negligible effect on the dynamics whenever κ is
small. Hence, we omit this term for simplicity.
21
The explicit numerical constants on the right-hand side of (1.17) are not sharp, but that is not important when κ is small.
22
The tensorial structure of these terms, which is expected to be important for treating the far-from-FLRW regime, is not
captured by our model system.
20
I. Rodnianski and J. Speck
11
by solutions to the linearized Einstein-scalar field equations of Prop. 3.2 (see (3.13b)-(3.13c)). Hence, for
illustration, we assume that (1.19) holds. Below, we highlight the role that it plays in deriving (1.17). We
remark that the signs of the terms in (1.19) are not important for the arguments that follow.
We now explain how to derive inequality (1.17) under the constraint (1.19). The inequality is heavily
based on the fact that there exists a good energy identity for solutions to equations (1.15b)-(1.15c) that
partially decouples from equation (1.15a). More precisely, the identity is partially coercive in ϕ and ν and
is as follows in divergence form:
(1.20)
∂t {(t∂t ϕ)2 + t2(1−q) (∂x ϕ)2 + t2(1−q) (∂x ν)2 + (1 − A2 )ν2 + κ(t∂t H)ν}
1
= ∂x { ∂t [t2(1−q) (∂x ν)2 ] + 2t1−2q (t∂t ϕ)∂x ϕ − t1−2q (∂x ν)ν − 2At1−2q (∂x ϕ)ν}
2
+ 2(1 − q)t1−2q (∂x ϕ)2 + t1−2q (∂x ν)2 + t−1 ν2 + 2At1−2q (∂x ϕ)∂x ν + κt−1 (t∂t H)ν.
The identity (1.20) follows from a tedious but straightforward calculation based on multiplying equation
(1.15c) by −2t∂t ϕ and also by 2Aν, multiplying (1.15b) by ν, and then differentiating by parts. We note
that a helpful intermediate step in deriving (1.20) is the identity
(1.21)
2A(t∂t ϕ)∂t ν = 2A∂t [(t∂t ϕ)ν] − 2A [∂t (t∂t ϕ)] ν
1
= ∂t { t2(1−q) ∂x [(∂x ν)2 ] − t2(1−q) (∂x ν)2 + (A2 − 1)ν2 − κ(t∂t H)ν}
2
− 2A∂x {t1−2q (∂x ϕ)ν} + 2A2 t−1 ν2 + 2At1−2q (∂x ϕ)∂x ν,
which follows from (1.15b)-(1.15c) and the multiplications/differentiations by parts mentioned above.
Note that the perfect ∂x derivative term on the right-hand side of (1.20) will vanish when we integrate over
x ∈ T. Next, we use Young’s inequality in the form 2A(∂x ϕ)∂x ν ≥ −(∂x ϕ)2 − A2 (∂x ν)2 ≥ −(∂x ϕ)2 −
2
2
3 (∂x ν) and the fact that q is near 1/3 whenever κ is small in order to completely absorb the next-to-last
term on the right-hand side of (1.20) into the first two. After absorbing, the remaining available positivity
1
1
coming from the first two terms is ≥ ( − Cκ) t1−2q (∂x ϕ)2 + (∂x ν)2 . We note that we still have to
3
3
address the non-coercive term κ(t∂t H)ν appearing on the left-hand side of (1.20) and the non-coercive
term κt−1 (t∂t H)ν appearing on the right-hand side of (1.20). Note that the strength of these terms is
proportional to κ.
To obtain inequality (1.17), we will add the identity (1.20) to a suitable multiple of a related but simpler
identity for solutions H to the metric wave equation (1.15a). To this end, we multiply (1.15a) by −2t∂t H
and differentiate by parts, which yields the identity
(1.22)
∂t {(t∂t H)2 + t2(1−q) (∂x H)2 } = ∂x {2t2 (∂t H)∂x H} + 2(1 − q)t1−2q (∂x H)2
+ 4t1−2q (∂x2 ν)(t∂t H) − 2κt−1 (t∂t H)ν.
By Young’s inequality, the last product on the right-hand side of (1.22) is ≥ −κt−1 ν2 − κt−1 (t∂t H)2 . At the
end of the proof, we explain how we can absorb the term −κt−1 ν2 . In contrast, we are not able to absorb
the term −κt−1 (t∂t H)2 , and it generates the last term on the right-hand side of (1.17); this is the term that
is responsible for the singular factor t−cκ on the right-hand side of (1.18).
To obtain inequality (1.17), we must use the constraint (1.19) to handle the product 4t1−2q (∂x2 ν)(t∂t H)
from the right-hand side of (1.22). We now explain why this is the case. Without the constraint, we would
have to control this product by first using equation (1.15b) to rewrite23 it as −4t−1 (2A2 − 1)(t∂t H)ν +
23
In higher dimensions, the analog of this step would be an elliptic estimate for the second spatial derivatives of ν.
A Regime of Linear Stability for the Einstein-Scalar Field System
12
8At−1 (t∂t ϕ)(t∂t H) + 4κt−1 (t∂t H)2 . The downside of this approach is that there are no positive definite terms available for absorbing the factors in the product 8At−1 (t∂t ϕ)(t∂t H). This would result
1
in the presence of an error integral c ∫ s−1 ∫ (s∂t ϕ)(s∂t H) dx ds on the right-hand side of (1.17),
s=t
Σs
where c is not of small size κ. To circumvent this difficulty and to parallel our analysis of the linearized
Einstein-scalar field system, we first differentiate by parts on the term as follows: 4t1−2q (∂x2 ν)(t∂t H) =
−4t1−2q (∂x ν)(t∂x ∂t H)+4t1−2q ∂x {∂x ν(t∂t H)}. From Young’s inequality and the constraint (1.19), we deduce that the right-hand side of of the previous expression is ≥ −2(1 + P κ)t1−2q (∂x ν)2 − 2A2 t1−2q (∂x ϕ)2 −
2P κt1−2q (∂x H)2 plus a perfect ∂x derivative term. Note that for κ sufficiently small relative to P , the
term −2P κt1−2q (∂x H)2 can be completely absorbed into the positive second term on the right-hand
side of (1.22), and that after absorbing, the remaining available positivity coming from this term is
≥ { 34 − C(1 + P )κ} t1−2q (∂x H)2 .
We now combine the above results. That is, under the caricature constraint (1.19), we integrate (1.20)
and (1.22) over (s, x) ∈ [t, 1] × T with respect to dxds. We then add the integrated version of (1.20)
to θ times the integrated version of (1.22). Choosing θ = θ∗ to be sufficiently small and positive and
using Young’s inequality in the form ab ≥ − 12 a2 − 12 b2 , we see that when κ is sufficiently small (depending on P ), most terms can be absorbed into the positive definite terms on the left-hand and right-hand
sides of (1.20) and (1.22). In fact, the only term that cannot be absorbed in this way is θ∗ times the
term κt−1 (t∂t H)2 mentioned just below equation (1.22), which leads to a spacetime error integral that
1
2
is ≤ κ ∫s=t s−1 E(T
(s) ds. In total, from this line of reasoning and straightforward calculations, we
otal);θ∗
conclude that for t ∈ (0, 1], we have the desired estimate (1.17) (see Footnote 21).
We close this section by highlighting two aspects of our analysis of the linearized Einstein-scalar field
equations that are not captured by the model system.
● In contrast to the scalar equation (1.15a), the principal spatial part of the metric evolution equations (3.15a)-(3.15b) (which is contained in the term t(h)Ricij ) is not an an elliptic operator. This
suggests that one cannot derive energy estimates for the metric components in our gauge. However, we are able to overcome this difficulty by combining an integration by parts argument with
the momentum constraints (3.13b)-(3.13c); see also Remark 5.4.
● In deriving the improved estimates at the lower derivative levels and the convergence results of
Theorem 5.2, we must use the alternate version (3.14b) of the lapse equation, which has no analog
in our model system. From the linearized Hamiltonian constraint (3.13a), we see that equation
(3.14b) is equivalent to the equation (3.14a) used for deriving high-order energy estimates. The
advantage of equation (3.14b) is that it involves pure spatial derivatives, which are smaller than
time derivatives near {t = 0}.
1.5. Comments on other matter models, higher dimensions, and the far-from-trace-free case. Scalar
field matter has two important properties, described just below, that allow us to prove the stability results of
the present paper and those of [55]. We anticipate that other matter models with similar properties might
allow for proofs of similar results. See [30] for a class of candidate matter models, where the authors
used Fuchsian techniques to construct families of non-spatially homogeneous solutions with Big Bang
singularities to various nonlinear Einstein-matter systems. We note that their construction also applied to
the Einstein-vacuum equations in 10 or more spatial dimensions and thus yielded rigorous examples of the
solutions that were heuristically argued to exist in [31]. The existence of these spatially inhomogeneous
Kasner-like vacuum solutions is relevant for the discussion two paragraphs below. The first important
property of the scalar field matter model is simply that it allows for the existence of spatially isotropic and
I. Rodnianski and J. Speck
13
nearly spatially isotropic Kasner solutions to the Einstein-matter system. We recall that nearly spatially
isotropic Kasner solutions have second fundamental forms with trace-free part that blow up at the rate
κt−1 with κ small (see (1.13)), and that this blow-up rate ultimately leads to the mild energy blow-up rate
(1.6). We now contrast this against the case of the Einstein-vacuum equations in three space dimensions.
In vacuum, we have A = 0 in (1.9b) and thus (1.12b)
√ and (1.13) imply that the trace-free part of the Kasner
second fundamental form blows up at the rate 23 t−1 . Combining this blow-up rate with the methods of
this paper, one would
only able to derive energy estimates in the spirit of (1.6) showing that the energy
√
2/3
−c
as t ↓ 0. Unfortunately, such a bound for the energy does not appear to be useful for
blows up like t
controlling error terms in the nonlinear problem. One crucial difficulty is that such a blow-up rate does not
even allow us to derive our second main result (Theorem 5.2) showing that lower-order spatial derivative
terms enjoy improved behavior in t. As we have mentioned, we need these improved estimates to control
nonlinear error terms when proving our stable singularity formation result [55] for the nonlinear equations.
We revisit this issue in more detail in the next paragraph. The second important property of the scalar field
matter model is that its time derivatives do not appear in the evolution equations for the metric (equations
(3.6a)-(3.6b)) nor in the elliptic PDE for the lapse (equation (3.9)). This property is closely tied to the
fact that the characteristics of the scalar field agree with those of the Einstein field equations (that is, the
principal part of the equations is the same). This property also plays a critically important role in allowing
us to prove our second main result (Theorem 5.2) because, at least at the lower orders, the time derivatives
of the scalar field are much larger than its spatial derivatives. We note that fluids verifying the equation
of state p = c2s ρ with 0 < cs < 1 do not have this property, even if the fluid is irrotational (roughly because
the sound cones are necessarily distinct from the gravitational null cones). This is consistent with the
oscillatory behavior for solutions to the Euler-Einstein system observed by Ringström [52] in the Bianchi
IX symmetry class when 0 < cs < 1 (see the discussion in Sect. 1.1).
We now describe some of the obstacles to deriving stability results for the Einstein-scalar field system
in the case of far-from-FLRW Kasner backgrounds (when κ is no longer small). Although our methods
could be used to obtain estimates for solutions to the linearized systems, they do not seem to be strong
enough to allow for a proof of stable blow-up in the nonlinear problem. To keep the discussion short, we
use the model problem of Sect. 1.4 to illustrate the difficulties; essentially the same difficulties arise in
the Einstein-scalar field system. Our goal is to highlight the reasons that for parameters corresponding to
far-from-spatially isotropic backgrounds, our methods do not allow us to prove that ∣t∂t H∣ remains uniformly bounded over the interval t ∈ (0, 1]. Recalling that ∂t H is an analog of the trace-free part of the
second fundamental form, we see that an inability to prove uniform boundedness of ∣t∂t H∣ is analogous
to not even being able to recover (in the context of a bootstrap argument) the Kasner second fundamental form trace-free blow-up rate of t−1 for Einstein-scalar field solutions. In the nonlinear problem, such
a bad estimate would lead (by a Gronwall estimate) to energy estimates that are drastically worse than
(1.6): the top-order energy estimates would be allowed to blow-up faster than data × t−C for all constants
C > 0. Consequently, our entire approach to the nonlinear problem would break down, and we would not
even be able to show that the solution exists near {t = 0}. In the remaining discussion in this paragraph,
A, κ, and q are the parameters (1.14) from the model problem of Sect. 1.4. The ensuing argument is
based on revisiting equation (1.15a) and allowing a loss of derivatives. We remark that similar analysis
based on allowing loss of derivatives forms the crux of the proof of Theorem 5.2. To keep the discussion
short, we keep only the two terms on the left-hand side of (1.15a), and we isolate the term ∂t (t∂t H).
That is, we consider the inequality ∣∂t (t∂t H)∣ ≤ ∣t1−2q ∂x2 H∣ + ⋯, where we ignore ⋯ from now on (see,
however, Footnote 24). We assume that we have already proved the energy bound (1.18) for the linear
A Regime of Linear Stability for the Einstein-Scalar Field System
14
solution (see Remark 1.2) and also the higher-order versions corresponding to commuting the linear equations up to two times with ∂x derivatives. Using (1.18), examining the strength of the energies (1.16),
and using Sobolev embedding in one spatial dimension to control L∞ (Σt ) norms by the energies, we
deduce the pointwise bound ∣t1−q ∂x2 H∣ ≲ data × t−cκ where data denotes terms that are controlled by the
data and its first two ∂x derivatives (and the data are given at t = 1). For illustration, we first consider
parameters corresponding to nearly spatially isotropic backgrounds. That is, we assume that κ ≈ 0 and
+
q ≈ 1/3. Hence, in this case, we have ∣t1−2q ∂x2 H∣ ≲ data × t−(1/3 ) . The important point is that the righthand side of the previous inequality is integrable in t over the interval (0, 1]. Thus, we may integrate
+
the estimate ∣∂t (t∂t H)∣ ≲ data × t−(1/3 ) with respect to time starting from time 1 to deduce that24 for
(t, x) ∈ (0, 1] × T, we have the uniform bound25 ∣∂t (t∂t H)∣ ≲ data. We now explain why, for parameters
corresponding to far-from-spatially isotropic backgrounds,
√our analysis does not yield a uniform bound
for ∣∂t (t∂t H)∣. Specifically, we consider parameters
κ ≈ 2/3 and q ≈ 1. Then reasoning as above, we
√
−
2/3)
−(1+c
1−2q
2
. Since the right-hand side of this inequality is not
find that ∣∂t (t∂t H)∣ ≤ ∣t
∂x H∣ ≲ data × t
integrable in time over t ∈ (0, 1], we are not able to derive an estimate showing that ∣t∂t H∣ is uniformly
bounded.
Finally, we make some comments on extending our stability results to higher dimensions. For brevity,
we limit our discussion to the Einstein-scalar field and Einstein-vacuum systems. For the Einstein-scalar
field system in any number of spatial dimensions, we expect that our linear stability results for nearFLRW Kasner backgrounds and the stable singularity formation results of [55] will go through without
any significant changes. Moreover, in the case of the Einstein-vacuum equations in n spatial dimensions
with n sufficiently large, there exists a class of Kasner solutions for which we expect that sufficiently
strong versions of the linear stability results of the present paper, suitable for deriving nonlinear stable
blow-up results as in [55], should remain valid. As we mentioned above, the existence (but not stability)
of non-spatially homogeneous solutions to the Einstein-vacuum equations with Big Bang singularities has
already been shown in [30] when n ≥ 10. We now provide some justification for the above remarks. First
we note that it is possible to derive an approximate L2 monotonicity identity for the linearized Einsteinvacuum equations that parallels the results for the Einstein-scalar field model derived in Props. 5.1 and 7.5.
More precisely, in the vacuum case, one can combine the metric energy estimates and the lapse estimates
in a way that leads to the kinds of special cancellations that are found in Props. 5.1 and 7.5; we do not
provide these computations in the present article, though we remark that they are not too different than
our computations for the Einstein-scalar field model. However, one faces the difficulty that in vacuum, the
trace-free part of the Kasner second fundamental form has large size (1 − 1/n)1/2 t−1 , a fact that follows
from the fact that in vacuum, the Kasner exponent constraints (1.9a)-(1.9b) become
n
(1.23)
n
∑ qi = 1,
∑ qi2 = 1.
i=1
i=1
The expression (1−1/n)1/2 t−1
suggests that the energy blow-up rate for solutions to the linearized Einsteinvacuum equations becomes worse as n → ∞, which seems to be an obstacle to stable blow-up. To
Note that we have ignored the term κt−1 ν on the right-hand side of (1.15a). If we kept this term, then it would in fact
invalidate the uniform estimates for H sketched here. The reason is that the model lapse equation (1.15b) is strongly coupled
to ∂t H and ∂t ϕ, which have quite singular behavior as t ↓ 0. In the linearized Einstein-scalar field system, we are able to
circumvent this difficulty by deriving improved estimates for the lapse at the lower derivative levels via the alternate lapse
equation (3.14b); see also the very last point in Sect. 1.4 for further comments.
