The initial value problem in general relativity

Transcription

1 LC Physics Colloquium, Spring 2015

2 Abstract In 1915, Einstein introduced equations describing a theory of gravitation known as general relativity. The Einstein equations, as they are now called, are at once elegant and extremely complicated. Thus it was not until the middle of the 20th century that Yvonne Choquet-Bruhat showed they permit an initial value problem i.e. that if the state of a system is specified at an initial time, then there exists a corresponding solution to the equations specifying the state at a later time. In this talk we first discuss initial value problems in classical physics, before describing important features of the initial value problem in general relativity. We outline some of the challenges in studying the initial value problem, some recent progress, and list some important unsolved problems in this exciting area of research.

3 Shameless advertising Math 490: Curvature of Space & Time Prof. Stavrov MTΘF 11:30-12:20

4 Caveat (emptor) I am a mathematician... mathematical general relativity! Other important topics: Data & observation Numerical simulation Theoretical physics Getting these communities of people talking to one another! Please do not hesitate to ask questions throughout!

5 Initial value problem (IVP) Given state of system now what happens in the future? Ingredients Questions Evolution equations Initial conditions Short-time questions: Existence? Uniqueness, Continual dependence on initial conditions? Global questions: Behavior of solutions?

6 Classical dynamics Dynamical equations d x dt = p Principle of least action Conservation law Minimize tf t i d p dt = V ( x) { } 1 2 p 2 V ( x) dt H = 1 2 p 2 + V ( x) is conserved Free evolution V = 0 d p dt = 0 straight line trajectory

7 Initial value problem theory Fundamental theorem of ODEs For any initial x 0, p 0 there exists a unique short-time solution. Global behavior Determined by conservation of energy H.

8 Example: Simple Harmonic Oscillator Potential p V (x) = 1 2 x 2 x Equations H V dx dt = p dp dt = x x

9 Example: Electromagnetism Equations Energy t E = B 4πJ E = 4πρ t B = E B = 0 H = 1 ( E 2 + B 2) dv 2 Partial differential equations, but linear Constraints: satisfied initially preserved by evolution Energy does not give point-wise control

10 Doing electromagnetism First pass Given a charge configuration, what is E? What is B? Given E and B, what is trajectory of a test particle? Second pass Electromagnetic waves Reformulation using potentials, gauge condition wave equation Initial value problem Given E and B now, how do they evolve? Initial E and B must satisfy constraints Wave equation formulation is mathematically well-behaved

11 General relativity: Spacetime diagrams Particles, fields, etc. all defined in space and time View a space time from vantage point of an observer r me = 0 Dr. S Dr. O t me = 2 t me = 1 t me = 0

12 General relativity: Spacetime diagrams Particles, fields, etc. all defined in space and time View a space time from vantage point of an observer r me = 0 Dr. S Dr. O t K = 0 t me = 2 t me = 1 t me = 0 c = 1 special relativity (Also: G = 1)

13 Classical IVP General relativity IVP in GR General relativity: Geometry, scaling, maps I Which lines are straight? I Metric length scale at each point of map projections/

14 General relativity: Geometry and spacetime Einstein (and Hilbert, Poincare, et al.) Spacetime metric g (spacetime length scale) Gravitational model: particles obey principle of least action, with respect to length determined by g Needs to be same for any observer ( geometric ) Einstein s equation Ric 1 2 R g = 8πT }{{}}{{} Geometry Matter fields Ric, R are notions of curvature, depend on derivatives of g T also involves g

15 General relativity: Simple example a small planet r = 0 Dr. S: crash & burn land on planet t = 2 t = 1 Qualitatively Newtonian dynamics... t = 0

16 General relativity: Simple example a small planet r = 0 Dr. S: crash & burn land on planet Dr. O: angular momentum! t = 2 t = 1 Qualitatively Newtonian dynamics... t = 0

17 General relativity: Large mass black hole region Schwarzchild 1915; r = 2m r = 3m r = 4m region with mass m What about inside?

18 General relativity: Large mass black hole region Schwarzchild 1915; Kruskal 1960 region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m 3m Let s go on an adventure...

19 General relativity: Large mass black hole region Schwarzchild 1915; region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m Dr. S s fate...? 3m

20 General relativity: Large mass black hole region Schwarzchild 1915; region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m 3m

21 Lessons learned from Schwarzchild solution Observations Some coordinate systems behave better than others. Interesting features (e.g. BH) may be regions of space time. Singularities may form; due to non-linearity. Questions Can interesting features form dynamically? Are singularities typically hidden? Weak cosmic censorship conjecture. Need an initial value formulation.

22 Towards an initial value problem Recall initial value problem framework Specify initial conditions ( state at t = 0), satisfying constraints if applicable Use equations to evolve in time (need good formulation/theory) 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?

23 The short-time initial value problem (I) Local perspective (Choquet-Bruhat et al., 1950 s ): Focus on a small region; choose wave-adapted coordinates Einstein s equation becomes a (non-linear) wave equation, which can be solved Patch together little pieces to form a spacetime Wave-like behavior, including gravity waves Maximally-extended nice spacetime

24 The short-time initial value problem (II) Hamiltonian perspective (A.-D.-M. et al., 1960 s ): Choose an arbitrary time function Decompose equations analogous to E&M Clearly illustrates constraint and evolution equations: T g = Nk 0 = R + k 2 (trk) 2 T k = 2 g = k (trk) Conserved quantity: Energy-momentum Ongoing research: Understanding & constructing solutions to constraint equations Ongoing research: Solutions to evolution equations

25 Beyond short-time existence Lot s of fun questions... Isolated systems Singularity formation Black holes & weak cosmic censorship Stability of black holes Cosmology Stability of symmetric models Structure formation

26 Dynamical formation of singularities Incompleteness (Hawking & Penrose; 1970) Expansion θ satisfies dθ dt < 1 3 θ2 1 Thus θ < 1 3 t + θ 1 0 If θ 0 < 0, paths collide. θ 0 > 0 θ 0 < 0 Curvature singularities Understood in some symmetric situations Lots of work yet to be done

27 Formation of BHs & weak cosmic censorship conjecture Generic scenario (?) θ gravitational collapse Singularity BH region Horizon Singularity formation Hidden inside BH region cosmic censorship Outside observer Many examples... few theorems... Preskill-Thorne Hawking bets

28 Stability problems Stability Compare a symmetric solution to small, nearby configurations Important for theoretical and physical reasons Famous results Minkowski space time (Christodoulou-Klainerman) Rapidly expanding space times (Friedrich) Current hot topic Stability of Schwarzschild space time

29 Structure formation Expectations Small inhomogeneities radiated away Large inhomogeneities large-scale structure Known results Linear approximations Lots of heuristics Lots of good data coming in! Early stages... The math is exceedingly difficult new ideas needed!

30 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?

31 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 by Choquet-Bruhat Partial Differential Equations in General Relativity by Rendall Also books by Ellis & Hawking, Wald, etc.