The Stability of the Euler-Einstein
System with a Positive Cosmological
Constant
Jared Speck
jspeck@math.princeton.edu
University of Cambridge
April 7, 2010
Einstein equations (for a spacetime (M, gµν )
of signature (-,+,+,+))
(fluid)
Gµν + Λgµν = Tµν
Λ>0
µν
Bianchi identities =⇒ Dµ T(fluid)
=0
Our spacetimes: [0, T ] × T3
Red-shift effect
Light emitted by distant objects is red-shifted
Hubble-Humason “law" (1929): galaxies are receding from
Earth, and their velocities are proportional to their
distances from it ; “suggests" an expanding universe .
“Expansion of the universe”
a(t) is the scale factor
g = −dt 2 + a2 (t)
3
X
(dx k )2 ,
k =1
ȧ(t) > 0
Leads to red-shift effect
Why Λ > 0?
1990’s: experimental data (type IA supernovae, ...)
suggest accelerated expansion.
Λ > 0 =⇒ ∃ solutions with accelerated expansion. e.g.
P
g = −dt 2 + e2Ht 3a=1 (dx a )2 ,
p
H = Λ/3.
Stability
Background solution (Ours will be FLRW type)
Initial value problem formulation; wave coordinates
(de Donder 1921, Choquet-Bruhat 1952)
Goal: show that if we slightly perturb the background
data, then the resulting solution exists for all t ≥ 0
and that the spacetime is future causally geodesically
complete
We show convergence as t → ∞
Proofs based on “energy" estimates for quasilinear
wave equations and the Euler equations
Previous stability results
For Λ > 0 using the conformal method
Vacuum Einstein (Friedrich, 1986)
Einstein-Maxwell & Einstein-Yang-Mills (Friedrich,
1991)
Vacuum Einstein in (1 + n) dimensions, n odd
(Anderson, 2005)
For Λ = 0 using wave coordinates
Einstein-scalar field with a nonlinearity V (Φ) in
(1 + n) dimensions (Ringström, 2008)
Φ = V 0 (Φ)
V (Φ) emulates Λ > 0 : V (0) > 0, V 0 (0) = 0, V 00 (0) > 0
Previous stability results cont.
Newtonian cosmological models with Λ > 0 (Brauer,
Rendall, & Reula, 1994)
Irrotational Euler-Einstein with Λ > 0 (Rodnianski &
Speck, 2009)
Indices
Latin (spatial) indices: a, b, i, j, k ∈ {1, 2, 3}
Greek (spacetime) indices: α, β, κ, µ, ν ∈ {0, 1, 2, 3}
def
Summation convention: ua u a = u1 u 1 + u2 u 2 + u3 u 3 , etc.
∂t = ∂0
∂ = (∂t , ∂1 , ∂2 , ∂3 )
∂¯ = (∂1 , ∂2 , ∂3 )
The relativistic Euler equations
µν
Dµ T(fluid)
= 0,
Dµ (nu µ ) = 0,
µν
T(fluid)
= (ρ + p)g µν + pg µν ,
u 0 > 0.
uα u α = −1,
To close the system: combined first and second law of
thermodynamics:
ρ+p =n
dρ
,
dn
and an equation of state: p = cs2 ρ.
Friedmann-Lemaître-Robertson-Walker
background solutions
Ansatz:
e = −dt 2 + a2 (t)
g
3
X
(dx a )2 ,
a(0) = 1,
a=1
e=p
e(t),
p
def
e(0) =
p
p̄ > 0.
Plug into Einstein equations, and get:
s
p̄
d
a = a H + 2 3(1+c 2 ) ,
s
dt
3cs a
r
Λ
H=
,
3
2
e(t) = a−3(1+cs ) p̄.
p
Cosmetics and Psychology
Cosmetics:
def
Ω(t) = ln a(t) ≈ Ht,
ȧ(t)
def d
≈ H.