25
Note that we cannot recover such a uniform bound using only the energy estimate (1.18) and Sobolev embedding whenever
κ > 0. The same remark applies in the nonlinear Einstein-scalar field problem, even for near-FLRW solutions; see Footnote 42.
24
I. Rodnianski and J. Speck
15
overcome this difficulty, at least in a certain regime, we make the following observation: the energy blowup rate can be sharpened compared to inequality (1.5). More precisely, the blow-up rate can be controlled
by the eigenvalues of the trace-free part of the second fundamental form.26 That is, a more careful analysis,
not carried out in this article,27 shows that the blow-up rate of the energies can be bounded by ≲ data×t−cα ,
n
where c > 0 is a universal constant independent of n and α ∶= max{∣qi ∣}. It is not difficult to see that there
i=1
exists a family of Kasner solutions such that α ↓ 0 as n → ∞. In total, we anticipate that one might be able
to prove that Kasner solutions with α sufficiently small28 are nonlinearly stable in a neighborhood of the
Big Bang by using the methods of [55]. We note that for fixed large n, only a small portion of the vacuum
Kasner solutions could be treated in this fashion. We also note that it would be interesting to discover the
threshold value of n beyond which the stable Kasner solutions exist; it could be that the threshold value
n ≥ 10 from [30], which is sufficient for the existence of non-spatially homogeneous solutions, is not large
enough to imply stability via the methods of [55].
1.6. A related instance in which monotonicity led to global results. We now describe the work [5] by
Andersson and Moncrief, in which they proved global existence results for the Einstein-vacuum equations
using techniques that have some overlap (see the next paragraph) with the ones used in the present article
and in [55]. Specifically, they proved a future-global existence theorem (in the expanding direction) for
perturbations of spatially compact versions of FLRW-like vacuum spacetimes in 1 + m dimensions for
t2
m ≥ 3. The background solutions were of the form −dt2 + 2 γ, where the spatial metric γ verifies the
m
m−1
Einstein condition Ric = − 2 γ, where Ric is the Ricci curvature of γ. Readers may also consult [4, 49]
m
for proofs of the results of [5] in the case m = 3, where unlike in [5] and the present article, the latter two
works rely on curvature-based energies constructed from the Bel-Robinson tensor. To derive their results
[5], Andersson and Moncrief made some technical assumptions on γ, notably one29 that they called being
“stable.” This condition states that the eigenvalues of the operator hij → −∆γ hij −2Riaj b hab , which appears
in linearized versions of the evolution equations, are non-negative. Here, hij is a symmetric type (02) tensor
and Riaj b is the Riemann curvature tensor of γ. In our proof of stable blow-up [55], terms like 2Riaj b hab
also appear, but we are able to treat them as lower-order error terms (that is, we do not have to work with
combinations such as −∆γ hij − 2Riaj b hab ). In [5], the authors also proved that a rescaled version of the
perturbed spatial metric converges to an element of the moduli space of γ. In the case m = 3, the Einstein
condition implies that γ has constant negative sectional curvature, and Mostow’s rigidity theorem implies
that the moduli space is trivial. Hence, the rescaled solution in fact converges to the background solution.
In contrast, in our work [55], there is a large family of possible end states corresponding to the asymptotic
behavior of the solution near the Big Bang. The family of course includes members of the Kasner family
(1.8), but it also includes30 a much larger family of “x−dependent” Kasner-like states. As stated in [5],
26
Specifically, we mean the version of the second fundamental form with one index up and one down.
See inequality (5.30) for a foreshadowing of the role that the eigenvalues play in deriving energy estimates.
28
One can think of 1/n as a parameter that can be chosen to be sufficiently small to close the estimates.
29
The authors also made additional assumptions. Specifically, they assumed that either the moduli space of γ is trivial or
that γ is contained in an integrable moduli space of Einstein structures.
30
More accurately, we do not rigorously prove in [55] that the family includes x−dependent end states. However, we recall
here the work [7] described in Sect. 1.1, in which Andersson and Rendall constructed solutions with end states that are analytic
in x with non-trivial x dependence. Based on this work and the results of [55], we expect that it may be possible to remove the
analyticity assumption (perhaps only in the near-FLRW regime), thus further enlarging the known set of possible end states;
see also Sect. 5.4.
27
A Regime of Linear Stability for the Einstein-Scalar Field System
16
the work of Andersson and Moncrief was motivated by Fisher’s and Moncrief’s work [38], in which the
linearized stability analysis was carried out. It was also motivated by a different form of monotonicity
discovered by Fisher and Moncrief. Specifically, they found a reduced Hamiltonian description of the
Einstein-vacuum flow [33–38] which applied to a family of spacetimes containing CMC hypersurfaces
(see also the result [6] of Andersson-Moncrief-Tromba in two spatial dimensions). Their Hamiltonian was
the volume functional of constant-time hypersurfaces Σt , where, as in the present article, the Σt were CMC
hypersurfaces. They showed that the Hamiltonian is monotonic along the flow of their reduced equations,
t2
that its critical points are precisely the metrics −dt2 + 2 γ from above that verify the Einstein condition,
m
and that its second variation is positive semidefinite for a large subset of these metrics.
The analysis in [5] has some features in common with the present work, such as the use of a constantmean-curvature foliation to reveal monotonicity, studying the solution at the level of the metric, and exploiting the key signed terms in the evolution equations that drive the monotonic behavior. One important
difference with our work here is that Andersson and Moncrief were able to close their proof by bounding the lapse in terms of the second fundamental form via standard elliptic estimates. In contrast, in the
present work, our estimate for the lapse is coupled to the energy estimate for the scalar field in a precise
fashion that results in the cancellation of dangerous linear terms and the emergence of favorably signed
terms; see Props. 5.1 and 7.5. The analog of this step in the model problem of Sect. 1.4 is the divergence
identity (1.20). This step plays a critical role in our derivation of energy estimates whose blow-up rate is
sufficiently mild to be of use in the nonlinear problem. Another notable difference is that unlike our work
here, the work [5] relies on spatial harmonic coordinates; see Remark 5.4 for additional comments about
those coordinates.
1.7. Paper outline.
● In Sect. 2, we introduce some notation and conventions that we use throughout the article.
● In Sect. 3, we provide the Einstein-scalar field equations in CMC-transported spatial coordinates.
We also linearize the equations around members of the (generalized) Kasner family.
● In Sect. 4, we provide the norms and energies that we use in our analysis.
● In Sect. 5, we state and prove our two main theorems showing a strong version of linear stability
when the Kasner backgrounds are near-FLRW.
● In Sect. 6, we briefly outline the role that our linear stability estimates play in our proof [55] of the
nonlinear stability of the FLRW solution near the Big Bang singularity.
● In Sect. 7, we introduce our family of parabolic lapse gauges and derive linear stability results in
these gauges.
2. N OTATION AND C ONVENTIONS
In this section, we summarize some notation and conventions that we use throughout the article.
2.1. Indices. Greek “spacetime” indices α, β, ⋯ take on the values 0, 1, 2, 3, while Latin “spatial” indices
a, b, ⋯ take on the values 1, 2, 3. Repeated indices are summed over (from 0 to 3 if they are Greek,
and from 1 to 3 if they are Latin). When working with the nonlinear equations in CMC-transported
spatial coordinates gauge or the parabolic lapse gauges, spatial indices are lowered and raised with the
Riemannian 3−metric gij and its inverse g ij . When working with the linearized equations, we will always
explicitly raise and lower indices with the background Kasner 3−metric g̊ij and its inverse (g̊ −1 )ij .
I. Rodnianski and J. Speck
17
2.2. Spacetime tensorfields and Σt −tangent tensorfields. We denote spacetime tensorfields Tν1 ⋯νnµ1 ⋯µm
in bold font. In the nonlinear equations, we denote the g−orthogonal projection of Tν1 ⋯νnµ1 ⋯µm onto the
constant-time hypersurfaces Σt ∶= {(s, x) ∈ R × T3 ∣ s = t} in non-bold font: Tb1 ⋯bna1 ⋯am . We also denote
general Σt −tangent tensorfields in non-bold font.
2.3. Coordinate systems and differential operators. We often work in a fixed standard local coordinate
system (x1 , x2 , x3 ) on T3 . The vectorfields ∂j ∶= ∂x∂ j are globally well-defined even though the coordinates themselves are not. Hence, in a slight abuse of notation, we use {∂1 , ∂2 , ∂3 } to denote the globally
defined vectorfield frame. We denote the corresponding dual frame by {dx1 , dx2 , dx3 }. As we described
in Sect. 1.3, the spatial coordinates can be transported along the unit normal to Σt , thus producing a local coordinate system (x0 , x1 , x2 , x3 ) on manifolds-with-boundary of the form (T, 1] × T3 , and we often
write t instead of x0 . The corresponding vectorfield frame on (T, 1] × T3 is {∂0 , ∂1 , ∂2 , ∂3 }, and the corresponding dual frame is {dx0 , dx1 , dx2 , dx3 }. Relative to this frame, the Kasner metrics g̊ are of the form
(1.8). The symbol ∂µ denotes the frame derivative ∂x∂ µ , and we often write ∂t instead of ∂0 and dt instead
of dx0 . Most of our equations and estimates are stated relative to the frame {∂µ }µ=0,1,2,3 and dual frame
{dxµ }µ=0,1,2,3 .
We use the notation ∂f to denote the spatial coordinate gradient of the function f . Similarly, if Θ is a
Σt − tangent one-form, then ∂Θ denotes the Σt −tangent type (02) tensorfield with components ∂i Θj relative
to the frame described above.
If I⃗ = (n1 , n2 , n3 ) is a triple of non-negative integers, then we define the spatial multi-indexed differen⃗ ∶= n1 + n2 + n3 denotes the order of I.
⃗
tial operator ∂I⃗ by ∂I⃗ ∶= ∂1n1 ∂2n2 ∂3n3 . The notation ∣I∣
Throughout, D denotes the Levi-Civita connection of g. We write
m
n
r=1
r=1
(2.1) Dν Tν1 ⋯νnµ1 ⋯µm = ∂ν Tν1 ⋯νnµ1 ⋯µm + ∑ Γνµαr Tν1 ⋯νnµ1 ⋯µr−1 αµr+1 ⋯µm − ∑ Γνανr Tν1 ⋯νr−1 ανr+1 ⋯νnµ1 ⋯µm
to denote a component of the covariant derivative of a tensorfield T (with components Tν1 ⋯νnµ1 ⋯µm ) defined on M. The Christoffel symbols of g, which we denote by Γµαν , are defined by
1
Γµλν ∶= (g−1 )λσ {∂µ gσν + ∂ν gµσ − ∂σ gµν } .
2
We use similar notation to denote the covariant derivative of a Σt −tangent tensorfield T (with components Tb1 ⋯bna1 ⋯am ) with respect to the Levi-Civita connection ∇ of the Riemannian metric g. The
Christoffel symbols of g, which we denote by Γji k , are defined by
(2.2)
(2.3)
1
Γji k ∶= g ia {∂j gak + ∂k gja − ∂a gjk } .
2
2.4. Integrals and L2 norms. Throughout this section, f denotes a scalar function defined on the hypersurface Σt = {(s, x) ∈ R × T3 ∣ s = t}. We define
(2.4)
1
2
3
∫Σ f dx ∶= ∫T3 f (t, x , x , x ) dx.
t
Above, the notation “ ∫T3 f dx” denotes the integral of f over T3 with respect to the measure corresponding to the volume form of the standard Euclidean metric E on T3 , which has the components
Eij = diag(1, 1, 1) relative to the coordinate frame described above. All of our Sobolev norms are built out
A Regime of Linear Stability for the Einstein-Scalar Field System
18
of the (spatial) L2 norms of scalar quantities (which may be the components of a tensorfield). We define
the standard L2 norm ∥ ⋅ ∥L2 over Σt as follows:
(2.5)
∥f ∥L2 = ∥f ∥L2 (t) ∶= (∫ f 2 dx)
1/2
.
Σt
For integers N ≥ 0, we define the standard H N norm ∥ ⋅ ∥H N over Σt as follows:
(2.6)
∥f ∥H N
⎞
⎛
2
= ∥f ∥H N (t) ∶= ∑ ∥∂I⃗f ∥L2 (t)
⎠
⎝∣I∣≤N
⃗
1/2
.
2.5. Constants. We use C and c to denote positive numerical constants that are free to vary from line to
line. If A and B are two quantities, then we often write
A≲B
(2.7)
to mean that “there exists a constant C > 0 such that A ≤ CB.”
3. T HE E INSTEIN -S CALAR F IELD E QUATIONS IN CMC-T RANSPORTED S PATIAL C OORDINATES
AND THE L INEARIZED E QUATIONS
In this section, we provide the standard formulation of the Einstein-scalar field equations relative to
CMC-transported spatial coordinates. We then linearize the equations around a Kasner solution (1.8).
3.1. Preliminary discussion. We begin by stating some basic facts concerning this formulation of the
equations. The fundamental unknowns are g, k, n, and φ, where g and n are as in (1.7), and k is the second
fundamental form of the hypersurfaces Σt . More precisely, the Σt −tangent type (02) tensorfield k is defined
by requiring that following relation holds for all vectorfields X, Y tangent to Σt ∶
(3.1)
g(DX N̂, Y ) = −k(X, Y ),
where D is the Levi-Civita connection of g and
(3.2)
N̂ ∶= n−1 ∂t
is the future-directed normal to Σt . It is a standard fact that k is symmetric:
(3.3)
k(X, Y ) = k(Y, X).
Let ∇ denote the Levi-Civita connection of g. The action of the spacetime connection D can be decomposed into the action of ∇ and k as follows:
(3.4)
DX Y = ∇X Y − k(X, Y )N̂.
Remark 3.1 (The mixed form of k verifies equations with a good structure). When working with the
components of k, we will always write it in the mixed form k ij ∶= g ia kaj with the first index upstairs and
the second one downstairs. The reason is that the nonlinear evolution and constraint equations verified by
the components k ij have a more favorable structure than the corresponding equations verified by kij .
I. Rodnianski and J. Speck
19
3.2. The Einstein-scalar field equations in CMC-transported spatial coordinates. In the following
proposition, we formulate the Einstein-scalar field equations (1.1a)-(1.1b) relative to CMC-transported
spatial coordinates.
Proposition 3.1 (The Einstein-scalar field equations in CMC-transported spatial coordinates). In
CMC-transported spatial coordinates normalized by k aa (t, x) ≡ −t−1 , the Einstein-scalar field system
consists of the following equations.
The Hamiltonian and momentum constraint equations verified by gij , k ij , and φ are respectively:
2T(N̂,N̂)
(3.5a)
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
R − k ab k ba + (k aa )2 = (n−1 ∂t φ)2 + g ab ∇a φ∇b φ,
²
−2
t
(3.5b)
∇a k ai
− ∇i k aa = −n−1 ∂t φ∇i φ .
² ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
−T(N̂,∂i )
0
The metric evolution equations verified by gij and k ij are:
(3.6a)
∂t gij = −2ngia k aj ,
(3.6b)
∂t k ij = −g ia ∇a ∇j n + n{Ricij + k aa k ij −g ia ∇a φ∇j φ },
¯
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
−1
−t
−T ij +(1/2)I ij T
where R = Ricaa denotes the scalar curvature of gij , Ricij denotes the Ricci curvature of gij (see (3.17)),
I ij = diag(1, 1, 1) denotes the identity transformation, and T ∶= (g−1 )αβ Tαβ denotes the trace of the
energy-momentum tensor (1.2).
√
The volume form factor detg verifies the auxiliary equation31
√
n−1
(3.7)
∂t ln (t−1 detg) =
.
t
The scalar field wave equation is:
−trkDN̂ φ
−DN̂ DN̂ φ
³¹
¹ ¹ ¹ ¹ ¹ ¹ ·¹¹ ¹ ¹ ¹ ¹ ¹ µ
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
1 −1
−1
ab
−1
(3.8)
−n ∂t (n ∂t φ) +g ∇a ∇b φ = n ∂t φ −n−1 g ab ∇a n∇b φ.
t
32
The elliptic lapse equation is:
(3.9)
g ab ∇a ∇b (n − 1) = (n − 1){R + (k aa )2 −g ab ∇a φ∇b φ}
²
−2
t
+ R − g ∇a φ∇b φ + (k aa )2 − ∂t (k aa ) .
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
ab
0
The CMC gauge condition and the constraints (3.5a)-(3.5b) are preserved by the flow of the remaining
equations if they are verified by the data.
This equation, which we do not use in the present article, is implied by (3.6a) and the CMC condition k aa = −t−1 .
Below, when we linearize the equations, we will view n − 1 as a linearly small quantity. Hence, we prefer to write (3.9) as
an equation in n − 1.
