Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The Global Stability of the Minkowski
Spacetime Solution to the Einstein-Nonlinear
Electromagnetic System in Wave Coordinates
Jared Speck
jspeck@math.mit.edu
Department of Mathematics
Massachusetts Institute of Technology
October 12, 2011
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The “Electromagnetic Divergence Problem”
Electric field surrounding an electron: E(Maxwell) = q |r|r̂ 2
R Electrostatic energy = R3 E(Maxwell) |2 dVol
R∞
= 4q 2 π r =0 r12 dr = ∞
Lorentz force: F(Lorentz) = qE(Maxwell) = Undefined
(at the electron’s location)
Analogous difficulties appear in QED
M. Born: “The attempts to combine Maxwell’s
equations with the quantum theory (...) have not
succeeded. One can see that the failure does not lie
on the side of the quantum theory, but on the side of
the field equations, which do not account for the
existence of a radius of the electron (or its finite
energy=mass).”
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The “Electromagnetic Divergence Problem”
Electric field surrounding an electron: E(Maxwell) = q |r|r̂ 2
R Electrostatic energy = R3 E(Maxwell) |2 dVol
R∞
= 4q 2 π r =0 r12 dr = ∞
Lorentz force: F(Lorentz) = qE(Maxwell) = Undefined
(at the electron’s location)
Analogous difficulties appear in QED
M. Born: “The attempts to combine Maxwell’s
equations with the quantum theory (...) have not
succeeded. One can see that the failure does not lie
on the side of the quantum theory, but on the side of
the field equations, which do not account for the
existence of a radius of the electron (or its finite
energy=mass).”
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Towards a possible resolution
Idea (Born, Infeld, Bialynicki-Birula, Gibbons, Carley,
Kiessling, Tahvildar-Zadeh, · · · ): nonlinear field equations
Example: the Born-Infeld model ( =⇒ finite energy
electrons)
Lorentz force remains undefined; Kiessling → new law of
motion for point charges
All physically reasonable models ∼ Maxwell model in the
weak-field limit
No physically reasonable model should destroy the stability of
Minkowski spacetime (in absence of point charges). This is
what my work confirms.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Towards a possible resolution
Idea (Born, Infeld, Bialynicki-Birula, Gibbons, Carley,
Kiessling, Tahvildar-Zadeh, · · · ): nonlinear field equations
Example: the Born-Infeld model ( =⇒ finite energy
electrons)
Lorentz force remains undefined; Kiessling → new law of
motion for point charges
All physically reasonable models ∼ Maxwell model in the
weak-field limit
No physically reasonable model should destroy the stability of
Minkowski spacetime (in absence of point charges). This is
what my work confirms.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The 1 + 3 dimensional Einstein-nonlinear
electromagnetic equations
1
Ricµν − gµν R = Tµν ,
2
(dF)λµν = 0,
(dM)λµν = 0.
gµν is the spacetime metric; Ricµν = Ricµν (g, ∂g, ∂ 2 g) is
def
the Ricci tensor; R = (g −1 )κλ Ricκλ is the scalar curvature
Tµν is the energy-momentum tensor
Fµν is the Faraday tensor
Mµν is the Maxwell tensor
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Previous mathematical results in 1 + 3 dimensions
Fritz John: blow-up for nonlinear wave equations in three