ω(t) = Ω(t) =
dt
a(t)
P
e = −dt 2 + e2Ω(t) 3a=1 (dx a )2 exponentially grows;
g
e = p̄e−3(1+cs2 )Ω(t) exponentially decays.
p
Rescale:
def
hjk = e−2Ω(t) gjk ,
def
2
P = e3(1+cs )Ω(t) p.
Coordinate gauge
Wave coordinates
Γµ = e
Γµ = 3ω(t)δ0µ ≈ 3Hδ0µ
Use to “modify” the Einstein equations (without
changing them!)
Γµ |t=0 = e
Γµ |t=0 =⇒ Γµ ≡ e
Γµ
Proof: (≈ Choquet-Bruhat)
Γµ − e
Γµ satisfies a system of wave equations with 0
data
The modified Euler-Einstein system
def
def
2
def
def
hjk = e−2Ω gjk , P = e3(1+cs )Ω p W = 3cs2 , H 2 =
def
ˆg =
g αβ ∂α ∂β ; “friction", “error", “dangerous"
Λ
3
ˆ g (g00 + 1) = 5H∂t g00 + 6H 2 (g00 + 1) + 400 ,
ˆ g g0j = 3H∂t g0j + 2H 2 g0j − Hg ab Γajb + 40j ,
ˆ g hjk = 3H∂t hjk + 4jk ,
−1 Pua ∂t u a + (1 + cs2 )P∂a u a = 4,
u ν ∂ν (P − p̄) + (1 + cs2 )
u0
u ν ∂ν u j +
cs2
Πjα ∂α (P − p̄) = (W − 2)ωu j + 4j .
(1 + cs2 )P
Norms (N ≥ 3, 0 < q 1,
Ω(t) ≈ Ht, hjk = e−2Ωgjk )
¯ 00 kH N + eqΩ kg00 + 1kH N ,
Sg00 +1;N = eqΩ k∂t g00 kH N + e(q−1)Ω k∂g
3 X
def
¯ 0j kH N + e(q−1)Ω kg0j kH N ,
e(q−1)Ω k∂t g0j kH N + e(q−2)Ω k∂g
Sg0∗ ;N =
def
j=1
def
Sh∗∗ ;N =
3 X
¯ jk kH N + k∂h
¯ jk kH N−1 ,
eqΩ k∂t hjk kH N + e(q−1)Ω k∂h
j,k =1
v
u 3
uX
def
UN= t
e2(1+q)Ω ku j k2H N ,
j=1
def
S N = eΩ ku j kH N + kP − p̄kH N ,
def
QN =
Sg00+1 ;N + Sg0∗ ;N + Sh∗∗ ;N + UN−1 + SN
.
|
{z
}
Measures the deviation from the FLRW solution
The global existence theorem
Theorem
(J.S. 2010) Assume that N ≥ 3 and 0 < cs2 < 1/3. Then
there exist 0 > 0 and C > 1 such that for all ≤ 0 , if
QN (0) ≤ C −1 , then there exists a global future causal
geodesically complete solution to the modified
Euler-Einstein system. Furthermore,
QN (t) ≤ holds for all t ≥ 0.
def
QN (t) = supτ ∈[0,t] Q N (τ ).
Asymptotics of the solution
Theorem
(J.S. 2010) Under the assumptions of the global existence
theorem, with the additional assumption N ≥ 5, there exist
(∞)
q > 0, a smooth Riemann metric gjk with corresponding
(∞)
jk
Christoffel symbols Γijk and inverse g(∞)
on T3 such that
(∞)
ke−2Ω gjk (t, ·) − gjk kH N ≤ Ce−qHt ,
jk
ke2Ω g jk (t, ·) − g(∞)
kH N ≤ Ce−qHt ,
(∞)
ke−2Ω ∂t gjk (t, ·) − 2ωgjk kH N ≤ Ce−qHt ,
(∞)
ab
kg0j (t, ·) − H −1 g(∞)
Γajb kH N−3 ≤ Ce−qHt ,
k∂t g0j (t, ·)kH N−3 ≤ Ce−qHt ,
Asymptotics cont.