31
32
A Regime of Linear Stability for the Einstein-Scalar Field System
20
Remark 3.2. In (3.6b) and throughout, ∂t k ij ∶= ∂t (k ij ).
Proof of Prop. 3.1. It is well-known that equations (3.5a)-(3.5b) follow from (1.1a); see, for example,
[65, Ch. 10], and note that our k has the opposite sign convention of the one in [65]. It is also well-known
that equations (3.6a)-(3.8) follow from (1.1a)-(1.1b); see, for example, [58, Section 6.2] or [60, Section
10 of Chapter 18]. To derive (3.9), we take the trace of (3.6b) and use the CMC condition k aa = −t−1 .
The preservation of the gauge condition and constraints is a standard result that can be derived from a
straightforward modification of the argument presented in [3, Theorem 4.2].
3.3. Linearized quantities. In deriving the L2 approximate monotonicity, we work with the linearized
quantities defined just below in Def. 3.1. In the definition, g denotes the (Riemannian) 3 metric from
Prop. 3.1, g̊ denotes the 3 metric corresponding to the Kasner solution (1.8), and similarly for the other
quantities.
Definition 3.1 (Linearly small quantities). We define (for a, b, i, j = 1, 2, 3 and α = 0, 1, 2, 3)
(3.10a)
(3.10b)
(3.10c)
(3.10d)
(3.10e)
(3.10f)
(3.10g)
hij ∶= gij − g̊ij ,
1
(h) i
Γa b ∶= (g̊ −1 )ic {∂a hcb + ∂b hac − ∂c hab } ,
2
1
(h)
R ∶= − (g̊ −1 )ab (g̊ −1 )ef ∂e ∂f hab + (g̊ −1 )ef ∂a (h)Γeaf ,
2
1
1
1
(h)
Ricij ∶= − (g̊ −1 )ia (g̊ −1 )ef ∂e ∂f hja + (g̊ −1 )ef ∂j (h)Γei f + (g̊ −1 )ia g̊jb (g̊ −1 )ef ∂a (h)Γebf ,
2
2
2
i
i
i
K j ∶= tk j − tk̊ j ,
∂α ϕ ∶= ∂α φ − ∂α φ̊,
ν ∶= n − 1.
Remark 3.3 (Justification of Def. 3.1). The main point is that for solutions to the nonlinear equations
that are near the Kasner solution (1.8), all of the quantities defined in Def. 3.1 are linearly small.
It is convenient to work with the following time-rescaled Kasner tensorfields, which have constantvalued components.
Definition 3.2 (Time-rescaled mixed second fundamental form of the Kasner solution). We define
(for i, j = 1, 2, 3)
(3.11a)
K̊ ij ∶= tk̊ ij ,
1
̂
K̊ ij ∶= tk̊ ij + I ij ,
3
1
i
−1
ia
where k̊ j = − 2 (g̊ ) ∂t g̊aj is the mixed second fundamental form of the Kasner solution (1.8) and I ij ∶=
diag(1, 1, 1) is the identity transformation.
(3.11b)
Remark 3.4. Above and throughout, T̂ denotes the trace-free part of the Σt −tangent tenor T . Note that
the identity K = K̂ follows from definition (3.10e) and the CMC condition k aa (t, x) ≡ −t−1 .
In the next proposition, we linearize the equations of Prop. 3.1 around a fixed Kasner solution (1.8).
That is, we derive linearized equations for the linearly small quantities (h, K, ϕ, ν) by expanding, in the
nonlinear equations of Prop. 3.1, the nonlinear quantities as the corresponding Kasner quantities plus the
linearly small quantities of Def. 3.1 and discarding all quadratic and higher-order terms.
I. Rodnianski and J. Speck
21
Proposition 3.2 (The linearized Einstein-scalar field equations in CMC-transported spatial coordinates). Consider the equations of Prop. 3.1 linearized around the Kasner solution (1.8). The linearized
equations in the unknowns (h, K, ϕ, ν) take the following form (see Def. 3.1 for the definitions of some of
the quantities).
The linearized constant mean curvature condition is:
(3.12)
K aa = 0.
The linearized versions of the Hamiltonian and momentum constraint equations (3.5a)-(3.5b) are:
̂
(3.13a) t2(h)R − 2K̊ ab K ba − 2At∂t ϕ + 2A2 ν = 0,
̂
̂
(3.13b)
∂a K ai = −A∂i ϕ − (h)Γaab K̊ bi + (h)Γab i K̊ ab ,
̂
̂
(g̊ −1 )ab ∂a K ib = −A(g̊ −1 )ia ∂a ϕ − (g̊ −1 )ab(h)Γai c K̊ cb + (g̊ −1 )ab(h)Γac b K̊ ic ,
(3.13c)
√
where the constant 0 ≤ A ≤ 2/3 is defined by (1.9b).
The linearized version of the lapse equation (3.9) can be expressed in either of the following two forms:
(3.14a)
̂
2At∂t ϕ + 2K̊ ab K ba = t2 (g̊ −1 )ab ∂a ∂b ν + (2A2 − 1)ν,
(3.14b)
t2 (g̊ −1 )ab ∂a ∂b ν − ν = t2(h)R.
Equation (3.13a) can be used to show that (3.14a) is equivalent to (3.14b).
The linearized versions of the metric evolution equations (3.6a)-(3.6b) are:
(3.15a)
∂t hij = −2t−1 K̊ aj hia − 2t−1 g̊ia K aj − 2t−1 g̊ia K̊ aj ν,
(3.15b)
∂t K ij = −t(g̊ −1 )ia ∂a ∂j ν − t−1 K̊ ij ν + t(h)Ricij .
The linearized version of the scalar field wave equation (3.8) is:
(3.16)
−∂t (t∂t ϕ) + t(g̊ −1 )ab ∂a ∂b ϕ = −A∂t ν + At−1 ν.
Remark 3.5. Equation (3.13b) is the linearized version of ∇a k ai = −n−1 ∂t φ∇i φ, while equation (3.13c) is
the linearized version of ∇a k ia = −n−1 ∂t φ∇i φ. We use both of these equations when deriving estimates.
Remark 3.6 (Propagation of L2 regularity). In deriving the equations of Prop. 3.2, we have linearized
a version of the Einstein-scalar field system written relative to a dynamic system of coordinates that is
adapted to the nonlinear flow. It is for this reason that the energy identities for the linearized equations,
which we derive below in Propositions 5.1 and 5.2, should be viewed as providing relevant information
about the L2 regularity of the nonlinear solution. In particular, the proofs of these propositions can be
modified in a straightforward fashion to yield a coercive energy identity for the nonlinear equations.
Proof of Prop. 3.2. In the equations of Prop. 3.1, we expand the Riemannian metric g as an order 0 term
and a linearly small term as: gij = g̊ij + hij , and similarly for (tk ij , φ, n). We then discard all quadratic
and higher-order terms, which yields the proposition. We remark that it is straightforward to see that in
Def. 3.1, (h)Γai b is the linearization of the Christoffel symbol Γai b (see (2.3)) around the Kasner solution,
and similarly for (h)Ricij and (h)R. We have obtained latter two linearizations from the standard expression
(3.17)
Ricij = g ic ∂a Γcaj − g ic ∂c Γjaa + g ic Γaab Γcbj − g ic Γcab Γab j
for the Ricci curvature of g in terms of its Christoffel symbols (2.3) and the definition R ∶= Ricaa .
A Regime of Linear Stability for the Einstein-Scalar Field System
22
4. N ORMS AND E NERGIES
In this short section, we define the norms and energies that play a role in our analysis.
4.1. Pointwise norms. We will use the following two norms.
a1 ⋯am
)
.
Definition 4.1 (Pointwise norms). Let T be a type (m
n Σt −tangent tensor with components Tb1 ⋯bn
Then ∣T ∣F rame denotes a norm of the components of T ∶
3
(4.1a)
3
3
3
∣T ∣2F rame ∶= ∑ ⋯ ∑ ∑ ⋯ ∑ ∣Tb1 ⋯bna1 ⋯am ∣ .
a1 =1
am =1 b1 =1
2
bn =1
∣T ∣g̊ denotes the g̊−norm of T , where g̊ is the background Kasner metric:
(4.1b)
∣T ∣2g̊ ∶= g̊a1 a′1 ⋯g̊am a′m (g̊ −1 )b1 b1 ⋯(g̊ −1 )bn bn Tb1 ⋯bna1 ⋯am Tb′ ⋯b′n
′
′
a′1 ⋯a′m
.
1
4.2. Sobolev norms. We will use the following two Sobolev norms. The norms ∥ ⋅ ∥Hg̊M are “more geometric” and naturally arise in our energy estimates. The norms ∥ ⋅ ∥HFMrame involve components relative
to the transported coordinate frame; we use them when we derive improved estimates for the lower-order
derivatives of the solution by analyzing their frame components.
a1 ⋯am
)
Definition 4.2 (Sobolev norms). Let T be a type (m
.
n Σt −tangent tensor with components Tb1 ⋯bn
We define
(4.2a)
(4.2b)
∥T ∥HFMrame = ∥T ∥HFMrame (t) ∶= ∑ ∥∣∂I⃗T (t, ⋅)∣F rame ∥L2 ,
⃗
∣I∣≤M
∥T ∥Hg̊M = ∥T ∥Hg̊M (t) ∶= ∑ ∥∣∂I⃗T (t, ⋅)∣g̊ ∥L2 ,
⃗
∣I∣≤M
where ∥f ∥L2 is defined in (2.5), I⃗ denotes a spatial coordinate derivative multi-index (see Sect. 2.3), and
(∂I⃗T )b1 ⋯bna1 ⋯am ∶= ∂I⃗(Tb1 ⋯bna1 ⋯am ).
(4.3)
We often use the notation ∥T ∥L2 in place of ∥T ∥HF0 rame and the notation ∥T ∥L2g̊ in place of ∥T ∥Hg̊0 .
Definition 4.3 (Solution norms). The specific norms relevant for the solutions under study are as follows:
(4.4) H(F rame);M (t) ∶= ∥K∥H M
F rame
+ ∥∂h∥HFMrame + ∥t∂t ϕ∥H M
F rame
2
+ t2/3 ∥∂ϕ∥HFMrame + ∑ t(2/3)p ∥ν∥H M +p .
p=0
4.3. Energies. The monotonicity inequalities involve the following energies for the linearized variables.
I. Rodnianski and J. Speck
23
Definition 4.4 (Energies). For t ∈ (0, 1], we define E(M etric) [⋅](t) ≥ 0, ⋯, E(T otal);θ [⋅](t) ≥ 0 as follows:
(4.5a)
1
2
2
2
E(M
etric) [K, ∂h](t) ∶= ∫ ∣K∣g̊ + ∣t∂h∣g̊ dx,
4
Σt
(4.5b)
2
E(Scalar)
[∂t ϕ, ∂ϕ](t) ∶= ∫ (t∂t ϕ)2 + ∣t∂ϕ∣2g̊ dx,
Σt
2
E(∂Lapse)
[∂ν](t) ∶= ∫ ∣t∂ν∣2g̊ dx,
(4.5c)
Σt
2
E(Lapse)
[ν](t) ∶= ∫ ν2 dx,
(4.5d)
(4.5e)
2
E(T
otal);θ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](t)
Σt
2
2
∶= θE(M
etric) [K, ∂h](t) + E(Scalar) [∂t ϕ, ∂ϕ](t)
2
2
+ E(∂Lapse)
[∂ν](t) + E(Lapse)
[ν](t).
Above, θ is a small positive constant that we choose below in order to exhibit the desired monotonicity.
We will also use the following up-to-order M energy:
(4.6)
2
2
E(T
otal);θ;M (t) ∶= ∑ E(T otal);θ [∂I⃗K, ∂∂I⃗h, ∂t ∂I⃗ϕ, ∂∂I⃗ϕ, ∂∂I⃗ν, ∂I⃗ν](t).
⃗
∣I∣≤M
In Lemma 4.3, we compare the strength of the energies to the strength of the norms. The analysis is
straightforward and amounts to tracking powers of t. We first provide the following lemma, whose simple
proof we omit.
Lemma 4.1 (Basic properties of the spatial part of the Kasner metric). Let κ ≥ 0 be as defined in
(1.12b). The components of g̊ and g̊ −1 verify the following estimates for (t, x) ∈ (0, 1] × T3 , (i, j = 1, 2, 3):
∣g̊ij ∣ ≤ t2/3−2κ ,
(4.7a)
∣(g̊ −1 )ij ∣ ≤ t−2/3−2κ ,
(4.7b)
where κ is defined in (1.12b).
Furthermore, the 3 × 3 matrices g̊ij and (g̊ −1 )ij have the following positive definiteness properties:
(4.8a)
(4.8b)
t2/3+2κ δab X a X b ≤ g̊ab X a X b ≤ t2/3−2κ δab X a X b ,
t−2/3+2κ δ ab ξa ξb ≤ (g̊ −1 )ab ξa ξb ≤ t−2/3−2κ δ ab ξa ξb ,
∀X ∈ R3 ,
∀ξ ∈ R3 ,
where δab and δ ab are standard Kronecker deltas.
Furthermore, the Kasner metric g̊ defined in (1.8) verifies
(4.9)
∂t g̊ij = −2t−1 g̊ia K̊ aj ,
∂t (g̊ −1 )ij = 2t−1 (g̊ −1 )ja K̊ ia ,
where K̊ ij = diag(−q1 , −q2 , −q3 ).
Before comparing the strength of the energies and the norms, we first provide the following simple
elliptic estimate, which allows us to recover estimates for the top-order derivatives of the linearized lapse.
Lemma 4.2 (Top-order estimate for ν). If ν verifies equation (3.14a), then the following33 elliptic inequality holds:
̂
t2 ∥∂ 2 ν∥L2g̊ ≲ ∣(2A2 − 1)∣∥ν∥L2 + 2A∥t∂t ϕ∥L2 + 2∣K̊∣g̊ ∥K∥L2g̊ .
(4.10)
We note that ∥∂ 2 ν∥2L2 = ∫Σt (g̊ −1 )ab (g̊ −1 )ef ∂a ∂e ν∂b ∂f ν dx.
g̊
33
A Regime of Linear Stability for the Einstein-Scalar Field System
24
Proof. We multiply equation (3.14a) by t2 (g̊ −1 )ef ∂e ∂f ν, integrate by parts, and use Cauchy-Schwarz and
Young’s inequality as well as the simple estimate ∥(g̊ −1 )ef ∂e ∂f ν∥L2 ≲ ∥∂ 2 ν∥L2g̊ .
Lemma 4.3 (Energy-norm comparison lemma). Let N ≥ 0 be an integer and let κ ≥ 0 be as defined
in (1.12b). Under the assumptions of Lemma 4.2, there exist constants34 C, c > 0 depending on θ such
that the following comparison estimates hold for the norm (4.4) and the total energy (4.6) on the interval
t ∈ (0, 1] ∶
(4.11a)
(4.11b)
E(T otal);θ;N (t) ≤ Ct−cκ H(F rame);N (t),
H(F rame);N (t) ≤ Ct−cκ E(T otal);θ;N (t).
Proof. Lemma 4.3 follows easily from Lemma 4.1, Lemma 4.2 (which allows us to bound the top-order
linearized lapse term t4/3 ∥ν∥H N +2 from (4.4) in terms of the remaining terms), and the definitions of the
quantities involved.
5. T HE T WO L INEARIZED S TABILITY T HEOREMS
In this section, we state and prove our two main theorems, which collectively provide our stability
results for solutions to the linearized equations of Prop. 3.2.
5.1. Statement of the two theorems. We first state our main approximate L2 monotonicity theorem. The
proof is located in Sect. 5.3.1.
Theorem 5.1 (Approximate L2 monotonicity for solutions to the linearized equations). Consider a
solution to the linear equations of Prop. 3.2 corresponding to the data (K(1), h(1), ∂t ϕ(1), ∂ϕ(1)) (given
on Σ1 ), where ν(1) is determined by the elliptic PDEs (3.14a)-(3.14b). There exist a small constant θ∗ > 0
and constants C, c > 0 such that if κ ≥ 0 is sufficiently small (see definition 1.12b) and H(F rame);0 (1) < ∞
(see definition (4.4)), then the solution verifies the following energy inequality35 for t ∈ (0, 1], where
E(T otal);θ∗ is defined by (4.5e):
(5.1)
2
2
E(T
otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](t) ≤ CE(T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](1)
1
1 1
1
− θ∗ ∫ s−1 ∫ ∣s∂h∣2g̊ dx ds − ∫ s−1 ∫ ∣s∂ϕ∣2g̊ dx ds
6
6 s=t
s=t
Σs
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
past-favorable sign
1
1
1
1
− ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds − ∫ s−1 ∫ ν2 dx ds
6 s=t
2 s=t
Σs
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
past-favorable sign
1
2
+ cκ ∫ s−1 E(T
otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](s) ds .
s=t
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
error integral that can create energy growth
In addition, if N ≥ 0 is an integer and H(F rame);N (1) < ∞ (see definition (4.4)), then the total energy
(4.6) verifies the following inequality for t ∈ (0, 1] ∶
(5.2)
34
E(T otal);θ∗ ;N (t) ≤ CE(T otal);θ∗ ;N (1)t−cκ .
