General relativity and the Einstein equations

Transcription

1 April 23, 2013

2 Special relativity 1905 Let S and S be two observers moving with velocity v relative to each other along the x-axis and let (t, x) and (t, x ) be the coordinate systems used by these observers. Before 1905 almost no one doubted that: t = t, x = x vt. t = γ(t vx c 2 ), x = γ(x vt). Here c is the speed of light and γ = 1. 1 v2 c 2

3 Special relativity This theory unifies space and time. Hermann Minkowski 1908: Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Thus Space and Time Spacetime In special relativity spacetime is flat, it is the Minkowski spacetime ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2

4 Some consequences of special relativity Time dilation Length contraction E = Mc 2 These consequences are facts, i.e., they have been verified by experiments.

5 General relativity The classical gravitational theory by Newton cannot easily be modified to capture the theory of special relativity. In November 1915, Einstein gave a talk at the Prussian Academy of Science where he presented his General theory of relativity, which is a theory for gravity. Spacetime is a four dimensional manifold M equipped with a Lorentzian metric g ab, a, b = 0, 1, 2, 3. The metric g ab is determined through the Einstein field equations (below c = G = 1) G ab := R ab 1 2 g abr + Λg ab = 8πT ab In this talk we take the cosmological constant Λ = 0.

6 The left hand side of Einstein s equations The Einstein equations written in this form look easy: G ab = 8πT ab However, the left hand side of the equations ( made of marble ) written out in local coordinates is very complicated since G ab := R ab 1 2 g abr where R ab = R d adb, and R = Ra a, and where R d abc = x b Γd ac x a Γd bc + Γe acγ d be Γe bc Γd ae, Γ c ab = 1 2 g cd( g bd x a + g ad x b g ab x d ).

7 The right hand side of G ab = 8πT ab The right hand side ( made of wood ) is the energy momentum tensor of the matter. We have to make a choice of matter model: fluid matter models (dust, a perfect fluid,...) kinetic matter models (Vlasov or Boltzmann matter) elastic matter models field theoretical models (a scalar field, a YM field,...) Some models are used by astrophysicists... In addition to the system G ab = 8πT ab, equations for the evolution of the matter must often be included. Let us consider the Einstein-Vlasov system as an example. This will be the system under consideration in the second part of this talk.

8 Kinetic theory The Einstein-Vlasov system is one of many so called kinetic equations. Examples of kinetic equations are: The Boltzmann equation (collisional neutral gases) The Vlasov-Maxwell system (collisionless plasmas) The Vlasov-Fokker-Planck system (collisional plasmas) The Vlasov-Poisson system (collisionless Newtonian gravity) The Einstein-Vlasov system (collisionless general relativity) All these examples model an ensemble of particles (atoms, molecules, ions, stars, galaxies) with density f on phase space, i.e., f = f (t, x, p), t R, x R 3, p R 3.

9 The Einstein-Vlasov system Let (x α, p α ) be local coordinates on the tangent bundle of the spacetime (M, g). The mass shell PM = {g αβ p α p β = m 2 := 1, p 0 > 0} TM, is invariant under geodesic flow ẋ α = p α, ṗ α = Γ α βγ pβ p γ. Note that p 0 can be expressed in terms of p a, a = 1, 2, 3 by the mass shell condition. On PM we thus use coordinates (t, x a, p a ), a = 1, 2, 3.

10 The Einstein-Vlasov system The Vlasov equation for f = f (t, x a, p a ) on PM reads t f + pa p 0 x af 1 p 0 Γa βγ pβ p γ p af = 0. Define the energy momentum tensor by T αβ := g The Einstein-Vlasov system reads p α p β f dp1 dp 2 dp 3 p 0. R αβ 1 2 Rg αβ = 8πT αβ. It has nice mathematical properties!

11 Some highlights of general relativity The gravitational redshift (gravitational time dilation) The bending of light The perihelion advance (of Mercury) Predicting black holes (Schwarzschild 1916 but...) Predicting big bang (Friedmann 1922, Lemaitre 1927 but...) Predicting that the universe is expanding (Friedmann 1922, Lemaitre 1927, confirmed by Hubble 1929, the expansion is accelerating, Nobel Prize 2011) The Penrose-Hawking singularity theorems ( ) Numerical breakthrough 2005 by Franz Pretorius Formation of black holes purely by gravitational waves by Christodoulou 2008 (Shaw prize with Hamilton 2011) General relativity is necessary for GPS!