Theorem
(Continued)
kg00 + 1kH N ≤ Ce−qHt ,
k∂t g00 kH N ≤ Ce−qHt ,
(∞)
ke−2Ω Kjk (t, ·) − ωgjk kH N ≤ Ce−qHt .
In the above inequality, Kjk is the second fundamental form of
the hypersurface {t = const}.
Similar results hold for the fluid variables.
Fundamental idea of proof
Proposition
The norm QN (t) has the following property: if the solution
blows up at a time Tmax , then limt↑Tmax QN (t) = ∞.
To prove global existence: we show that QN (t) remains
uniformly bounded for all t ≥ 0.
The energies
Basic idea of analysis: analyze “energies" that are ≈ to the
norms. Energies are defined using the vectorfield method:
vectorfield
z }| {
e ν [v , v ]X µ ,
J ν [v , v ] = T
µ
∂ν (J ν [v , v ]) = lower order.
e ν [v , v ] = canonical stress corresponding to v ;
T
µ
encodes “almost-"conservation laws
e ν [v , v ]) = Qµ [v , v ]; (have to use the equations)
∂ν (T
µ
X
=
suitable timelike vectorfield
R
0
J
[v , v ]d 3 x is the square of the “energy" of v
T3
Time derivative of energy is controlled by the
divergence theorem
Two sources of stress
Energy-norm equivalence
Lemma
If QN is small enough, then the energies and norms are
equivalent:
Eg00 +1;N
Eg0∗ ;N
Eh∗∗ ;N
EN + UN−1
QN
≈ Sg00 +1;N ,
≈ Sg0∗ ;N ,
≈ Sh∗∗ ;N ,
≈ SN ,
≈ QN .
Gronwall estimates for energies
Lemma
The following system of inequalities (for t ≥ t1 ) holds for the modified
equations. Furthermore, it features small-data global bounds.
U 2N−1 (t)≤
U 2N−1 (t1 )
<0
t
Z
}|
{
z
2(W − 1 + q) e(1+q)Hτ U 2N−1 + CQN U N−1 dτ ,
+
E 2N (t)≤ E 2N (t1 ) + C
Z
τ =t1
t
e−qHτ Q2N dτ ,
t1
E2g00 +1;N (t)
E2g0∗ ;N (t)
E2h∗∗ ;N (t)
≤
≤
≤
Z
E2g00 +1;N (t1 )
+
E2g0∗ ;N (t1 )
Z
E2h∗∗ ;N (t1 )
t
t1
−4qHE2g00 +1;N + Ce−qHτ QN Eg00 +1;N dτ,
t
−4qHE2g0∗ ;N + CEh∗∗ ;N Eg0∗ ;N + Ce−qHτ QN Eg0∗ ;N dτ,
+
t1
Z
t
+
t1
He−qHτ E2h∗∗ ;N + Ce−qHτ QN Eh∗∗ ;N dτ.