As we have mentioned, C and c are free to vary from line to line.
Footnote 21 applies to inequality (5.1) as well.
35
I. Rodnianski and J. Speck
25
Furthermore, if N ≥ 0 is an integer and H(F rame);N (1) < ∞, then the total solution norm (4.4) verifies
the following inequality for t ∈ (0, 1] ∶
(5.3)
H(F rame);N (t) ≤ CH(F rame);N (1)t−cκ .
Remark 5.1. Theorem 5.1 should be viewed as relevant for estimating the high derivatives in the nonlinear
problem, while Theorem 5.2 below should be viewed as relevant for estimating the low derivatives; see
Sect. 6.
Remark 5.2. The proof of Theorem 5.1 essentially amounts to combining an intricate collection of integration by parts identities in the right way. The model problem of Sect. 1.4 captures some of the main
ideas behind the argument. Certain aspects of our proof somewhat remind us of arguments used in [12],
in which Bartnik gave a new proof of the positive mass theorem of Schoen-Yau [56, 57] and Witten [66].
His proof was simpler than the previous proofs but was valid only under the assumption that the metric
is near-Euclidean and required the use of spatial harmonic coordinates. Like our proof, his involved expressing the scalar curvature of the Riemannian 3−metric in terms of Christoffel symbols, integrating with
respect to the measure corresponding to the Euclidean metric, and absorbing all of the unsigned quadratic
terms into favorably signed quadratic terms (whose coefficients happened to be sufficiently large).
The next theorem shows that the lower-order derivatives of the linear solution enjoy improved estimates
in t and that certain components converge as t ↓ 0. As we outline in Sect. 6, the improved behavior is
essential for proving the nonlinear stable blow-up results of [55]. The proof of the theorem is based on
revisiting the linearized equations and treating them as transport equations (with derivative-losing error
terms) at the lower-order levels. Elliptic estimates for the lapse also play a role. The main difficulty is
finding a suitable order in which to prove the estimates. In essence, this amounts to finding effective
dynamic decoupling in the equations. The proof of the theorem is located in Sect. 5.3.2.
Theorem 5.2 (Improved estimates for the lower derivatives and convergence). Assume the hypotheses and conclusions of Theorem 5.1. There exist constants C, c > 0 such that if N ≥ 2 is an integer,
∥h∥L2
(1), H(F rame);N (1) < ∞ (see definition (4.4)), and κ > 0 is sufficiently small (see definition
F rame
1.12b), then the linear solution verifies the following estimates for t ∈ (0, 1] ∶
(5.4a)
∥∂t K∥H N −1
(5.4b)
∥K∥H N −1
(5.4c)
∥h∥L2
(5.4d)
∥∂h∥H N −1
(5.4e)
F rame
F rame
F rame
F rame
∥∂t (t∂t ϕ − Aν)∥H N −1
≤ CH(F rame);N (1)t−1/3−cκ ,
≤ CH(F rame);N (1),
1
(1) + H(F rame);N (1)} t2/3−cκ ,
F rame
κ
≤ C {∥h∥L2
C
H(F rame);N (1)t2/3−cκ ,
κ
≤ CH(F rame);N (1)t−1/3−cκ ,
≤
(5.4f)
∥t∂t ϕ∥H N −1 ≤ CH(F rame);N (1),
(5.4g)
∥∂ϕ∥H N −2
(5.4h)
(5.4i)
(5.4j)
F rame
≤ CH(F rame);N (1) {1 + ∣ ln(t)∣} ,
∥ν∥H N ≤ CH(F rame);N (1)t−cκ ,
∥ν∥H N −1 ≤ CH(F rame);N (1)t2/3−cκ ,
C
∥ν∥H N −2 ≤ H(F rame);N (1)t4/3−cκ .
κ
A Regime of Linear Stability for the Einstein-Scalar Field System
26
−1
(T3 ), a type (11) tensorfield
Convergence. There a exist a symmetric type (02) tensorfield hRegular ∈ HFNrame
−1
KBang ∈ HFNrame
(T3 ) verifying (KBang )aa ≡ 0, and a function ΨBang ∈ H N −1 (T3 ) such that
(5.5a)
∥t−2qj hij + 2 ln(t)(KBang )ij − (hRegular )ij ∥H N −1 ≤ CH(F rame);N (1)t2/3−cκ , (if qi = qj ),
(5.5b) ∥t−2qj hij +
(5.5c)
1 2(qi −qj )
t
(KBang )ij − (hRegular )ij ∥
≤ CH(F rame);N (1)t2/3−cκ , (if qi ≠ qj ),
q i − qj
N
−1
H
∥K − KBang ∥H N −1
F rame
(5.5d)
(5.5e)
≤ CH(F rame);N (1)t2/3−cκ ,
∥t∂t ϕ − ΨBang ∥H N −1 ≤ CH(F rame);N (1)t2/3−cκ ,
∥∂ϕ − ln(t)∂ΨBang ∥H N −2
F rame
≤ CH(F rame);N (1),
and
(5.6a)
∥hRegular − h(1)∥H N −1
≤ CH(F rame);N (1),
∥KBang − K(1)∥H N −1
≤ CH(F rame);N (1),
F rame
(5.6b)
F rame
(5.6c)
∥ΨBang − ∂t ϕ(1)∥H N −1 ≤ CH(F rame);N (1).
In addition, the same estimates hold in the case κ = 0 with all factors of
1
replaced by 1 + ln t.
κ
● The convergence estimates of Theorem 5.2 are of a hybrid nature and suggest phenomena lying
somewhere in between orbital and asymptotic stability; see also Sect. 5.4. For example, (5.5c)
shows that the variable K converges (a form of stability), but it does not imply that it decays 0.
In fact, one already knows that in the nonlinear problem, perturbations do not generally decay
because all members of the Kasner family are solutions.
● The improved behavior in t provided by (5.4a)-(5.4j) is of critical importance in closing the nonlinear problem; see Sect. 6.
5.2. Energy identities verified by solutions to the linearized equations. In this section, we derive the
energy identities that form the crux the proof of our two main theorems. Unlike the theorems, the propositions proved in this section are valid for the solutions to the linearized equations independent of the Kasner
solution (g̊, φ̊) around which we linearize.
The most important ingredient in the proof of Theorem 5.1 is the energy identity provided by the following proposition. The analog of the proposition in the model problem of Sect. 1.4 is the identity (1.20).
I. Rodnianski and J. Speck
27
Proposition 5.1 (Energy identity for the linearized scalar field plus bonus control over the linearized
lapse). Solutions to the linearized equations of Prop. 3.2 verify the following identity for t ∈ (0, 1] ∶
(5.7)
2
2
2
2
2
∫Σ (t∂t ϕ) + ∣t∂ϕ∣g̊ dx + ∫Σ ∣t∂ν∣g̊ dx + (1 − A ) ∫Σ ν dx
t
t
t
= ∫ (∂t ϕ)
2
Σ1
+ ∣∂ϕ∣2g̊ dx + ∫
̂
− ∫ C1 (K̊, K, ν) dx
Σ
∣∂ν∣2g̊ dx + (1 − A2 ) ∫
Σ1
Σ1
ν2 dx + ∫
Σ1
̂
C1 (K̊, K, ν) dx
t
1
− 2∫
−∫
−∫
s=t
1
s=t
1
s=t
s−1 ∫
s−1 ∫
s−1 ∫
Σs
∣s∂ϕ∣2g̊ + C2 (K̊, s∂ϕ, s∂ϕ) dx ds
Σs
∣s∂ν∣2g̊ dx ds − A ∫
Σs
ν2 dx ds − ∫
where the constant 0 ≤ A ≤
√
1
s=t
1
s=t
s−1 ∫
s−1 ∫
Σs
Σs
Q1 (s∂ϕ, s∂ν) dx ds
̂
C1 (K̊, K, ν) dx ds,
2/3 is defined by (1.9b) and
̂
̂
C1 (K̊, K, ν) ∶= 2K̊ ab K ba ν,
(5.8a)
C2 (K̊, s∂ϕ, s∂ϕ) ∶= s2 (g̊ −1 )ab K̊ cb ∂a ϕ∂c ϕ,
(5.8b)
Q1 (s∂ϕ, s∂ν) ∶= 2s2 (g̊ −1 )ab ∂a ϕ∂b ν.
(5.8c)
Remark 5.3. The negative definite integrals on the right-hand side of (5.7) encourage some components
of the solution to decrease towards the past. The surprising aspect of (5.7) is the presence of the spacetime
integrals that are negative definite in ν and ∂ν. In Sect. 7, we show that a version of (5.7) also holds when
the CMC gauge is replaced with a parabolic lapse gauge.
Proof of Prop. 5.1. The proof involves combining three integration by parts identities. Throughout, we
silently use Lemma 4.1. To obtain the first identity, we multiply both sides of the linearized lapse equation
(3.14a) by ν and integrate by parts over T3 to deduce that
(5.9)
̂
2A ∫ t∂t ϕν dx = − ∫ ∣t∂ν∣2g̊ dx + (2A2 − 1) ∫ ν2 dx − 2 ∫ K̊ ab K ba ν dx.
Σt
Σt
Σt
Σt
The second identity is an energy estimate for the linearized scalar field wave equation. Specifically, we
replace t with the integration variable s in equation (3.16), multiply by −2s∂t ϕ, and integrate by parts over
(s, x) ∈ [t, 1] × T3 (we stress that t ≤ 1) to deduce that the following identity holds for t ∈ (0, 1] ∶
(5.10)
2
2
2
2
∫Σ (t∂t ϕ) + ∣t∂ϕ∣g̊ dx = ∫Σ (∂t ϕ) + ∣∂ϕ∣g̊ dx
t
1
− 2∫
1
s=t
− 2A ∫
s−1 ∫
Σs
1
s=t
∫Σ
s
∣s∂ϕ∣2g̊ + s2 (g̊ −1 )ab K̊ cb ∂a ϕ∂c ϕ dx ds
s∂t ϕ∂t ν dx ds + 2A ∫
1
s=t
s−1 ∫
Σs
Next, we note that equation (3.16) implies the identity
(5.11)
1
t∂t ϕ∂t ν = ∂t (t∂t ϕν) − A∂t (ν2 ) − tν(g̊ −1 )ab ∂a ∂b ϕ + At−1 ν2 .
2
s∂t ϕν dx ds.
A Regime of Linear Stability for the Einstein-Scalar Field System
28
To obtain the third identity, we now replace t with the integration variable s in equation (5.11), multiply
by 2A, and integrate by parts over (s, x) ∈ [t, 1] × T3 to deduce that
(5.12)
−2A ∫
1
s=t
2
2
∫Σ s∂t ϕ∂t ν dx ds = −2A ∫Σ ∂t ϕν dx + A ∫Σ ν dx
s
1
1
+ 2A ∫ t∂t ϕν dx − A ∫ ν2 dx
Σ
Σ
2
t
t
− 2A ∫
=∫
Σ1
1
s=t
s−1 ∫
Σs
s2 (g̊ −1 )ab ∂a ϕ∂b ν dx ds − 2A2 ∫
∣∂ν∣2g̊ dx + (1 − A2 ) ∫
Σ1
ν2 dx + 2 ∫
Σ1
1
s=t
s−1 ∫
̂
K̊ ab K ba ν dx
ν2 dx ds
Σs
̂
− ∫ ∣t∂ν∣2g̊ dx − (1 − A2 ) ∫ ν2 dx − 2 ∫ K̊ ab K ba ν dx
Σt
Σt
− 2A ∫
1
s=t
s−1 ∫
Σs
Σt
s2 (g̊ −1 )ab ∂a ϕ∂b ν dx ds − 2A2 ∫
1
s=t
s−1 ∫
ν2 dx ds,
Σs
where to obtain the second equality, we substituted the right-hand side of (5.9) for the integrals 2A ∫Σ1 ∂t ϕν dx
and 2A ∫Σt t∂t ϕν dx. We now use the identity (5.9) with t replaced by s to substitute for the integral
2A ∫Σs s∂t ϕν dx in the last spacetime integral on the right-hand side (5.10). Finally, we substitute the
right-hand side of (5.12) for the next-to-last spacetime integral on the right-hand side of (5.10). In total,
these steps lead to the identity (5.7).
In the next proposition, we derive an energy identity for the linearized metric solution variables.
Proposition 5.2 (Energy identity for the linearized metric variables). Solutions to the linearized equations of Prop. 3.2 verify the following identity for t ∈ (0, 1] ∶
(5.13)
1
1
2
2
2
2
∫Σ ∣K∣g̊ + 4 ∣t∂h∣g̊ dx = ∫Σ ∣K∣g̊ + 4 ∣∂h∣g̊ dx
t
1
1 1
− ∫ s−1 ∫ ∣s∂h∣2g̊ + C3 (K̊, s∂h, s∂h) dx ds
2 s=t
Σs
1
1
̂
̂
+ ∫ s−1 ∫ C4 (K̊, K, K) dx ds + ∫ s−1 ∫ C5 (K̊, s∂h, s∂h) dx ds
s=t
Σ
s=t
Σ
s
+∫
1
s=t
+ A∫
s−1 ∫
1
s=t
Σs
s−1 ∫
s
̂
C6 (K̊, s∂h, s∂ν) dx ds + ∫
1
s=t
Σs
Q2 (s∂ϕ, s∂ν) dx ds − A ∫
s−1 ∫
1
s=t
Σs
s−1 ∫
̂
C7 (K̊, K, ν) dx ds
Σs
Q3 (s∂h, s∂ϕ) dx ds,
I. Rodnianski and J. Speck
29
where
(5.14a) C3 (K̊, s∂h, s∂h) ∶= s2 (g̊ −1 )ab (g̊ −1 )ij (g̊ −1 )cf K̊ ec ∂e hai ∂f hbj ,
̂
̂
̂
C4 (K̊, K, K) ∶= 2g̊ic (g̊ −1 )ab K̊ cj K ia K jb − 2g̊ij (g̊ −1 )ac K̊ bc K ia K jb ,
(5.14b)
̂
̂
̂
(5.14c) C5 (K̊, s∂h, s∂h) ∶= s2 g̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ ac (h)Γicj (h)Γebf − s2 g̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ cj (h)Γiac (h)Γebf
̂
̂
+ s2 (g̊ −1 )ef K̊ ac (h)Γac b (h)Γebf − s2 (g̊ −1 )ef K̊ cb (h)Γaac (h)Γebf ,
̂
̂
̂
(5.14d) C6 (K̊, s∂h, s∂ν) ∶= 2s2 (g̊ −1 )ij K̊ bi (h)Γaab ∂j ν − 2s2 (g̊ −1 )ij K̊ ab (h)Γab i ∂j ν
(5.14e)
+ s2 (g̊ −1 )ij (g̊ −1 )ef K̊ aj ∂e hai ∂f ν,
̂
̂
C7 (K̊, K, ν) ∶= 2g̊ab (g̊ −1 )ij K̊ ai K bj ν,
(5.14f)
Q2 (s∂ϕ, s∂ν) ∶= 2s2 (g̊ −1 )ij ∂i ϕ∂j ν,
(5.14g)
Q3 (s∂h, s∂ϕ) ∶= 2s2 (g̊ −1 )ef (h)Γeaf ∂a ϕ.
Remark 5.4 (No need for spatial harmonic coordinates). Prop. 5.2 shows in particular that we can
derive energy estimates for the Einstein equations36 directly in CMC-transported spatial coordinates. Remarkably, we have not seen this observation made in the literature. Previous authors (see, for example,
[3, 5]) have instead chosen to impose the spatial harmonic coordinate condition ∆g xi = 0 to “reduce”
the Ricci tensor Rij of g to an elliptic operator acting on the components gij . That is, in spatial harmonic coordinates, we have Rij = − 21 g ab ∂a ∂b gij + fij (g, ∂g), which eliminates the last two products on the
right-hand side of (3.10d) and leads to a simpler proof of a basic L2 −type energy identity. In the proof
of Prop. 5.2, we handle these two products through a procedure involving integration by parts and the
constraint equations; see equations (5.22) and (5.23). The spatial harmonic coordinate condition, though
it may have advantages in certain contexts, introduces additional complications into the analysis. The
complications arise from the necessity of including a non-zero “shift vector” X i in the spacetime metric
g ∶ g = −n2 dt2 + gab (dxa + X a dt)(dxb + X b dt). To enforce the spatial harmonic coordinate condition, the
components X i must verify a system of elliptic PDEs that are coupled to the other solution variables.
