On the notion of quasilocal mass in general relativity
Mu-Tao Wang
Columbia University
Quantum Field Theory and Gravity
Regensburg, September 2010 Study quasilocal mass of a closed spacelike 2-surface Σ in spacetime by Hamilton-Jacobi method.
A new mass for Σ is defined using isometric embeddings into the Minkowski space and a canonical gauge.
The new definition has the desired “positivity” and “rigidity” properties, in addition to other natural ones.
Discuss the case for gravitational energy (mass).
2 A new mass for Σ is defined using isometric embeddings into the Minkowski space and a canonical gauge.
The new definition has the desired “positivity” and “rigidity” properties, in addition to other natural ones.
Discuss the case for gravitational energy (mass).
Study quasilocal mass of a closed spacelike 2-surface Σ in spacetime by Hamilton-Jacobi method.
3 The new definition has the desired “positivity” and “rigidity” properties, in addition to other natural ones.
Discuss the case for gravitational energy (mass).
Study quasilocal mass of a closed spacelike 2-surface Σ in spacetime by Hamilton-Jacobi method.
A new mass for Σ is defined using isometric embeddings into the Minkowski space and a canonical gauge.
4 Discuss the case for gravitational energy (mass).
Study quasilocal mass of a closed spacelike 2-surface Σ in spacetime by Hamilton-Jacobi method.
A new mass for Σ is defined using isometric embeddings into the Minkowski space and a canonical gauge.
The new definition has the desired “positivity” and “rigidity” properties, in addition to other natural ones.
5 gµν: Lorentzian metric, gravitational field.
Φ: matter fields.
Einstein’s equation is obtained by taking the variation of
1 Z Z R + L(g, Φ) 16π
where R is the scalar curvature of gµν and L is the Lagrangian of matter coupled to gravity.
In general relativity, the spacetime is a Lorentzian manifold with
6 Einstein’s equation is obtained by taking the variation of
1 Z Z R + L(g, Φ) 16π
where R is the scalar curvature of gµν and L is the Lagrangian of matter coupled to gravity.
In general relativity, the spacetime is a Lorentzian manifold with gµν: Lorentzian metric, gravitational field.
Φ: matter fields.
7 In general relativity, the spacetime is a Lorentzian manifold with gµν: Lorentzian metric, gravitational field.
Φ: matter fields.
Einstein’s equation is obtained by taking the variation of
1 Z Z R + L(g, Φ) 16π where R is the scalar curvature of gµν and L is the Lagrangian of matter coupled to gravity.
8 Ω bounded spacelike region in spacetime.
Question: How do we measure the total energy contained in Ω, counting contributions from both matter fields and gravitational field?
Einstein’s equation 1 R − Rg = 8πT . µν 2 µν µν where Tµν is the energy momentum tensor of matter density.
Relation between matter fields and gravitational field.
9 Einstein’s equation 1 R − Rg = 8πT . µν 2 µν µν where Tµν is the energy momentum tensor of matter density.
Relation between matter fields and gravitational field.
Ω bounded spacelike region in spacetime.
Question: How do we measure the total energy contained in Ω, counting contributions from both matter fields and gravitational field?
10 Given any tν constant (Killing) timelike unit vector, Z µ ν Tµνu t Ω represents energy intercepted by Ω (observed by tν).
ν ν t Killing implies Tµνt is divergence free.
Energy depends only on ∂Ω! Conservation law.
3,1 µ Special relativity in R , Ω a spacelike hypersurface and u its timelike unit normal.
11 ν ν t Killing implies Tµνt is divergence free.
Energy depends only on ∂Ω! Conservation law.
3,1 µ Special relativity in R , Ω a spacelike hypersurface and u its timelike unit normal.
Given any tν constant (Killing) timelike unit vector, Z µ ν Tµνu t Ω represents energy intercepted by Ω (observed by tν).
12 Energy depends only on ∂Ω! Conservation law.
3,1 µ Special relativity in R , Ω a spacelike hypersurface and u its timelike unit normal.
Given any tν constant (Killing) timelike unit vector, Z µ ν Tµνu t Ω represents energy intercepted by Ω (observed by tν).
ν ν t Killing implies Tµνt is divergence free.
13 3,1 µ Special relativity in R , Ω a spacelike hypersurface and u its timelike unit normal.
