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Classical IVP General relativity IVP in GR
The initial value problem in general relativity
Paul T. Allen
Lewis & Clark College
Willamette Physics Colloquium, Spring 2014
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
AbstractIn 1915, Einstein introduced equations describing a theory of
gravitation known as general relativity. The Einstein equations, asthey are now called, are at once elegant and extremelycomplicated. Thus it was not until the middle of the 20th centurythat Yvonne Choquet-Bruhat showed they permit an initial valueproblem – i.e. that if the state of a system is specified at an initialtime, then there exists a corresponding solution to the equationsspecifying the state at a later time.
In this talk we first discuss initial value problems in classicalphysics, before describing important features of the initial valueproblem in general relativity. We outline some of the challenges instudying the initial value problem, some recent progress, and listsome important unsolved problems in this exciting area of research.
This talk is to be accessible to any student who has completedthe introductory physics courses PHYS 221-222.
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Caveat (emptor)
I I am a mathematician. . . mathematical general relativity!I Other important topics:
I Data & observationI Numerical simulationI Theoretical physicsI Getting these communities of people talking to one another!
I Please do not hesitate to ask questions throughout!
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Initial value problem (IVP)
Given “state of system now” what happens in the future?
Ingredients
I Evolution equationsI Initial conditions
Questions
I Short-time questions: Existence? Uniqueness,Continual dependence on initial conditions?
I Global questions: Behavior of solutions?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Classical dynamics
Dynamical equations
d~x
dt= ~p
d~p
dt= −~∇V (~x)
Principle of least action
Minimize
∫ tf
ti
{1
2|~p|2 − V (~x)
}dt
Conservation law
H =1
2|~p|2 + V (~x) is conserved
Free evolution
V = 0 d~p
dt= 0 straight line trajectory
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Initial value problem theory
Fundamental theorem of ODEsFor any initial ~x0, ~p0 there exists a unique short-timesolution.
Global behaviorDetermined by conservation of energy H.
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Example: Simple Harmonic Oscillator
I Potential
V (x) =1
2x2
I Equations
dx
dt= p
dp
dt= −x
x
p
x
HV
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Example: Electromagnetism
Equations
∂
∂tE = ∇× B − 4πJ ∇ · E = 4πρ
∂
∂tB = −∇× E ∇ · B = 0
Energy
H =
∫∫∫1
2
(|E |2 + |B|2
)dV
I Partial differential equations, but linear
I Constraints: satisfied initially preserved by evolution
I Energy does not give point-wise control
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
“Doing” electromagnetism
First pass (222, 345,. . . )
I Given a charge configuration, what is E ? What is B?
I Given E and B, what is trajectory of a test particle?
Second pass (222, 342, 345,. . . )
I Electromagnetic waves
I Reformulation using potentials, gauge condition waveequation
Initial value problem
I Given E and B now, how do they evolve?
I Initial E and B must satisfy constraints
I Wave equation formulation is mathematically well-behaved
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Space time diagramsI Particles, fields, etc. all defined in a space timeI View a space time from vantage point of an observer
tme = 0
tme = 1
tme = 2
rme = 0 Dr. W Dr. K
tK = 0
I c = 1 special relativity (Also: G = 1)
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Space time diagramsI Particles, fields, etc. all defined in a space timeI View a space time from vantage point of an observer
tme = 0
tme = 1
tme = 2
rme = 0 Dr. W Dr. K
tK = 0
I c = 1 special relativity (Also: G = 1)
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Geometry, scaling, maps
I Which lines are ‘straight’?
I Metric ↔ length scale at each pointhttp://en.wikipedia.org/wiki/List of map projections/
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Geometry and space time
Einstein (and Hilbert, Poincare, et al.)
I Space-time metric g (space-time length scale)
I Gravitational model: particles obey principle of least action,with respect to length determined by g
I Needs to be same for any observer (“geometric”)
Einstein’s equation
Ric − 1
2R g︸ ︷︷ ︸
Geometry
= 8πT︸︷︷︸Matter fields
I Ric , R are notions of curvature
I T also involves g
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Simple example – a small planet
t? = 0
t? = 1
t? = 2
r? = 0
Dr. W: crash & burn
Dr. K: angular momentum!
Qualitatively Newtonian dynamics. . .Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Simple example – a small planet
t? = 0
t? = 1
t? = 2
r? = 0
Dr. W: crash & burn
Dr. K: angular momentum!
Qualitatively Newtonian dynamics. . .Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Large mass black hole region
Schwarzchild 1915;
Kruskal 1960
regi
onw
ith
mas
sm
r F=
2m
r F=
3m
r F=
4m
r F=
m
m
2m
3m
4m
What about inside?