Proof of Prop. 5.2. The proof involves combining a collection of integration by parts identities. Throughout, we silently use Lemma 4.1. To begin, we use the evolution equation (3.15b) to deduce that
(5.15)
∂t (∣K∣2g̊ ) = −2t−1 g̊ic (g̊ −1 )ab K̊ cj K ia K jb + 2t−1 g̊ij (g̊ −1 )ac K̊ bc K ia K jb
+ 2g̊ab (g̊ −1 )ij K ai {−t(g̊ −1 )bc ∂c ∂j ν − t−1 νK̊ bj + t(h)Ricbj } .
Note that we can express the first line of the right-hand side of (5.15) as
(5.16)
ˆ
ˆ
−2t−1 g̊ic (g̊ −1 )ab K̊ cj K ia K jb + 2t−1 g̊ij (g̊ −1 )ac K̊ bc K ia K jb
because the terms corresponding to the pure trace part of K̊ cancel. Furthermore, using equation (3.12),
we can express the second product on the second line of the right-hand side of (5.15) as
(5.17)
36
̂
−2t−1 g̊ab (g̊ −1 )ij K̊ ai K bj ν = −2t−1 g̊ab (g̊ −1 )ij K̊ ai K bj ν.
Although the proposition addresses only the linearized equations, essentially the same argument can be used to derive an
energy identity for the nonlinear equations.
A Regime of Linear Stability for the Einstein-Scalar Field System
30
Similarly, using the evolution equation (3.15a), we deduce that37
(5.18)
1
1
1
∂t (∣t2 ∂h∣2g̊ ) = t∣∂h∣2g̊ + t(g̊ −1 )bc (g̊ −1 )ij (g̊ −1 )ef K̊ ac ∂e hai ∂f hbj + t(g̊ −1 )ab (g̊ −1 )ij (g̊ −1 )cf K̊ ec ∂e hai ∂f hbj
4
2
2
1
+ (g̊ −1 )ab (g̊ −1 )ij (g̊ −1 )ef ∂e hai ∂f {−2thbc K̊ cj − 2tg̊bc K cj − 2tg̊bc K̊ cj ν} .
2
For convenience, in the remainder of this proof, we denote terms that can be expressed as perfect spatial
derivatives by “⋯.” These terms will vanish when we integrate the identities over T3 . First, we use equation
(3.13b) and differentiation by parts to express the first product on the second line of the right-hand side of
(5.15) as
(5.19)
−2tg̊ab (g̊ −1 )bc (g̊ −1 )ij K ai ∂c ∂j ν = 2t(g̊ −1 )ij ∂a K ai ∂j ν + ⋯
= −2At(g̊ −1 )ij ∂i ϕ∂j ν
̂
̂
− 2t(g̊ −1 )ij (h)Γaab K̊ bi ∂j ν + 2t(g̊ −1 )ij (h)Γab i K̊ ab ∂j ν
+ ⋯.
Next, we use equation (3.10d) to express the third product on the second line of the right-hand side of
(5.15) as
(5.20)
2tg̊ab (g̊ −1 )ij K ai (h)Ricbj = −t(g̊ −1 )ij (g̊ −1 )ef K ai ∂e ∂f hja
+ tg̊ab (g̊ −1 )ij (g̊ −1 )ef K ai ∂j (h)Γebf + tK ab (g̊ −1 )ef ∂a (h)Γebf .
Next, we use differentiation by parts to express the first product on the right-hand side of (5.20) as
(5.21)
−t(g̊ −1 )ij (g̊ −1 )ef K ai ∂e ∂f hja = t(g̊ −1 )ij (g̊ −1 )ef ∂e K ai ∂f hja + ⋯.
Next, we use equation (3.13c) and differentiation by parts to express the second product on the righthand side of (5.20) as
(5.22)
tg̊ab (g̊ −1 )ij (g̊ −1 )ef K ai ∂j (h)Γebf = −tg̊ab (g̊ −1 )ij (g̊ −1 )ef ∂j K ai (h)Γebf + ⋯
= At(g̊ −1 )ef ∂a ϕ(h)Γeaf
̂
̂
+ tg̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ cj (h)Γiac (h)Γebf − tg̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ ac (h)Γicj (h)Γebf
+ ⋯.
Next, we use equation (3.13b) and differentiation by parts to express the third product on the right-hand
side of (5.20) as
(5.23)
tK ab (g̊ −1 )ef ∂a (h)Γebf = −t(g̊ −1 )ef ∂a K ab (h)Γebf + ⋯
̂
̂
= At(g̊ −1 )ef ∂b ϕ(h)Γebf + t(g̊ −1 )ef K̊ cb (h)Γaac (h)Γebf − t(g̊ −1 )ef K̊ ac (h)Γac b (h)Γebf + ⋯.
37
We recall that ∣∂h∣2g̊ = (g̊ −1 )ab (g̊ −1 )ij (g̊ −1 )ef ∂e hai ∂f hbj .
I. Rodnianski and J. Speck
31
Combining (5.15)-(5.23) and carrying out straightforward computations, we deduce that
(5.24)
1
1
1
∂t (∣K∣2g̊ ) + ∂t (t2 ∣∂h∣2g̊ ) = t∣∂h∣2g̊ + t(g̊ −1 )ab (g̊ −1 )ij (g̊ −1 )cf K̊ ec ∂e hai ∂f hbj
4
2
2
̂
̂
− 2t−1 g̊ic (g̊ −1 )ab K̊ cj K ia K jb + 2t−1 g̊ij (g̊ −1 )ac K̊ bc K ia K jb
̂
̂
+ tg̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ cj (h)Γiac (h)Γebf − tg̊ab (g̊ −1 )ef (g̊ −1 )ij K̊ ac (h)Γicj (h)Γebf
̂
̂
+ t(g̊ −1 )ef K̊ cb (h)Γaac (h)Γebf − t(g̊ −1 )ef K̊ ac (h)Γac b (h)Γebf
̂
̂
− 2t(g̊ −1 )ij K̊ bi (h)Γaab ∂j ν + 2t(g̊ −1 )ij K̊ ab (h)Γab i ∂j ν − t(g̊ −1 )ij (g̊ −1 )ef K̊ aj ∂e hai ∂f ν
̂
− 2At(g̊ −1 )ij ∂i ϕ∂j ν − 2t−1 g̊ab (g̊ −1 )ij K̊ ai K bj ν + 2At(g̊ −1 )ef ∂a ϕ(h)Γeaf
+ ⋯.
To conclude (5.13), we have only to replace t with the integration variable s in the identity (5.24) and
integrate by parts over (s, x) ∈ [t, 1] × T3 (we stress that t ≤ 1).
5.3. Proof of Theorems 5.1 and 5.2. We now prove Theorems 5.1 and 5.2.
5.3.1. Proof of Theorem 5.1. Below we will prove that the following two inequalities hold for t ∈ (0, 1] ∶
(5.25)
1
2
2
2
[ν](t)
E(Scalar)
[∂t ϕ, ∂ϕ](t) + E(∂Lapse)
[∂ν](t) + ( − Cκ) E(Lapse)
3
1
̂
2
2
2
≤ E(Scalar)
[∂t ϕ, ∂ϕ](1) + E(∂Lapse)
[∂ν](1) + ( − Cκ) E(Lapse)
[ν](1) + ∫ C(K̊, K, ν) dx
3
Σ1
̂
+ ∫ C(K̊, K, ν) dx
Σt
1
1
− ( − Cκ) ∫ s−1 ∫ ∣s∂ϕ∣2g̊ dx ds
3
s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
1
1
1
− ( − Cκ) ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds − ∫ s−1 ∫ ν2 dx ds
3
s=t
Σs
s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
past-favorable sign
past-favorable sign
+ 2∫
1
s=t
s−1 ∫
Σs
̂
C(K̊, ν, K) dx ds,
A Regime of Linear Stability for the Einstein-Scalar Field System
32
(5.26)
2
2
E(M
etric) [K, ∂h](t) ≤ E(M etric) [K, ∂h](1)
1
1
− ( − Cκ) ∫ s−1 ∫ ∣s∂h∣2g̊ dx ds
3
s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
past-favorable sign
+∫
1
s=t
1
+∫
+∫
s=t
1
s=t
s−1 ∫
s−1 ∫
s−1 ∫
1
Σs
Q(s∂h, s∂ϕ) dx ds + ∫
Σs
̂
C(K̊, s∂h, s∂h) dx ds + ∫
Σs
̂
C(K̊, K, K) dx ds + ∫
s=t
1
s=t
s−1 ∫
1
s=t
Σs
s−1 ∫
s−1 ∫
Σs
Q(s∂ϕ, s∂ν) dx ds
Σs
̂
C(K̊, s∂h, s∂ν) dx ds
̂
C(K̊, K, ν) dx ds,
where Q(X, Y ) denotes a non-negative quadratic term that can be pointwise bounded by Q(X, Y ) ≲
∣X∣g̊ ∣Y ∣g̊ , and C(X, Y, Z) denotes a non-negative cubic term that can be pointwise bounded by C(X, Y, Z) ≲
∣X∣g̊ ∣Y ∣g̊ ∣Z∣g̊ .
Once we have shown (5.25) and (5.26), in order to prove (5.1), we add inequality (5.25) plus θ times
inequality (5.26). If we assume that κ ≥ 0 is sufficiently small, then we can choose a small positive constant
θ = θ∗ , independent of all small κ, such that, by virtue of Cauchy-Schwarz and Young’s inequality, we can
absorb all error integrals into the past-favorably-signed integrals, with the exception of two terms of the
form
(5.27)
Cκ ∫ ∣K∣2g̊ dx,
Σt
1
(5.28)
cκ ∫
s=t
s−1 ∫
Σs
∣K∣2g̊ dx ds.
The term (5.28) generates the last error integral on the right-hand side of (5.1). The term (5.27) can be
absorbed back into the left-hand side whenever Cκ is sufficiently small relative to θ∗ , at the expense of
increasing the constants C and c on the right-hand side of (5.1). In total, these steps lead to inequality
(5.1).
To deduce inequality (5.2), we commute the linearized equations with ∂I⃗, derive inequality (5.1) for the
differentiated linearized variables, and use Gronwall’s inequality to deduce that
(5.29)
2
2
−cκ
E(T
.
otal);θ∗ [∂I⃗K, ⋯, ∂I⃗ν](t) ≤ CE(T otal);θ∗ [∂I⃗K, ⋯, ∂I⃗ν](1)t
⃗ ≤ N , we arrive at (5.2).
Summing the estimates (5.29) for ∣I∣
Inequality (5.3) then follows from (5.2) and Lemma 4.3.
It remains for us to prove (5.25) and (5.26). To prove (5.25), we first bound the integrand on the third
line of the right-hand side of (5.7) as follows:
(5.30)
4
−2 {∣s∂ϕ∣2g̊ + C2 (K̊, s∂ϕ, s∂ϕ)} ≤ −2(1 − qM ax )∣s∂ϕ∣2g̊ ≤ − ( − 2κ) ∣s∂ϕ∣2g̊ ,
3
an estimate that follows easily from (1.12b) and the fact that the eigenvalues of K̊ ij are ≥ −qM ax ≥
− { 31 + κ}. To estimate the second integral on the fourth line of the right-hand side of (5.7), we bound
its integrand in magnitude via the pointwise estimate
(5.31)
2
A ∣Q1 (s∂ϕ, s∂ν)∣ ≤ ∣s∂ϕ∣2g̊ + A2 ∣s∂ν∣2g̊ ≤ ∣s∂ϕ∣2g̊ + ∣s∂ν∣2g̊ ,
3
I. Rodnianski and J. Speck
33
√
where we have used the simple inequality A ≤ 23 . Note that for sufficiently small κ, the right-hand of
(5.31) side does not fully exhaust the strength of the negative terms provided by the right-hand side of
(5.30) and by the integrand −∣s∂ν∣2g̊ on the right-hand side of (5.7). The desired inequality (5.25) thus
follows easily from (5.7), (5.30), and (5.31).
To prove inequality (5.26), we bound the integrand − 12 {∣s∂h∣2g̊ + C3 (K̊, s∂h, s∂h)} on the right-hand of
(5.13) side via the estimate
1
1
1
− {∣s∂h∣2g̊ + C3 (K̊, s∂h, s∂h)} ≤ − (1 − qM ax )∣s∂h∣2g̊ ≤ − ( − κ) ∣s∂h∣2g̊ ,
(5.32)
2
2
3
which follows from the aforementioned fact that the eigenvalues of K̊ ij are ≥ − { 13 + κ}. The desired
√
estimate (5.26) now easily follows from (5.13), (5.32), and the simple estimate A ≤ 23 mentioned above.
5.3.2. Proof of Theorem 5.2. We give the proof only in the case κ > 0. The case κ = 0 can be handled
by straightforward modifications of the case κ > 0. Throughout the proof, we silently use Lemma 4.1,
Lemma 4.3, and the t−weights inherent in Def. 4.3.
Proof of (5.4h) and (5.4i): We commute equation (3.14b) with ∂I⃗, multiply by ∂I⃗ν, and integrate by parts
over Σt to deduce that
(5.33)
t∥∂∂I⃗ν∥L2g̊ + ∥∂I⃗ν∥L2 ≤ Ct2 ∥∂I⃗(h)R∥L2 .
⃗ ≤ N −1, we have ∥∂ ⃗(h)R∥L2 ≤ Ct−4/3−cκ H(F rame);N (1).
From (3.10c) and (5.3), we deduce that whenever ∣I∣
I
The estimates (5.4h) and (5.4i) now readily follow.
Proof of (5.4a): We first deduce from equation (3.15b) that
(5.34)
∥∂t K∥HFNrame
≤ t1/3−cκ ∥ν∥HFNrame
+ Ct−1 ∥ν∥HFNrame
+ Ct∥(h)Ric∥HFNrame
.
−1
+1
−1
−1
From (3.10d), (5.3), and (5.4i), we conclude that the right-hand side of (5.34) is ≤ CH(F rame);N (1)t−1/3−cκ
as desired.
Proof of (5.4b), (5.5c), and (5.6b): These inequalities are straightforward consequences of inequality
(5.4a) and the integrability of the right-hand side of (5.4a) over the interval t ∈ (0, 1] whenever κ is sufficiently small.
Proof of (5.4c) and (5.4d): We give the details only for (5.4c) since the proof of (5.4d) is essentially the
same. To proceed, we first split K̊ aj into its pure trace and trace-free parts (using (3.11a)-(3.11b)) and use
equation (3.15a) to deduce that
(5.35)
ˆ
∂t (t−2/3 hij ) = −2t−1 (t−2/3 hia )K̊ aj − 2t−5/3 g̊ia K aj − 2t−5/3 g̊ia K̊ aj ν.
From equation (5.35), we deduce that
(5.36)
ˆ
∥∂t (t−2/3 h)∥L2F rame ≤ Ct−1 ∣K̊∣F rame ∥t−2/3 h∥L2F rame + Ct−5/3 ∣g̊∣F rame ∥K∥L2F rame
+ Ct−5/3 ∣g̊∣F rame ∣K̊∣F rame ∥ν∥L2 .
A Regime of Linear Stability for the Einstein-Scalar Field System
34
From inequality (5.3), we deduce that the right-hand side of (5.36) is
≤ cκt−1 ∥t−2/3 h∥L2F rame + CH(F rame);N (1)t−1−cκ .
Using this estimate and integrating (5.36) in time, we deduce that
1
C
(5.37) t−2/3 ∥h∥L2F rame (t) ≤ ∥h∥L2F rame (1) + H(F rame);N (1)t−cκ + cκ ∫ s−1 {s−2/3 ∥h∥L2F rame (s)} ds.
κ
s=t
Hence, from (5.37) and Gronwall’s inequality in the quantity t−2/3 ∥h∥L2F rame (t), we arrive at the desired
inequality (5.4c).
Proof of (5.4j): We need only to revisit the proof of (5.4i) and use the fact that the improved estimate
⃗ ≤ N − 2, we have ∥∂ ⃗(h)R∥L2 ≤ C H(F rame);N (1)t−2/3−cκ .
(5.4d) allows us to deduce that whenever ∣I∣
I
κ
Proof of (5.4e): We first deduce from equation (3.16) that
(5.38)
∥∂t (t∂t ϕ − Aν)∥H N −1 ≤ Ct∥(g̊ −1 )ab ∂a ∂b ϕ∥H N −1 + Ct−1 ∥ν∥H N −1 .
From (5.3) and (5.4i), we deduce that the right-hand side of (5.38) is ≤ the right-hand side of (5.4e) as
desired.
Proof of (5.4f), (5.5d), and (5.6c): These inequalities are straightforward consequences of inequality
(5.4e), inequality (5.4i), and the integrability of the right-hand side of (5.4e) over the interval t ∈ (0, 1]
whenever κ is sufficiently small.
Proof of (5.4g) and (5.5e): These inequalities are straightforward consequences of inequality (5.4f), inequality (5.5d), inequality (5.6c), and the integrability of the right-hand side of (5.4f) over the interval
t ∈ (0, 1] whenever κ is sufficiently small.
