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Stable cosmologies with collisionless charged matter

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Abstract

It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.
arXiv:2012.14241v1 [math-ph] 28 Dec 2020
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER
Hamed Barzegar,David Fajman
Abstract
It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlin-
early stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles
of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline
terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The
effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter
fields.
1. Introduction
Studying the global dynamics of non-vacuum solutions to the Einstein equations is a major effort
in general relativity with the aim to draw conclusions on isolated self-gravitating systems and
cosmology. In recent years several results have contributed substantially to the understanding
of non-vacuum dynamics with self-gravitating matter models, which provide realistic features of
matter in the actual universe such as relativistic fluids of various types and kinetic matter models.
While fluids are known to require expansion of spacetime to avoid shock formation (cf. [34]) Vlasov
matter shows a more regular behavior. When coupled to the Einstein equations, it is expected
to exhibit only those types of singularity formation which are caused by gravity. Indeed, most
nonlinear stability results for the vacuum Einstein equations have recently been generalized to
the Einstein–Vlasov system, most prominently for the de Sitter type spacetimes [33], Minkowski
spacetime [13, 23, 28, 36] and the Milne mo del [1] and lower dimensional analogues [18, 20, 21, 22].
The Einstein-Vlasov system describes spacetimes containing ensembles of self-gravitating collision-
less particles and provides a realistic description of the large scale structure of spacetime. It admits
various steady states modelling isolated self-gravitating matter configurations such as galaxies and
galaxy clusters [5,6, 31] and similarly a variety of matter dominated cosmological models [8,30]. In
contrast to those mo dels, in the stability analysis of [1, 13, 23, 28, 33, 36] the matter distribution is
considered to be small and disperses in the course of the evolution while the spacetime geometry
asymptotes to the background vacuum geometry.
While the Einstein–Vlasov system models purely gravitative effects there exist generalizations,
which include more detailed physical phenomena such as charged collisionless particles modeled by
the Einstein–Vlasov–Maxwell system on which we focus in the following.
1.1. The Einstein–Vlasov–Maxwell system. The Einstein–Vlasov–Maxwell system (EVMS)
(1.1)
Ric[h]1
2R[h]·h= 8πT,
Lh,F f= 0 ,
dF= 0 ,
d⋆ F =⋆J ,
describes spacetimes containing Maxwell fields and ensembles of collisionless charged particles,
which interact via gravity and electromagnetism [14, 17]. Here, hdenotes a Lorentzian metric on a
given 4-manifold M;T=V
T+MTis the total energy-momentum tensor with V
Tand MTdenoting
Date: December 29, 2020.
1
2H. Barzegar,D. Fajman
the energy-momentum tensors of the Vlasov matter and the Maxwell field, respectively; Lh,F is
the Liouville–Vlasov operator which consists of the geodesic spray and the Maxwell term; dand
denote the exterior derivative and the Hodge star operator on M, respectively; Fis the Faraday
tensor, and Jis the matter current 1-form.
When setting the Faraday tensor Fand the charge to zero, (1.1) reduces to the Einstein–Vlasov
system while setting the distribution function fto zero it reduces to the Einstein–Maxwell system.
There exist only few results on the EVMS concerning stationary solutions [4, 37, 38] and evolution
in spherical symmetry [29]. In particular, stability results have not been established yet in the class
of solutions to the EVMS.
1.2. Background spacetimes. We consider a class of cosmological vacuum spacetimes which are
Lorentz cones over a negative closed Einstein space (M, γ) of dimension 3 with Einstein constant
κ=2
9, i.e., Ric[γ] = 2
9γ, where the value of κis chosen for convenience. Then, the Milne model
(1.2) (0,)×M, dt2+t2
9·γ
is a solution to the vacuum Einstein equations. In contrast to the class of exponentially expand-
ing spacetimes or those with power law inflation, the Milne model does not exhibit accelerated
expansion. Its linear scale factor constitutes the threshold between accelerated and deccelerated
expansion, which makes the model particularly interesting from the perspective of the regularizing
effect of its expansion on various matter models [25]. Future stability of (1.2) under the Einstein
flow has been established in the vacuum case in [3] and for different types of matter in [1, 15, 26].
1.3. Main result. It is the purpose of the present paper to establish the first nonlinear stability
result for the EVMS. In particular, we prove future nonlinear stability for the Milne model in the
set of solutions to the EVMS. The main challenge in contrast to earlier stability results on the
Milne model consists in the direct mutual coupling of the matter fields, which does induce various
slowly decaying terms in the system. We discuss these aspects further below in detail. The main
theorem reads as follows.
Theorem 1.1. Let (M, γ)be a compact, 3-dimensional negative Einstein manifold without bound-
ary and Einstein constant κ=2
9, and let ǫ>0. Then, there exists a δ > 0such that for a
rescaled initial data (g0,Σ0, f0, ω0,˙ω0)H6(M)×H5(M)×HVl,5,4,c(T M )×H6(M)×H5(M)at
t=t0with compact momentum support of the initial particle distribution and
(1.3) (g0,Σ0, f0, ω0,˙ω0)B6,5,5,6,5
δ(γ, 0,0,0,0) ,
the corresponding solution to the rescaled Einstein–Vlasov–Maxwell system is future-global in time
and future complete. Moreover, the rescaled metric and trace-free part of the second fundamental
form converge as
(1.4) (g, Σ) (γ, 0) for τր0,
with decay rates determined by ǫas in (10.8) below. In particular, the Milne model is future
asymptotically stable for Einstein–Vlasov–Maxwell system in the class of initial data given above.
We will provide precise definitions of the objects in Theorem 1.1 further below. However, to give an
overview on the theorem, we briefly summarize them. We use (g, Σ, f , ω, ˙ω) to denote the rescaled
variables: the Riemannian metric, the trace-free part of the second fundamental, the distribution
function, the spatial vector potential, and its time-derivative, respectively. τ < 0 denotes the mean
curvature and is related to the time variable in (1.2) by t=3τ1. Therefore, τր0 corresponds
to t→ ∞.Hk(M) with k0 denotes the L2-based Sobolev norm and HVl,5,4,c(T M ) denotes
the space of distribution functions of compact momentum support on T M corresponding to the
standard L2-Sobolev norms (cf. [20]). We denote by B6,5,5,6,5
δ(·,·,·,·,·) a ball of radius δcentered
at its arguments in the set of H6(M)×H5(M)×HVl,5,4,c(T M )×H6(M)×H5(M).
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 3
Remark 1.2. The main theorem is related to foregoing works and more general settings in the
following sense.
(1) Theorem 1.1 in particular implies Theorem 1 of [1] (the Einstein-Vlasov case) and Theo-
rem 7.1 of [15] (Einstein-Maxwell case).
(2) We consider the EVMS with particles of identical rest mass m > 0 and identical charge q.
The present result however directly generalizes to a collection of ensembles with different
masses and charges (also of opposite signs). For simplicity of the presentation we restrict
ourselves to this specific case.
While there exists a number of stability results for relativistic Vlasov-Maxwell systems [10, 11, 12]
the present result is to our knowledge the first stability result for the EVMS.
1.4. Aspects of the proof. The proof of Theorem 1.1 is partially based on the techniques devel-
oped to control the distribution function in [19,20] and [1] and the techniques to control the Maxwell
fields derived in [15]. However, the present problem poses various new difficulties as Maxwell fields
and distribution function are directly coupled via the Maxwell term in the transport equation and
the particle-current term in the Maxwell equation. In fact, both terms are principal terms in the
sense that they decay a priori at the slowest rate when compared with other perturbative terms
in the respective equation. Moreover, the decay rates of these terms induce a loss in decay for the
corresponding energies of the Maxwell fields and distribution function, respectively. We refer to
those terms in the following as borderline terms.
In the Einstein equations a borderline term enters in the lapse equation (2.9c) via the rescaled
pressure η. This has already been observed and resolved in [1] in conjunction with the corresponding
borderline term () in the transport equation (2.25). Their mutual coupling was controlled in [1] by
using the continuity equation to obtain a sharp estimate for the energy density of the distribution
function.
In the presence of charges an additional borderline term enters the transport equation. It is caused
by the Maxwell field and denoted by (∗∗) in (2.25). It eventually yields a small ε-growth for the
energy of the distribution function similar to the borderline term () in the same equation. To
establish this behavior it is necessary that the energy controlling the Maxwell field, which appears
in this term, is uniformly bounded in time. A small loss for that energy would prevent the estimates
from closing.