Given any tν constant (Killing) timelike unit vector, Z µ ν Tµνu t Ω represents energy intercepted by Ω (observed by tν).
ν ν t Killing implies Tµνt is divergence free.
Energy depends only on ∂Ω! Conservation law.
14 (1) No Killing vector field in general spacetime (Komar mass in the presence of Killing field by Noether’s theorem).
(2) It only accounts for energy contribution from matters.
But gravitation energy must have contribution.
Schwarzschild is a vacuum solution and binding energy depending on distance apart.
The above construction does not work in general relativity because
15 (2) It only accounts for energy contribution from matters.
But gravitation energy must have contribution.
Schwarzschild is a vacuum solution and binding energy depending on distance apart.
The above construction does not work in general relativity because
(1) No Killing vector field in general spacetime (Komar mass in the presence of Killing field by Noether’s theorem).
16 But gravitation energy must have contribution.
Schwarzschild is a vacuum solution and binding energy depending on distance apart.
The above construction does not work in general relativity because
(1) No Killing vector field in general spacetime (Komar mass in the presence of Killing field by Noether’s theorem).
(2) It only accounts for energy contribution from matters.
17 Schwarzschild is a vacuum solution and binding energy depending on distance apart.
The above construction does not work in general relativity because
(1) No Killing vector field in general spacetime (Komar mass in the presence of Killing field by Noether’s theorem).
(2) It only accounts for energy contribution from matters.
But gravitation energy must have contribution.
18 The above construction does not work in general relativity because
(1) No Killing vector field in general spacetime (Komar mass in the presence of Killing field by Noether’s theorem).
(2) It only accounts for energy contribution from matters.
But gravitation energy must have contribution.
Schwarzschild is a vacuum solution and binding energy depending on distance apart.
19 Equivalence principle: Gravitation has no density, can at best be measured quasilocally.
Gravitational energy depends on underlying geometry, which has been distorted by the field itself. More nonlinear than matter fields.
In general, no “background” unless gravitation is weak (or asymptotic symmetry).
Possible in the asymptotically flat case (ADM and Bondi-Sachs total energy). Definition depends on asymptotic coordinates.
Difficulties in measuring gravitational energy:
20 Gravitational energy depends on underlying geometry, which has been distorted by the field itself. More nonlinear than matter fields.
In general, no “background” unless gravitation is weak (or asymptotic symmetry).
Possible in the asymptotically flat case (ADM and Bondi-Sachs total energy). Definition depends on asymptotic coordinates.
Difficulties in measuring gravitational energy:
Equivalence principle: Gravitation has no density, can at best be measured quasilocally.
21 In general, no “background” unless gravitation is weak (or asymptotic symmetry).
Possible in the asymptotically flat case (ADM and Bondi-Sachs total energy). Definition depends on asymptotic coordinates.
Difficulties in measuring gravitational energy:
Equivalence principle: Gravitation has no density, can at best be measured quasilocally.
Gravitational energy depends on underlying geometry, which has been distorted by the field itself. More nonlinear than matter fields.
22 Possible in the asymptotically flat case (ADM and Bondi-Sachs total energy). Definition depends on asymptotic coordinates.
Difficulties in measuring gravitational energy:
Equivalence principle: Gravitation has no density, can at best be measured quasilocally.
Gravitational energy depends on underlying geometry, which has been distorted by the field itself. More nonlinear than matter fields.
In general, no “background” unless gravitation is weak (or asymptotic symmetry).
23 Difficulties in measuring gravitational energy:
Equivalence principle: Gravitation has no density, can at best be measured quasilocally.
Gravitational energy depends on underlying geometry, which has been distorted by the field itself. More nonlinear than matter fields.
In general, no “background” unless gravitation is weak (or asymptotic symmetry).
Possible in the asymptotically flat case (ADM and Bondi-Sachs total energy). Definition depends on asymptotic coordinates.
24 1 1 On each end, gij − δij = O( r ), pij = O( r 2 ), etc. Z 1 i E = lim (gij,j − gjj,i )ν dSr r→∞ 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr . r→∞ 8π Sr
(E, P1, P2, P3) ADM energy momentum.
Future timelike 4-vector (positive mass) under dominant energy condition by PMT (Schoen-Yau, Witten).
(Ω, gij , pij ) asymptotically flat spacelike hypersurface in spacetime. Ω\cptset diffeomorphic to union of ends (complements of balls in 3 R ).