spatial dimensions (1981)
Christodoulou, Klainerman: global existence for nonlinear
wave equations and the null condition (1984)
Friedrich: “stability of the Minkowski spacetime solution” to
the Einstein-vacuum equations for restricted data using the
conformal method (1986)
Christodoulou-Klainerman: “ ” using an invariant framework
(1993)
Zipser: “ ” to the Einstein-Maxwell system (2000)
Lindblad-Rodnianski: “ ” to the Einstein-scalar field system
in wave coordinates (2005, 2010)
Loizelet: “ ” to the Einstein-scalar field-Maxwell system in
Lorenz-gauge + wave coordinates (2008)
Bieri: “ ” to the Einstein-vacuum equations for slowly
decaying data (2009)
JS: “ ” to the Einstein-nonlinear electromagnetic system in
wave coordinates (2010)
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The initial data on Σ0 = R3
g̊ = Riemannian metric
K̊ = symmetric type 02 tensor
D̊ = electric displacement
B̊ = magnetic induction
Constraint equations (j = 1, 2, 3)
2
R̊ − K̊ab K̊ ab + (g̊ −1 )ab K̊ab = 2T (N̂, N̂)|Σ0 ,
∂
(g̊ −1 )ab D̊ a K̊bj − (g̊ −1 )ab D̊ j K̊ab = T (N̂, j )|Σ0 ,
∂x
(g̊ −1 )ab D̊ a D̊b = 0,
(g̊ −1 )ab D̊ a B̊b = 0
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Positive mass theorem of Schoen-Yau, Witten
Decay assumptions on the data (κ > 0, M > 0)
(0)
(1)
• g̊ jk = δjk + h̊jk + h̊jk ,
(0)
2M
δjk ,
r
= o(r −1−κ ),
• h̊jk = χ(r )
(1)
• h̊jk
• K̊jk = o(r −2−κ ),
• D̊j = o(r
−2−κ
),
• B̊j = o(r −2−κ ),
as r → ∞,
as r → ∞,
as r → ∞,
as r → ∞
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Splitting the spacetime metric
gµν
Minkowski metric
z}|{
=
mµν
+hµν
(0)
(1)
hµν
+
hµν
|{z}
|{z}
Schwarzschild tail remainder piece
= χ rt χ(r ) 2M
r δµν
hµν =
(0)
hµν
χ ∈ C ∞,
χ ≡ 1 for z ≥ 3/4,
χ ≡ 0 for z ≤ 1/2.
Note: χ0 ( rt ) is supported in the interior region
{[1/2 ≤ r /t ≤ 3/4]}
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Assumptions on the electromagnetic Lagrangian
1
L = − (1) +O(|((1) , (2) )|2 )
| 2{z }
?
?L
def
(A1)
(Maxwell)
(1) = 21 (g −1 )κµ (g −1 )λν Fκλ Fµν ' “|B|2 − |E|2 , ”
def
(2) = 41 (g −1 )κµ (g −1 )λν Fκλ ?F µν = 18 #κλµν Fκλ Fµν '
“E · B”
Tκλ X κ Y λ ≥ 0
(Dominant energy condition)
(A2)
whenever
X , Y are both timelike (i.e., gκλ X κ X λ < 0, gκλ Y κ Y λ < 0)
X , Y are future-directed
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Example: the Born-Infeld Lagrangian
L (BI) = 1 −
?
q
1 + (1) − 2(2)
1
= − (1) +O(|((1) , (2) )|2 )
| 2{z }
?L
(Maxwell)
Null frame
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The Maxwell tensor and the energy-momentum tensor
• Mµν = 2
=
∂ ?L ?
∂ ?L
F µν −
Fµν
∂(1)
∂(2)
∗
+ O(|h||F|) + O(|F|3 ; h),
F µν
|{z}
Minkowskian Hodge dual
∂ ?L −1 κλ
∂ ?L
(g ) Fµκ Fνλ − (2)
gµν + gµν ?L
∂(1)
∂(2)
1
= (m−1 )κλ Fµκ Fνλ − mµν (m−1 )κη (m−1 )λζ Fκλ Fηζ
4
|
{z
}
• Tµν = −2
Minkowskian/linear energy-momentum tensor
+ O(|h||F|2 ) + O(|F|3 ; h)
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
The wave coordinate gauge
Wave coordinates (à la Choquet-Bruhat &
Lindblad-Rodnianski):
• (g −1 )κλ Γκµλ = 0
1
e g gµν + Nµν (g, ∇g)
=⇒ Ricµν = − 2
2
e g = (g −1 )κλ ∇κ ∇λ
2
∇ = Minkowski connection
Null frame
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The reduced equations
(1)
(0)
e g hµν
e g hµν
2
= Hµν − 2
,
∇λ Fµν + ∇µ Fνλ + ∇ν Fλµ = 0,
N #µνκλ ∇µ Fκλ = Iν .
