An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2nd, 2007 Wen-ling Huang Department of Mathematics University of Hamburg, Germany Structuring 1. Introduction of space-time 2. Einstein's field equations 3. ADM Energy Initial data sets • Positive mass theorem • Jang equation • Schoen-Yau's proof of the positive mass theorem • Structuring 1. Introduction of space-time 2. Einstein's field equations 3. ADM Energy Initial data sets • Positive mass theorem • Jang equation • Schoen-Yau's proof of the positive mass theorem • Introduction of space-time (1) Lorentzian manifold (M; g) M: a smooth, Hausdorff, paracompact manifold g: Lorentzian metric for M, i.e.: g is a smooth symmetric tensor field of type (0; 2) on M such that for each point p M the tensor 2 gp : Tp(M) Tp(M) R × ! is a non-degenerate inner product of signature ( ; +; : : : ; +) − Introduction of space-time (2) A non-zero tangent vector v Tp(M), p M, is said to be 2 2 timelike (resp. non-spacelike, null, spacelike) if gp(v; v) < 0 (resp. 0, = 0, > 0). The zero tangent vector is spacelike. ≤ A vector field X on M is said to be timelike if gp X(p); X(p) < 0 for all points p M. 2 In general, a Lorentzian manifold (M; g) does not necessarily have a globally defined continuous timelike vector field. If there exists such a timelike vector field, then (M; g) is said to be time- orientable. Introduction of space-time (3) Space-time (M; g): A connected, n-dimensional Lorentzian manifold together with the Levi-Civita connection (n 2). ≥ Four-dimensional space-times are the mathematical model in rel- ativity theory for the universe we live in. The manifold M is assumed to be connected, since we would not have any knowl- edge of other components. In this talk: n = 4. α; β = 0; 1; 2; 3 and i; j = 1; 2; 3. Introduction of space-time (4) Conformal mapping, isometry : Let (M; g) and (M~ ; g~) be two Lorentzian manifolds and let f : M M~ be a diffeomorphism. If there is a smooth function M! R l : >0 satisfying l(p)gp(u; v) = g~f(p) (f )p(u); (f )p(v) ! ∗ ∗ for all p M and u; v Tp(M) then f iscalled a conformal 2 2 mapping from (M; g) to (M~ ; g~). If l = 1 then f is said to be an isometry. ~ 1 If f : M M is a conformal mapping (resp. isometry) then f − is also a !conformal mapping (resp. isometry), and the manifolds (M; g) and (M~ ; g~) are called conformal (resp. isometric). Introduction of space-time (5) Examples for space-times Minkowski space-time • g = dt2 + dx2 + dy2 + dz2 Mink − Schwarzschild space-time • 2 2m 2 dr 2 2 2 2 gSchw = 1 dt + 2 + r dθ + sin θdφ − − r 1 m − r Kerr space-time • 2 2mr 2 4mar sin θ Σ 2 gKerr = 1 dt dtdφ + dr − − Σ − Σ 2 2 4 2mra sin θ +Σdθ2 + r2 + a2 + sin2 θdφ2 Σ where Σ r2 + a2 cos2 θ, r2 2mr + a2. ≡ 4 ≡ − Introduction of space-time (6) Examples for space-times Robertson-Walker space-times • g = dt2 + S2(t)dσ2; − where dσ2 is the metric of a three-space of curvature 1. Robertson-Walker space-times are spatially homogeneous, isotropic solutions. de-Sitter space-time is the hyperboloid • v2 + w2 + x2 + y2 + z2 = α2 − in flat five-dimensional Minkowski space g = dv2 + dw2 + dx2 + dy2 + dz2: − Its constant scalar curvature R is positive. Structuring 1. Introduction of space-time 2. Einstein's field equations 3. ADM Energy Initial data sets • Positive mass theorem • Jang equation • Schoen-Yau's proof of the positive mass theorem • Einstein's field equations (1) What is the connection between the energy-momentum tensor of matter Tαβ and the metric gαβ ? Einstein's field equations (2) The first version of the field equations was simply Rαβ = κ Tαβ: However, Einstein also looked for a conservation law of energy and momentum: @ 1 @gµν p g g T µν p g T µν = 0; ν µλ λ @x − − 2 − @x in modern notation αβ T ;β = 0: Einstein's field equations (3) αβ Conservation equation T ;β=0 Special relativity: The total flux over a closed surface of the flow of energy and momentum is zero. General relativity: We choose a suitable neighborhood of a point α P with normal coordinates x , such that the components gαβ f g γ of the metric are flat and the components Γαβ of the connection are zero. Then we obtain approximate conservation of energy, momentum and angular momentum in a small region of space- time. Einstein's field equations (4) The first version of field equations was simply Rαβ = κ Tαβ; αβ but the conservation T ;β = 0 as a physical