The fluid energy currents
def
Given any variations (Ṗ, u̇ 1 , u˙2 , u˙3 ), we set u̇ 0 = − u10 ua u̇ a
and define the fluid energy current:
def
J̇ µ =
uµ
(1 + cs2 )Pu µ
2
µ
Ṗ
+
2
u̇
Ṗ
+
gαβ u̇ α u̇ β
(1 + cs2 )P
cs2
J̇ 0 = “density of energy"
J̇ 0 ≈ Ṗ 2 + e2Ω δab u̇ a u̇ b
The divergence of J̇
h
i
uµ
(1 + cs2 )Pu µ i
2
Ṗ + ∂µ
gαβ u̇ α u̇ β
(1 + cs2 )P
cs2
(1 + cs2 )Pu a
(1 + cs2 )Pu 0
α β
+
(∂
g
)
u̇
u̇
+
(∂t g00 )(u̇ 0 )2
a
αβ
cs2
cs2
2(1 + cs2 )Pu 0
2(1 + cs2 )P(u 0 − 1)
0 α
+
(∂
g
)
u̇
u̇
+
(∂t gab )u̇ a u̇ b
t
0α
cs2
cs2
h i
ua
(1 + cs2 )P
− 2 ∂t
u̇ a Ṗ +
(∂t gab − 2ωgab )u̇ a u̇ b
u0
cs2
2(1 + cs2 )P
2F
2(1 + cs2 )P
+
(W − 1)ωgab u̇ a u̇ b +
Ṗ +
gαβ Gα u̇ β
2
2
cs
(1 + cs )P
cs2
|
{z
}
h
∂µ J̇ = ∂µ
µ
<0
Control of k∂µJ̇ µkL1
Using the formula for ∂µ (J̇ µ ) plus bootstrap assumption
QN (t) ≤ , we can show:
k∂µ J̇ µ kL1 ≤ Ce−qHt QN 2
Simple consequence:
d
dt
Z
0
3
Z
J̇ d x =
T3
T3
∂µ J̇ µ d 3 x ≤ Ce−qHt QN 2 .
Comparison with flat spacetime
Christodoulou’s monograph “The Formation of
Shocks in 3-Dimensional Fluids" shows that on the
Minkowski spacetime background, shock singularities
can form in solutions to the Euler equations arising
from data that are arbitrarily close to that of a uniform
quiet state.
Conclusion: Exponentially expanding
spacetimes can stabilize fluids.
Topologies beyond T3
The study of wave equations arising from metrics
featuring accelerated expansion is a “very local"
problem.
A patching argument can “very likely" be used to
allow for many of the topologies considered by
Ringström: Unimodular Lie Groups different from
SU2; H3 ; H2 × R; · · ·
Future directions
p = 0 : preliminary analysis suggests that it is stable
The case p = cs2 ρ with cs2 ≥ 1/3 is unclear; numerical
evidence could be helpful
Other equations of state
Sub-exponential expansion rates
Thank you
gµν building block energies
If α > 0, β ≥ 0, and φ is a solution to
ˆ g φ = αH∂t φ + βH 2 φ + F ,
then we control solutions to this equation using an energy
Z
def 1
2
E(γ,δ)
[φ, ∂φ] =
{−g 00 (∂t φ)2 + g ab (∂a φ)(∂b φ) − 2γHg 00 φ∂t φ + δH 2 φ2 } d 3 x
2 T3
γ, δ can be adjusted so that ∃η > 0 such that
Z n
o
d 2
2
(E(γ,δ) [φ, ∂φ]) ≤ −ηHE(γ,δ)
[φ, ∂φ] +
− (∂t φ + γHφ)F + 4E;(γ,δ) [φ, ∂φ] d 3 x,
dt
T3
4E;(γ,δ) [φ, ∂φ] = −γH(∂a g ab )φ∂b φ − 2γH(∂a g 0a )φ∂t φ − 2γHg 0a (∂a φ)(∂t φ)
1
− (∂a g 0a )(∂t φ)2 − (∂a g ab )(∂b φ)(∂t φ) − (∂t g 00 )(∂t φ)2
2
1
ab
ab
+
∂t g + Hg
(∂a φ)(∂b φ) − γH(∂t g 00 )φ∂t φ − γH(g 00 + 1)(∂t φ)2 .
2
Energies for gµν
def
E2g00 +1;N =
X
2
e2qΩ E(γ
[∂α~ (g00 + 1), ∂(∂α~ (g00 + 1))]
00 ,δ00 )
|~
α|≤N
E2g0∗ ;N
def
=
3
X X
2
e2(q−1)Ω E(γ
[∂α~ g0j , ∂(∂α~ g0j )]
0∗ ,δ0∗ )
|~
α|≤N j=1
E2h∗∗ ;N
def
=
3
X X
|~
α|≤N
j,k =1
2
[0, ∂(∂α~ hjk )]
e2qΩ E(0,0)
1
+
2
Z
2
3
cα~ H (∂α~ hjk ) d x
T3