(1;h)
(2;h)
• Hµν = P(∇µ h, ∇ν h) + Qµν
(∇h, ∇h) + Qµν
(F, F) + H4
µν ,
ν
ν
• Iν = Q(2;F
) (∇h, F) + I4 ,
1 ∂ 2?L
1 ∂ ?L #µνκλ
+
2 ∂Fµν ∂Fκλ 2 ∂(2)
1 n −1 µκ −1 νλ
(m ) (m ) − (m−1 )µλ (m−1 )νκ
=
2
• N #µνκλ = −
o
#µνκλ
− 2hµκ (m−1 )νλ − 2(m−1 )µκ hνλ + N4
.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The energy of the reduced solution
•
2
E`;γ;µ
(t)
def
= sup
0≤τ ≤t
XZ
|I|≤` Στ
n
o
|∇∇IZ h(1) |2 + |LIZ F|2 w(q) d 3 x,
def
q = |x| − t,
• w = w(q) =
1 + (1 + |q|)1+2γ , if q > 0,
,
1 + (1 + |q|)−2µ , if q < 0,
0 < γ < 1/2, 0 < µ < 1/2,
∂
∂
∂ def ∂
, xµ ν − xν µ , x κ κ 0≤µ<ν≤3
• Z=
µ
∂x
∂x
∂x
|
{z∂x
}
11 Minkowskian conformal Killing fields
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Continuation principle
Lemma
If the solution blows up at time Tmax > 0, then one of the
following blow-up scenarios must occur:
The hyperbolicity of the equations breaks down as t ↑ Tmax
limt↑Tmax E`;γ;µ (t) = ∞
Moral conclusion: You can show global existence by proving
that the energy E`;γ;µ (t) never blows up.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Main results
Theorem
Let ` ≥ 8 and assume the wave coordinate condition. There
exists a constant ε` > 0 such that if E`;γ;µ (0) + M ≤ ε ≤ ε` , then
the reduced solution exists for (t, x) ∈ (−∞, ∞) × R3 and the
resulting spacetime is geodesically complete. Furthermore,
there exist constants c` > 0, e
c` > 0 such that
E`;γ;µ (t) ≤ c` ε(1 + |t|)ec` ε
for t ∈ (−∞, ∞).
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The bootstrap argument
Goal: assume that there exist constants 0 < ε and 0 < δ < 1/4
such that for t ∈ [0, T )
• E`;γ;µ (t) ≤ ε(1 + t)δ ,
0 ≤ k ≤ `,
and prove that for 0 ≤ k ≤ ` and t ∈ [0, T ) :
Z t
2
(0) + M 2 + ε3 + ε (1 + τ )−1 Ek2;γ;µ (τ ) dτ
Ek2;γ;µ (t) . E`;γ;µ
0
Z t
+ ε (1 + τ )−1+Cε Ek2−1;γ;µ (τ ) dτ
0
Conclusion (Gronwall’s
lemma): 2
2
E`;γ;µ (t) ≤ c` E`;γ;µ (0) + M 2 + ε3 (1 + t)ec` ε
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Fundamental difficulty
The behavior of Ek2;γ;µ (t) is not directly accessible. We only
know how to study Ek2;γ;µ (t) through the use of auxiliary
quantities, e.g. energy currents.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The EOV and the canonical stress
The equations of variation (EOV)
for us
z}|{
= İλµν ,
0
∇λ Ḟµν + ∇µ Ḟνλ + ∇ν Ḟλµ
N #µνκλ [h, F]∇µ Ḟκλ = İν
• To estimate F, we can use (g −1 )κλ Dκ Tµλ = 0.
• But how do we obtain useful differential identities for Ḟ?
• Answer: the canonical stress tensor (Christodoulou)
1
def
Q̇ µν = N #µζκλ Ḟκλ Ḟνζ − δνµ N #ζηκλ Ḟζη Ḟκλ
4
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The EOV and the canonical stress
The equations of variation (EOV)
for us
z}|{
= İλµν ,
0
∇λ Ḟµν + ∇µ Ḟνλ + ∇ν Ḟλµ
N #µνκλ [h, F]∇µ Ḟκλ = İν
• To estimate F, we can use (g −1 )κλ Dκ Tµλ = 0.
• But how do we obtain useful differential identities for Ḟ?
• Answer: the canonical stress tensor (Christodoulou)
1
def
Q̇ µν = N #µζκλ Ḟκλ Ḟνζ − δνµ N #ζηκλ Ḟζη Ḟκλ
4
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The EOV and the canonical stress
The equations of variation (EOV)
for us
z}|{
= İλµν ,
0
∇λ Ḟµν + ∇µ Ḟνλ + ∇ν Ḟλµ
N #µνκλ [h, F]∇µ Ḟκλ = İν
• To estimate F, we can use (g −1 )κλ Dκ Tµλ = 0.