However, the Maxwell equations themselves have a borderline term marked by () in (2.33). This
term is caused by the presence of charged particles. It can be controlled by the energy of the
distribution function but would in this case pick up its small growth and prevent the bootstrap
argument from closing. At this point it is important to use the fact that the respective term in the
Maxwell equations is determined by the matter current, which, in turn is determined to leading
order by the energy density, which, due to the continuity equation does not have a loss in decay in
comparison with the energy of the distribution function. This observation is the key that enables
us to close the hierarchy of estimates and thereby the bootstrap argument.
1.5. Organization of the paper. In Section 2 notations, geometric setup and energies are intro-
duced. In Section 3, bounds on the energy-momentum tensor in terms of the energies are given.
In Section 4 we provide the estimate for the momentum support. Section 5 provides the energy
estimates for the distribution function and the Maxwell fields. The energy density is estimated in
Section 6. Lapse and shift vector are estimated in Section 7. The energy estimates for the spatial
geometry are given in Section 8. In Section 9 we provide the energy estimate for the total energy
incorporating all previously derived estimates. Section 10 provides the proof of the main theorem
based on the foregoing sections.
Acknowledgements. This work was supported in part by the Austrian Science Fund (FWF) via
the project Geometric transport equations and the non-vacuum Einstein flow (P 29900-N27).
4H. Barzegar,D. Fajman
2. Preliminaries
In this section we recall various notations and facts on the geometric setup from [1, 15].
2.1. Notation. In this paper, M=R×M, with Mbeing a three-dimensional compact Riemannian
manifold, denotes a four-dimensional Lorentzian manifold equipped with Lorentzian metrics hand
h, and the associated covariant derivatives e
and , respectively. Further, Riemannian metrics
on Mwill be denoted by γ, ˜g,g, the associated Christoffel symbols by b
Γ[γ], e
Γ[˜g], Γ[g], and the
associated covariant derivatives by b
D,D,D, respectively. We will also denote the determinants of
a generic metric by |·|. The Laplacian of gis then defined as ∆ = trgD2. Moreover, gwill stand
for the Riemannian measure induced on Mby g. The Riemannian inner products on a tangent
space TxMat a point xis given by ,·igand ,·i˜g, respectively. The Hodge-Laplacian acting on
differential forms on Mwill be denoted by ∆H=dd+ddwhere dand ddenote the exterior
derivative and the codiffernetial with respect to the metric gon M, respectively. LYwill represent
the Lie-derivative in the direction of a vector field Y. We occasionally will use the notations
b
N:= N/31 and b
X:= X/N. The Greek indices will stand for the spacetime coordinates on M
and the Latin indices will denote the coordinates on M, whereas the coordinates on the tangent
bundle T M of Mwill be denoted by the bold Latin letters a,b,c,... ∈ {1,...,6}. Furthermore,
we denote the standard (L2-based) Sobolev norm with respect to the fixed metric γof order 0
by k·kH(M)for all functions and symmetric tensor fields on M. For brevity we write HH(M).
Throughout this paper, Cdenotes any positive constant which is uniform in the sense that it does
not depend on the solution of the system once a smallness parameter δfor the initial data and
the initial time T0are chosen. Moreover, if δis decreased or T0is increased, Cwill keep its value.
Nevertheless, the actual value of Cmay change from line to line.
2.2. Background geometry. Throughout the rest of the paper, we consider the Einstein space
(M, γ ) with Ric[γ] = 2
9γ. Then, the Einstein operator ∆E(cf. [9] and [27]) associated with
γacting on symmetric 2-tensors is defined by ∆E≡ −b
γ2˚
Rγ, where b
γ:= γij b
Dib
Djand
(˚
Rγu)ij := Riem[γ]ikjℓ ukℓ for an arbitrary symmetric 2-tensor u. The lowest positive eigenvalue of
the Einstein operator λ0obeys 9λ01 in the present setting (cf. [1]), which is relevant for the
energy estimate for metric and second fundamental form. In addition, ker ∆E={0}holds in the
present setting (cf. [3] and [1]), which assures the coercivity of the same energy.
2.3. Spacetime, gauges and rescaled variables. We consider the spacetime (M,h) and write
the unrescaled Lorenztian metric hin ADM formalism as
h=e
N2dt dt + ˜gij dxi+e
Xidtdxj+e
Xjdt,
where e
Nand e
Xare the lapse function and the shift vector field. Let e
Σ and τbe the trace-free part
and the trace of the second fundamental form of the hypersurfaces {t= const.}which we assume
that all have constant mean curvature τ.
2.3.1. CMCSH gauge. In the spacetime setting introduced above, the constant mean curvature
spatial harmonic (CMCSH) gauge (cf. [2]) is achieved by
(2.1)
t=τ ,
˜gij e
Γ[˜g]
ij b
Γ[γ]
ij = 0 .
2.3.2. Rescaled variables. We further define rescaled quantities g,N,X, and Σ by
(2.2) gij := τ2˜gij , gij := τ2˜gij , N := τ2e
N , Xi:= τe
Xi,Σij := τe
Σij ,
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 5
and we introduce a rescaled time Tby τ:= τ0·eTfor T(−∞,) and some fixed τ0<0.
Note that T=ττand =τ dT . We occasionally will use the dot notation to denote the
T-derivative.
2.3.3. Rescaled geometry. In the new time coordinate and with the rescaled variables the Lorentzian
metric reads
(2.3) h= (τ0)2e2TN2dT 2+gijdxiXidT dxjXjdT =: τ2h .
Remark 2.1. The Milne solution with the choice of the time coordinate τand τ0=3 reads
hMilne =e2TdT 2+1
9γ.
Now, let Π be the second fundamental form of the hypersurfaces {T= const.}with respect to the
metric h. Then, it is readily seen that
(2.4) Π = Σ + N11N
3g .
One therefore finds that the future-directed timelike unit normal e0of the hypersurfaces {T=
const.}with respect to his
(2.5) e0=N1(T+X).
With this definition the following lemma holds.
Lemma 2.2 ( [15], Lemma 2.1).For any fC(M)and ξC(R,1(M))
Le0g=,
[Le0,g]f= 2hΠ, D2fig+hDlog N, De0fig+hS, Df ig
+ 2hΠ, D log NDf igtrgΠhDlog N, Df ig,
[Le0,divg]ξ= 2hΠ, Dξig+hDlog N, Le0ξig+hS, Df ig
+ 2hΠ, D log NξigtrgΠhDlog N, ξig,
where S:= 2divgΠDtrgΠ.
Throughout this paper, the index ˆ
0 refers to the vector field e0whereas 0 refers to the time-function
τ. For the later use we summarize the connection coefficient with respect to the Lorentzian metric
h, i.e. Γ[h]α
βγ , using the Koszul formula
(2.6) Γ[h]ˆ
0
ˆ
0ˆ
0= 0 ,
Γ[h]i
ˆ
0ˆ
0=gij N1jN ,
Γ[h]ˆ
0
iˆ
0= 0 ,
Γ[h]i
jˆ
0=gikΠkj ,
Γ[h]ˆ
0
ij =Πij ,
Γ[h]i
jk = Γ[g]i
jk .
2.3.4. Slice-adapted gauge. On a vector potential A1(M) the Lorenz gauge d⋆ A = 0 is
usually imposed. As discussed in [15], it turns out to be difficult to work with the Lorenz gauge
in the present context. Instead, we impose the slice-adapted gauge, as introduced in [15], which is
adapted to the hypersurfaces of the foliation. This gauge determines Auniquely by requiring that
the spatial components of Aassociated to the foliation, denoted by the 1-forms ωC(R,1(M))
with ω(i) = A(i), are divergence-free and orthogonal to the kernel of the Hodge Laplacian, and
the component of Ain direction of the vector field e0, denoted by Ψ := A(e0)C(M) has
vanishing integral on each spatial slice. This is indeed achievable:
Lemma 2.3 ( [15], Lemma 5.4).Let F2(M)be exact. Then, there exists a unique form
A1(M)with dA=Fsuch that
(2.7) divgω= 0 , ω ker(∆H),ZM
Ψg= 0 .
6H. Barzegar,D. Fajman
Remark 2.4. We consider ωC(R,1(M)) as an element in Ω1(M) by requiring ω(e0) = 0.
With this choice one could write A=ω+ Ψe
0where e
01(M) is dual to e0.
2.4. Rescaled Einstein equations. In this subsection we use the (3 + 1)-dimensional ADM
formalism to establish the rescaled Einstein equations.