Proof of (5.5a), (5.5b), and (5.6a): Throughout this paragraph, we do not sum over i. Recall that g̊ii = t2qi ,
that K̊ ii = −qi , and that the off-diagonal components of these tensorfields are 0. Multiplying equation
(3.15a) by t−2qj , we deduce the equation ∂t (t−2qj hij ) = −2t−1+2(qi −qj ) K ij + 2qi δij t−1 ν. From this equation,
the estimates (5.4i) and (5.5c), and the simple estimate ∣qi − qj ∣ ≤ 2κ (see (1.12b)), we deduce that for
i, j = 1, 2, 3, we have
(5.39)
∥∂t {t−2qj hij − 2 (∫
1
s=t
s−1+2(qi −qj ) ds) (KBang )ij }∥
H N −1
≤ CH(F rame);N (1)t−1/3−cκ .
The desired results (5.5a), (5.5b), and (5.6a) follow easily from (5.39), the integrability of the right-hand
side of (5.39) over the interval t ∈ (0, 1] whenever κ is sufficiently small, and (5.6b).
5.4. Comments on realizing “end states”. The results of Theorem 5.2 show that for some time-rescaled
versions of the linearized solution variables, there is a well-defined map from their “initial state” along
the initial data hypersurface Σ1 to their “end state” along Σ0 . For example, the estimate (5.5d) exhibits
this fact for t∂t ϕ, in which case the end state is ΨBang and the map is from H N to H N −1 . It is natural to
wonder whether or not one can realize a given end state (more precisely, one in which time derivative terms
in the equations dominate) by finding suitable initial data that lead to it. Although we do not give a proof
that one can “realize all end states in which time derivative terms dominate” in solutions to the linearized
equations of Prop. 3.2, we do point to some evidence in this direction by discussing some relevant results
I. Rodnianski and J. Speck
35
in a simplified context. Our discussion here is closely connected to the work described in Sect. 1.1 in
which authors used Fuchsian methods to construct singular solutions to various Einstein-matter systems
under symmetry or analyticity assumptions. We consider a model equation in 1 + 1 dimensions, obtained
from the linearized scalar field equation (3.16) in the case g̊ = gF LRW = t2/3 ∑3i=1 (dxi )2 by dropping the
linearized lapse terms and making the symmetry assumption that the solution depends on only a single
spatial variable. We caution that ignoring the lapse and its elliptic PDE is tantamount to sidestepping
new difficulties not found in the standard Fuchsian framework, which applies to hyperbolic equations.
Specifically, our model equation in ϕ = ϕ(t, x) on the domain (t, x) ∈ (0, 1] × T is
−∂t (t∂t ϕ) + t1/3 ∂x2 ϕ = 0.
(5.40)
We have made the symmetry assumption only for convenience; the arguments we sketch below remain
valid without it. The methods of [15, 16] (see also the many other related works cited in Sect. 1.1), can be
used to show that given an asymptotic expansion for the end state of the form ln tΨ1 (x) + Ψ2 (x) (where
the Ψi have sufficient Sobolev regularity), one can construct a solution ϕ to (5.40) existing on a slab of
the form (0, 1] × T such that ϕ = ln tΨ1 (x) + Ψ2 (x) + R(t, x). Furthermore, there is a suitably strong
t−dependent Sobolev norm on the time slices Σt such that the norm of the remainder term R vanishes as
t ↓ 0. Moreover, R becomes negligible relative to ln tΨ1 (x) + Ψ2 (x) as t ↓ 0. We now sketch the proof
of these claims by following the approach outlined in Sect. 1.1. We note that our analysis involves much
simpler t weights in the energies compared to the weights of [15, 16] because we are treating a simple
linear scalar equation. We recall that the overall strategy of the proof is to construct a sequence of standard
initial value problems that approximate the “singular initial value problem with vanishing Cauchy data for
R given along Σ0 .” To begin our sketch of a proof, we deduce the following equation for R(t, x) from
(5.40):
(5.41)
−∂t (t∂t R) + t1/3 ∂x2 R = −t1/3 ln t∂x2 Ψ1 (x) − t1/3 ∂x2 Ψ2 (x).
We now derive an estimate for the energy E 2 [R](t) defined by
E 2 [R](t) ∶= ∫ (t1/3 ∂t R)2 + (∂x R)2 dx.
(5.42)
Σt
A straightforward integration by parts argument based on multiplying equation (5.41) by t−1/3 ∂t R yields
that for 0 < t1 < t2 ≤ 1, we have
(5.43)
E [R](t2 ) ≤ E [R](t1 ) + {∥∂x2 Ψ1 ∥L2 + ∥∂x2 Ψ2 ∥L2 } ∫
≤ E [R](t1 ) + C
t2
(1 + ln s)s−1/3 ds
s=t1
2
2
{∥∂x Ψ1 ∥L2 + ∥∂x Ψ2 ∥L2 } {tp2
− tp1 } ,
where p is a constant chosen to be slightly smaller than 2/3. Inequality (5.43) is the main ingredient
that one needs to deduce the desired existence result and estimates for R. Note that the estimate (5.43)
loses one derivative relative to Ψ1 and Ψ2 . In a detailed proof of the desired results (see the methods of
[16]), one considers a sequence {Rn }∞
n=0 of solutions to (5.41), where Rn has 0 Cauchy data on Σtn (and
thus E [Rn ](tn ) = 0) and is a classical solution on [tn , 1]. Here {tn }∞
n=0 is a sequence of times in (0, 1]
decreasing to 0 as n → ∞. An argument similar to the one used to prove (5.43) yields that for m < n, we
have
(5.44)
sup E [Rn − Rm ](t) ≤ C {∥∂x2 Ψ1 ∥L2 + ∥∂x2 Ψ2 ∥L2 } {tpm − tpn } .
t∈[tm ,1]
A Regime of Linear Stability for the Einstein-Scalar Field System
36
38
It follows from (5.44) that for any > 0, {Rn }∞
n=0 is Cauchy in the norm
f → sup {∥t1/3 ∂t f (t)∥L2 + ∥∂x f (t)∥L2 }
t∈(,1]
and thus converges39 to the desired solution R.
Remark 5.5. We could have instead derived energy estimates by multiplying equation (5.41) by t−P ∂t R
for any choice of P ∈ [1/3, 5/3), and a similar argument would yield a uniform bound for the energy
1/3−P
1−P
2
2
∫Σt (t 2 ∂t R) + (t 2 ∂x R) dx for t ∈ (0, 1]. We could even have allowed P to mildly depend on
x. This illustrates the freedom (mentioned in Sect. 1.1) in choosing viable t−weights in the Fuchsian
approach.
It is not difficult to modify the above arguments so that they apply if one includes the semilinear term40
t1/3 (∂x ϕ)2 on the right-hand side of (5.40); this term is a model for the kinds of semilinear terms that one
finds in the Einstein-scalar field system. It would be interesting to know to what extent the arguments can
be extended to apply to the full linearized system of Prop. 3.2 and the full nonlinear Einstein-scalar field
system in three spatial dimensions. The framework of [2] provides a possible starting point for establishing
such an extension. However, that framework applies only to symmetric hyperbolic Fuchsian systems and
thus it would need to be modified to treat the Einstein-scalar field system in gauges involving an elliptic
or parabolic lapse PDE.
6. C OMMENTS ON THE NONLINEAR PROBLEM
We now provide some justification for why in the near-FLRW regime, our linear stability results are
strong enough to control the nonlinear terms in the proof of stable singularity formation [55]. The discussion here by no means constitutes a complete proof of the results of [55]. Rather, it is an overview of the
central role that the linear stability results play in the nonlinear problem. Our discussion here applies to
solutions of the nonlinear equations of Prop. 3.1 that are near the FLRW solution gF LRW = −dt2 + gF LRW
(see (1.11)). In closing the nonlinear problem, we assume the small-data condition
(6.1)
∥g − gF LRW ∥L2
F rame
(1) + H(F rame);N (1) ≤ ,
where is a small positive parameter and N is a sufficiently large positive integer. In [55], we show that
N = 8 is large enough to allow for closure of all nonlinear estimates at all orders.
Our aim here is to sketch why versions of the linear energy inequalities (5.1) and (5.2) also hold for
near-FLRW solutions to the nonlinear equations of Prop. 3.1; such a priori estimates constitute the main
step in the proof of stable singularity formation in [55]. We begin by noting that the correct nonlinear
analog of the linearized energy (4.5a) for the ∂I⃗−differentiated metric variables is
(6.2)
2
2
E(M
(t) ∶= ∫ {∣t∂I⃗k̂∣g +
etric);I⃗
Σt
1
2
∣t∂I⃗∂g∣g } dx.
4
Higher-order energy estimates for the {Rn }∞
n=0 may be obtained in a similar fashion.
39
In the fully detailed construction of the analog of R for the nonlinear problems treated in [15, 16], the authors extend the
Rn to be 0 on [0, tn ) and show that this extension implies that Rn is a weak solution on an interval [0, δ).
40
More precisely, when the term t1/3 (∂x ϕ)2 is present, one can show that the remainder term R exists and verifies estimates
similar to the ones derived above on a small slab (0, δ]×T, where δ > 0 depends on a Sobolev norm of Ψ1 and Ψ2 . This argument
requires higher-order energy estimates because of the nonlinearity.
38
I. Rodnianski and J. Speck
37
Similarly, the correct nonlinear analog of the total energy (4.5e) for the ∂I⃗−differentiated variables is the
2
quantity E(T
(t) defined by everywhere replacing the variables in (4.5e) with their ∂I⃗−differentiated
otal);θ;I⃗
nonlinear versions and with the norms ∣ ⋅ ∣g̊ replaced by ∣ ⋅ ∣g .
⃗ ≤ N , compare the strength of the nonlinearities
To sketch the desired energy estimates, we will, for ∣I∣
2
in the ∂I⃗−commuted equations to the strength of E(T otal);θ;I⃗(t). To keep the discussion short, we address
only the nonlinear terms from the evolution equation (3.6b) for k ij that are quadratic in the derivatives
of g, that is, the quadratic terms coming from41 the Ricci tensor of g. We will sketch a proof of the fact
that for θ = θ∗ > 0 well-chosen as before, these nonlinear terms generate an error integral of the form
1
2
C ∫s=t s−1/3−cκ E(T
(s) ds. That is, for t ∈ (0, 1], we have the following analog of the linear energy
otal);θ∗ ;I⃗
inequality (5.1):
(6.3)
1
2
C ∫ s−1/3−cκ E(T
(s) ds
+integrals as in (5.1)
otal);θ∗ ;I⃗
s=t
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
error integral generated by the nonlinearities
+ additional nonlinear error integrals not treated here.
2
2
E(T
(t) ≤ E(T
(1) +
otal);θ∗ ;I⃗
otal);θ∗ ;I⃗
Because the function s−1/3−cκ is integrable over the interval s ∈ (0, 1] for sufficiently small κ, the error
1
2
integral C ∫s=t s−1/3−cκ E(T
(s) ds poses no obstacle to deriving the desired priori energy estimate
otal);θ ;I⃗
∗
2
E(T
(t) ≤ C2 t−cκ ,
otal);θ∗ ;I⃗
(6.4)
as in the linear estimate42 (5.2).
We now explain the origin of the error integral explicitly listed on the right-hand side of (6.3). For our
purposes here, it is convenient to perform computations relative to the following frame43 {e′(A) }3A=1 and
′
dual frame {θ (A) }3A=1 ∶
(6.5)
e′(A) ∶= t−1/3 ∂A ,
′
θ (A) ∶= t1/3 dxA .
The appeal of the frame {e′(A) }3A=1 is that it is orthonormal as measured by the background metric gF LRW ,
and, as we will explain, it is approximately orthonormal (in a sense that we make precise below) for the
perturbed metric g. The perturbed metric and its inverse can be expanded as
(6.6)
′
′
g = gAB θ (A) ⊗ θ (B) ,
g −1 = (g −1 )AB e′(A) ⊗ e′(B) .
The estimates (5.4c) and (5.4d) and Sobolev embedding suggest that in the nonlinear problem, we can
⃗ ≤N −3∶
prove the following coordinate component estimates for ∣I∣
(6.7)
41
∣∂I⃗ {gij − (gF LRW )ij }∣ ≲ t2/3−cκ ,
∣∂I⃗ {g ij − (gF−1LRW )ij }∣ ≲ t−2/3−cκ .
These terms generate cubic error integrals in the metric energy estimates that are similar to other error integrals that are
generated by integration by parts. For example, similar cubic error integrals arise from the nonlinear analog of (5.21).
42
In the nonlinear problem, κ is not equal to the size of tk̂F LRW , a tensor which in fact vanishes. Instead, κ should be
viewed as a small constant representing an upper bound for the size of the trace-free part of the dynamic quantity tk, which
will be controlled by a bootstrap argument. Justification for this approach is provided by inequality (5.4b).
43
In (6.5) and the remainder of Sect. 6, ∂A ∶= ∂x∂A , with {xA }A=1,2,3 denoting the transported spatial coordinates. Moreover,
∂I⃗ is still as defined in Sect. 2.3.
A Regime of Linear Stability for the Einstein-Scalar Field System
38
Indeed, as part of the bootstrap argument in [55], we show that the estimates (6.7) hold, and we shall take
them for granted here. Contracting inequalities (6.7) against the frame/dual frame, we find that they are
⃗ ≤ N − 3):
approximately orthonormal for the metric g in the following weak sense (for ∣I∣
(6.8)
∣∂I⃗ {gAB − δAB }∣ ≲ t−cκ ,
∣∂I⃗ {g AB − δ AB }∣ ≲ t−cκ ,
where δAB and δ AB are standard Kronecker deltas.
The connection coefficients of the frame relative to g are given by44
∇e′(A) e′(B) = (g −1 )CD γADB e′(C) ,
(6.9)
where since the vectorfield commutators [e′(A) , e′(B) ] vanish, we have45
1 ′
{e (gCB ) + e′(B) (gAC ) − e′(C) (gAB )} .
2 (A)
′
Next, we note the following standard expression for the Ricci curvature of g ∶ Ric = RicAB e′(A) ⊗ θ (B) ,
where
γACB =
(6.10)
(6.11)
RicAB = (g −1 )AE (g −1 )CF e′(C) (γEF B ) − (g −1 )AE (g −1 )CF e′(E) (γBCF )
+ (g −1 )AE (g −1 )CF (g −1 )DH γCF D γEHB − (g −1 )AE (g −1 )CF (g −1 )DH γEF D γCHB .
We now examine the evolution equation (3.6b). In view of (6.11), we see that relative to the frame/dual
frame (6.5), the ∂I⃗−commuted evolution equation (3.6b) reads
(6.12)
∂t (t∂I⃗k AB ) = (g −1 )AE (g −1 )CF e′(C) (∂I⃗γEF B ) − (g −1 )AE (g −1 )CF e′(E) (∂I⃗γBCF )
+ (g −1 )AE (g −1 )CF (g −1 )DH γCF D ∂I⃗γEHB +⋯,
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
top-order term
where, for illustration, we have kept only the linear Ricci terms and a single quadratic term involving a
top-order factor. Revisiting the proof of Prop. 5.2, we find that the quadratic term contributes the following
“new” cubic integral to the right-hand side of the energy inequality (6.3):
1
(6.13)
∣∫ ∫ (g −1 )AB (g −1 )CF (g −1 )DH γCF D (s∂I⃗γEHB )(s∂I⃗k̂ EA ) dx ds∣
s=t Σ
s
≲∫
≲∫
1
s=t
1
s=t
∫Σ ∣γ∣g ∣s∂I⃗γ∣g ∣s∂I⃗k̂∣g dx ds
s
2
∥∣γ∣g ∥L∞ (s)E(T
(s) ds.
otal);θ∗ ;I⃗
Hence, to justify the structure of the error integral on the right-hand side of (6.3), it remains only for us to
show that for t ∈ (0, 1], we have
(6.14)
∥∣γ∣g ∥L∞ (t) ≲ t−1/3−cκ .
⃗ = 1 implies that
To this end, we first note that (6.7) with ∣I∣
(6.15)
∣γACE ∣ ≲ t−1/3−cκ ,
Recall that ∇ denotes the Levi-Civita connection of g.
We are using here the standard notation X(f ) to denote the derivative of the scalar function f in the direction of the
vectorfield X.
44
45
I. Rodnianski and J. Speck
39
where in deriving (6.15), we have incurred three factors of t−1/3 relative to the estimate (6.7), one for each
contraction against a frame vector belonging to {e′(A) }3A=1 . We therefore deduce from (6.8) and (6.15) that
∣γ∣g = (g −1 )AB (g −1 )CD (g −1 )EF γACE γBDF ≲ t−2/3−cκ ,
2
(6.16)
which yields (6.14).