• But how do we obtain useful differential identities for Ḟ?
• Answer: the canonical stress tensor (Christodoulou)
1
def
Q̇ µν = N #µζκλ Ḟκλ Ḟνζ − δνµ N #ζηκλ Ḟζη Ḟκλ
4
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Expansion of the canonical stress
Q̇ µν
linear energy-momentum tensor in flat spacetime
z
}|
{
1
=
Ḟ µζ Ḟνζ − δνµ Ḟζη Ḟ ζη
4
corrections to linear theory arising from h
z
}|
{
1
η
−hµκ Ḟκζ Ḟνζ − hκλ Ḟ µκ Ḟνλ + δνµ hκλ Ḟκη Ḟλ
2
1 µ #ζηκλ
#µζκλ
+ N4
Ḟκλ Ḟνζ − δν N4
Ḟζη Ḟκλ .
4
|
{z
}
quartic error terms
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
An expression for ∇µ Q̇ µν
Lemma
If Ḟµν verifies the equations of variation with inhomogeneous
terms İνκλ and İη , then
1
∇µ Q̇ µν = − N #ζηκλ Ḟζη İνκλ + Ḟνη İη
2
1
− (∇µ hµκ )Ḟκζ Ḟνζ − (∇µ hκλ )Ḟ µκ Ḟνλ + (∇ν hκλ )Ḟκη Ḟλη
2
1
#µζκλ
#ζηκλ
+ (∇µ N4
)Ḟκλ Ḟνζ − (∇ν N4
)Ḟζη Ḟκλ
4
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
A current for Ḟ
Set X ν = w(q)δ0ν and define
µ
µ
µ ν
J̇(h,F
) [Ḟ] = −Q̇ ν X = −w(q)Q̇ 0
def
def
w(q) =
1 + (1 + |q|)1+2γ , if q > 0,
1 + (1 + |q|)−2µ , if q < 0,
0 < γ < 1/2, 0 < µ < 1/2.
Lemma
0
2
J̇(h,F
) [Ḟ] ≈ |Ḟ| w(q).
Null frame
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
The divergence theorem
Lemma
Let Ḟµν be a solution to the electromagnetic equations of
variation with inhomogeneous terms İν . Assume that “h . ε” is
suitably small. Then
Z
Σt
2
3
|Ḟ| w(q) d x +
.
Z
2
Σ0
+
0
3
Στ
|Ḟ| w(q) d x + ε
Z tZ
0
Z tZ
Στ
|α̇|2 + ρ̇2 + σ̇ 2 w 0 (q) d 3 x dτ
Z tZ
0
Στ
|Ḟ0κ İκ |w(q) d 3 x dτ
|Ḟ|2
w(q) d 3 x dτ
1+τ
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Weighted Klainerman-Sobolev inequality (with
def
q = |x| − t)
Lemma
(1 + t + |x|)[(1 + |q|)w(q)]1/2 |φ(t, x)| .
1/2
(1 + t + |x|)[(1 + |q|)w(q)]
2
1/2
(1 + t + |x|) [(1 + |q|)w(q)]
|F(t, x)| .
|∇φ(t, x)| .
X
w 1/2 ∇IZ φ(t, ·) 2 ,
L
|I|≤2
X
w 1/2 LIZ F(t, ·) 2 ,
L
|I|≤2
X
w 1/2 ∇IZ φ(t, ·) 2
L
|I|≤3
Not sufficient to close the estimates; we need to upgrade
these inequalities using the special null structure of the
equations.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Weighted Klainerman-Sobolev inequality (with
def
q = |x| − t)
Lemma
(1 + t + |x|)[(1 + |q|)w(q)]1/2 |φ(t, x)| .
1/2
(1 + t + |x|)[(1 + |q|)w(q)]
2
1/2
(1 + t + |x|) [(1 + |q|)w(q)]
|F(t, x)| .
|∇φ(t, x)| .
X
w 1/2 ∇IZ φ(t, ·) 2 ,
L
|I|≤2
X
w 1/2 LIZ F(t, ·) 2 ,
L
|I|≤2
X
w 1/2 ∇IZ φ(t, ·) 2
L
|I|≤3
Not sufficient to close the estimates; we need to upgrade
these inequalities using the special null structure of the
equations.
