An introduction to General Relativity the positive mass

An introduction to General Relativityand

the positive mass theorem

National Center for Theoretical Sciences, Mathematics DivisionMarch 2nd, 2007

Wen-ling HuangDepartment 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



3. ADM Energy



• Jang equation


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 suchthat for each point p ∈ M the tensor

gp : Tp(M) × Tp(M) → R

is a non-degenerate inner product of signature (−, +, . . . , +)


A non-zero tangent vector v ∈ Tp(M), p ∈ M, is said to betimelike (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 necessarilyhave a globally defined continuous timelike vector field. If thereexists such a timelike vector field, then (M, g) is said to be time-orientable.


Space-time (M, g):A connected, n-dimensional Lorentzian manifold together withthe 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 isassumed 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.


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 functionl : M → R>0 satisfying l(p)gp(u, v) = gf(p)

(

(f∗)p(u), (f∗)p(v))

for all p ∈ M and u, v ∈ Tp(M) then f is called a conformal

mapping from (M, g) to (M, g). If l = 1 then f is said to be anisometry.

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).


Examples for space-times

• Minkowski space-time

gMink = −dt2 + dx2 + dy2 + dz2

• Schwarzschild space-time

gSchw = −(

1 − 2m

r

)

dt2 +dr2

1 − 2mr

+ r2

(

dθ2 + sin2 θdφ2

)

• Kerr space-time

gKerr = −(

1 − 2mr

Σ

)

dt2 − 4mar sin2 θ

Σdtdφ +

Σ

4dr2

+Σdθ2 +(

r2 + a2 +2mra2 sin2 θ

Σ

)

sin2 θdφ2

where Σ ≡ r2 + a2 cos2 θ, 4 ≡ r2 − 2mr + a2.


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, isotropicsolutions.

• 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



3. ADM Energy



• Jang equation


Einstein’s field equations (1)

What is the connection between

the energy-momentum tensor of matter Tαβ

and the metric gαβ ?


The first version of the field equations was simply

Rαβ = κTαβ.

However, Einstein also looked for a conservation law of energyand momentum:

∂

∂xν

(√−g gµλ Tµν) − 1

2

√−g∂gµν

∂xλTµν = 0,

in modern notationTαβ

;β = 0.


Conservation equation Tαβ;β=0

Special relativity: The total flux over a closed surface of the flowof energy and momentum is zero.

General relativity: We choose a suitable neighborhood of a pointP with normal coordinates xα, such that the components gαβ

of the metric are flat and the components Γγαβ of the connection

are zero. Then we obtain approximate conservation of energy,momentum and angular momentum in a small region of space-

time.


The first version of field equations was simply

Rαβ = κTαβ,

but the conservation Tαβ;β = 0 as a physical postulate would

implyRαβ

;β = 0,

which restricts the freedom of the choice of the space-time metric.


Contracting the Bianchi identity twice, we get

(Rαβ − R

2gαβ);β = 0.

Einstein guessed that the quantity Rαβ − R2 gαβ is the energy-

momentum tensor.

We obtain Einstein’s field equations:

Rαβ − R2 gαβ = 8π Tαβ

where

Rαβ : Ricci curvature,

R : scalar curvature,

Tαβ : energy-momentum tensor of matter.


Examples for vacuum (Tαβ = 0) solutions of Einstein’sfield equations


gMink = −dt2 + dr2 + r2

(

dθ2 + sin2 θdφ2

)


gSchw = −(

1 − 2m

r

)

dt2 +dr2

1 − 2mr

+ r2

(

dθ2 + sin2 θdφ2

)

• Kerr space-time

gKerr = −(

1 − 2mr

Σ

)


Σdtdφ +

Σ

4dr2

+Σdθ2 +(


Σ

)

sin2 θdφ2


Einstein’s field equaitons (7)

In Schwarzschild space-time, the metric has singularities at r = 0

and r = 2m. However, RαβγδRαβγδ = 48m2

r6 implies

(i) At r = 0: curvature (space-time) singularity;

(ii) At r = 2m: coordinate singularity, which can be removed bya coordinate transformation (isotropic coordinates):Let r = ρ(1 + 2m

ρ )2, then

gSchw = −(1 − m

2ρ)2

(1 + m2ρ)

2dt2 +

(

1 +m

2ρ

)4(

dρ2 + ρ2(dθ2 + sin2 θdφ2))

.

• r = 2m iff ρ = m2

Structuring



3. ADM Energy



• Jang equation


ADM Energy (1)

T00: local energy densityT0i: local momentum density

For a large domain with gravitational sources there does not exista globally defined covariant quantity “energy”.

But in asymptotically flat space-times, at large distance from thesource, the gravitational effects become less important. Energycan 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 timeat a rate depending on the flux of radiation escaping betweensuccessive null surfaces.

Asymptotical structure of spatial infinity(Arnowitt, Deser and Misner):Set of conditions for asymptotic flatness and an expression forthe total energy-momentum (ADM energy-momentum) in termsof 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.

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

M4: space-time manifold; t = x0, x1, x2, x3: coordinates

The sub-manifolds M 3(t) defined by t = constant are calledspatial slices of the coordinates system.

These spatial slices are spatial in the sense that 〈X,X〉 > 0 forany nonzero tangent vector to M 3(t).

Structuring



3. ADM Energy



• Jang equation


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:

=⇒ local energy density: T00 =1

16π

(

R + (hii)

2 − hij hij).

