An introduction to General Relativity and the positive mass theorem

An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 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 g p : T p (M) T p (M) R is a non-degenerate inner product of signature (, +,..., +)

Introduction of space-time (2) A non-zero tangent vector v T p (M), p M, is said to be timelike (resp. non-spacelike, null, spacelike) if g p (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 g p ( X(p), X(p) ) < 0 for all points p M. 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 timeorientable.

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 relativity theory for the universe we live in. The manifold M is assumed to be connected, since we would not have any knowledge 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 l : M R >0 satisfying l(p)g p (u, v) = g f(p) ( (f ) p (u), (f ) p (v) ) for all p M and u, v T p (M) then f is called a conformal mapping from (M, g) to ( M, g). If l = 1 then f is said to be an isometry. If f : M M is a conformal mapping (resp. isometry) then f 1 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 Mink = dt 2 + dx 2 + dy 2 + dz 2 Schwarzschild space-time ( g Schw = 1 2m r Kerr space-time g Kerr ( = 1 2mr ) Σ ( +Σdθ 2 + )dt 2 + dr2 1 2m r dt 2 4mar sin2 θ Σ r 2 + a 2 + 2mra2 sin 2 θ Σ + r 2 ( dθ 2 + sin 2 θdφ 2 ) dtdφ + Σ dr2 where Σ r 2 + a 2 cos 2 θ, r 2 2mr + a 2. ) sin 2 θdφ 2

Introduction of space-time (6) Examples for space-times Robertson-Walker space-times g = dt 2 + S 2 (t)dσ 2, where dσ 2 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 v 2 + w 2 + x 2 + y 2 + z 2 = α 2 in flat five-dimensional Minkowski space g = dv 2 + dw 2 + dx 2 + dy 2 + dz 2. 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 T αβ and the metric g αβ?

Einstein s field equations (2) The first version of the field equations was simply R αβ = κ T αβ. However, Einstein also looked for a conservation law of energy and momentum: ( g x ν gµλ T µν) 1 g µν g 2 x λ T µν = 0, in modern notation T αβ ;β = 0.

Einstein s field equations (3) Conservation equation T αβ ;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 P with normal coordinates {x α }, such that the components g αβ of the metric are flat and the components Γ γ αβ of the connection are zero. Then we obtain approximate conservation of energy, momentum and angular momentum in a small region of spacetime.

Einstein s field equations (4) The first version of field equations was simply R αβ = κ T αβ, but the conservation T αβ ;β = 0 as a physical postulate would imply R αβ ;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 αβ R 2 gαβ ) ;β = 0. Einstein guessed that the quantity R αβ R 2 gαβ is the energymomentum tensor. We obtain Einstein s field equations: where R αβ : R : T αβ : R αβ R 2 g αβ = 8π T αβ Ricci curvature, scalar curvature, energy-momentum tensor of matter.

Einstein s field equations (6) Examples for vacuum (T αβ = 0) solutions of Einstein s field equations Minkowski space-time g Mink = dt 2 + dr 2 + r 2 ( dθ 2 + sin 2 θdφ 2 ) Schwarzschild space-time ( g Schw = 1 2m r Kerr space-time g Kerr ( = 1 2mr ) Σ ( +Σdθ 2 + )dt 2 + dr2 1 2m r dt 2 4mar sin2 θ Σ r 2 + a 2 + 2mra2 sin 2 θ Σ + r 2 ( dθ 2 + sin 2 θdφ 2 ) dtdφ + Σ dr2 where Σ r 2 + a 2 cos 2 θ, r 2 2mr + a 2. ) sin 2 θdφ 2

Einstein s field equaitons (7) In Schwarzschild space-time, the metric has singularities at r = 0 and r = 2m. However, R αβγδ R αβγδ = 48m2 r 6 implies (i) At r = 0: curvature (space-time) singularity; (ii) At r = 2m: coordinate singularity, which can be removed by a coordinate transformation (isotropic coordinates): Let r = ρ(1 + 2m ρ )2, then g Schw = (1 m 2ρ )2 (1 + m 2ρ )2dt2 + r = 2m iff ρ = m 2 ( 1 + m ) 4 ( ) dρ 2 + ρ 2 (dθ 2 + sin 2 θdφ 2 ). 2ρ

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

T 00 : local energy density T 0i : local momentum density ADM Energy (1) 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 behavior 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.

Spacelike hypersurfaces ADM Energy (3) Let M be a 3-dimensional submanifold in a space-time M 4. M is a spacelike timelike null hypersurface, if the induced metric on M is positive definite Lorentz degenerate.