postulate would imply αβ R ;β = 0; which restricts the freedom of the choice of the space-time metric. Einstein's field equations (5) Contracting the Bianchi identity twice, we get R (Rαβ gαβ) = 0: − 2 ;β αβ R αβ Einstein guessed that the quantity R 2 g is the energy- momentum tensor. − We obtain Einstein's field equations: R R g = 8π T αβ − 2 αβ αβ where Rαβ : Ricci curvature; R : scalar curvature; Tαβ : energy-momentum tensor of matter. Einstein's field equations (6) Examples for vacuum (Tαβ = 0) solutions of Einstein's field equations Minkowski space-time • 2 2 2 2 2 2 gMink = dt + dr + r dθ + sin θdφ − Schwarzschild space-time • 2 2m 2 dr 2 2 2 2 gSchw = 1 dt + 2 + r dθ + sin θdφ − − r 1 m − r Kerr space-time • 2 2mr 2 4mar sin θ Σ 2 gKerr = 1 dt dtdφ + dr − − Σ − Σ 2 2 4 2mra sin θ +Σdθ2 + r2 + a2 + sin2 θdφ2 Σ where Σ r2 + a2 cos2 θ, r2 2mr + a2. ≡ 4 ≡ − Einstein's field equaitons (7) In Schwarzschild space-time, the metric has singularities at r = 0 2 and r = 2m. However, RαβγδR = 48m implies αβγδ r6 (i) At r = 0: curvature (space-time) singularity; (ii) At r = 2m: coordinate singularity, which can be removed by a coordinate transformation (isotropic coordinates): 2m 2 Let r = ρ(1 + ρ ) , then 2 m 4 (1 2 ) m − ρ 2 2 2 2 2 2 gSchw = m 2 dt + 1 + dρ + ρ (dθ + sin θdφ ) : −(1 + 2ρ) 2ρ r = 2m iff ρ = m • 2 Structuring 1. Introduction of space-time 2. Einstein's field equations 3. ADM Energy Initial data sets • Positive mass theorem • Jang equation • Schoen-Yau's proof of the positive mass theorem • ADM Energy (1) T00: local energy density T0i: local momentum density For a large domain with gravitational sources there does not exist a globally defined covariant quantity \energy". But in asymptotically flat space-times, at large distance from the source, the gravitational effects become less important. Energy can be defined in the region which is far away from the sources. ADM Energy (2) There are two distinct regimes in which the asymptotic be- havior of the gravitational field has been found to yield useful information concerning the structure of a gravitation system. Asymptotical structure of null infinity (Bondi, Van der Burg, Metzner and Sachs): The total energy, measured at null infinity, decreases with time at a rate depending on the flux of radiation escaping between successive null surfaces. Asymptotical structure of spatial infinity (Arnowitt, Deser and Misner): Set of conditions for asymptotic flatness and an expression for the total energy-momentum (ADM energy-momentum) in terms of the asymptotic behavior of the gravitational field. ADM Energy (3) Spacelike hypersurfaces Let M be a 3-dimensional submanifold in a space-time M 4. spacelike positive definite M is a 8 timelike hypersurface, if the induced metric on M is 8 Lorentz . < null < degenerate : : ADM Energy (4) Example of spacelike hypersurfaces: spatial slices M4: space-time manifold; t = x0; x1; x2; x3: coordinates The sub-manifolds M 3(t) defined by t = constant are called spatial slices of the coordinates system. These spatial slices are spatial in the sense that X; X > 0 for h i any nonzero tangent vector to M 3(t). Structuring 1. Introduction of space-time 2. Einstein's field equations 3. ADM Energy Initial data sets • Positive mass theorem • Jang equation • Schoen-Yau's proof of the positive mass theorem • ADM Energy (5) Initial data set Let (M; g; h) be a spacelike hypersurface in a space-time, where M is a 3-dimensional manifold g is the Riemannian metric of M h is the second fundamental form of M. (M; g; h) is usually called an initial data set. It obeys the constraint equations which come from Gauss equa- tion and Codazzi equations. ADM Energy (6) Gauss equation: 1 i 2 ij = local energy density: T00 = R + (h i) hij h : ) 16π − The sum of the intrinsic and extrinsic curvatures of a spatial section is a measure of non-gravitational energy density of the space-time (J.A. Wheeler). Codazzi equations: 1 = local moment density: T i = (hij hk gij): ) 0 8πrj − k R: scalar curvature of M, : Levi-Civita connection of M. r ADM Energy (7) Energy conditions Weak energy condition: The energy-momentum tensor at each p M obeys the inequal- α β 2 ity T W W 0 for any time like vector W Tp. αβ ≥ 2 Dominant energy condition: αβ T W is non-spacelike for any timelike vector W Tp. β 2 ADM Energy (8) (M; g; h) is time-symmetric if hij = 0.