25 Z 1 i E = lim (gij,j − gjj,i )ν dSr r→∞ 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr . r→∞ 8π Sr
(E, P1, P2, P3) ADM energy momentum.
Future timelike 4-vector (positive mass) under dominant energy condition by PMT (Schoen-Yau, Witten).
(Ω, gij , pij ) asymptotically flat spacelike hypersurface in spacetime. Ω\cptset diffeomorphic to union of ends (complements of balls in 3 R ).
1 1 On each end, gij − δij = O( r ), pij = O( r 2 ), etc.
26 (E, P1, P2, P3) ADM energy momentum.
Future timelike 4-vector (positive mass) under dominant energy condition by PMT (Schoen-Yau, Witten).
(Ω, gij , pij ) asymptotically flat spacelike hypersurface in spacetime. Ω\cptset diffeomorphic to union of ends (complements of balls in 3 R ).
1 1 On each end, gij − δij = O( r ), pij = O( r 2 ), etc. Z 1 i E = lim (gij,j − gjj,i )ν dSr r→∞ 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr . r→∞ 8π Sr
27 (Ω, gij , pij ) asymptotically flat spacelike hypersurface in spacetime. Ω\cptset diffeomorphic to union of ends (complements of balls in 3 R ).
1 1 On each end, gij − δij = O( r ), pij = O( r 2 ), etc. Z 1 i E = lim (gij,j − gjj,i )ν dSr r→∞ 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr . r→∞ 8π Sr
(E, P1, P2, P3) ADM energy momentum.
Future timelike 4-vector (positive mass) under dominant energy condition by PMT (Schoen-Yau, Witten).
28 “The search for a definition of quasilocal mass” is the first one in Penrose’s (1983) list of major unsolved problems in classical general relativity.
There are many reasons to search for such a concept. Many important statements in general relativity make sense only with the presence of a good definition of quasilocal mass. For example, binding energy of two bodies rotating around each other.
More importantly, a good definition of quasilocal mass may help us to control the dynamics of the gravitational field. Generalize the energy method in hyperbolic equations?
Σ a closed spacelike 2-surface in spacetime. Define quasilocal mass and quasilocal energy momentum 4-vector for Σ.
29 There are many reasons to search for such a concept. Many important statements in general relativity make sense only with the presence of a good definition of quasilocal mass. For example, binding energy of two bodies rotating around each other.
More importantly, a good definition of quasilocal mass may help us to control the dynamics of the gravitational field. Generalize the energy method in hyperbolic equations?
Σ a closed spacelike 2-surface in spacetime. Define quasilocal mass and quasilocal energy momentum 4-vector for Σ.
“The search for a definition of quasilocal mass” is the first one in Penrose’s (1983) list of major unsolved problems in classical general relativity.
30 More importantly, a good definition of quasilocal mass may help us to control the dynamics of the gravitational field. Generalize the energy method in hyperbolic equations?
Σ a closed spacelike 2-surface in spacetime. Define quasilocal mass and quasilocal energy momentum 4-vector for Σ.
“The search for a definition of quasilocal mass” is the first one in Penrose’s (1983) list of major unsolved problems in classical general relativity.
There are many reasons to search for such a concept. Many important statements in general relativity make sense only with the presence of a good definition of quasilocal mass. For example, binding energy of two bodies rotating around each other.
31 Σ a closed spacelike 2-surface in spacetime. Define quasilocal mass and quasilocal energy momentum 4-vector for Σ.
“The search for a definition of quasilocal mass” is the first one in Penrose’s (1983) list of major unsolved problems in classical general relativity.
There are many reasons to search for such a concept. Many important statements in general relativity make sense only with the presence of a good definition of quasilocal mass. For example, binding energy of two bodies rotating around each other.
More importantly, a good definition of quasilocal mass may help us to control the dynamics of the gravitational field. Generalize the energy method in hyperbolic equations?
32 1. Right asymptotics for both large sphere and small sphere limits.
2. Positivity under dominant energy condition for a large class of surfaces.
3,1 3. Rigidity: vanishes for surfaces in R .
Various attempts to define quasilocal mass (Hawking, Geroch, Bartnik, Penrose, Dougan-Mason, Huisken, etc.), will focus on the Hamilton-Jacobi analysis approach in this lecture.