7. A S ECOND P ROOF OF L INEARIZED S TABILITY VIA PARABOLIC L APSE G AUGES
In this section, we introduce a new family of gauges for the Einstein-scalar field system and show that a
version of the approximate L2 monotonicity inequality (5.1) can also be derived in solutions to linearized
(around the Kasner backgrounds) versions of the corresponding equations. The gauges involve a parabolic
equation for the lapse variable n that depends on a real number λ. The approximate monotonicity is
present for near-FLRW Kasner backgrounds when 3 ≤ λ < ∞. As we will see, in this parameter range,
the parabolic lapse PDEs are locally well-posed only in the past direction. Formally, λ = ∞ corresponds
to the CMC lapse equation. However, our proof of approximate monotonicity in the parabolic gauges is
somewhat different than in the CMC case and does not allow us to directly recover the results of Sect. 5
by taking a limit λ → ∞.
7.1. Choice of a gauge and the corresponding formulation of the Einstein-scalar field equations. In
formulating the nonlinear Einstein-scalar field equations in the new gauges, we continue to use transported
spatial coordinates and to decompose g = −n2 dt2 + gab dxa dxb as in (1.7). We now fix the lapse gauge.
Definition 7.1 (Choice of a parabolic lapse gauge). Let λ ≠ 0 be a real number. We now impose the
following relation, which fixes the lapse gauge:
λ−1 (n − 1) = tk aa + 1.
(7.1)
Remark 7.1. Note that the CMC-transported spatial coordinates gauge of Sect. 3 corresponds to λ = ∞.
We now provide the (nonlinear) Einstein-scalar field equations relative to the gauge (7.1) with transported spatial coordinates.
Proposition 7.1 (The Einstein-scalar field equations in the gauge (7.1) with transported spatial coordinates). Under the gauge condition (7.1) and with transported spatial coordinates, the Einstein-scalar
field system consists of the following equations.
The Hamiltonian and momentum constraint equations verified by gij , k ij , and φ are respectively:
2T(N̂,N̂)
(7.2a)
R − k ab k ba +
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
= (n−1 ∂t φ)2 + g ab ∇a φ∇b φ,
(k aa )2
²
t−2 {λ−1 (n−1)−1}2
(7.2b)
∇a k ai − ∇i k aa = −n−1 ∂t φ∇i φ .
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
²
−1 −1
λ
t
∇i n
−T(N̂,∂i )
The evolution equations verified by gij and k ij are:
(7.3a)
∂t gij = −2ngia k aj ,
(7.3b)
∂t k ij = −g ia ∇a ∇j n + n{Ricij +
k aa k ij
²
−g ia ∇a φ∇j φ },
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
t−1 {λ−1 (n−1)−1}kij −T ij +(1/2)I ij T
A Regime of Linear Stability for the Einstein-Scalar Field System
40
where R denotes the scalar curvature of gij , Ricij denotes the Ricci curvature of gij , I ij = diag(1, 1, 1)
denotes the identity transformation, and T ∶= (g−1 )αβ Tαβ denotes the trace of the energy-momentum
tensor (1.2).
√
The volume form factor detg verifies the auxiliary equation46
√
n−1
∂t ln (t−1 detg) = (1 − λ−1 )
(7.4)
.
t
The scalar field wave equation is:
−trkDN̂ φ
−DN̂ DN̂ φ
³¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹
¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
1
(7.5)
−n−1 ∂t (n−1 ∂t φ) +g ab ∇a ∇b φ = n−1 {1 − λ−1 (n − 1)} ∂t φ −n−1 g ab ∇a n∇b φ.
t
The parabolic lapse equation is:
1
1
(7.6) λ−1 ∂t (n − 1) + g ab ∇a ∇b (n − 1) = (n − 1){ 2 (1 − λ−1 ) + R − g ab ∇a φ∇b φ}
t
t
1
1
+ λ−1 (λ−1 − 2) 2 (n − 1)2 + λ−2 2 (n − 1)3 + R − g ab ∇a φ∇b φ.
t
t
When λ > 0, the gauge condition (7.1) and the constraints (7.2a)-(7.2b) are preserved by the past flow
of the remaining equations if they are verified by the data.
Remark 7.2. We are primarily interested in the gauge (7.1) when λ ≥ 3. Note that when λ > 0, the
parabolic equation (7.4) is locally well posed only in the past direction.
Proof of Prop. 7.1. The proposition can be proved by making simple modifications to the standard arguments that yield Prop. 3.1.
7.2. Linearizing around Kasner. In the next proposition, we linearize the equations of Prop. 7.1 around
the Kasner solution (1.8). As in Prop. 3.2, the linearized equations can be derived from straightforward
algebraic calculations and hence we do not provide a detailed derivation.
Proposition 7.2 (The linearized Einstein-scalar field equations in the gauge (7.1) with transported
spatial coordinates). Consider the equations of Prop. 7.1 linearized around the Kasner solution (1.8). The
linearized equations in the unknowns (h, K, ϕ, ν) take the following form (see Def. 3.1 for the definitions
of some of the quantities).
The linearized parabolic gauge condition (7.1) is:
(7.7)
K aa = λ−1 ν.
The linearized versions of the Hamiltonian and momentum constraint equations (7.2a)-(7.2b) are:
(7.8a)
(7.8b)
(7.8c)
46
̂
t2(h)R − 2K̊ ab K ba − 2At∂t ϕ + 2(A2 − λ−1 )ν = 0,
̂
̂
∂a K ai = λ−1 ∂i ν − A∂i ϕ − (h)Γaab K̊ bi + (h)Γab i K̊ ab ,
(g̊ −1 )ab ∂a K ib = λ−1 (g̊ −1 )ia ∂a ν − A(g̊ −1 )ia ∂a ϕ
̂
̂
− (g̊ −1 )ab(h)Γai c K̊ cb + (g̊ −1 )ab(h)Γac b K̊ ic .
This equation, which we do not use in the present article, is implied by (3.6a) and the gauge condition (7.1).
I. Rodnianski and J. Speck
41
The linearized version of the lapse equation (7.6) can be expressed in either of the following two forms:
(7.9a)
(7.9b)
̂
2At∂t ϕ + 2K̊ ab K ba = λ−1 t∂t ν + t2 (g̊ −1 )ab ∂a ∂b ν + (2A2 − 1 − λ−1 )ν,
λ−1 t∂t ν + t2 (g̊ −1 )ab ∂a ∂b ν − (1 − λ−1 )ν = t2(h)R.
Equation (7.8a) can be used to show that (7.9a) is equivalent to (7.9b).
The linearized versions of the metric evolution equations (7.3a)-(7.3b) are:
(7.10a)
∂t hij = −2t−1 K̊ aj hia − 2t−1 g̊ia K aj − 2t−1 g̊ia K̊ aj ν,
(7.10b)
∂t K ij = −t(g̊ −1 )ia ∂a ∂j ν − (1 − λ−1 )t−1 K̊ ij ν + t(h)Ricij .
The linearized version of the scalar field wave equation (7.5) is:
(7.11)
−∂t (t∂t ϕ) + t(g̊ −1 )ab ∂a ∂b ϕ = −A∂t ν + A(1 − λ−1 )t−1 ν.
Proof. The proof is essentially the same as that of Prop. 3.2. In the gauge (7.1) and thus in Prop. 7.2, the
linearly small quantities are the same as the ones from Def. 3.1 except that K aa ∶= tk aa − tk̊ aa = λ−1 (n − 1) =
λ−1 ν is now linearly small rather than completely vanishing as it did in Prop. 3.2.
7.3. Statement of the main monotonicity theorem. We now state our main monotonicity theorem for
solutions to the linear equations of Prop. 7.2. It is a direct analog of Theorem 5.1. The theorem involves
the following energies and norms, which provide slightly different estimates for the lapse compared to
the CMC gauge. The main point is that we are no longer able to obtain control over the highest-order
analog of ∥∂ 2 ν∥L2g̊ because of the nature of parabolic energy estimates. We are, however, able to control
a spacetime integral of the highest-order analog of ∂ 2 ν, which is provided by the highest-order analog of
the first term on the second line of the right-hand side of (7.20).
Definition 7.2 (Energies). In terms of the quantities defined in Def. 4.4, we define the following energy
E(Almost T otal);θ (t) ≥ 0 for t ∈ (0, 1] ∶
(7.12)
2
2
2
E(Almost
T otal);θ [K, ∂h, ∂t ϕ, ∂ϕ, ν](t) ∶= θE(M etric) [K, ∂h](t) + E(Scalar) [∂t ϕ, ∂ϕ](t)
2
+ E(Lapse)
[ν](t).
As in Sect. 5, θ is a small positive constant that we will choose below in order to exhibit the monotonicity.
We will also use the following up-to-order M energy E(Almost T otal);θ;M (t) ≥ 0:
(7.13)
2
2
E(Almost
T otal);θ;M (t) ∶= ∑ E(Almost T otal);θ [∂I⃗K, ∂∂I⃗h, ∂t ∂I⃗ϕ, ∂∂I⃗ϕ, ∂I⃗ν](t).
⃗
∣I∣≤M
Definition 7.3 (Solution norms). In terms of the Sobolev norms of Def. 4.2, we define the solution norms
(7.14)
H(P arabolic F rame);M (t) ∶= ∥K∥H M
F rame
+ ∥∂h∥HFMrame + ∥t∂t ϕ∥H M
F rame
1
+ t2/3 ∥∂ϕ∥HFMrame + ∑ t(2/3)p ∥ν∥H M +p .
p=0
2
Remark 7.3. Note that H(P arabolic F rame);0 controls one derivative of ν while E(Almost
[K, ∂h, ∂t ϕ, ∂ϕ, ν]
T otal);θ
does not.
A Regime of Linear Stability for the Einstein-Scalar Field System
42
Theorem 7.1 (Approximate L2 monotonicity for solutions to the linearized equations corresponding
to the parabolic lapse gauge). Consider a solution to the linear equations of Prop. 7.2 corresponding
to the data (K(1), h(1), ∂t ϕ(1), ∂ϕ(1), ν(1)) (given on Σ1 ). There exist a small constant θ∗ > 0 and
constants C, c > 0 such that if the parabolic gauge parameter verifies λ ≥ 3, if κ ≥ 0 is sufficiently small
(see definition 1.12b), and if H(P arabolic F rame);0 (1) < ∞ (see definition (7.14)), then the solution verifies
the following energy inequality47 for t ∈ (0, 1], where E(Almost T otal);θ∗ is defined by (7.12):
(7.15)
2
2
E(Almost
T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ν](t) ≤ CE(Almost T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](1)
1
1
1
1
− θ∗ ∫ s−1 ∫ ∣s∂h∣2g̊ dx ds − ∫ s−1 ∫ ∣s∂ϕ∣2g̊ dx ds
6
12 s=t
Σs
Σs
s=t
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
past-favorable sign
1
1
2 1
− ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds − ∫ s−1 ∫ ν2 dx ds
21 s=t
9 s=t
Σs
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
past-favorable sign
1
2
+ cκ ∫ s−1 E(Almost
T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ∂ν, ν](s) ds .
s=t
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
error integral that can create energy growth
Furthermore, the following inequality holds for t ∈ (0, 1] ∶
(7.16)
E(Almost T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ν](t) ≤ CE(Almost T otal);θ∗ [K, ∂h, ∂t ϕ, ∂ϕ, ν](1)t−cκ .
In addition, if N ≥ 0 is an integer and H(P arabolic F rame);N (1) < ∞ (see definition (7.14)), then the total
energy (4.6) verifies the following inequality for t ∈ (0, 1] ∶
(7.17)
C
H(P arabolic F rame);N (1)t−cκ
E(T otal);θ∗ ;N (t) ≤ { κ
CH(P arabolic F rame);N (1)(1 + ∣ ln t∣)
if κ ≠ 0,
if κ = 0.
In addition, if N ≥ 0 is an integer and H(P arabolic F rame);N (1) < ∞, then the total solution norm (7.14)
verifies the following inequality for t ∈ (0, 1] ∶
(7.18)
C
H(P arabolic F rame);N (1)t−cκ
if κ ≠ 0,
H(P arabolic F rame);N (t) ≤ { κ
CH(P arabolic F rame);N (1)(1 + ∣ ln t∣) if κ = 0.
Remark 7.4. The assumption λ ≥ 3 can be improved to λ > 2. We avoid providing the messy algebraic
details of the improvement because the constants C and c as well as the allowable smallness of κ depend
on λ in the full range λ > 2.
Remark 7.5. Based on the estimates of Theorem 7.1, we could also prove an analog of Theorem 5.2 in
the parabolic lapse gauges for λ ≥ 3. Since the results and proof of the theorem would closely resemble
those of Theorem 5.2, we do not provide the details.
47
Footnote 21 applies to inequality (7.15) as well.
I. Rodnianski and J. Speck
43
7.4. Preliminary estimates and identities towards the proof of Theorem 7.1. In our proof of Theorem 7.1, we use the following comparison lemma, which can be proved by using arguments similar to the
ones we used to prove Lemma 4.3 (except that clearly we no do not use the elliptic estimate provided by
Lemma 4.2); we omit the simple proof.
Lemma 7.3 (Parabolic energy-norm comparison lemma). Let N ≥ 0 be an integer and let κ ≥ 0 be
as defined in (1.12b). There exist constants C, c > 0 depending on θ such that the following comparison
estimates hold for the norm (7.14) and the total energy (4.6) on the interval t ∈ (0, 1] ∶
E(T otal);θ;N (t) ≤ Ct−cκ H(P arabolic F rame);N (t),
(7.19a)
H(P arabolic F rame);N (t) ≤ Ct−cκ E(T otal);θ;N (t).
(7.19b)
We will also use the following simple parabolic energy estimate, which can be used to derive top-order
L2 estimates for the linearized lapse variable.
Lemma 7.4 (Parabolic energy estimate for ν). There exists a constant C > 0 such that if the parabolic
gauge parameter (7.1) verifies λ ≥ 3 and κ ≥ 0 (see definition 1.12b), then solutions ν to the linear
parabolic equation (7.9a) verify the following inequality for t ∈ (0, 1] ∶
(7.20)
λ−1 ∫ ∣t∂ν∣2g̊ dx ≤ λ−1 ∫
Σt
1
Σ1
∣∂ν∣2g̊ dx
1
4
∣s2 ∂ 2 ν∣2g̊ dx ds − λ−1 ( − 2κ) ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds
3
s=t
Σs
s=t
Σs
1
1
1
̂
+ C ∫ s−1 ∫ (K̊ ab K ba )2 dx ds + C ∫ s−1 ∫ (s∂t ϕ)2 dx ds + C ∫ s−1 ∫ ν2 dx ds.
s=t
Σ
s=t
Σ
s=t
Σ
−∫
s−1 ∫
s
Proof. Integrating by parts over
s
[t, 1] × T3
λ−1 ∫ ∣t∂ν∣2g̊ dx = λ−1 ∫
(7.21)
Σt
+ 2∫
Σ1
1
s
(we stress that t ≤ 1) we deduce (without using any equation)
∣∂ν∣2g̊ dx − 2λ−1 ∫
1
s=t
s−1 ∫
Σs
∣s∂ν∣2g̊ + s2 (g̊ −1 )ab K̊ cb ∂a ν∂c ν dx ds
s(g̊ −1 )ef ∂e ∂f ν(λ−1 s∂t ν) dx ds.
∫
s=t Σ
s
Using equation (7.9a) to substitute for λ−1 s∂t ν in (7.21) and integrating by parts over Σs on the term
2
{(g̊ −1 )ef ∂e ∂f ν} , we deduce
(7.22)
λ−1 ∫ ∣t∂ν∣2g̊ dx = λ−1 ∫
Σt
∣∂ν∣2g̊ dx
Σ1
− 2λ−1 ∫
− 2∫
1
s=t
1
s=t
s−1 ∫
s−1 ∫
Σs
Σs
∣s∂ν∣2g̊ + s2 (g̊ −1 )ab K̊ cb ∂a ν∂c ν dx ds
∣s2 ∂ 2 ν∣2g̊ dx ds
− (2A2 − 1 − λ−1 ) ∫
1
s=t
+ 4A ∫
−1 ef
∫Σ sν(g̊ ) ∂e ∂f ν dx ds
s
1
̂
s2 ∂t ϕ(g̊ −1 )ef ∂e ∂f ν dx ds + 4 ∫ ∫ sK̊ ab K ba (g̊ −1 )ef ∂e ∂f ν dx ds.
∫
s=t Σ
s=t Σ
1
s
s
A Regime of Linear Stability for the Einstein-Scalar Field System
44
Using inequality (5.30), we estimate the second integral on the right-hand side of (7.22) as follows:
1
4
∣s∂ν∣2g̊ + s2 (g̊ −1 )ab K̊ cb ∂a ν∂c ν dx ds ≤ −λ−1 ( − 2κ) ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds.