ADM Energy (4) Example of spacelike hypersurfaces: spatial slices M 4 : space-time manifold; t = x 0, x 1, x 2, x 3 : 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 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 equation and Codazzi equations.

ADM Energy (6) Gauss equation: = local energy density: T 00 = 1 ( R + (h i 16π i ) 2 h ij h ij). 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: = local moment density: T i 0 = 1 8π j(h ij h k k gij ). R: scalar curvature of M, : Levi-Civita connection of M.

Energy conditions ADM Energy (7) Weak energy condition: The energy-momentum tensor at each p M obeys the inequality T αβ W α W β 0 for any time like vector W T p. Dominant energy condition: T αβ W β is non-spacelike for any timelike vector W T p.

ADM Energy (8) (M, g, h) is time-symmetric if h ij = 0. (M, g, h) is a maximal slice if the mean curvature vanishes, h i i = 0. From Gauss equation, the weak energy condition reduces in these cases to R 0.

ADM Energy (9) Choose a coordinate frame, and take W = (1, 0, 0, 0). Then we obtain from the dominant energy condition that T αβ W β = (T0 0, T 0 1, T 0 2, T 0 3 ) is non-spacelike. This implies T 00 T0 it 0i. and 1 ( R+(h i 2 i )2 h ij h ij) ( j h ij h l l gij ) g ik ( j h kj h l l gkj ).

ADM Energy (10) Asymptotically flat initial data set An initial data set (M, g, h) is said to be asymptotically flat if, outside a compact subset, M is diffeomorphic to R 3 \B r and g and h satisfy g ij = δ ij + O ( 1), r k g ij = O ( 1 ), r 2 l k g ij = O ( 1 ), r 3 h ij = O ( 1 ), r 2 k h ij = O ( 1 ). r 3 where r is the Euclidian distance.

ADM Energy (11) Examples for asymptotically flat space-times Minkowski space-time g Mink = dt 2 + dr 2 + r 2 ( dθ 2 + sin 2 θdφ 2 ) Schwarzschild space-time ( g Schw = 1 2m r Kerr space-time g Kerr ( = 1 2mr ) Σ ( +Σdθ 2 + )dt 2 + dr2 1 2m r dt 2 4mar sin2 θ Σ r 2 + a 2 + 2mra2 sin 2 θ Σ + r 2 ( dθ 2 + sin 2 θdφ 2 ) dtdφ + Σ dr2 where Σ r 2 + a 2 cos 2 θ, r 2 2mr + a 2. ) sin 2 θdφ 2

ADM Energy (12) The following space-times are not asymptotically flat. Robertson-Walker space-times g = dt 2 + S 2 (t)dσ 2, where dσ 2 is the metric of a three-space of curvature ±1. Robertson-Walker space-times are homogeneous isotropic solutions. de-sitter space-time is the hyperboloid v 2 + w 2 + x 2 + y 2 + z 2 = α 2 in flat five-dimensional space with metric g = dv 2 + dw 2 + dx 2 + dy 2 + dz 2. Its constant scalar curvature R is positive.

ADM Energy (13) Examples for asymptotically flat initial data sets Example 1. Schwarzschild space-time: M = {t = constant}, ( g = 1 + m ) 4 ( ) dρ 2 + ρ 2 (dθ 2 + sin 2 θdφ 2 ), h = 0. 2ρ Example 2. Kerr space-time: M = {t = constant}, g = Σ ( dr2 + Σdθ 2 + r 2 + a 2 + 2mra2 sin 2 θ ) sin 2 θdφ 2, Σ h ij = O ( 1 ), k r 3 h ij = O ( 1 ). r 4

ADM Energy (14) Arnowitt-Deser-Misner 1961: Let (M, g, h) be an asymptotically flat initial data set. The total energy E and the total linear momentum P k are defined as E = 1 16π r lim ( j g ij i g jj )ds i, S r P k = 1 8π r lim (h ki g ki h jj )ds i, S r where S r is the sphere of radius r and 1 k 3.

ADM Energy (15) For any spatial-slice of the Minkowski space-time, E = 0, P 1 = P 2 = P 3 = 0. For any spatial-slice of the Schwarzschild and Kerr solution, E = m, P 1 = P 2 = P 3 = 0. Reference: S. Arnowitt, S. Deser, C. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961), 997-1006. 1986, Bartnik proved that E is independent on the choice of asymptotic coordinates. Reference: R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 36 (1986), 661-693.

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 (16) Positive mass conjecture For any asymptotically flat initial data set which obeys the dominant energy condition, its ADM energy is always positive (except for initial data set in flat Minkowski space-time, which has zero energy). This conjecture was studied by various mathematicians and physicists in 1960 s and 1970 s. It was proved by Richard Schoen and Shing-Tung Yau in 1979 and 1981. We are going to give a brief description of the idea of the proof.

