On the mathematical framework in general relativity · 12/4/2012 · Cauchy problem in general relativity Global results and recent developments Summary Historical overview Initial

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



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1 Motivation




5 Summary




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Historical Overview

Classical mechanics (Newton)static geometric backgroundnot accurate on large-scale

Einstein’s theory of relativity 1915dynamic "spacetime"describes astronomical observations accurately




Summary

Eddington’s experiment (deflection of light)

dynamic spacetime

→ matter deforms thegeometry ofspacetime

→ the geometry ofspacetimesdetermines howmatter moves




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General relativity

Key idea of general relativity

Gravitation is not a force but a geometric property ofspace and time.

Description via the Einstein equations.




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Lorentzian geometryGeneral relativity

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1 Motivation




5 Summary




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Manifolds

ManifoldsM are the main object in differential geometry. Theyare equipped with

topologylocal chartscompatibility

Examples: open subsetsof Rn, sphere ...




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Manifolds







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Manifolds







Summary


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.




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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




Summary


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




Summary


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.




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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.




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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




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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




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Adaption

Conservation of energy: divT = 0but: divRic 6= 0

DefinitionThe Einstein tensor of a spacetime is

G = Ric− 12

Sg.

now: divG = 0




Summary


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.




Summary

Historical overviewInitial data constraintsReduced Einstein equations

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1 Motivation




5 Summary




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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




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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!




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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




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Solving the constraint equations

The constraint equations can be decomposed in different waysand transformed to elliptic equations and solved.

→ get initial data (non-unique!)




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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




Summary


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)




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1 Motivation




5 Summary




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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




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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.




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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.




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Outline

1 Motivation




5 Summary




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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.


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.


Documents

On the mathematical framework in general relativity · 12/4/2012 · Cauchy problem in general relativity Global results and recent developments Summary Historical overview Initial