2.4.1. Matter quantities. The matter quantities which appear in the ADM formulation of Einstein
equations read (cf., e.g. [32])
(2.8) ˜ρ:= e
N2T00 ,˜
ji:= e
NT0
i,e
Sij := 8πTij 1
2(trhT) ˜gij ,
where ˜ρand ˜
jiare the unrescaled energy density and the matter current, respectively.
2.4.2. Einstein equations. The Einstein flow in CMCSH gauge reads
R− |Σ|2
g+2
3= 4τ·ρ ,(2.9a)
DiΣij =τ2jj,(2.9b) g1
3N=N|Σ|2
g+τ·η
|{z}
()1,(2.9c)
gXi+RijXj= 2DjNΣij DiN
31+ 2Nτ 2ji
2NΣjk DjXkΓi
jk b
Γi
jk ,(2.9d)
Tgij = 2NΣij + 2N
31gij LXgij ,(2.9e)
TΣij =ij NRij +2
9gij +D2
ij N+ 2NΣikΣk
j
1
3N
31gij N
31Σij LXΣij +Nτ Sij .(2.9f)
Here, we use the notations Rij := Ric[g]ij ,R:= trgRic[g] and the rescaled matter quantities
(2.10) ρ:= 4π˜ρ·τ3, η := 4π˜ρ+ ˜gij Tij ·τ3, ji:= 8π˜
ji·τ5, Sij := 8πe
Sij ·τ1.
Remark 2.5. The term () is a borderline term, which only yields no loss on the decay for the
gradient of the lapse, when the decomposition for the pressure ηbelow is used, which identifies
the energy density as the leading order term in η. This has already been observed in [1] and is in
particular not due to the coupling between Maxwell and Vlasov part.
We decompose the rescaled energy density as
(2.11) η=ρ+τ2η,
where
(2.12) η= 4π˜gij Tij ·τ5.
For the later use we also denote Tab := τ7Tab and we consider the following lemma which is
proved in [2] (cf. also [3]).
Lemma 2.6 ( [3], Lemma 6.2).In the CMCSH gauge the following identity holds
(2.13) Rij +2
9gij =1
2Lg,γ (gγ)ij +Jij ,
where
Lg,γ (gγ) := g(gγ)2˚
Rγ(gγ),with g,γ (gγ)ij := 1
p|g|b
Dkhgkℓp|g|b
D(gγ)ij i,
and
kJkHk1Ckgγk2
Hk.
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 7
An important fact is the uniform positivity of the lapse function, which is used throughout the
remainder of the paper.
Lemma 2.7. The lapse function Nis uniformly positive. In particular, one has
(2.14) 0 < N 3.
Proof. The proof follows from applying the maximum principle to the elliptic equation for the lapse
function.
2.5. Vlasov matter. In this subsection we give a quick introduction to the Vlasov matter and
then rescale the momentum and finally derive the rescaled transport equation.
2.5.1. The mass-shell relation. Throughout this paper we assume that all particles have the same
positive mass m= 1 modeled by a distribution function with the domain
(2.15) P={(x, e
p) : |e
p|2
h=1,˜p0<0} ⊂ TM,
where e
p:= ˜pµµwith 0=τand ˜pµbeing the canonical coordinates on the tangent bundle of
M. One can associate an energy-momentum tensor to a distribution function ˜
f:P[0,) by
(2.16) V
Tαβ [˜
f](x) := ZPx
˜
f˜pα˜pβPx,
where Pxis the Riemannian measure induced on Pxby the Lorentzian metric hat a given point
x, and is given by
Px:= p|h|
˜p0
d˜p1d˜p2d˜p3=e
N
˜p0
˜p,
where
˜p:= |˜g|1
2d˜p1d˜p2d˜p3.
We consider the projection map pr : PTMwhich does (t, xi, p0, pi)7→ (t, xi, pi). Then, instead
of using ˜
fwe deal with the function f:= ˜
fpr1which we refer to as distribution function for the
remainder of the paper.
2.5.2. The rescaled momentum. We rescale the momentum vector field as
(2.17) ˜pa=τ2pa.
As a result we have ˜pi=τ2pi. The unrescaled mass-shell relation in (2.15) gives (cf., e.g., [35])
(2.18) ˜p0=e
N2− | e
X|2
˜g1he
X, ˜pi˜g+qhe
X, ˜pi2
˜g+e
N2− | e
X|2
˜g(1 + |˜p|2
˜g).
Rescaling the variables in the previous equation and denoting p0:= τ2˜p0, we find
(2.19) p0=N11− | b
X|2
g1τhb
X, pig+bp,
or equivalently,
p0=1 + τ2|p|2
g
Nbpτhb
X, pig,
where
(2.20) bp:= qτ2hb
X, pi2
g+1− | b
X|2
g(1 + τ2|p|2
g).
Furthermore, we find
(2.21) ˜p0=h0α˜pα=e
Nbp ,
which, in particular, implies that Px=bp1˜p. We further use the notation p:= Np0.
8H. Barzegar,D. Fajman
Remark 2.8. Note that (2.19) reduces to p0=q1 + |p|2
gwhen X= 0 and N= 1. These are the
values that correspond to the background geometry.
2.5.3. The transport equation. The transport equation in the presence of the electromagnetic field
reads
(2.22) ˜pµµfe
Γi
µν ˜pµ˜pν+q˜pαFi
α˜pif= 0 ,
where e
Γα
βγ e
Γ[h]α
βγ . We wish to rewrite (2.22) in terms of the rescaled variables. We start with
the unrescaled Christoffel symbols of h(cf. [1])
(2.23) e
Γ[h]a
bc = Γ[g]a
bc +N1Σbc +1
3gbcXa,
e
Γ[h]a
00 =τ2Γa,
e
Γ[h]a
0b=τ1(δa
b+ Γa
b),
where
Γa:= TXaXa2
3(N3)Xa+XbDbXa2NΣa
bXb+NDaN
+hN1TNN1XbDbN+N1Σbc +1
3gbcXbXciXa,(2.24a)
Γa
b:= NΣa
b+1
3δa
b(3 N) + DbXaN1XaDbN+N1Σbc +1
3gbcXcXa.(2.24b)
We use notations Γand Γ
when we want to suppress the indices of the above two objects. The
rescaled transport equation finally takes the following form, using the natural horizontal and vertical
derivatives on T M ,Aa:= apiΓk
aiBkand Ba:= pa,
Tf=τN pa
pAafτ1p
NΓaBaf
|{z }
()
+2paBaf2pcΓa
cBaf
τΣbc +1
3gbcXapbpc
pBaf+τqFaBaf
|{z }
(∗∗)
,
(2.25)
where
(2.26) Fi:= hijF0j+pa
p0hij Faj +τ
NFa0b
Xi,
with hij =gij N2XiXj, which can be read off from the metric h.
Remark 2.9. The terms marked by () and (∗∗) are borderline terms. In particular, the term
marked by (∗∗) originates from the Maxwell field and the slow decay is caused by the first term in
(2.26). In combination with the factor τthe term F0jappears in the energy for the Maxwell field
(2.49), which itself is ε-small but does not decay. As a consequence the term (∗∗) in the Vlasov
equation yields a growth for the energy of the distribution function of the type exp(C εT ), however,
only if we obtain sharp estimates for the energy of F. Therefore, we need to avoid that the loss for
the energy of the distribution function couples back into the equation for the Maxwell field.
2.6. Maxwell equation. We start with the the Maxwell equation which in Heaviside–Lorentz
units reads
(2.27) hλµ e
λFµα =qZf˜pαPx=: Jα.
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 9
Recall that the index 0 refers to the vector field τ. Rescaling the result according to (2.2) and
(2.17) yields
Jα=qZf˜pα
bp˜p=qτ 3Zf˜pα
bpp,
where we used
˜p=τ3|g|1
2dp1dp2dp3=: τ3p.
Thus, by (2.21) we have
(2.28) J0=τqN Zf dµp=: τJ,
and using
(2.29) ˜pa
bp=τ1Xap0+gabpb
bp=: NPa
we find
(2.30) Ji=qNτ 3ZfPip=: τ3Ji.
The definition of Pais motivated by the relation Bap0=τ2Pa(cf. Appendix E). Keeping (2.3)
in mind, for the left-hand side of (2.27) we get
hλµ e
λFµα =hλµ hλFµα e
Γβ
µλ Γβ
µλFβα e
Γβ
αλ Γβ
αλFµβ i=hλµλFµα =τ2hλµλFµα ,
where we used the facts from Appendix A. Hence,
(2.31) hλµλFµα =τ2˜
Jα.