What properties qualify for a valid definition?
33 2. Positivity under dominant energy condition for a large class of surfaces.
3,1 3. Rigidity: vanishes for surfaces in R .
Various attempts to define quasilocal mass (Hawking, Geroch, Bartnik, Penrose, Dougan-Mason, Huisken, etc.), will focus on the Hamilton-Jacobi analysis approach in this lecture.
What properties qualify for a valid definition?
1. Right asymptotics for both large sphere and small sphere limits.
34 3,1 3. Rigidity: vanishes for surfaces in R .
Various attempts to define quasilocal mass (Hawking, Geroch, Bartnik, Penrose, Dougan-Mason, Huisken, etc.), will focus on the Hamilton-Jacobi analysis approach in this lecture.
What properties qualify for a valid definition?
1. Right asymptotics for both large sphere and small sphere limits.
2. Positivity under dominant energy condition for a large class of surfaces.
35 Various attempts to define quasilocal mass (Hawking, Geroch, Bartnik, Penrose, Dougan-Mason, Huisken, etc.), will focus on the Hamilton-Jacobi analysis approach in this lecture.
What properties qualify for a valid definition?
1. Right asymptotics for both large sphere and small sphere limits.
2. Positivity under dominant energy condition for a large class of surfaces.
3,1 3. Rigidity: vanishes for surfaces in R .
36 What properties qualify for a valid definition?
1. Right asymptotics for both large sphere and small sphere limits.
2. Positivity under dominant energy condition for a large class of surfaces.
3,1 3. Rigidity: vanishes for surfaces in R .
Various attempts to define quasilocal mass (Hawking, Geroch, Bartnik, Penrose, Dougan-Mason, Huisken, etc.), will focus on the Hamilton-Jacobi analysis approach in this lecture.
37 Hamilton-Jacobi analysis of gravitational action (ADM, Brown-York, Hawking-Horowitz, Kijowski, etc).
Apply to the time history of a spatially bounded region in spacetime.
The Hamiltonian is a 2-surface integral at terminal time which depends on (tµ, uµ) along Σ
38 Apply to the time history of a spatially bounded region in spacetime.
The Hamiltonian is a 2-surface integral at terminal time which depends on (tµ, uµ) along Σ
Hamilton-Jacobi analysis of gravitational action (ADM, Brown-York, Hawking-Horowitz, Kijowski, etc).
39 The Hamiltonian is a 2-surface integral at terminal time which depends on (tµ, uµ) along Σ
Hamilton-Jacobi analysis of gravitational action (ADM, Brown-York, Hawking-Horowitz, Kijowski, etc).
Apply to the time history of a spatially bounded region in spacetime.
40 Hamilton-Jacobi analysis of gravitational action (ADM, Brown-York, Hawking-Horowitz, Kijowski, etc).
Apply to the time history of a spatially bounded region in spacetime.
The Hamiltonian is a 2-surface integral at terminal time which depends on (tµ, uµ) along Σ
41 tµ is a future timelike unit vector field.
uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω.
pµν is the second fundamental form of Ω in spacetime.
v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
42 uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω.
pµν is the second fundamental form of Ω in spacetime.
v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
tµ is a future timelike unit vector field.
43 tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω.
pµν is the second fundamental form of Ω in spacetime.
v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
tµ is a future timelike unit vector field. uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
44 k is the mean curvature of Σ as boundary of Ω.
pµν is the second fundamental form of Ω in spacetime.
v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
tµ is a future timelike unit vector field. uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where
45 pµν is the second fundamental form of Ω in spacetime.
v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
tµ is a future timelike unit vector field. uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω.
46 v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
tµ is a future timelike unit vector field. uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω. pµν is the second fundamental form of Ω in spacetime.
47 tµ is a future timelike unit vector field. uµ a future timelike unit normal vector (unit normal of spacelike hypersurface Ω bounded by Σ).
tµ = Nuµ + Nµ The surface Hamiltonian is Z µ µ 1 µ λ ν H(t , u ) = − Nk − N (pµν − pλgµν)v 8π Σ where k is the mean curvature of Σ as boundary of Ω. pµν is the second fundamental form of Ω in spacetime. v ν is the outward unit spacelike normal along Σ (orthogonal to uν).
48 Reference Hamiltonian should come from data associated with isometric embedding of the time history of the boundary into a reference spacetime.
