paul t. allen · paul t. allen lewis & clark college willamette physics colloquium, spring 2014...

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



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.




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!




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?




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




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.




Example: Simple Harmonic Oscillator

I Potential

V (x) =1

2x2

I Equations

dx

dt= p

dp

dt= −x

x

p

x

HV




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




“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




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)




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/




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




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



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. . . ?





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. . . ?





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. . . ?





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. . . ?




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.




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?




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




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




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




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




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




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




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!




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?




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 · paul t. allen lewis & clark college willamette physics colloquium, spring 2014...

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