We compute the following for Proposition below, which gives the rescaled Maxwell equations
(2.32)
F0i=τ1Tωi+iNΨXjωj,
Fˆ
0i=N1Tωi+XjjωiiΨ,
Fij =iωjjωi,
and
τ2˜
Jˆ
0=τ2˜
J(e0) = τ2N1˜
JT+b
Xj˜
Jj=τ2τN 1˜
J0+b
Xj˜
Jj=N1J+τb
XjJj,
where we used ˜
JT=˜
J(T) = ˜
J(ττ) = τ˜
J0.
Proposition 2.10. Let F2(M)be exact which solves (2.31) and let A1(M)be a vector
potential for Fwhich satisfies the slice-adapted gauge conditions in Lemma 2.3 with the same Ψ
and ωgiven there. Then, we have
gΨ = divg·Dlog(N)) [Le0,divg]ωN1J
|{z}
()
+τb
XjJj,(2.33a)
(Le0(Le0ω))k+ ∆Hωk=k(e0Ψ) + e0Ψ·kN
N+ Ψ ·k(e0log(N))
+gij iN
NFjk + trgΠ·Fˆ
0k+ 2gij ΠikFjˆ
0τJk.(2.33b)
Remark 2.11. The term () is borderline in the following sense. As defined in (2.28) it is given up
to a factor by the integral of the distribution function. A straightforward estimate by the energy
of the distribution function would induce a small growth for the field Ψ, which would prevent the
estimates from closing. To obtain a uniform bound on Ψ it is crucial to observe that the leading
order term in () is in fact the energy density, which is established in Lemma 5.3.
10 H. Barzegar,D. Fajman
Proof of Proposition 2.10. From (2.27) and using the slice-adapted gauge, we get (cf. Eqs. (5.16)–
(5.18) in [15])
gij DiFjˆ
0= ∆gΨ + divg·Dlog(N)) + [Le0,divg]ω ,(2.34)
gij DiFjk =Hωk+ trgΠ·Fˆ
0k+gij ΠikFjˆ
0,(2.35)
Dˆ
0Fˆ
0k= (Le0(Le0ω))kk(e0Ψ) e0Ψ·kN
NΨ·k(e0log(N))
gij iN
NFjk +gij Πik Fˆ
0j.(2.36)
Inserting the results into (2.31) and using (2.28) and (2.30) finishes the proof.
2.7. Local existence. A local-existence-theory for the system (2.9), (2.25) and (2.33) can be
derived based on the ideas of [15,19].
Proposition 2.12. Consider CMC-initial data
(2.37) (g0,Σ0, f0, ω0,˙ω0)B6,5,5,6,5
δ(γ, 0,0,0,0)
with δ > 0sufficiently small. Let IRbe a compact interval with T0I. Then, there exists
aT > 0and a unique solution (g, k, f , ω, ˙ω)to the Einstein–Vlasov–Maxwell system with J=
(T0T, T0+T)launched by this initial data. Tdepends continuously on the H6(M)-, H5(M)-,
HVl,5,4,c(T M )-, H6(M)-, and H5(M)-norm of g0,Σ0,f0,ω0, and ˙ω0, respectively. The following
regularity properties hold
(g, k),(ω, ˙ω)Cb(J, H 6×H5)C1
b(J, H5×H4),
fCb(J, HVl,5,4,c(T M )) .
Moreover, the solution is either global in time, i.e. T=or
lim sup
tրT
(kgγkH6+kΣkH5+kωkH6+k˙ωkH5+k|fk|5,2)2δ.
Proof. The local existence theory can be established similar to the corresponding theorems for the
Vlasov case in [1] and the Maxwell case in [15]. When considering the Maxwell equations in Lorenz
gauge we obtain wave-type equation for the 4-potential in the form
H,hAα=τ2Jα,
where H,h denotes the Hodge wave operator of the metric h.
Complementing the Einstein-Vlasov system with Maxwell terms by this equations does not change
the structure of the elliptic-hyperbolic system with respect to the Einstein-Vlasov case considered
in [19]. The additional wave-type equation can be treated at the same order of regularity as the
evolution equations for the spatial metric g (which is decomposed into first order equations). A
local-existence theorem analogue to Theorem 4.2 of [19] follows by a similar proof, where the vector
potential is controlled in the same regularity class as the metric. In consequence we obtain a local-
existence theory for the EVMS in CMCSH-Lorenz gauge.
Then, as shown in [15], we perform a gauge-transformation for the vector potential of the Maxwell
field to obtain a potential obeying the slice-adapted gauge while conserving the regularity of the
potential and thereby of the solution. We conclude at this point that we have existence and
uniqueness of a local-in-time solution of the EVMS in CMCSH-slice-adapted gauge in the respective
regularity class.
It remains to prove the continuation criterion. For this purpose we need to ensure that a solution,
which is small in CMCSH-slice-adapted gauge remains small when its vector potential is transformed
to the Lorenz gauge. We therefore consider the gauge transformation of the vector potential
(2.38) Aµ=A
µ+µΛ,
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 11
such that A
µfulfils the Lorenz gauge. Then Λ solves the wave equation
(2.39) hΛ = e0ΨN1(1 hij hij) (e0ΛΨ) b
Xib
XjDiωj+hijΠij Ψ + N1gij jωi,
with trivial initial data. Here, we used (2.6) and relations in Appendix A. From this follows that
smallness of the solution in slice-adapted gauge in the considered regularity class implies smallness
of the solution in Lorenz gauge in the corresponding regularity class on short time intervals, as
Λ is determined by the right-hand side of the wave equation. Note therefore that e0Ψ fulfils the
elliptic equation (2.33a), which improves its regularity by ellipticity of the equation. This implies
the continuation criterion in slice-adapted gauge.
2.8. Coupling of the equations and consequences for the long-time behavior. Having
derived all equations we give a brief summary on the coupling issues in the remark below.
Remark 2.13. The rescaled EVMS consists of systems of equations (2.9), (2.25), and (2.33). In this
rescaled system we have particular matter-electromagnetic coupling terms, i.e., τ qFiBifand N1J,
which appear as source terms in the Maxwell equations and in the Vlasov equations, respectively.
Those terms are not present in the respective individual cases, i.e., pure electromagnetism in [15]
or pure uncharged matter in [1]. In addition, they constitute principal terms in the sense of their
decay properties. Both terms decay slower or at the same rate as the slowest decaying neighbouring
source terms and thereby constitute potential obstacles when analyzing the rescaled equations. The
major observation of the following bootstrap analysis is that those terms can indeed be handled
in the existing bootstrap hierarchy of the previous works and in turn yield the same asymptotic
behaviour as the respective cases.
2.9. Energy-momentum tensor. We compute and collect all relevant terms from the energy-
momentum tensor in the following. The unrescaled energy-momentum tensor is given by
T=V
T+MT.
Accordingly, we have for the matter variables defined in (2.8),
˜ρ˜ρV+ ˜ρM,˜
ja˜
ja
V+˜
ja
M, η ηV+ηM,
where scripts V and M stand for Vlasov and Maxwell matter fields, respectively.
Then, the rescaled Vlasov matter quantities take the following form
ρV= 4πN 2Zf(p0)2
bpp,(2.40a)
ja
V= 8πN Zfp0pa
bpp,(2.40b)
ηV= 4πZf|p+τ1p0X|2
g
bpp,(2.40c)
V
Tab = 8πZfpapb
bpp.(2.40d)
For the Maxwell field we start with the unrescaled energy-momentum tensor (again in the Heaviside–
Lorentz units)
(2.41) MTµν =Fα
µFνα 1
4hµν Fαβ Fαβ .
We first compute Fαβ Fαβ in terms of the rescaled variables
(2.42) FαβFαβ =2τ6N2gij F0iF0j+ 2τ5N2XigjkF0kFij +τ4hikhj Fij Fkℓ .
12 H. Barzegar,D. Fajman
Thus, in terms of rescaled variables MT00 takes the form
(2.43) MT00 =τ2hijF0iF0j+τ4
41− | b
X|2
gN2Fαβ Fαβ .
For the off-diagonal components of the energy-momentum tensor we have
(2.44) MT0i=τ2hjkF0kFij 1
4τ3XiFαβ Fαβ .