3 Isometric embedding of Σ into R has been used to define Brown-York mass and Liu-Yau mass (Kijowski, Booth-Mann, Epp, Lau) with uµ = tµ (i.e. N = 1, Nµ = 0) to be specified. They satisfy nice positivity property (Shi-Tam, Liu-Yau), but not the rigidity property.
3 Unique isometric embedding into R for metrics with positive Gauss curvature (Weyl, Nirenberg, Pogorelov).
The energy is defined to be
physical Hamiltonian − reference Hamiltonian
49 3 Isometric embedding of Σ into R has been used to define Brown-York mass and Liu-Yau mass (Kijowski, Booth-Mann, Epp, Lau) with uµ = tµ (i.e. N = 1, Nµ = 0) to be specified. They satisfy nice positivity property (Shi-Tam, Liu-Yau), but not the rigidity property.
3 Unique isometric embedding into R for metrics with positive Gauss curvature (Weyl, Nirenberg, Pogorelov).
The energy is defined to be
physical Hamiltonian − reference Hamiltonian
Reference Hamiltonian should come from data associated with isometric embedding of the time history of the boundary into a reference spacetime.
50 3 Unique isometric embedding into R for metrics with positive Gauss curvature (Weyl, Nirenberg, Pogorelov).
The energy is defined to be
physical Hamiltonian − reference Hamiltonian
Reference Hamiltonian should come from data associated with isometric embedding of the time history of the boundary into a reference spacetime.
3 Isometric embedding of Σ into R has been used to define Brown-York mass and Liu-Yau mass (Kijowski, Booth-Mann, Epp, Lau) with uµ = tµ (i.e. N = 1, Nµ = 0) to be specified. They satisfy nice positivity property (Shi-Tam, Liu-Yau), but not the rigidity property.
51 The energy is defined to be
physical Hamiltonian − reference Hamiltonian
Reference Hamiltonian should come from data associated with isometric embedding of the time history of the boundary into a reference spacetime.
3 Isometric embedding of Σ into R has been used to define Brown-York mass and Liu-Yau mass (Kijowski, Booth-Mann, Epp, Lau) with uµ = tµ (i.e. N = 1, Nµ = 0) to be specified. They satisfy nice positivity property (Shi-Tam, Liu-Yau), but not the rigidity property.
3 Unique isometric embedding into R for metrics with positive Gauss curvature (Weyl, Nirenberg, Pogorelov).
52 New definition of quasilocal energy (W-Yau)
3,1 3,1 For an isometric embedding X :Σ → R and T0 ∈ R , we define the quasilocal energy to be
µ µ µ µ E(Σ, X , T0) = H(t , u ) − H(t0 , u0 ) µ where t0 = T0.
µ µ µ Prescription for u0 , t , and u ?
53 3,1 3,1 For an isometric embedding X :Σ → R and T0 ∈ R , we define the quasilocal energy to be
µ µ µ µ E(Σ, X , T0) = H(t , u ) − H(t0 , u0 ) µ where t0 = T0.
µ µ µ Prescription for u0 , t , and u ?
New definition of quasilocal energy (W-Yau)
54 µ µ µ µ E(Σ, X , T0) = H(t , u ) − H(t0 , u0 ) µ where t0 = T0.
µ µ µ Prescription for u0 , t , and u ?
New definition of quasilocal energy (W-Yau)
3,1 3,1 For an isometric embedding X :Σ → R and T0 ∈ R , we define the quasilocal energy to be
55 µ µ µ Prescription for u0 , t , and u ?
New definition of quasilocal energy (W-Yau)
3,1 3,1 For an isometric embedding X :Σ → R and T0 ∈ R , we define the quasilocal energy to be
µ µ µ µ E(Σ, X , T0) = H(t , u ) − H(t0 , u0 ) µ where t0 = T0.
56 New definition of quasilocal energy (W-Yau)
3,1 3,1 For an isometric embedding X :Σ → R and T0 ∈ R , we define the quasilocal energy to be
µ µ µ µ E(Σ, X , T0) = H(t , u ) − H(t0 , u0 ) µ where t0 = T0.
µ µ µ Prescription for u0 , t , and u ?
57 µ We take u0 to be the unit normal future timelike unit vector field µ µ ν µ in the direction of the normal part of t0 , i.e. t0 = Nu0 + N where Nµ is tangent to Σ.
