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 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

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 ∈ 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 ∈ ∈ 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. ∈ 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 relativity 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 ∈ ∈ 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

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 √ g g T µν √ 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 { } 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

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 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. ADM Energy (3)

Spacelike hypersurfaces

Let M be a 3-dimensional submanifold in a space-time M 4.

spacelike positive deﬁnite M is a  timelike hypersurface, if the induced metric on M is  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

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: 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 8π∇j − k R: scalar curvature of M, : Levi-Civita connection of M. ∇ ADM Energy (7)

Energy conditions

Weak energy condition: The energy-momentum tensor at each p M obeys the inequal- α β ∈ ity T W W 0 for any time like vector W Tp. αβ ≥ ∈ Dominant energy condition: αβ T W is non-spacelike for any timelike vector W Tp. β ∈ ADM Energy (8)

(M, g, h) is time-symmetric if hij = 0.

(M, g, h) is a maximal slice if the mean curvature vanishes, i hi = 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). αβ Then we obtain from the dominant energy condition that T Wβ = 0 1 2 3 (T0 , T0 , T0 , T0 ) is non-spacelike. This implies i T00 T T0i. ≥ q 0 and 1 i 2 ij ij l ij kj l kj R+(h i) hij h ( jh h g ) gik ( jh h g ). 2 − ≥ q ∇ − l ∇ − l ADM Energy (10)

Asymptotically ﬂat initial data set

An initial data set (M, g, h) is said to be asymptotically ﬂat if, 3 outside a compact subset, M is diﬀeomorphic to R Br and g and h satisfy \ 1 g = δ + O , ij ij r 1 ∂ g = O , k ij r2 1 ∂ ∂ g = O , l k ij r3 1 h = O , ij r2 1 ∂khij = O 3 . r where r is the Euclidian distance. ADM Energy (11)

Examples for asymptotically ﬂat space-times 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 ≡ − ADM Energy (12)

The following space-times are not asymptotically flat. 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 homogeneous isotropic solutions. de-Sitter space-time is the hyperboloid • v2 + w2 + x2 + y2 + z2 = α2 − in flat five-dimensional space with metric g = dv2 + dw2 + dx2 + dy2 + dz2. − 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 , { } m 4 g = 1 + dρ2 + ρ2(dθ2 + sin2 θdφ2) , h = 0. 2ρ Example 2. Kerr space-time: M = t = constant , { } Σ 2mra2 sin2 θ g = dr2 + Σdθ2 + r2 + a2 + sin2 θdφ2, Σ 4 1 1 hij = O 3 , ∂khij = O 4 . r r 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 Pk are deﬁned as 1 E = lim (∂ g ∂ g )dSi, 16π r I j ij − i jj →∞ Sr 1 P = lim (h g h )dSi, k 8π r I ki − ki jj →∞ Sr where Sr is the sphere of radius r and 1 k 3. ≤ ≤ ADM Energy (15)

For any spatial-slice of the Minkowski space-time,

E = 0, P1 = P2 = P3 = 0. For any spatial-slice of the Schwarzschild and Kerr solution,

E = m, P1 = P2 = P3 = 0.

Reference: S. Arnowitt, S. Deser, C. Misner, Coordinate invariance and energy expres- sions 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

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 physi- cists 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

Jang’s equation

i j ij f f f,ij g hij = 0. − 1 + f 2 1 + f 2 − |∇ | p |∇ | Under the suitable boundary condition f = o(1/r), the metric |∇ | g¯ = g + f f ∇ ⊗ ∇ is asymptotically ﬂat. Reference: P.S. Jang, On the positivity of energy in general relativity, J. Math. Phys. 19(5) (1978), 1152-1155 ADM Energy (18)

Sketch of Jang’s idea 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 ij fij h = and gij = δij f,if,j. √1+ f 2 − |∇ | ADM Energy (19)

Sketch of Jang’s idea 2. Introduce Jang’s equation: i j ij f f f,ij g hij = 0. − 1 + f 2 1 + f 2 − |∇ | p |∇ | Deﬁne ¯ij ij fij h = h and g¯ij = gij + f,if,j. − √1+ f 2 |∇ | g¯ = 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(g¯) associated to the metric g¯ is the same as the ADM energy E(g) associated to the metric g, and E(g¯) is zero only if g¯ is ﬂat and h¯ = 0. The case g¯ 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

Schoen-Yau’s positive mass theorem Schoen and Yau (1979): The positive mass conjecture for the • initial data sets with hij = 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 Gen- • eral 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 hij = 0. Step 2. Jang’s equation has a solution for the initial data sets without apparent horizons, and E(g¯) = E(g) 0. ≥ – Ω: suﬃciently large compact subdomain of M f solution of Jang’s equation with f ∂Ω = 0. ⇒ ∃ | – Ω Ω . . . M: large domains with M = Ω . 1 ⊂ 2 ⊂ ⊂ i i – fi: solutions of Jang’s equation over Ωi. S – fi converges to a global solution f of Jang’s equation. – g¯ = g + f f E(g) = E(g¯). ij ij ,i ,j ⇒ ADM Energy (24)

Sketch of Schoen-Yau’s proof Step 2. – Conformally transform the metric g¯ to ϕ4g¯, where 1 A 1 ∆ϕ = Rϕ, ϕ = 1 + + O( ). 8 r r3 – ϕ4g¯ is asymptotically flat and the scalar curvature is zero. – E(ϕ4g¯) = E(g¯) + 1A 0. 2 ≥ – A 0 and A = 0 iff g¯ is flat, ≤ 1 1 1 A = R ϕ (det g¯ij)2 dx. −4π ZM 8 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 def- inition 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 for- • mulation 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 mass- energy momentum of asymptotical de Sitter universes ? Thank you for your attention !