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Heuristics
Moral reason for stability: the “worst possible”
quadratic terms are absent from the equations
weak null condition
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Important surfaces in Minkowski space
t ∈ R, x ∈ R3
Cs− = {(τ, y ) | |y | + τ = s} are the ingoing null cones
def
def
Cq+ = {(τ, y ) | |y | − τ = q} are the outgoing null cones
def
Σt = {(τ, y ) | τ = t} are the constant time slices
def
Sr ,t = {(τ, y ) | τ = t, |y | = r } are the Euclidean spheres
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
The Minkowskian null frame
t
L = ∂t − ∂ r
t−
L = ∂t + ∂ r
r=
co
ns t
angular directions
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Minkowskian null frame and null coordinates
def
def
def
Null frame: N = {L, L, e1 , e2 }, T = {L, e1 , e2 }, L = {L}
def
L = ∂t − ∂r is tangent to the ingoing cones
def
L = ∂t + ∂r is tangent to the outgoing cones
e1 , e2 are orthonormal, & tangent to the spheres
Null coordinates (useful for expressing decay rates)
q = r − t (constant on outgoing cones)
s = r + t (constant on ingoing cones)
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Null decompositions
def
With 6 mµν = mµν +
1
2
def
Lµ Lν + Lµ Lν , 6 υ µν = 12 [µνκλ]Lκ Lλ ,

def
αµ =6 mµν Fνλ Lλ = BAD,



def

αµ =6 mµν Fνλ Lλ = good,
The 6 components of F
def

ρ
= 12 Fκλ Lκ Lλ = good,



def
σ
= 21 6 υ κλ Fκλ = good.
6 mµν projects m−orthogonally onto the Sr ,t
π µν projects m−orthogonally onto the Cq+
def
∇µ = π µκ ∇κ are the good derivatives
Upgraded decay
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Upgraded decay estimates via wave coordinates
Null p
decompose the wave coordinate condition
|det g|(g −1 )µν = 0 to obtain (Lindblad-Rodnianski)
∇µ
|∇h|LT . |∇h| + |h||∇h|
Conclusion: |h|LT decays better than “expected”
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Upgraded decay estimates for F
def
def
With αµ =6 mµν Fνλ Lλ , Λ = L + 14 hLL L,
Null decomposing the electromagnetic equations =⇒
r −1 |∇Λ (r α)| . r −1 |h|LL |α| +
+
X
|I1 |+|I2 |≤1
+
X
|I|≤1
r
−1
X
|I|≤1
r −1 |LIZ F|LN + |LIZ F|T T
|∇IZ1 h||LIZ2 F|
(1 + |q|)−1 |h| |LIZ F|LN + |LIZ F|T T
+ cubic error terms.
Conclusion: r α satisfies an “ODE” with favorable sources
Integration yields |α| . (1 + t)−1
Klainerman-Sobolev yields only |α| . (1 + t)−1+δ
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Post-upgraded-decay-estimate integral inequalities
XZ tZ
|I|≤k
0
Στ
.ε
n
o
I
(LZ F0ν ) N #µνκλ ∇µ LIZ Fκλ − L̂IZ N #µνκλ ∇µ Fκλ w(q) d 3 x dτ
XZ tZ
|I|≤k
+ ε
0
|
+ ε
0
Στ
(1 + τ )−1 |∇[∇IZ h(1) ]|2 w(q) d 3 x dτ
XZ tZ
|I|≤k
+ ε
(1 + τ )−1 |LIZ F|2 w(q) d 3 x dτ
XZ tZ
|I|≤k
+ ε
Στ
X
0
n
o
|∇[∇IZ h(1) ]|2 + |LIZ F|2LN + |LIZ F|2T T w 0 (q) d 3 x dτ
Στ
Z tZ
|J 0 |≤k −2 0
Στ
X Z tZ
|J|≤k −1 0
0
Στ
(1 + τ + |q|)−1+Cε |∇[∇JZ h(1) ]|2 w(q)d 3 x dτ
{z
absent if k = 1
(1 + τ + |q|)−1+Cε |LJZ F|2 w(q)d 3 x dτ + ε3
}
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Null frame
Upgraded decay
Future directions
Introduction to the theory of regularly hyperbolic PDEs
(with Willie Wong)
Formation of singularities in nonlinear electromagnetic
equations
Intro
Einstein equations
Reduced equations
Main theorem
EOV
Decay estimates
Thank you
Null frame
Upgraded decay