Finally, the spatial part reads
(2.45) MTij =τ4N2F0iF0jτ3N2Xk(F0iFjk +F0jFik) + τ2hkFik Fj 1
4τ2Fαβ Fαβ .
Then, the rescaled Maxwell quantities can be expressed by the components of the energy-momentum
tensor of Maxwell field computed above as
ρM= 4πτ N2MT00 2τ1Xi(MT0i) + τ2XiXj(MTij),(2.46a)
ji
M= 8πN 1hN2Xi(MT00 ) + τ1b
Xib
Xj+hij MT0j+τ2hikXj(MTij )i,(2.46b)
ηM= 4πτ 3gij (MTij).(2.46c)
2.10. Norms for matter fields. We recall in the following briefly the norms used to control
distribution function and Faraday tensor as introduced in [20] and [15], respectively.
2.10.1. L2-Sobolev energy of the distribution function. L2-Sobolev energies of the distribution func-
tion can be defined based on the Sasaki metrics with respect to γand g. For γ,γγijdxidxj+
γij b
Dpib
Dpj, where b
Dpi:= dpi+b
Γi
jk dpjdpk, defines a metric on T M . We denote the associated co-
variant derivative by b
D. The volume form induced by γis then given by γ:= −|γ|Q3
i=1 dxidpi.
A weighted metric on T M is defined by γ:= γijdxidxj+hpi2
γγij b
Dpib
Dpj,where hpiγ:=
q1 + |p|2
γ. We then define the L2-Sobolev energy of the distribution function with respect to the
associated Sasaki metric of γby
(2.47) |||f|||ℓ,µ := sX
kZT M hpi2µ+4(k)
γ|b
Dkf|2
γγ
The function space associated with the above norm is denoted by HVl,ℓ,µ(T M ). Moreover, pointwise
estimates are taken with respect to the following L
xL2
p-norm
|||f|||,ℓ,µ := sup
xM
sX
kZTxMhpi2µ+4(k)
γ|b
Dkf|2
γdbµp
,
where dbµp:= |γ|1
2dp1dp2dp3. Then, the following lemma holds.
Lemma 2.14. For fsufficiently regular, there exists a constant Csuch that
|||f|||,ℓ,µ C|||f|||+2,µ
holds.
In addition, we consider the L2-Sobolev energy associated with the dynamical metric g. The Sasaki
metric with respect to gis ggij dxidxj+gij DpiDpj,where Dpi:= dpi+Γi
jk dpjdpk. The asso-
ciated covariant derivative and the connection coefficient are denoted by Dand Γ(cf. Appendix C),
respectively, and the volume form induced by gon T M is given by g:= −|g|Q3
i=1 dxidpi.
Then, we define analogously a weighted version of the Sasaki metric associated with gby g:=
gij dxidxj+hpi2gij DpiDpj,where hpi:= q1 + |p|2
g. We finally define the L2-Sobolev energy
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 13
of the distribution function with respect to the Sasaki metric associated with the dynamical metric
gby
(2.48) Eℓ,µ(f) := sX
kZT M hpi2µ+4(k)|Dkf|2
gg
Under suitable smallness assumptions, E5,4(f) and |||f|||5,4are equivalent.
2.10.2. L2-Sobolev norm of the Faraday tensor. In the following we define the L2-Sobolev norm of
the Faraday tensor which measures the perturbation of the Maxwell field. It is defined as follows
(cf. [15])
|||F|||2
H:= X
kZMτ2gij Di1···DikF0iDi1···DikF0j
+gij gabDi1···DikFia Di1···DikFjbg.(2.49)
The factor τ2in the first term compensates the the growth of the τ-component coming from
Aτ=dT
AT=τ1AT, relative to the spatial components of F.
2.11. Smallness. To determine the final estimates we use a standard bootstrap argument. We
define a set of smallness conditions for the dynamical quantities which serves as the bootstrap
assumptions. We define
B6,5,5,6,5
δ,τ (γ, 0,0,0,0) := n(g, Σ, f, ω, ˙ω)H6×H5×HVl,5,4×H6×H5
|τ|1
2(kgγkH6+kΣkH5) + kρkH4+|τ|1
2· |||f|||5,4+|||F|||H5< δo.(2.50)
We say that (g(τ),Σ(τ), f(τ), ω(τ),˙ω(τ)) is δ-small if (g, Σ, f , ω, ˙ω)B6,5,5,6,5
δ,τ (γ, 0,0,0,0). To
refer to δ-small data we also use the term smallness assumptions. A direct consequence of the
smallness assumption is stated in the following lemma.
Lemma 2.15. For δ-small data (2.50) with δsufficiently small, the energies E5,4(f)and |||f|||5,4
are equivalent.
Proof. The proof follows straightforwardly from the definitions and the smallness assumption.
The smallness assumptions imply the smallness of the perturbation for the lapse function and shift
vector. The following result is a corollary of Proposition 7.1.
Corollary 2.16. For any δ > 0there exists a ¯
δ > 0such that (g, Σ, f , ω, ˙ω)B6,5,5,6,5
δ,τ (γ, 0,0,0,0)
implies
(2.51) |τ|1(kN3kH6+kXkH6)<¯
δ .
3. Estimating the energy-momentum tensor
In this section we will estimate the components of the energy-momentum tensor of the Maxwell
field and other matter quantities by the norms defined in the previous section.
Lemma 3.1. For k > 3/2we have the following estimate
(3.1) kMT00kHk+|τ|1· kMT0ikHk+|τ|2· kMTkHkC|||F|||2
Hk,
where MTdenotes the spatial part of MTand C=CkNkHk,kNkL,kN1kL,kXkHk.
Proof. The statement of the lemma follows directly from the expressions (2.43)–(2.45) (cf. also
Lemma 4.2 in [15]).
We summarize the estimates of the rescaled matter quantities in the following proposition.
14 H. Barzegar,D. Fajman
Proposition 3.2. For k > 3/2the following estimates hold
(3.2)
kρkHkCh|τ| · |||F|||2
Hk+Ek,3(f)i,
kjkHkCh|||F|||2
Hk+Ek,3(f)i,
kηkHkCh|τ|1· |||F|||2
Hk+Ek,4(f)i,
kSkHkCh|τ| · |||F|||2
Hk+τ2Ek,4(f) + kρkHki,
where C=CkNkHk,kNkL,kN1kL,kXkHk.
Proof. Combining Lemma 13 in [1] with the Lemma 3.1 result in the estimates of the proposition.
3.1. Pointwise estimates on the momentum variables. In this subsection we provide some
useful auxiliary estimates used further below.
Lemma 3.3. The following estimates hold for large T, using the smallness of the lapse function
and shift vector, and provided that the momenta have compact support
|p|g
bp.|τ|1,or |p|g
bp.|p|g,(3.3)
p0
bp.1
N1− | b
X|2
g,(3.4)
|P|g.|τ|1· | b
X|g+N1|p|g.(3.5)
Proof. The proof follows straightforwardly from the explicit expression for the respective quantities.
4. Control of the momentum support
Based on the characteristic system associated to the transport equation, we derive an estimate on
the radius of the momentum support of the distribution function.
The characteristic system associated with the rescaled transport equation reads
dxa
dT =τpa
p0,(4.1)
dpa
dT =τ1Γap02pa+ 2Γa
bpb+τΓa
bc +N1Σbc +1
3gbcXapbpc
p0τqFa.(4.2)
With the definition of the auxiliary quantity (cf. [1])
(4.3) G(T, x, p) := |p|2
g,
one finds its derivative along a characteristic
(4.4) dG
dT =|p|2
˙g+ 2τ1hΓ, pigp0+ 4gikΓi
jpjpk+2τ
NΣbc +1
3gbcpbpc
p0hX, pig4τ qhF, pig,
which implies the following result.
Lemma 4.1. For δ-small data (2.50) with δsufficiently small, the following estimate holds for any
characteristic
(4.5)
dG
dT C[|˙g|g+|Γ
|g+ (1 + |Σ|g)|X|g]G+C|τ|1· |Γ|g+| | · |F|gG.
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 15
Defining the supremum of the values of Gin the support of fat a fixed time Tanalogous to [1] as
(4.6) G[T] := sup npG(T, x, p)|(x, p)supp f(T,.,.)o,
we derive an estimate to control the momentum support in the following proposition.