The sum of the intrinsic and extrinsic curvatures of a spatialsection is a measure of non-gravitational energy density of thespace-time (J.A. Wheeler).

Codazzi equations:

=⇒ local moment density: T i0 =

1

8π∇j(h

ij − hkk gij).

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-ity TαβWαWβ ≥ 0 for any time like vector W ∈ Tp.

Dominant energy condition:TαβWβ is non-spacelike for any timelike vector W ∈ Tp.

ADM Energy (8)

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

(M, g, h) is a maximal slice if the mean curvature vanishes,hi

i = 0.

From Gauss equation, the weak energy condition reduces in thesecases 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 αβWβ =

(T 00 , T 1

0 , T 20 , T 3

0 ) is non-spacelike. This implies

T00 ≥√

T i0T0i.

and1

2

(

R+(hii)

2−hij hij)

≥√

(∇jhij − hll gij) gik (∇jhkj − hl

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 R3\Br and gand h satisfy

gij = δij + O(1

r

)

,

∂kgij = O( 1

r2

)

,

∂l∂kgij = O( 1

r3

)

,

hij = O( 1

r2

)

,

∂khij = O( 1

r3

)

.

where r is the Euclidian distance.

ADM Energy (11)

Examples for asymptotically flat space-times


gMink = −dt2 + dr2 + r2

(

dθ2 + sin2 θdφ2

)


gSchw = −(

1 − 2m

r

)

dt2 +dr2

1 − 2mr

+ r2

(

dθ2 + sin2 θdφ2

)

• Kerr space-time

gKerr = −(

1 − 2mr

Σ

)


Σdtdφ +

Σ

4dr2

+Σdθ2 +(


Σ

)

sin2 θdφ2


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 so-lutions.

• 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,

g =(

1 +m

2ρ

)4(

dρ2 + ρ2(dθ2 + sin2 θdφ2))

, h = 0.

Example 2. Kerr space-time: M = t = constant,

g =Σ

4dr2 + Σdθ2 +(


Σ

)

sin2 θdφ2,

hij = O( 1

r3

)

, ∂khij = O( 1

r4

)

.

ADM Energy (14)

Arnowitt-Deser-Misner 1961:

Let (M, g, h) be an asymptotically flat initial data set. The totalenergy E and the total linear momentum Pk are defined as

E =1

16πlim

r→∞

∮

Sr

(∂jgij − ∂igjj)dSi,

Pk =1

8πlim

r→∞

∮

Sr

(hki − gkihjj)dSi,

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 ofasymptotic coordinates.

Reference:

R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure

Appl. Math. 36 (1986), 661-693.

Structuring



3. ADM Energy



• Jang equation


ADM Energy (16)

Positive mass conjecture

For any asymptotically flat initial data set which obeysthe dominant energy condition, its ADM energy isalways positive (except for initial data set in flatMinkowski 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 1979and 1981.

We are going to give a brief description of the idea of the proof.

Structuring



3. ADM Energy



• Jang equation


ADM Energy (17)

Jang’s equation

(

gij − f if j

1 + |∇f |2)( f,ij

√

1 + |∇f |2− hij

)

= 0.

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 flatinitial data set (M, g, h) is that of Minkowski space-time:There exists a scalar function f defined on M , such that

hij = f ij√1+|∇f |2

and gij = δij − f,if,j.

ADM Energy (19)


2. Introduce Jang’s equation:(

gij − f if j

1 + |∇f |2)( f,ij

√

1 + |∇f |2− hij

)

= 0.

Define

hij = hij − f ij√1+|∇f |2

and gij = gij + f,if,j.

g = g + ∇f ⊗∇f

is asymptotically flat, if |∇f | = o(1/r).

ADM Energy (20)


3. Under the assumption that Jang’s equation has a solutionf .Jang proved the ADM energy E(g) associated to the metricg is the same as the ADM energy E(g) associated to the met-ric g, and E(g) is zero only if g is flat and h = 0. The case gis flat and h = 0 is equivalent that (M, g, h) is an initial dataset 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



3. ADM Energy



• Jang equation


ADM Energy (21)

Schoen-Yau’s positive mass theorem

• Schoen and Yau (1979): The positive mass conjecture for theinitial data sets with hij = 0.Jang’s equation has solutions when the initial data sets haveno apparent horizon.

• 1981: Schoen and Yau study the general case.

The difficulty occurs when apparent horizons exist in the initialdata set. Under a suitable conformal mapping, Schoen andYau 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.

– Ω1 ⊂ Ω2 ⊂ . . . ⊂ M : large domains with M =⋃

i Ωi.

– fi: solutions of Jang’s equation over Ωi.

– fi converges to a global solution f of Jang’s equation.

– gij = gij + f,if,j ⇒ E(g) = E(g).

ADM Energy (24)

Sketch of Schoen-Yau’s proof

Step 2. – Conformally transform the metric g to ϕ4g, where

∆ϕ =1

8Rϕ, ϕ = 1 +

A

r+ O(

1

r3).

– ϕ4g is asymptotically flat and the scalar curvature is zero.

– E(ϕ4g) = E(g) + 12A ≥ 0.

– A ≤ 0 and A = 0 iff g is flat,

A = − 1

4π

∫

M

1

8R ϕ (det gij)

12 dx.

Step 3. Initial data sets with apparent horizons: Under a suitableconformal 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 momentumTotal 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 initialdata 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-Sitteruniverse is a good model. What is the definition of total mass-energy momentum of asymptotical de Sitter universes ?

Thank you for your attention !

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An introduction to General Relativity the positive mass