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 (17) Jang s equation (g ij f i f j )( f ),ij 1 + f 2 1 + f 2 h ij = 0. Under the suitable boundary condition f = o(1/r), the metric is asymptotically flat. ḡ = g + f f Reference: P.S. Jang, On the positivity of energy in general relativity, J. Math. Phys. 19(5) (1978), 1152-1155

Sketch of Jang s idea ADM Energy (18) 1. First look at the asymptotically flat initial data sets (M, g, h) of Minkowski space-time. Necessary and sufficient condition that an asymptotically flat initial data set (M, g, h) is that of Minkowski space-time: There exists a scalar function f defined on M, such that h ij f = ij and g 1+ f 2 ij = δ ij f,i f,j.

ADM Energy (19) Sketch of Jang s idea 2. Introduce Jang s equation: (g ij f i f j )( f ),ij 1 + f 2 1 + f 2 h ij = 0. Define h ij = h ij f ij 1+ f 2 and ḡ ij = g ij + f,i f,j. ḡ = g + f f is asymptotically flat, if f = o(1/r).

ADM Energy (20) Sketch of Jang s idea 3. Under the assumption that Jang s equation has a solution f.jang proved the ADM energy E(ḡ) associated to the metric ḡ is the same as the ADM energy E(g) associated to the metric g, and E(ḡ) is zero only if ḡ is flat and h = 0. The case ḡ is flat and h = 0 is equivalent that (M, g, h) is an initial data set for Minkowski space-time. Reference: P.S. Jang, On the positivity of energy in general relativity, J. Math. Phys. 19(5) (1978), 1152-1155

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 (21) Schoen-Yau s positive mass theorem Schoen and Yau (1979): The positive mass conjecture for the initial data sets with h ij = 0. Jang s equation has solutions when the initial data sets have no apparent horizon. 1981: Schoen and Yau study the general case. The difficulty occurs when apparent horizons exist in the initial data set. Under a suitable conformal mapping, Schoen and Yau close these apparent horizons.

ADM Energy (22) References R. Schoen, S.T. Yau, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65 (1979), 45-76. R. Schoen, S.T. Yau, Positivity of the Total Mass of a General Space-Time, Phys. Rev. Lett. 43 (1979), 159-183. R. Schoen, S.T. Yau, Proof of the positive mass theorem II, Commun. Math. Phys. 79 (1981), 231-260.

ADM Energy (23) Sketch of Schoen-Yau s proof Step 1. The positive mass conjecture is true for initial data sets which satisfy h ij = 0. Step 2. Jang s equation has a solution for the initial data sets without apparent horizons, and E(ḡ) = E(g) 0. Ω: sufficiently large compact subdomain of M f solution of Jang s equation with f Ω = 0. Ω 1 Ω 2... M: large domains with M = i Ω i. f i : solutions of Jang s equation over Ω i. f i converges to a global solution f of Jang s equation. ḡ ij = g ij + f,i f,j E(g) = E(ḡ).

ADM Energy (24) Sketch of Schoen-Yau s proof Step 2. Conformally transform the metric ḡ to ϕ 4 ḡ, where ϕ = 1 8 Rϕ, ϕ = 1 + A r + O( 1 r 3). ϕ 4 ḡ is asymptotically flat and the scalar curvature is zero. E(ϕ 4 ḡ) = E(ḡ) + 1 2 A 0. A 0 and A = 0 iff ḡ is flat, A = 1 4π M 1 8 R ϕ (det ḡ ij) 1 2 dx. Step 3. Initial data sets with apparent horizons: Under a suitable conformal mapping one may close these apparent horizons.

ADM Energy (25) Witten s proof uses the method of Dirac operator. Reference E. Witten, A new proof of the positive energy theorem, Commun. Math. Phys. 80 (1981), 381-402.

ADM Energy (26) Total angular momentum Total angular momentum is also fundamental quantity in physics. The Kerr solution describes a rotating black hole. For the definition of total angular momentum in asymptotically flat initial data set see, e.g., Ashtekar, Penrose, etc.: Conformal compactness formulation (1979). T. Regge, C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys. 88(1974), 286-318. X. Zhang, Angular momentum and positive mass theorem, Commun. Math. Phys. 206 (1999), 137-155. Solutions of Einstein s field equations with positive cosmological constant describe universe with dark energy and the de-sitter universe is a good model. What is the definition of total massenergy momentum of asymptotical de Sitter universes?

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