Proposition 4.2. For δ-small data (2.50) with δsufficiently small, the following estimate holds
G|TG|T0+CZT
T0
(eskΓkH2+|||F|||H2)ds
×exp CZT
T0kΣkH2+kN3kH2+kXkH2+kΓ
kH2+|q|es|||F|||H2ds.(4.7)
Proof. From Lemma 4.1 and the definition of Fi, we get
d
dT GC(kΣkH2+kN3kH2+kXkH2+kΓ
kH2+| | · |||F|||H2)G
+C|τ|1kΓkH2+|||F|||H2.
Applying Gr¨onwall’s lemma completes the proof.
5. Energy estimates
5.1. Energy estimate of the distribution function. We first give the energy estimate for the
distribution function.
Before we proceed, we prove the following lemma:
Lemma 5.1. Assume that δ-small data (2.50) with δsufficiently small holds. Then, for a generic
sufficiently regular function φon T M , the following inequality holds
(5.1) X
mkZ|DmZφ dµp|2
gg.X
mkZZ¯p|Dmφ|gp2
g.
Proof. To obtain this inequality, we use (cf. (4.26) in [1])
(5.2) DaZφ dµp=ZAaφ dµp.
For higher covariant derivatives one gets terms of the form
(5.3) piRiem[g]Dφ ,
where m1 and denotes various abstract contractions of tensor indices. Then, the smallness
assumption on the metric yields
X
mkZDmZφdµp
2
g.X
mkZZ|Dmφ|gp+Z|p|g|Dm1φ|gp2
g
.X
mkZZ(1 + |p|g)|Dmφ|gp2
g
.X
mkZZ¯p|Dmφ|gp2
g.(5.4)
16 H. Barzegar,D. Fajman
Proposition 5.2. Let fbe a solution of the transport equation (2.25). For δ > 0sufficiently small
and (g, Σ, f, ω, ˙ω)B6,5,5,6,5
δ,τ (γ, 0,0,0,0) the estimate
TE2
k,µ(f)ChkN3kHk+kΣkHk+kXkHk+1 +|τ|·kN1TXkHk+|τ|1· kN1ΓkHk
+kΓ
kHk+k(Σ + g)XkHk+|τ|G+|q| · |||F|||HkiE2
k,µ(f),(5.5)
holds for k > 5/2and µ3, provided that |τ|Gis bounded by a constant.
Proof. Recall the following notation. Using the introduced bold Latin indices we define the frame
{θa}a6:= {A1,A2,A3,B1,B2,B3}. That is, the indices a∈ {1,2,3}correspond to the horizontal
directions whereas the indices a∈ {4,5,6}correspond to the vertical directions. The proof is
identical to the proof of Proposition 12 in [1] except for the last term. However, for the last term
which comes from the Maxwell part of the transport equation (2.25) one can proceed similarly.
Indeed; we consider
(5.6) Da1···DakTf ,
which arises from T|Dkf|2
gwhen applying Tto the L2-energy (2.48). For all the other terms
involving Twe refer to Proposition 12 in [1]. Therefore, we will focus on the Maxwell part of the
transport equation in (5.6)
Da1···Dak(τqFaBaf) = τ qθa1Da2···DakX
2jk
Γc
aja1Da2···Dac···Dak(FaBaf)
=τqθa1···θak(FaBaf) + ...+τ q(1)k1Γc1
aka1Γc2
c1a2···Γck1
ck2ak1(FaBaf),
where we suppressed the mixed terms. After commuting the operator FaBato the front by using
the relations in Appendix D one obtains
(5.7) τqFaBa(Da1···Dakf),
and other mixed terms which are of the form (cf. (5.3))
(5.8) τqDk1FapiDk2Riemk3Dk4f ,
with Piki=k. From (5.7) and after integration by parts when evaluating
τq X
kZT M hpi2µ+4(k)ga1b1···gakbkDb1···Dbkf·Da1···DakTf dµg,
one finds the corresponding term |q| · |||F|||Hkin (5.5). All the mixed terms are of lower order and
can be absorbed in the latter term.
5.2. Estimate of current density. As argued previously, the specific form of the effective terms
arising from the current, Jand Jiis essential to obtain sufficient bounds on the Maxwell fields.
The following lemma establishes these bounds.
Lemma 5.3. The following estimates
kJkHkC|q| · kρVkHk,(5.9)
kJkHkC|q||τ|1kXkHkkρVkHk+kjVkHk,(5.10)
hold, provided that |τ|Gis bounded by a constant and δ-small data (2.50) with δsufficiently small
holds.
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 17
Proof. To obtain the estimates above we need following inequalities
sup
(x,p)supp f(T,.,.)bpC(1 + |τ|G),sup
(x,p)supp f(T,.,.)bp
p2C(1 + |τ|G)3,
and
sup
(x,p)supp f(T,.,.)
1
pC(1 + |τ|G),
where we applied the smallness conditions. Then,
kJkHk=
qN Zf p
Hk
C|q|(1 + |τ|G)3kN1ρVkHk,(5.11)
from which the first estimate follows. Note here that the expression bpand p0appear via Rf µp=
Rf(p0)2/bp·bp/(p0)2µpto generate ρV. When the factor bp/(p0)2is hit by derivatives of the norm,
the generated terms can be estimated by a uniform constant invoking the smallness assumptions.
All those terms are absorbed into the constant C.
For the second estimate we have
kJkHk=
1XiZfp0
bpp+qZfpi
bpp
Hk
C|q|(1 + |τ|G)|τ|1kXkHkkρVkHk+kjVkHk,(5.12)
where ρVand jVare defined in (2.40). Then, the assumption on the boundedness of |τ|Gfinishes
the proof.
Remark 5.4. Estimate (5.9) enables us to estimate Jby the energy density of the distribution
function and eventually by the full energy density, which, as shown below, is uniformly bounded.
This prevents this term to pick up the loss from the Vlasov energy.
5.3. Energy estimate for the vector potential. As explained in Section 5.1 of [15], the struc-
ture of the system (2.33a)–(2.33b) suggests the following energy
(5.13)
Ek(ω) :=
k1
X
=0 ZMh(∆H)Le0ω, Le0ωig+h(∆H)+1ω, ωigg
≃ kLe0ωk2
Hk1(M[g]) +kωk2
Hk(M[g])
with k1. In the following we omit the argument ωand write EkEk(ω) for simplicity.
Remark 5.5. The gauge condition ωker(∆H) allows us to control the L2-norm of ωby Ek.
Remark 5.6. Under smallness assumptions the L2-Sobolev norm of the Faraday tensor (2.49) is
equivalent to the energy (5.13) and the norm of Ψ in the following sense
(5.14) |||F|||Hk≃ kΨkHk+1 +pEk+1 .
Remark 5.7. Both terms on the right-hand side, as shown below, are uniformly bounded.
Lemma 5.8. Let Fbe a solution of (2.31) and let A1(M)be a gauged vector potential for F,
Ψ, and ωas in Lemma 2.3. Then, for k > 5/2and assuming kNkHkis bounded by some constant,
we have
TEkC(kDlog NkHk1+kΠkHk1+kSkHk1)Ek
+C[ke0ΨkHk+ (ke0log NkHk+kΠkHk1)kΨkHk+|τ| · kJ kHk1]pEk.(5.15)
18 H. Barzegar,D. Fajman
Proof. The lemma is proved along the same lines as the proof of Lemma 5.7 in [15], and by using
(2.33b) and (cf. Eq. (5.37) in [15])
(5.16) kie0FkHk1CkDΨkHk1+kΨkHk1kDlog NkHk1+pEk,
where iY: Ω(M)1(M) is the interior product for any vector field YX(M). Here, by
abuse of notation, we consider the Sobolev norm of ie0FC(R,1(M)).
Lemma 5.9. As long as kDlog NkHkis small enough the following estimate holds
kΨkHk+1 ChkΠkHk1+ (1 + kΠkHk1)kDlog NkHk1+kSkHk1ipEk
+CkJkHk1
|{z }
()
+|τ| · kXkHk1kJkHk1.(5.17)
Remark 5.10. The term () is borderline, as it would yield a growth for the norm of Ψ. Uniform
boundedness is achieved due to the estimated using the energy density as outlined above.
Proof of Lemma 5.9. By elliptic regularity and the fact that (2.33a) has a unique solution, we find
kΨkHk+1 CkΨkHkkDlog NkHk+k[Le0,divg]ωkHk1
+kN1kHk1kJkHk1+|τ|k b
XkHk1kJkHk1.
And from Lemma 2.2 it follows
k[Le0,divg]ωkHk1C[kΠkHk1+ (1 + kΠkHk1)kDlog NkHk1+kSkHk1]pEk.