µ µ This defines the reference Hamiltonian H(t0 , u0 ) which is shown to be equal to Z − kˆ Σˆ where Σˆ is the projection of Σ onto the orthogonal complement of µ t0 .
3,1 We proved a unique isometric embedding theorem into R with convex shadows, i.e. Σˆ is a convex surface in the totally geodesic 3 R .
3,1 µ Consider Σ ⊂ R and t0 a constant future timelike unit vector.
58 Z − kˆ Σˆ where Σˆ is the projection of Σ onto the orthogonal complement of µ t0 .
3,1 We proved a unique isometric embedding theorem into R with convex shadows, i.e. Σˆ is a convex surface in the totally geodesic 3 R .
3,1 µ Consider Σ ⊂ R and t0 a constant future timelike unit vector.
µ We take u0 to be the unit normal future timelike unit vector field µ µ ν µ in the direction of the normal part of t0 , i.e. t0 = Nu0 + N where Nµ is tangent to Σ.
µ µ This defines the reference Hamiltonian H(t0 , u0 ) which is shown to be equal to
59 3,1 We proved a unique isometric embedding theorem into R with convex shadows, i.e. Σˆ is a convex surface in the totally geodesic 3 R .
3,1 µ Consider Σ ⊂ R and t0 a constant future timelike unit vector.
µ We take u0 to be the unit normal future timelike unit vector field µ µ ν µ in the direction of the normal part of t0 , i.e. t0 = Nu0 + N where Nµ is tangent to Σ.
µ µ This defines the reference Hamiltonian H(t0 , u0 ) which is shown to be equal to Z − kˆ Σˆ where Σˆ is the projection of Σ onto the orthogonal complement of µ t0 .
60 3,1 µ Consider Σ ⊂ R and t0 a constant future timelike unit vector.
µ We take u0 to be the unit normal future timelike unit vector field µ µ ν µ in the direction of the normal part of t0 , i.e. t0 = Nu0 + N where Nµ is tangent to Σ.
µ µ This defines the reference Hamiltonian H(t0 , u0 ) which is shown to be equal to Z − kˆ Σˆ where Σˆ is the projection of Σ onto the orthogonal complement of µ t0 .
3,1 We proved a unique isometric embedding theorem into R with convex shadows, i.e. Σˆ is a convex surface in the totally geodesic 3 R .
61 We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
62 µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
63 Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
64 µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
65 Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
66 We assume the mean curvature vector of Σ in spacetime is 3,1 spacelike. For a reference isometric embedding X :Σ → R and a µ µ t0 , we claim there exists a unique future timelike unit vector t along Σ ⊂ M such that
µ 3,1 The expansion of Σ along t0 in R is the same as the expansion of Σ along tµ in M.
Define uµ by tµ = Nuµ + Nµ along Σ ⊂ M for the same N and Nµ.
µ µ Thus t and t0 have the same lapse functions and shift vectors 3,1 along Σ ⊂ M and Σ ⊂ R , respectively.
Use this (tµ, uµ) on Σ ⊂ M to compute the physical Hamiltonian µ µ H(t , u ) and this defines our quasilocal energy E(Σ, X , T0).
67 68 For any spacelike 2-surface Σ in spacetime, the mean curvature vector is
H = −kv µ + puµ where k is the mean curvature of Σ in a spacelike hypersurface Ω with ∂Ω = Σ and p is the trace of the restriction of pij to Σ.
The definition of H is indeed independent of Ω and the choice of uµ and v µ.
It turns out E(Σ, X , T0) can be expressed in term of the mean curvature vector field H of Σ in M and τ = −hX , T0i (the observer time function).
69 The definition of H is indeed independent of Ω and the choice of uµ and v µ.
It turns out E(Σ, X , T0) can be expressed in term of the mean curvature vector field H of Σ in M and τ = −hX , T0i (the observer time function).
For any spacelike 2-surface Σ in spacetime, the mean curvature vector is
H = −kv µ + puµ where k is the mean curvature of Σ in a spacelike hypersurface Ω with ∂Ω = Σ and p is the trace of the restriction of pij to Σ.
70 It turns out E(Σ, X , T0) can be expressed in term of the mean curvature vector field H of Σ in M and τ = −hX , T0i (the observer time function).