3
s=t
Σs
s=t
Σs
√
2
−1 ef
2
Using the simple estimate A ≤
3 , Young’s inequality, and the simple estimate ∥(g̊ ) ∂e ∂f ν∥L ≲
2
∥∂ ν∥L2g̊ , we deduce that the four integrals on the third through fifth lines of the right-hand side of (7.22)
are collectively bounded by
(7.23) −2λ−1 ∫
(7.24)
≤ −∫
1
s−1 ∫
1
s=t
1
s−1 ∫
+C∫
s=t
Σs
s−1 ∫
Σs
∣s2 ∂ 2 ν∣2g̊ dx ds
ν2 dx ds + C ∫
1
s=t
s−1 ∫ (s∂t ϕ)2 dx ds + C ∫
Σs
1
s=t
̂
s−1 ∫ (K̊ ab K ba )2 dx ds.
Σs
The desired inequality (7.20) now follows easily from the identities (1.12b) and (7.22) and the inequalities
(7.23) and (7.24).
As in the proof of Theorem 5.1, the most important step in the proof of Theorem 7.1 is provided by an
energy identity for the linearized scalar field and lapse that simultaneously yields favorably signed (to the
past) integrals for both variables. We provide this identity in the next proposition.
Proposition 7.5 (Combined energy identity for the linearized scalar field and linearized lapse in the
parabolic lapse gauge). Assume that the parabolic gauge parameter verifies λ ≠ 0. Then solutions to the
linearized equations of Prop. 7.2 verify the following identity for t ∈ (0, 1] ∶
(7.25)
1 −1
−1
2
2
2
2
∫Σ (t∂t ϕ) + ∣t∂ϕ∣g̊ dx + {A + 2 λ (1 − λ )} ∫Σ ν dx − A ∫Σ Q4 (t∂t ϕ, ν) dx
t
t
t
1
= ∫ (∂t ϕ)2 + t2 ∣∂ϕ∣2g̊ dx + {A2 + λ−1 (1 − λ−1 )} ∫ ν2 dx − A ∫ Q4 (∂t ϕ, ν) dx
2
Σ1
Σ1
Σ1
−2 ∫
1
s=t
s−1 ∫
Σs
∣s∂ϕ∣2g̊ + C2 (K̊, s∂ϕ, s∂ϕ) dx ds
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
−(1 − λ−1 ) ∫
1
s=t
s−1 ∫
Σs
s−1 ∫
Σs
∣s∂ν∣2g̊ dx ds
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
past-favorable sign for λ > 1
−(1 − λ−2 ) ∫
1
s=t
ν2 dx ds
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign for λ > 1
− A∫
1
s=t
s−1 ∫
− (1 − λ−1 ) ∫
Σs
1
s=t
Q1 (s∂ϕ, s∂ν) dx ds
s−1 ∫
Σs
̂
C1 (K̊, K, ν) dx ds,
I. Rodnianski and J. Speck
45
√
where the constant 0 ≤ A ≤ 2/3 is defined in (1.9b), the cubic forms Ci (⋯) and the quadratic form
Q1 (⋯) are defined in (5.8a)-(5.8c), and
Q4 (t∂t ϕ, ν) ∶= 2t∂t ϕν.
(7.26)
Proof. The proof has some features in common with our proof of Prop. 7.5, but other aspects of it are
different. Again, the main idea is to combine three integration by parts identities in suitable proportions.
To obtain the first identity, we divide equation (7.9a) by t and then replace t with the integration variable
s, multiply by (1 − λ−1 )ν, and integrate by parts over (s, x) ∈ [t, 1] × T3 (we stress that t ≤ 1) to deduce
that
1
1 −1
(7.27)
λ (1 − λ−1 ) ∫ ν2 dx = λ−1 (1 − λ−1 ) ∫ ν2 dx
2
2
Σt
Σ1
1
− (1 − λ−1 ) ∫
s=t
s−1 ∫
Σs
∣s2 ∂ν∣2g̊ dx ds
+ (2A2 − 1 − λ−1 )(1 − λ−1 ) ∫
1
s=t
s−1 ∫
ν2 dx ds
Σs
− 2A(1 − λ−1 ) ∫
1
− 2(1 − λ−1 ) ∫
̂
s ∫ K̊ ab K ba ν dx ds.
Σ
s−1 ∫
s=t
1
−1
s=t
Σs
s∂t ϕν dx ds
s
To obtain the second identity, we replace t with the integration variable s in equation (7.11), multiply
by −s∂t ϕ, and integrate by parts over (s, x) ∈ [t, 1] × T3 (we stress that t ≤ 1) to deduce that
(7.28)
2
2
2
2
2
∫Σ (t∂t ϕ) + t ∣∂ϕ∣g̊ dx = ∫Σ (∂t ϕ) + ∣∂ϕ∣g̊ dx
t
1
− 2∫
1
s=t
− 2A ∫
s−1 ∫
Σs
∣s∂ϕ∣2g̊ + s2 (g̊ −1 )ab K̊ cb ∂a ϕ∂c ϕ dx ds
1
1
−1
∫ s∂t ϕ∂t ν dx ds + 2A(1 − λ ) ∫s=t ∫Σ s∂t ϕν dx ds.
s=t Σ
s
s
Next, we note that equation (7.11) implies
1
t∂t ϕ∂t ν = ∂t (t∂t ϕν) − A∂t (ν2 ) − tν(g̊ −1 )ab ∂a ∂b ϕ + A(1 − λ−1 )t−1 ν2 .
(7.29)
2
To obtain the third identity, we replace t with the integration variable s in equation (7.29), multiply by
2A, and integrate by parts over (s, x) ∈ [t, 1] × T3 to deduce that
(7.30)
− 2A ∫ t∂t ϕν dx + A2 ∫ ν2 dx
Σt
= −2A ∫
+ 2A ∫
Σ1
1
s=t
1
− 2A ∫
Σt
∂t ϕν dx + A ∫ ν2 dx
Σ
2
s=t
1
∫Σ s∂t ϕ∂t ν dx ds
s
s−1 ∫
Σs
s2 (g̊ −1 )ab ∂a ϕ∂b ν dx ds − 2A2 (1 − λ−1 ) ∫
1
s=t
s−1 ∫
ν2 dx ds.
Σs
1
Adding (7.27), (7.28), and (7.30), and noting the cancellation of the integrals ±2A ∫s=t ∫Σs s∂t ϕ∂t ν dx ds
1
and ±2A(1 − λ−1 ) ∫s=t s−1 ∫Σs s∂t ϕν dx ds, we arrive at the desired identity (7.25).
A Regime of Linear Stability for the Einstein-Scalar Field System
46
In the next proposition, we derive an energy identity for the linearized metric solution variables. It is an
analog of Prop. 5.2.
Proposition 7.6 (Energy identity for the linearized metric variables in the parabolic lapse gauge).
Assume that the parabolic gauge parameter verifies λ ≠ 0. Then solutions to the linearized equations of
Prop. 7.2 verify the following identity for t ∈ (0, 1] ∶
(7.31)
1
1
2
2
2
2
∫Σ ∣K∣g̊ + 4 ∣t∂h∣g̊ dx = ∫Σ ∣K∣g̊ + 4 ∣∂h∣g̊ dx
t
1
1 1
− ∫ s−1 ∫ ∣s∂h∣2g̊ + C3 (K̊, s∂h, s∂h) dx ds
2 s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
past-favorable sign
−2λ−1 ∫
1
s=t
s−1 ∫
Σs
∣s∂ν∣2g̊ dx ds
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign for λ > 0
+∫
+∫
1
s=t
1
s=t
+ A∫
s−1 ∫
s−1 ∫
1
s=t
+ λ−1 ∫
Σs
Σs
̂
C6 (K̊, s∂h, s∂ν) dx ds + (1 − λ−1 ) ∫
s−1 ∫
1
s=t
̂
C5 (K̊, s∂h, s∂h) dx ds
̂
C4 (K̊, K, K) dx ds + ∫
1
s=t
Σs
s−1 ∫
s−1 ∫
Q2 (s∂ϕ, s∂ν) dx ds − A ∫
Σs
Σs
1
s=t
1
s=t
s−1 ∫
Σs
s−1 ∫
Σs
̂
C7 (K̊, K, ν) dx ds
Q3 (s∂h, s∂ϕ) dx ds,
Q3 (s∂h, s∂ν) dx ds,
√
where the constant 0 ≤ A ≤ 2/3 is defined in (1.9b) and the cubic forms Ci (⋯) and the quadratic forms
Qi (⋯) are defined in (5.14a)-(5.14g).
Proof of Prop. 7.6. We repeat the proof of Prop. 5.2 and take into account the few differences between the
linearized equations of Prop. 3.2 and the linearized equations of Prop. 7.2. In particular, the identity (5.24)
̂
holds in the present context, but with the next-to-last term −2t−1 g̊ab (g̊ −1 )ij K̊ ai K bj ν multiplied by the factor
1 − λ−1 (coming from the second term on the right-hand side of (7.10b)) and two additional terms: i) the
term 2λ−1 t∣∂ν∣2g̊ coming from the analog of the step (5.19) and the presence of the term λ−1 ∂i ν on the
right-hand side of equation (7.8b) and ii) the cross term −2λ−1 t(g̊ −1 )ef (h)Γeaf ∂a ν coming from the analog
of steps (5.22) and (5.23) and the presence of the term λ−1 ∂i ν on the right-hand side of equation (7.8b)
and the term λ−1 (g̊ −1 )ia ∂a ν on the right-hand side of (7.8c).
I. Rodnianski and J. Speck
47
7.5. Proof of Theorem 7.1. We now prove Theorem 7.1. Below we will prove that the following two
inequalities hold for t ∈ (0, 1] ∶
(7.32)
2
2
E(Scalar)
[∂t ϕ, ∂ϕ](t) + E(Lapse)
[ν](t)
2
2
≤ CE(Scalar)
[∂t ϕ, ∂ϕ](1) + CE(Lapse)
[ν](1)
1
1
− ( − 2κ) ∫ s−1 ∫ ∣s∂ϕ∣2g̊ dx ds
6
s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign for sufficiently small κ
1
4 1
2
− ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds − ∫ s−1 ∫ ν2 dx ds
21 s=t
9 s=t
Σs
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
past-favorable sign
+C∫
1
s−1 ∫
s=t
Σs
̂
Q2 (K̊, K) dx ds,
2
2
(7.33) E(M
etric) [K, ∂h](t) ≤ E(M etric) [K, ∂h](1)
1
1
− (1 − Cκ) ∫ s−1 ∫ ∣s∂h∣g̊2 dx ds
3
s=t
Σs
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign for small κ
−2λ−1 ∫
1
s=t
s−1 ∫
Σs
∣s∂ν∣2g̊ dx ds
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
past-favorable sign
+∫
1
s=t
+ A∫
s−1 ∫
1
s=t
+ λ−1 ∫
+∫
+∫
+∫
1
s=t
1
s=t
1
s=t
Σs
s−1 ∫
1
s=t
Q(s∂h, s∂ν) dx ds
Σs
s−1 ∫
s−1 ∫
s−1 ∫
s−1 ∫
Q(s∂h, s∂ϕ) dx ds + A ∫
Σs
1
s=t
s−1 ∫
Q(s∂ϕ, s∂ν) dx ds
Q(s∂h, s∂ν) dx ds
Σs
̂
C(K̊, s∂h, s∂h) dx ds + ∫
Σs
̂
C(K̊, K, K) dx ds + (1 − λ−1 ) ∫
Σs
Σs
1
s=t
s−1 ∫
̂
C(K̊, s∂h, s∂ν) dx ds
Σs
1
−1
̂
s ∫ C(K̊, K, ν) dx ds
s=t
Σ
s
̂
C(K̊, ∂h, ∂ν) dx ds,
where Q(X, Y ) denotes a non-negative quadratic term that can be pointwise bounded by Q(X, Y ) ≲
∣X∣g̊ ∣Y ∣g̊ , and C(X, Y, Z) denotes a non-negative cubic term that can be pointwise bounded by C(X, Y, Z) ≲
∣X∣g̊ ∣Y ∣g̊ ∣Z∣g̊ .
The proofs of (7.15) and (7.16) then follow from (7.32)-(7.33) as straightforward adaptations of the
proofs of (5.1) and (5.2). Furthermore, if H(P arabolic F rame);N (1) < ∞, then by commuting the equations
A Regime of Linear Stability for the Einstein-Scalar Field System
48
⃗ ≤ N and arguing as in the proof of (7.16), we deduce that (see definition (7.13))
of Prop. 7.2 with ∂I⃗ for ∣I∣
(7.34)
2
2
−cκ
E(Almost
.
T otal);θ∗ ;N (t) ≤ CE(Almost T otal);θ∗ ;N (1)t
Inequality (7.17) will then follow from (7.34) and Lemma 7.3 at t = 1 after we estimate the top-order
2
terms in E(T otal);θ∗ ;N that are not present in E(Almost T otal);θ∗ ;N , namely, E(∂Lapse)
[∂∂I⃗ν](t) (see definition
⃗ = N . To this end, we consider the ∂ ⃗−commuted version of inequality (7.20) in which all
4.5c) when ∣I∣
I
linearized solution variables are replaced with their ∂I⃗−commuted counterparts. Inserting the estimates
implied by (7.34) into the last three integrals on the right-hand side of the ∂I⃗−commuted version of (7.20),
carrying out straightforward computations, and using Lemma 7.3 at t = 1, we conclude the desired estimate
(7.17).
Inequality (7.18) then follows from inequality (7.17) and Lemma 7.3.
It remains for us to prove (7.32) and (7.33). Inequality (7.32) follows from the identity (7.25) and a
few simple applications of Young’s inequality, which we now describe. First, the left-hand side of (7.25)
is positive definite in ∫Σt (t∂t ϕ)2 dx, ∫Σt t2 ∣∂ϕ∣2g̊ dx, and ∫Σt ν2 dx, as can easily be seen from the simple
pointwise estimate
(7.35)
∣AQ4 (∂t ϕ, ν)∣ ≤
1
A2
(t∂t ϕ)2 + {A2 + λ−1 (1 − λ−1 )} ν2 .
1 −1
−1
2
4
A + 4 λ (1 − λ )
Next, we note that since λ ≥ 3, the three spacetime integrals on the right-hand side of (7.25) with integrands
(7.36)
− 2 {∣s∂ϕ∣2g̊ + C2 (K̊, s∂ϕ, s∂ϕ)} ,
(7.37)
− (1 − λ−1 )∣s∂ν∣2g̊ ,
(7.38)
− AQ1 (s∂ϕ, s∂ν)
are collectively bounded from above by
1
1
6
4
7
≤ − { − 2κ − } ∫ s−1 ∫ ∣s∂ϕ∣g̊2 dx ds − (1 − ) (1 − λ−1 ) ∫ s−1 ∫ ∣s∂ν∣2g̊ dx ds
3
6 s=t
7
Σs
s=t
Σs
1
1
1
2
≤ − { − 2κ} ∫ s−1 ∫ ∣s∂ϕ∣2g̊ dx ds −
s−1 ∫ ∣s∂ν∣2g̊ dx ds,
6
21 ∫s=t
s=t
Σs
Σs
an estimate that follows from the pointwise estimate (5.30) and the estimate
(7.39)
(7.40)
7
A2
6
×
∣s∂ϕ∣2g̊ + (1 − λ−1 )∣s∂ν∣2g̊
−1
6 1−λ
7
7
6
≤ ∣s∂ϕ∣2g̊ + (1 − λ−1 )∣s∂ν∣2g̊ .
6
7
A ∣Q1 (s∂ϕ, s∂ν)∣ ≤ A {∣s∂ϕ∣2g̊ + ∣s∂ν∣2g̊ } ≤
In the last step of (7.40), we have used the simple inequality A2 ≤ 32 . To control the last integral on the
right-hand side of (7.25), we bound its integrand via the simple estimate
(7.41)
1
1 − λ−1 −1 ̂ a b 2
̂
(1 − λ−1 )s−1 ∣C1 (K̊, K, ν)∣ ≤ (1 − λ−2 )s−1 ν2 + 2
s (K̊ b K a ) ,
2
1 + λ−1
which allows us to soak the portion 21 (1−λ−2 )s−1 ν2 into the spacetime integral −(1−λ−2 ) ∫s=t s−1 ∫Σs ν2 dx ds
on the right-hand side of (7.25). In total, these estimates yield (7.32).
Inequality (7.33) can be derived from the identity (7.31) by a straightforward modification of the argument we used to prove (5.26).
1
I. Rodnianski and J. Speck
49
ACKNOWLEDGMENTS
The authors thank Mihalis Dafermos for offering enlightening comments on an earlier version of this
work. They also thank the anonymous referees for providing valuable feedback that helped improve the
exposition. IR gratefully acknowledges support from NSF grant # DMS-1001500. JS gratefully acknowledges support from NSF grants # DMS-1454419 and # DMS-1162211, from a Sloan Research Fellowship
provided by the Alfred P. Sloan Foundation, and from a Solomon Buchsbaum grant administered by the
Massachusetts Institute of Technology.
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