Combining the results and using Lemma 5.3 completes the proof.
We need to estimate the term ke0ΨkHkappearing in (5.15). This is done in Lemma 5.14 which
need further preparations:
Lemma 5.11. Let the integer k > 7/2and assume that kNkHk2is bounded. Then, we have
kiτFkHk2C|τ|1kNkHk2kΨkHk1+kΨkHk2kDlog NkHk2+pEk1
+C|τ|1kXkHk2pEk1.(5.18)
Proof. We recall that iτFC(R,1(M)). Hence,
(5.19) iτF(a) = Fa0=τ1N·ie0F(a) + τ1XiFai .
The first term can be estimated by (5.16) and for the second term we use (5.14). Then, invoking
the assumption on kNkHk2finishes the proof.
Lemma 5.12. Let k, µ Nsuch that k > 7/2and µ3. Further, assume that kNkHk2is
bounded. Then, the following estimate holds
(5.20) ke0JkHk2CkTNkHk2+kb
NkHk2+kXkHk1kJkHk2+C|q|Ek1(f),
provided that kΓkHk2is bounded.
Proof. We have
(5.21) e0J=N2TNJ+qZTfp+qZf T(p) + b
XkkJ.
The third term can be rewritten as
qZfT(p) = N13b
NDiXi+ 6J,
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 19
where we used the evolution equation for |g|1
2and the rescaling of the momentum according to
(2.17) to get
(5.22) T(p) = 3b
NDiXi+ 6p.
For the second term in (5.21), using the transport equation, we find
ZTf dµp=ZτN pi
pAifτ1p
NΓaBaf+ 2paBaf
a
cpcBafτΣbc +1
3gbcXapbpc
pBafτqFiBifp.(5.23)
The last term after integration by parts gives
τq ZFiBif dµp=τ q ZBiFif dµp
=τq Zfhgij b
Xib
XjFaj +τ
NFa0b
XiiBipa
p0p.(5.24)
The third term of the (5.23) cancels the third term of (5.22), after integration by parts. Hence,
qZTf dµp+qZfT(p) = qZτ N pi
pAifτ1p
NΓaBafa
cpcBaf
τΣbc +1
3gbcXapbpc
pBaf+τqBiFifp
=: q(I1+I2+I3+I4+I5).(5.25)
Invoking Lemmas 3.3 and 5.1 and the relations in Appendix E, integrating by parts whenever
possible, and imposing the smallness assumptions, leads to
(5.26)
kI1kHk2C· |τ| · Ek1(f),
kI2kHk2C· |τ| · kΓkHk2·Ek2(f),
kI3kHk2C· kΓ
kHk2·Ek2(f),
kI4kHk2C· |τ| · k(Σ + g)XkHk2·Ek2+1(f)
kI5kHk2C· |τq| · |||F|||Hk2·Ek2(f).
Note that Ek2+1(f)Ek1(f). Hence,
ke0JkHk2CkN2kHk2· kTNkHk2+kb
NkHk2+kXkHk1kJkHk2
+|q|
5
X
i=1 kIikHk2.
Assuming that kNkHk2and kΓkHk2are bounded, we arrive at the estimate of the lemma.
Lemma 5.13. Let the integers k > 7/2and let µ3and assume that kNkHk2is bounded. Then,
ke0JkHk2CkTNkHk2+kb
NkHk2+kXkHk1kJkHk2
+C|q|h1 + |τ|+|τ|1· kΓkHk2+kΓ
kHk2+kΣkHk2+kb
NkHk2+kXkHk1
+|τ|1(kXkHk2+kTXkHk2) + kTNkHk2+|q| · |||F|||Hk2iEk1(f).(5.27)
20 H. Barzegar,D. Fajman
Proof. We have
(5.28) e0Jk=N2TNJk+qZTfPkp+qZfTPkp+qZfPkT(p) + b
XiiJk.
Again, using (5.22), the fourth term can be rewritten as
(5.29) qZfPkT(p) = N13b
NDiXi+ 6Jk.
For the second and the fifth terms of (5.28) we repeat a similar calculation to the one in Lemma
5.12. The corresponding term which after integrating by parts cancels the fifth term of (5.28) is
2RPkpaBaf dµp. Thus,
(5.30) 2qZPkpaBaf dµp+ 6N1Jk=2qZBaPkpaf dµp.
Using (E.1g), under smallness assumptions, we find
(5.31)
qZBaPkpaf dµp
Hk2C|q|Ek2(f).
For the Maxwell part of the second term of (5.28) one gets, after integration by parts,
(5.32)
τq ZFiBifPkp
Hk2C|q|(1 + |τ|)|||F|||Hk2·Ek2(f).
Therefore,
qZTfPkp
Hk2C|q|h|τ|+|τ|11 + τ2kΓkHk2+kΓ
kHk2
+|τ| · k(Σ + g)XkHk2+|q| · |||F|||Hk2iEk1(f).(5.33)
Finally, putting (F.3) into the third term of (5.28) and making use of the relations in Appendix F,
one finds
qZfTPkp
Hk2C|q|h|τ|1(kXkHk2+kTXkHk2) + kTNkHk2
+kΣkHk2+kb
NkHk2iEk2(f).(5.34)
Inserting the results above and invoking the smallness assumptions yields the proof.
Lemma 5.14. Let integers k > 7/2and and µ3. Then, as long as kDlog NkHkis small enough
and kΠkHk2is bounded, the following holds
ke0ΨkHkCkΠkHk2+kSkHk2+kDlog NkHk2+kD∂e0log NkHk2
+kLe0SkHk2+kLe0ΠkHk2pEk+kΨkHk
+CkTNkHk2+kb
NkHk2+kXkHk1kJkHk2
+C1 + kTNkHk2+kTXkHk2+kb
NkHk2+kXkHk1|τ| · kXkHk2· kJ kHk2
+C|q|n|τ| · h1 + kΓkHk2+k(Σ + g)XkHk2+kXkHk2· k b
NkHk2
+kTXkHk2+kTNkHk2+|q| · |||F|||Hk2i+kΓ
kHk2oEk1(f).(5.35)
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 21
Proof. Differentiating (2.33a) in the direction of e0and using the elliptic regularity, we arrive at
(cf. (5.39) in [15])
ke0ΨkHkCk[Le0,gkHk2+ke0divgDlog N)kHk2
+ke0[Le0,divg]ωkHk2+ke0(N1J+τb
XjJj)kHk2.(5.36)
The first three terms are estimated in Lemma 5.9 in [15] by assuming that kDlog NkHkis small
enough and kΠkHk2is bounded:
k[Le0,gkHk2+ke0divgDlog N)kHk2+ke0[Le0,divg]ωkHk2
CkΠkHk2+kSkHk2+kDlog NkHk2+kD∂e0log NkHk2
+kLe0SkHk2+kLe0ΠkHk2pEk+kΨkHk+kDlog NkHk1ke0ΨkHk1.(5.37)
We estimate the last term of (5.36). To this end, we use the following relations:
e0N1=N3TN+XkkN,
e0τ=τN 1,
e0Xk=N1TXk+XjjXk,
e0(τb
XjJj) = τXkJk+τ e0b
XkJk+τb
Xke0Jk,
ke0N1JkHk2CkN3kHk2(kTNkHk2+kXkHk2kNkHk1)kJkHk2
+CkN1kHk2ke0JkHk2,
ke0b
XkHk2CkN3kHk2(kTNkHk2+kNkHk1kXkHk2)kXkHk2
+kNkHk2(kTXkHk2+kXkHk1kXkHk2),
ke0(τhb
X, Jig)kHk2C|τ|kN1kHk2· k b
XkHk2+ke0b
XkHk2kJkHk2
+C|τ|k b
XkHk2ke0JkHk2.
Then, by the smallness assumptions and boundedness of kNkHk2, one finds
(5.38)
ke0(N1J+τb
XjJj)kHk2
CkTNkHk2+kb
NkHk2+kXkHk1kJkHk2
+C1 + kTNkHk2+kTXkHk2+kb
NkHk2+kXkHk1|τ| · kXkHk2· kJ kHk2
+C|q|n|τ| · h1 + kΓkHk2+k(Σ + g)XkHk2+|q| · |||F|||Hk2+kXkHk2· k b
NkHk2
+kTXkHk2+kTNkHk2+|q| · |||F|||Hk2i+kΓ
kHk2oEk1(f).