Let’s go on an adventure. . .Dr. W’s fate. . . ?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Large mass black hole region
Schwarzchild 1915; Kruskal 1960
regi
onw
ith
mas
sm
r F=
2m
r F=
3m
r F=
4m
r F=
m
m
2m
3m
4m
What about inside?
Let’s go on an adventure. . .
Dr. W’s fate. . . ?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Large mass black hole region
Schwarzchild 1915;
Kruskal 1960
regi
onw
ith
mas
sm
r F=
2m
r F=
3m
r F=
4m
r F=
m
m
2m
3m
4m
What about inside?Let’s go on an adventure. . .
Dr. W’s fate. . . ?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
General relativity: Large mass black hole region
Schwarzchild 1915;
Kruskal 1960
regi
onw
ith
mas
sm
r F=
2m
r F=
3m
r F=
4m
r F=
m
m
2m
3m
4m
What about inside?Let’s go on an adventure. . .Dr. W’s fate. . . ?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Lessons learned from Schwarzchild solution
Observations
I Some coordinate systems behave better than others.
I Interesting features (e.g. BH) may be regions of space time.
I Singularities may form; due to non-linearity.
Questions
I Can interesting features form dynamically?
I Are singularities typically ‘hidden’?Weak cosmic censorship conjecture.
Need an initial value formulation.
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Towards an initial value problem
Recall initial value problem framework
I Specify initial conditions (“state” at t = 0), satisfyingconstraints if applicable
I Use equations to evolve in time (need goodformulation/theory)
I Verify constraints are preserved by evolution
General relativity
? Which time coordinate should we use?
? What if we choose another time coordinate?
? Are there constraints?
? Are the equations even solvable from an IVP perspective?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
The short-time initial value problem (I)
Local perspective (Choquet-Bruhat et al., 1950’s→):
I Focus on a small region; choose ‘wave-adapted’coordinates
I Einstein’s equation becomes a (non-linear) waveequation, which can be solved
I Patch together little pieces to form a spacetime Wave-like behavior, including gravity waves Maximally-extended “nice” spacetime
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
The short-time initial value problem (II)
Hamiltonian perspective (A.-D.-M. et al., 1960’s→):I Choose an ‘arbitrary’ time functionI Decompose equations analogous to E&MI Clearly illustrates constraint and evolution
equations:
∂
∂Tg = Nk 0 = R + |k |2 − (trk)2
∂
∂Tk = ∇2g + . . . 0 = ∇ · k −∇(trk)
I Conserved quantity: Energy-momentum Ongoing research: Understanding &
constructing solutions to constraint equations Recent work: Solutions to evolution equations
using special coordinates
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Beyond short-time existence
Lot’s of fun questions. . .
Isolated systems
I Singularity formationI Black holes & weak cosmic censorshipI Stability of black holes
Cosmology
I Stability of symmetric modelsI Structure formation
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Dynamical formation of singularities
Incompleteness (Hawking & Penrose; 1970)
I Expansion θ satisfies
dθ
dt< −1
3θ2
I Thus θ <1
13 t + θ−1
0I If θ0 < 0, paths collide.
θ0 > 0 θ0 < 0
Curvature singularities
I Understood in some symmetric situations
I Lots of work yet to be done
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Formation of BHs & weak cosmic censorship conjecture
‘Generic’ scenario (?)
θ → −∞ ↔ gravitationalcollapse
Singularity formation
‘Hidden’ inside BH region
‘cosmic censorship’Outside observer
Horizo
n
BH region
Sin
gula
rity
I Many examples. . . few theorems. . .
I Preskill-Thorne – Hawking bets
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Stability problems
Stability
I Compare a symmetric solution to ‘small, nearby’configurations
I Important for theoretical and physical reasons
Famous results
I Minkowski space time (Christodoulou-Klainerman)
I Rapidly expanding space times (Friedrich)
Current hot topic
I Stability of Schwarzschild space time
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Structure formation
Expectations
I Small inhomogeneities radiated away
I Large inhomogeneities large-scale structure
Known results
I Linear approximations
I Lots of heuristics
I Lots of good data coming in!
Early stages. . .The math is exceedingly difficult new ideas needed!
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Concluding remarks
General relativity is complicated. . . and fascinating!
We know many things. . .
. . . and a lot remains to be done!
Thank you for your attention.
Questions?
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity
Classical IVP General relativity IVP in GR
Resources
Introductory books
General Relativity by Woodhouse
A First Course in General Relativity by Schutz
Also books by Carroll, Hartle, etc.
More advanced
General Relativity and the Einstein Equations byChoquet-Bruhat
Partial Differential Equations in General Relativity by Rendall
Also books by Ellis & Hawking, Wald, etc.
Paul T. Allen Lewis & Clark College
The initial value problem in general relativity