on the mathematical framework in general relativity · 12/4/2012 · cauchy problem in general...
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On the mathematical frameworkin general relativity
Annegret Burtscher12
1Faculty of MathematicsUniversity of Vienna
2Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)
Colloquium for Master and PhD studentsDec 4, 2012
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical Overview
Classical mechanics (Newton)static geometric backgroundnot accurate on large-scale
Einstein’s theory of relativity 1915dynamic "spacetime"describes astronomical observations accurately
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Eddington’s experiment (deflection of light)
dynamic spacetime
→ matter deforms thegeometry ofspacetime
→ the geometry ofspacetimesdetermines howmatter moves
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
General relativity
Key idea of general relativity
Gravitation is not a force but a geometric property ofspace and time.
Description via the Einstein equations.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Manifolds
ManifoldsM are the main object in differential geometry. Theyare equipped with
topologylocal chartscompatibility
Examples: open subsetsof Rn, sphere ...
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Manifolds
ManifoldsM are the main object in differential geometry. Theyare equipped with
topologylocal chartscompatibility
Examples: open subsetsof Rn, sphere ...
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Manifolds
ManifoldsM are the main object in differential geometry. Theyare equipped with
topologylocal chartscompatibility
Examples: open subsetsof Rn, sphere ...
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Tangent space
Let p be a point of themanifoldM.
A tangent space TpM isa real vector spaceattached to p.
The tangent bundleTM is the disjoint union⋃
TpM of all tangentspaces.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Tensor fields
An (r , s) tensor field A ∈ T rs (M) onM defines a multilinear
map
A(p) : TpM∗ × . . .× TpM∗︸ ︷︷ ︸r−times
×TpM× . . .× TpM︸ ︷︷ ︸s−times
→ R
at each point p ∈M.
Examples: T 10 (M) are vector fields, T 0
1 (M) are 1-forms
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Lorentzian metric
We smoothly assign to each point a scalar product on thetangent space:
Lorentzian metricA Lorentzian metric tensor g on a manifoldM is asymmetric, non-degenerate (0,2)-tensor field onM with constant index 1.
Example on Rn+1: 〈vp,wp〉 = −v0w0 +∑n
i=1 v iw i
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Curvature (short)
For each Lorentzian manifoldthere exists the uniqueLevi-Civita connection ∇(covariant derivative).Parallel transport isdescribed in terms of covariantderivation.Failure of commutativity is ameasure for curvature.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Curvature (exact)
DefinitionLetM be a Lorentzian manifold with Levi-Civita connection ∇.The (1,3)-tensor field R, defined by
RXY (Z ) := ∇[X ,Y ]Z − [∇X ,∇Y ]Z
is called Riemann curvature tensor.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Ricci and scalar curvature
Contractions of Riemannian curvature yield simpler invariants:
Definition
Ricci curvature Ric is the C13 contraction of R: Rij =
∑Rm
ijm
Definition
Scalar curvature S is the contraction of Ric: S =∑
g ijRij
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Model
The Einstein equations give a relation between thecurvature of spacetime and the matter distributionof the universe
modeled by a 4-dimensional Lorentzian manifold (M,g)
gravitation is an effect of the curvature ofMmatter distribution is given by the energy-momentumtensor T
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Adaption
Conservation of energy: divT = 0but: divRic 6= 0
DefinitionThe Einstein tensor of a spacetime is
G = Ric− 12
Sg.
now: divG = 0
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Lorentzian geometryGeneral relativity
Einstein equations
Einstein equations
G = 8πT
General relativity is the study of the solutions of this system ofequations – a system of coupled nonlinear partialdifferential equations.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Historical Overview
1915: Einstein introduced his equations
1952: Choquet-Bruhat proved that Einstein equations can beformulated as an initial value problem and showedlocal existence of solutions
later: improvements on regularity
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Foliation
Initial data should consist ofa 3-dim. manifold Σ
with a Riemannian metric hand a symmetric tensor field Kon Σ
... but this is not enough!
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Initial data constraints
need compatibility of the curvature inM and the curvature in Σ
Gauß and Codazzi equations + (vacuum) Einstein equations→
S(h) = |K|2h − (trhK)2
∇jK jk −∇kK j
j = 0
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Solving the constraint equations
The constraint equations can be decomposed in different waysand transformed to elliptic equations and solved.
→ get initial data (non-unique!)
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Gauge fixing
By fixing coordinates, the Einstein equations can be simplified:
e.g. harmonic coordinates xµ such that �gxµ = 0
→ can write (vacuum) Einstein equations as a system ofquasilinear hyperbolic partial differential equations
−12
∑α,β
gαβ∂α∂βgµν + Fµν(g, ∂g) = 0
→ existence result using standard theory
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Historical overviewInitial data constraintsReduced Einstein equations
Local existence
TheoremSuppose h and K are smooth on Σ and the constraints aresatisfied, then the initial value problem has a smooth solution inthe neighborhood of Σ.
generalizations: h ∈ Hsloc(Σ), K ∈ Hs−1
loc (Σ) for s > 2(Klainerman–Rodnianski 2005)
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Global existence?
Can solutions be extendedglobally?
Intuitively: a singularity is aplace where the curvatureof spacetime becomesinfinite
→ but these points are in factmissing from the solution
→ spacetime is incomplete insome sense
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Singularity theorems
Theorem (Hawking and Penrose, 1970)
Spacetime (M,g) is not timelike and null geodesicallycomplete if:
RαβVαV β ≥ 0 for every non-spacelike vector VA generic condition for tangent vectors holds.There are no closed timelike curves.There exists at least one of the following: a compactachronal set without edges, a closed trapped surface, or apoint p such that null geodesics from p are focussed by thematter or curvature and start to reconverge.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Breakdown criteria
Theorem (Klainerman and Rodnianski, 2010)
Let (M,g) be a globally hyperbolic development of Σ foliatedby the CMC level hypersurfaces of a time function t < 0, suchthat Σ corresponds to the level surface t = t0. Assume that Σsatisfies the specific metric inequality. Then the first time T < 0,with respect to the t-foliation, of a breakdown is characterizedby the condition
lim supt→T−
(‖K(t)‖L∞ + ‖∇ log n(t)‖L∞) =∞.
More precisely the spacetime together with the foliation Σt canbe extended beyond any value T < 0 for which the above valueis finite.
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Outline
1 Motivation
2 BackgroundLorentzian geometryGeneral relativity
3 Cauchy problem in general relativityHistorical overviewInitial data constraintsReduced Einstein equations
4 Global results and recent developments
5 Summary
A. Burtscher On the mathematical framework in general relativity
MotivationBackground
Cauchy problem in general relativityGlobal results and recent developments
Summary
Summary
General relativity describes gravitation in terms ofcurvature.
Solutions to the Einstein equations with appropriate initialdata exist locally.
Global behavior is determined by the formation ofsingularities.
A. Burtscher On the mathematical framework in general relativity
Appendix For Further Reading
For Further Reading
S. Hawking and Ellis.The large scale structure of space-time.Cambridge University Press, 1973.
B. O’Neill.Semi-Riemannian Geometry (with applications to relativity).
Academic Press, 1983.
S. Klainerman and I. Rodnianski.On the breakdown criterion in general relativity.J. Amer. Math. Soc. 23 (2010), 345-382.
A. Burtscher On the mathematical framework in general relativity