Inserting (5.37) and (5.38) into (5.36) and assuming that kDlog NkHk2is small enough, one
obtains the claim of the lemma.
22 H. Barzegar,D. Fajman
Proposition 5.15. For the energy defined in (5.13) we have the estimate
(5.39)
TEkCkΣkHk1+kdivgΣkHk1+ke0NkHk+kN3kHk
+kLe0ΣkHk2+kLe0divgΣkHk2Ek
+CkTNkHk2+kN3kHk+ke0NkHk+kXkHk1+kΣkHk1
+kdivgΣkHk1+kLe0ΣkHk2+kLe0divgΣkHk2kJkHk2pEk
+C|τ| · kJ kHk2pEk+C|q||τ|+kΓ
kHk2Ek1(f)pEk,
as long as the norms in the brackets are bounded.
Proof. The boundedness of kΠkHk2and kDlog NkHk2together with Lemma 5.9 implies
(5.40) pEk+kΨkHkCpEk+kJkHk2+|τ| · kXkHk2kJkHk2.
Then, combining Lemmas 5.8 and 5.14, applying the smallness assumptions, and finally assuming
the boundedness of the norms appearing in the estimate yield the proof.
Remark 5.16. Note that all terms appearing in the right-hand side of the inequality in Proposi-
tion 5.15 have good decay behaviour, i.e., they do not cause problems in the final estimate.
6. Energy estimates from divergence identity
We define the L2-Sobolev energy of the rescaled energy density by
(6.1) ̺k:= sX
kZM|Dρ|2
gg.
We introduce
(6.2) b
T:= T+LX.
Then, the divergence identity reads (cf. (G.2))
(6.3) b
Tρ= (3 N)ρ+1
2τN 1Da(N2ja)1
6τ2NgabTab 1
2τ2NΣabTab .
Moreover, we need the following identities (cf. [16] and [1])
(6.4) b
T, DiDj1. . . Djmu=X
1am
Dj1. . . Dja1DbDja+1 . . . Djmu·b
TΓb
jai.
and
(6.5) TZM
u dµg=ZM
(3 N)u dµg+ZMb
Tu dµg
for any function uon M. Then, one finds the following estimate for ̺k.
Proposition 6.1. For integers k > 7/2the following estimate holds
(6.6)
T̺kC(kD(Nk)kHk2+kN3kHk)̺k
+Cτ2
Nab +1
3gab)Tab
Hk+C|τ| ·
N1divg(N2j)
Hk,
provided that |τ|Gis bounded by a constant.
Proof. The proof is analogous to the proof of Proposition 16 in [1].
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 23
7. Elliptic Estimates for the lapse function and the shift vector
In this section we obtain elliptic estimates for the lapse function and the shift vector and their
respective time derivatives.
Proposition 7.1. The following estimates hold for the lapse function and the shift vector
kN3kHkCkΣk2
Hk2+|τ| · kρkHk2+|τ|3· kηkHk2,(7.1)
kXkHkCkΣk2
Hk2+kgγk2
Hk1+|τ|·kρkHk3+|τ|3· kηkHk3+τ2kNjkHk2.(7.2)
Proof. The estimates follows directly by the elliptic regularity applied to the elliptic equations for
the lapse function and the shift vector.
7.1. Estimate of the time derivatives. Further, we can estimate the time derivatives of the
lapse function and shift vector by using the elliptic estimate.
Proposition 7.2. Let k, µ Nwith k > 9/2and µ3. For δ-small data (2.50) with δsufficiently
small the following estimates hold
kTNkHkChkb
NkHk+kXkHk+kΣk2
Hk1+kgγk2
Hk+|τ| · kSkHk2+|τ| · kρkHk1
+|τ|3· kηkHk2+τ2kjkHk1+|τ|3· kTkHk1+|τ|3Ek1+1(f) + τ2|||F|||2
Hk1i,(7.3)
kTXkHkChkb
NkHk1+kXkHk+kΣk2
Hk1+kgγk2
Hk1+|τ| · kSkHk3+|τ| · kρkHk2
+|τ|3· kηkHk3+τ2kjkHk1+|τ|3· kTkHk1+|τ|3Ek2+1(f) + τ2|||F|||2
Hk2i,(7.4)
where Tdenotes the spatial part of the rescaled energy-momentum tensor Tand integers µ3and
k > 7/2.
Proof. Differentiating the elliptic system for (N , X) and using the relations in Appendix B yields
(7.5)
1
3TN= 2NhD2N, Σig+ 2 b
NN− hD2N, LXgig
+h2Dj(NΣij )Dib
NXi+RijXjiDiN
+ 2N3N1
3|Σ|2
g+ 2h∇X, Σ,ΣigNhΣ,1
2Lg,γ (gγ) + Jig
+hΣ, D2Nig− hΣ,LXΣig+Nτ hΣ, Sig
+TN|Σ|2
g+τρ +τ3η+NT(τ ρ) + T(τ3η),
where h∇X, Σ,ΣigDiXjΣi
kΣk
j, and
∆(TXi) + RijTXj=[T,∆] XiTRijXj
+ 2Dj(TNij + 2DjNTΣij Tgij Djb
N1
3Di(TN)
+ 2TNτ 2ji+ 2NT(τ2ji)2TNΣjk +N TΣjk TgjDXk
DjTXkgjℓ TΓk
mℓXmΓi
jk b
Γi
jk 2NΣjk DjXkTΓi
jk .(7.6)
For the lapse function the elliptic regularity by using the smallness assumptions yields
(7.7) kTNkHkCkb
NkHk+kXkHk+kΣk2
Hk1+kgγk2
Hk+|τ| · kSkHk2
+|τ| · kρkHk2+|τ|3· kηkHk2+|τ| · kTρkHk2+|τ|3· kTηkHk2
+kΣk2
Hk2+|τ| · kρkHk2+|τ|3· kηkHk2kTNkHk2.
Using elliptic regularity iteratively in conjunction with the smallness assumptions for both equations
yields the estimates on the time-derivatives. The procedure is straighforward and follows the
analogue case in [1].
24 H. Barzegar,D. Fajman
8. Energy Estimate For Geometric Objects
In this section we mainly adapt the results on the energy for the geometric perturbation from [3]
and [1] to the present case.
8.1. Decomposing the evolution equations. We use the following form of the evolution equa-
tions.
Lemma 8.1. The evolution equations for gand Σare equivalent to the system
T(gγ) = 2NΣ + Fgγ,(8.1a)
T(6Σ) = 12Σ 3NLg,γ (gγ) + 6N τ S Xib
Di(6Σ) + FΣ,(8.1b)
where
kFgγkHkCkΣk2
Hk1+kgγk2
Hk+|τ| · kρkHk2+|τ|3kηkHk2+τ2kNjkHk1,
kFΣkHk1CkΣk2
Hk1+kgγk2
Hk+|τ| · kρkHk1+|τ|3kηkHk1+τ2kNjkHk2.
Proof. The proof is formally identical to [1].
8.2. Energy. We define an energy for the trace-free part of the second fundamental form and the
metric perturbation. This definition depends on the lowest eigenvalue λ0of the Einstein operator
of γ. We define the constant α=α(λ0, δα) by
(8.2) α:= 1 δα,
where
(8.3) δα:= (0, λ0>1/9,
p19(λ0ε), λ0= 1/9,
with 0 < ε 1. Next, we define the correction constant accordingly by
(8.4) cE:= 1 δ2
α.
Once εis fixed, in case λ0= 1/9, δαcan be made suitably small, independent of the other constants
which play role in the final estimate.
We are now able to define the energy for the geometric perturbation by
(8.5) Ek(gγ, Σ) := X
1mk
E(m)(gγ, Σ) := X
1mkE(m)(gγ, Σ) + cEΓ(m)(gγ, Σ),
where
(8.6) E(m)(gγ, Σ) := 18 ZMhΣ,Lm1
g,γ Σig+9
2ZMh(gγ),Lm
g,γ (gγ)ig,
Γ(m)(gγ, Σ) := 6 ZMhΣ,Lm1
g,γ (gγ)ig,
for integers m1. Note that here ,·i is defined for the symmetric covariant 2-tensors uand vby
hu, vi:= uij vkℓ γik γj.
Lemma 8.2. For the integers k > 5/2, there exists a constant C > 0such that for δ-small data
(2.50) with δsufficiently small the inequality
(8.7) kgγk2
Hk+kΣk2
Hk1CEk(gγ, Σ)
holds.
Proof. For the proof we refer to Lemma 19 in [1].
STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER