The Stability of the Irrotational
Euler-Einstein System with a Positive
Cosmological Constant
Jared Speck & Igor Rodnianski
jspeck@math.princeton.edu
University of Cambridge & Princeton University
October 26, 2009
Indices
Greek indices ∈ {0, 1, 2, 3}
Latin indices ∈ {1, 2, 3}
∂t = ∂0
∂ = (∂t , ∂1 , ∂2 , ∂3 )
∂¯ = (∂1 , ∂2 , ∂3 )
Einstein’s equations
Gµν + Λgµν = Tµν
We fix Λ > 0
Bianchi identities imply Dµ T µν = 0
Our spacetimes will have topology [0, T ] × T3
Einstein’s equations
Gµν + Λgµν = Tµν
We fix Λ > 0
Bianchi identities imply Dµ T µν = 0
Our spacetimes will have topology [0, T ] × T3
Einstein’s equations
Gµν + Λgµν = Tµν
We fix Λ > 0
Bianchi identities imply Dµ T µν = 0
Our spacetimes will have topology [0, T ] × T3
Einstein’s equations
Gµν + Λgµν = Tµν
We fix Λ > 0
Bianchi identities imply Dµ T µν = 0
Our spacetimes will have topology [0, T ] × T3
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
The notion of “stability"
Need a background solution (Ours will be FLRW type)
Initial value problem formulation of Einstein equations
We will use 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 → ∞
Our proofs are based on energy estimates for
quasilinear wave equations
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results
For Λ = 0 :
Vacuum Einstein using maximal foliation
(Christodoulou & Klainerman, 1993)
Einstein-Maxwell (Zipser, 2000)
Vacuum Einstein using double-null foliation
(Klainerman & Nicolò, 2003)
Einstein-scalar field using wave coordinates
(Lindblad & Rodnianski, 2004-2005)
Einstein-Maxwell-scalar field in 1 + n dimensions
(Loizelet, 2008)
Vacuum Einstein for more general data (Bieri, 2009)
Previous stability results cont.
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 potential 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.
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 potential 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.
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 potential 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.
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 potential 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.
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 potential 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.
Euler-Poisson with a cosmological constant (Brauer,
Rendall, & Reula, 1994)
Why Λ > 0?
Answer # 1: Einstein was the first to envision Λ : he was
looking for static solutions.
Answer # 2: In 1929, Hubble formulated an empirical “law,"
which was based on observations of the redshift effect,
and which suggested that the universe is expanding.
Hubble’s “law:" galaxies are receding from Earth, and their
velocities are proportional to their distances from it.
In the 1990’s: data from type Ia supernovae and the
Cosmic Microwave Background suggested accelerated
expansion.
Λ > 0 =⇒ ∃ solutions
with accelerated expansion. e.g.
√
P3
2( Λ/3)t
2
a 2
g = −dt + e
a=1 (dx )
Why Λ > 0?
Answer # 1: Einstein was the first to envision Λ : he was
looking for static solutions.
Answer # 2: In 1929, Hubble formulated an empirical “law,"
which was based on observations of the redshift effect,
and which suggested that the universe is expanding.
Hubble’s “law:" galaxies are receding from Earth, and their
velocities are proportional to their distances from it.
In the 1990’s: data from type Ia supernovae and the
Cosmic Microwave Background suggested accelerated
expansion.
Λ > 0 =⇒ ∃ solutions
with accelerated expansion. e.g.
√
P3
2( Λ/3)t
a 2
2
g = −dt + e
a=1 (dx )
Why Λ > 0?
Answer # 1: Einstein was the first to envision Λ : he was
looking for static solutions.
Answer # 2: In 1929, Hubble formulated an empirical “law,"
which was based on observations of the redshift effect,
and which suggested that the universe is expanding.
Hubble’s “law:" galaxies are receding from Earth, and their
velocities are proportional to their distances from it.
In the 1990’s: data from type Ia supernovae and the
Cosmic Microwave Background suggested accelerated
expansion.
Λ > 0 =⇒ ∃ solutions
with accelerated expansion. e.g.
√
P3
2( Λ/3)t
2
a 2
g = −dt + e
a=1 (dx )
Why Λ > 0?
Answer # 1: Einstein was the first to envision Λ : he was
looking for static solutions.
Answer # 2: In 1929, Hubble formulated an empirical “law,"
which was based on observations of the redshift effect,
and which suggested that the universe is expanding.
Hubble’s “law:" galaxies are receding from Earth, and their
velocities are proportional to their distances from it.
In the 1990’s: data from type Ia supernovae and the
Cosmic Microwave Background suggested accelerated
expansion.
Λ > 0 =⇒ ∃ solutions
with accelerated expansion. e.g.
√
P3
2( Λ/3)t
a 2
2
g = −dt + e
a=1 (dx )
Why Λ > 0?
Answer # 1: Einstein was the first to envision Λ : he was
looking for static solutions.
Answer # 2: In 1929, Hubble formulated an empirical “law,"
which was based on observations of the redshift effect,
and which suggested that the universe is expanding.
Hubble’s “law:" galaxies are receding from Earth, and their
velocities are proportional to their distances from it.
In the 1990’s: data from type Ia supernovae and the
Cosmic Microwave Background suggested accelerated
expansion.
Λ > 0 =⇒ ∃ solutions
with accelerated expansion. e.g.
√
P3
2( Λ/3)t
a 2
2
g = −dt + e
a=1 (dx )
The modified irrotational Euler-Einstein
system
def
def
def
def
ˆg =
hjk = e−2Ω gjk , W = 3/(1 + 2s), H 2 = Λ3 , g αβ ∂α ∂β
ˆ g (g00 + 1) = 5H∂t g00 + 6H 2 (g00 + 1) + 400
ˆ g g0j = 3H∂t g0j + 2H 2 g0j
− 2Hg ab Γajb + 40j
ˆ g hjk = 3H∂t hjk + 4jk
−∂t2 Φ + mjk ∂j ∂k Φ + 2m0j ∂t ∂j Φ = W ω∂t Φ + 4(∂Φ)
mµν is the reciprocal acoustic metric
The “4” terms are error terms
The appearance of zj in the fluid equation
−∂t2 Φ + mjk ∂j ∂k Φ + 2m0j ∂t ∂j Φ = wω∂t Φ + 4(∂Φ)
jk
m =
g jk − 4jk(m)
(1 + 2s) + 4(m)
2sg 00 eΩ g ja za + · · ·
m0j = −
(1 + 2s) + 4(m)
4(m) = (1 + 2s)(g 00 + 1)(g 00 − 1) + 2(1 + 2s)eΩ g 00 g 0a za + · · ·
4jk(m) = 2eΩ (g jk g 0a + 2sg 0k g aj )za + · · ·
def
zj =
e−Ω ∂j Φ
∂t Φ
The global existence theorem
Theorem
(I.R. & J.S. 2009) 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 reduced irrotational
Euler-Einstein system. Furthermore,
QN (t) ≤ holds for all t ≥ 0.
Asymptotics of the solution
Theorem
(I.R. & J.S. 2009) Under the assumptions of the global
existence theorem, with the additional assumption N ≥ 5,
(∞)
there exist q > 0, a smooth Riemann metric gjk with
(∞)
jk
corresponding Christoffel symbols Γijk and inverse g(∞)
on
¯ (∞) , Ψ(∞) on T3 such
T3 , and (time independent) functions ∂Φ
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.
kewΩ ∂t Φ(t, ·) − Ψ(∞) kH N−1 ≤ Ce−qHt
¯
¯ (∞) kH N−1 ≤ Ce−qHt .
k∂Φ(t,
·) − ∂Φ
Comparison with flat space
Christodoulou’s monograph “The Formation of
Shocks in 3-Dimensional Fluids" shows that on the
Minkowski space background, shock singularities can
form in solutions to the irrotational fluid equation
arising from data that are arbitrarily close to that of a
uniform quiet state.
Conclusion: Exponentially expanding
spacetimes can stabilize irrotational fluids.
Comparison with flat space
Christodoulou’s monograph “The Formation of
Shocks in 3-Dimensional Fluids" shows that on the
Minkowski space background, shock singularities can
form in solutions to the irrotational fluid equation
arising from data that are arbitrarily close to that of a
uniform quiet state.
Conclusion: Exponentially expanding
spacetimes can stabilize irrotational fluids.
Topologies beyond that of 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; · · ·
Topologies beyond that of 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
Other equations of state
Sub-exponential expansion rates
Non-zero vorticity : forthcoming article
Future directions
Other equations of state
Sub-exponential expansion rates
Non-zero vorticity : forthcoming article
Future directions
Other equations of state
Sub-exponential expansion rates
Non-zero vorticity : forthcoming article
Thank you
The norms
def
¯ 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
Sg0∗ ;N =
e(q−1)Ω k∂t g0j kH N + e(q−2)Ω k∂g
j=1
def
Sh∗∗ ;N =
3 X
¯ jk kH N−1
¯ jk kH N + k∂h
eqΩ k∂t hjk kH N + e(q−1)Ω k∂h
j,k =1
¯ HN
SN = kewΩ ∂t Φ − Ψ̄0 kH N + e(w−1)Ω k∂Φk
def
def
QN = Sg00 +1;N + Sg0∗ ;N + Sh∗∗ ;N + SN
gµν building block energies
If α > 0, β ≥ 0, and v is a solution to
ˆ g v = αH∂t v + βH 2 v + F ,
then we control solutions to this equation using an energy of the
form
def
2
E(γ,δ)
[v , ∂v ] =
Z
1
+
{−g 00 (∂t v )2 + g ab (∂a v )(∂b v ) − 2γHg 00 v ∂t v + δH 2 v 2 } d 3 x
2 T3
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
Energies for ∂Φ
E02
EN2
1
=
2
def
def
=
Z
(ewΩ ∂t Φ − Ψ̄0 )2 + e2wΩ mjk (∂j Φ)(∂k Φ) d 3 x
T3
E02
X 1Z
+
e2W Ω (∂t ∂α~ Φ)2 + e2wΩ mjk (∂j ∂α~ Φ)(∂k ∂α~ Φ) d 3 x.
2 T3
1≤|~
α|≤N
Equivalence of norms and energies
Lemma
If QN (t) is small enough, then the norms and energies are
equivalent.
The Euler-Einstein system
With R = g αβ Rαβ , Einstein’s field equations are
1
(fluid)
Rµν − Rgµν + Λgµν = Tµν
2
(fluid)
= (ρ + p)uµ uν + pgµν
Tµν
ρ = proper energy density, p = pressure, u = four-velocity
u is future-directed, and
uα u α = −1
The Euler-Einstein system
With R = g αβ Rαβ , Einstein’s field equations are
1
(fluid)
Rµν − Rgµν + Λgµν = Tµν
2
(fluid)
= (ρ + p)uµ uν + pgµν
Tµν
ρ = proper energy density, p = pressure, u = four-velocity
u is future-directed, and
uα u α = −1
The relativistic Euler equations
Consequence
of the Bianchi identities
1
Dµ R µν − 2 Rg µν = 0:
µν
Dµ T(fluid)
=0
(1)
Conservation of particle number:
Dµ (nu µ ) = 0
n = proper number density
(1) and (2) are the relativistic Euler equations.
(2)
The relativistic Euler equations
Consequence
of the Bianchi identities
1
Dµ R µν − 2 Rg µν = 0:
µν
Dµ T(fluid)
=0
(1)
Conservation of particle number:
Dµ (nu µ ) = 0
n = proper number density
(1) and (2) are the relativistic Euler equations.
(2)
The relativistic Euler equations
Consequence
of the Bianchi identities
1
Dµ R µν − 2 Rg µν = 0:
µν
Dµ T(fluid)
=0
(1)
Conservation of particle number:
Dµ (nu µ ) = 0
n = proper number density
(1) and (2) are the relativistic Euler equations.
(2)
Equation of state
To close the Euler system, we need more equations.
Fundamental thermodynamic relation:
ρ+p =n
∂ρ
∂n
Our equation of state:
p = cs2 ρ
r
0 < cs <
cs is the speed of sound
1
3
Equation of state
To close the Euler system, we need more equations.
Fundamental thermodynamic relation:
ρ+p =n
∂ρ
∂n
Our equation of state:
p = cs2 ρ
r
0 < cs <
cs is the speed of sound
1
3
Equation of state
To close the Euler system, we need more equations.
Fundamental thermodynamic relation:
ρ+p =n
∂ρ
∂n
Our equation of state:
p = cs2 ρ
r
0 < cs <
cs is the speed of sound
1
3
Fluid vorticity
where σ ≥ 0.
√
√ def ρ + p
σ=
,
n
σ is known as the enthalpy per particle.
The fluid vorticity is defined by
def
vµν = ∂µ βν − ∂ν βµ ,
where
√
def
βµ = − σuµ .
Fluid vorticity
where σ ≥ 0.
√
√ def ρ + p
σ=
,
n
σ is known as the enthalpy per particle.
The fluid vorticity is defined by
def
vµν = ∂µ βν − ∂ν βµ ,
where
√
def
βµ = − σuµ .
Irrotational fluids
An irrotational fluid is one for which
def
vµν = ∂µ βν − ∂ν βµ = 0
Local consequence:
√
def
βµ = − σuµ = −∂µ Φ
Φ is the fluid potential. We think of ∂Φ existing globally; Φ
itself may only exist locally (Poincaré’s lemma). Only ∂Φ
enters into the equations.
Irrotational fluids
An irrotational fluid is one for which
def
vµν = ∂µ βν − ∂ν βµ = 0
Local consequence:
√
def
βµ = − σuµ = −∂µ Φ
Φ is the fluid potential. We think of ∂Φ existing globally; Φ
itself may only exist locally (Poincaré’s lemma). Only ∂Φ
enters into the equations.
Irrotational fluids
An irrotational fluid is one for which
def
vµν = ∂µ βν − ∂ν βµ = 0
Local consequence:
√
def
βµ = − σuµ = −∂µ Φ
Φ is the fluid potential. We think of ∂Φ existing globally; Φ
itself may only exist locally (Poincaré’s lemma). Only ∂Φ
enters into the equations.
Irrotational fluids
An irrotational fluid is one for which
def
vµν = ∂µ βν − ∂ν βµ = 0
Local consequence:
√
def
βµ = − σuµ = −∂µ Φ
Φ is the fluid potential. We think of ∂Φ existing globally; Φ
itself may only exist locally (Poincaré’s lemma). Only ∂Φ
enters into the equations.
Irrotational fluids
An irrotational fluid is one for which
def
vµν = ∂µ βν − ∂ν βµ = 0
Local consequence:
√
def
βµ = − σuµ = −∂µ Φ
Φ is the fluid potential. We think of ∂Φ existing globally; Φ
itself may only exist locally (Poincaré’s lemma). Only ∂Φ
enters into the equations.
Scalar formulation of irrotational fluid
mechanics
Consequence: for irrotational fluids, the Euler equations
µν
Dµ T(fluid)
= 0, Dµ (nu µ ) = 0 are equivalent to a scalar
quasilinear wave equation that is the Euler-Lagrange
equation corresponding to a Lagrangian L(σ) :
Dα
∂L α
D Φ = 0.
∂σ
L=p
uµ = −σ −1/2 ∂µ Φ
σ = −g αβ (∂α Φ)(∂β Φ)
Scalar formulation of irrotational fluid
mechanics
Consequence: for irrotational fluids, the Euler equations
µν
Dµ T(fluid)
= 0, Dµ (nu µ ) = 0 are equivalent to a scalar
quasilinear wave equation that is the Euler-Lagrange
equation corresponding to a Lagrangian L(σ) :
Dα
∂L α
D Φ = 0.
∂σ
L=p
uµ = −σ −1/2 ∂µ Φ
σ = −g αβ (∂α Φ)(∂β Φ)
Energy-momentum tensor
Corresponding to L, we have the energy-momentum
tensor:
∂L
(∂µ Φ)(∂ν Φ) + gµν L
∂σ
= 0 for solutions to the E-L equation
def
(scalar )
=2
Tµν
µν
Dµ T(scalar
)
For an irrotational fluid,
(scalar )
(fluid)
Tµν
= Tµν
Energy-momentum tensor
Corresponding to L, we have the energy-momentum
tensor:
∂L
(∂µ Φ)(∂ν Φ) + gµν L
∂σ
= 0 for solutions to the E-L equation
def
(scalar )
=2
Tµν
µν
Dµ T(scalar
)
For an irrotational fluid,
(scalar )
(fluid)
Tµν
= Tµν
The case p = cs2ρ
For p = cs2 ρ, s =
1−cs2
; cs2
2cs2
=
1
,
2s+1
it follows that
L = p = (1 + s)−1 σ s+1
(scalar )
Tµν
= 2σ s (∂µ Φ)(∂ν Φ) + gµν (1 + s)−1 σ s+1
The fluid equation:
Dα (σ s D α Φ) = 0
The case p = cs2ρ
For p = cs2 ρ, s =
1−cs2
; cs2
2cs2
=
1
,
2s+1
it follows that
L = p = (1 + s)−1 σ s+1
(scalar )
Tµν
= 2σ s (∂µ Φ)(∂ν Φ) + gµν (1 + s)−1 σ s+1
The fluid equation:
Dα (σ s D α Φ) = 0
Background solution ansatz
e we make the
e, Φ,
To construct a background solution, g
ansatz
e = −dt 2 + a(t)2
g
3
X
(dx k )2 ,
k =1
from which it follows that the only corresponding non-zero
Christoffel symbols are
e
Γj 0k = e
Γk0j = aȧδjk
ȧ
e
Γj k0 = e
Γ0k j = δjk .
a
Background solution
Plugging this ansatz into the Euler-Einstein system, we deduce
that:
2
2
def
ρa3(1+A ) ≡ ρ̄å3(1+A ) = κ0
r
r
Λ ρ
Λ
κ0
ȧ = a
+ =a
+
3 3
3 3a3(1+A2 )
ρ̄ > 0, å = a(0)
r
e
Ω(t)
def
= a(t) ≈ e
−Ht
,
def
H=
Λ
3
e = (∂t Φ,
e 0, 0, 0)
∂Φ
s + 1 1/(2s+2)
3
e = κ0
∂t Φ
e−W Ω , W =
2s + 1
1 + 2s
Wave coordinates
To hyperbolize the Einstein equations, we use a variant of
the wave coordinates:
Γν = e
Γν = 3ωδ 0ν
r
def
ω = ∂t Ω ≈ H =
Λ
3
def
g φ = g αβ Dα Dβ φ
def
ˆ gφ =
g αβ ∂α ∂β φ
In wave coordinates,
ˆ gφ =
g φ = 0 ⇐⇒ 3ω∂t Φ
| {z }
dissipative term
Wave coordinates
To hyperbolize the Einstein equations, we use a variant of
the wave coordinates:
Γν = e
Γν = 3ωδ 0ν
r
def
ω = ∂t Ω ≈ H =
Λ
3
def
g φ = g αβ Dα Dβ φ
def
ˆ gφ =
g αβ ∂α ∂β φ
In wave coordinates,
ˆ gφ =
g φ = 0 ⇐⇒ 3ω∂t Φ
| {z }
dissipative term
Energy vs. norm comparison
Lemma
If Sg00 +1;N + Eg0∗ ;N + Eh∗∗ ;N + SN is sufficiently small, then
C −1 Eg00 +1;N ≤ Sg00 +1;N ≤ CEg00 +1;N
C −1 Eg0∗ ;N ≤ Sg0∗ ;N ≤ CEg0∗ ;N
C −1 Eh∗∗ ;N ≤ Sh∗∗ ;N ≤ CEh∗∗ ;N
C −1 EN ≤ SN ≤ CEN
Integral inequalities
t
Z
e−qHτ QN (τ ) dτ
EN (t) ≤ EN (t1 ) + C
τ =t1
Z
t
−2qHEg00 +1;N (τ ) + Ce−qHτ QN (τ ) dτ
Eg00 +1;N (t) ≤ Eg00 +1;N (t1 ) +
τ =t1
Z
t
−2qHEg0∗ ;N (τ ) + CEh∗∗ ;N (τ ) + Ce−qHτ QN (τ ) dτ
Eg0∗ ;N (t) ≤ Eg0∗ ;N (t1 ) +
τ =t1
Z t
Eh∗∗ ;N (t) ≤ Eh∗∗ ;N (t1 ) +
τ =t1
1 −qHτ
He
Eh∗∗ ;N (τ ) + Ce−qHτ QN (τ ) dτ.
2