For any spacelike 2-surface Σ in spacetime, the mean curvature vector is
H = −kv µ + puµ where k is the mean curvature of Σ in a spacelike hypersurface Ω with ∂Ω = Σ and p is the trace of the restriction of pij to Σ.
The definition of H is indeed independent of Ω and the choice of uµ and v µ.
71 For any spacelike 2-surface Σ in spacetime, the mean curvature vector is
H = −kv µ + puµ where k is the mean curvature of Σ in a spacelike hypersurface Ω with ∂Ω = Σ and p is the trace of the restriction of pij to Σ.
The definition of H is indeed independent of Ω and the choice of uµ and v µ.
It turns out E(Σ, X , T0) can be expressed in term of the mean curvature vector field H of Σ in M and τ = −hX , T0i (the observer time function).
72 Suppose H is spacelike, then its direction defines a connection one-form AH for the normal bundle of Σ.
Fact: the mean curvature vector H0 of the isometric embedding 3,1 X :Σ → R is H0 = ∆X .
∇ and ∆ are the gradient and Laplace operator for functions on Σ with respect to the induced metric, respectively.
73 Fact: the mean curvature vector H0 of the isometric embedding 3,1 X :Σ → R is H0 = ∆X .
∇ and ∆ are the gradient and Laplace operator for functions on Σ with respect to the induced metric, respectively.
Suppose H is spacelike, then its direction defines a connection one-form AH for the normal bundle of Σ.
74 ∇ and ∆ are the gradient and Laplace operator for functions on Σ with respect to the induced metric, respectively.
Suppose H is spacelike, then its direction defines a connection one-form AH for the normal bundle of Σ.
Fact: the mean curvature vector H0 of the isometric embedding 3,1 X :Σ → R is H0 = ∆X .
75 Suppose H is spacelike, then its direction defines a connection one-form AH for the normal bundle of Σ.
Fact: the mean curvature vector H0 of the isometric embedding 3,1 X :Σ → R is H0 = ∆X .
∇ and ∆ are the gradient and Laplace operator for functions on Σ with respect to the induced metric, respectively.
76 The quasilocal energy E(Σ, X , T0) with respect to (X , T0) is 1 Z 1 Z q kˆ − [ |H|2(1 + |∇τ|2) + (∆τ)2 8π ˆ 8π Σ Σ . −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p1 + |∇τ|2|H|
where Z Z q 2 2 2 kˆ = [ |H0| (1 + |∇τ| ) + (∆τ) Σˆ Σ −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p 2 0 1 + |∇τ| |H0|
77 Z Z q 2 2 2 kˆ = [ |H0| (1 + |∇τ| ) + (∆τ) Σˆ Σ −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p 2 0 1 + |∇τ| |H0|
The quasilocal energy E(Σ, X , T0) with respect to (X , T0) is 1 Z 1 Z q kˆ − [ |H|2(1 + |∇τ|2) + (∆τ)2 8π ˆ 8π Σ Σ . −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p1 + |∇τ|2|H| where
78 The quasilocal energy E(Σ, X , T0) with respect to (X , T0) is 1 Z 1 Z q kˆ − [ |H|2(1 + |∇τ|2) + (∆τ)2 8π ˆ 8π Σ Σ . −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p1 + |∇τ|2|H| where Z Z q 2 2 2 kˆ = [ |H0| (1 + |∇τ| ) + (∆τ) Σˆ Σ −1 ∆τ − ∆τ sinh ( ) − AH (∇τ)] p 2 0 1 + |∇τ| |H0|
79 Quasilocal mass is defined to be the infimum of quasilocal energy E(Σ, X , T0) among all “admissible” (X , T0) (quasilocal observers).
m(Σ) = inf E(Σ, X , T0)
We prove:
(1) Positivity: m(Σ) ≥ 0 under dominant energy condition on spacetime and convexity assumptions on Σ.
3,1 (2) Rigidity: m(Σ) = 0 if Σ is in R .
80 (1) Positivity: m(Σ) ≥ 0 under dominant energy condition on spacetime and convexity assumptions on Σ.
3,1 (2) Rigidity: m(Σ) = 0 if Σ is in R .
Quasilocal mass is defined to be the infimum of quasilocal energy E(Σ, X , T0) among all “admissible” (X , T0) (quasilocal observers).
m(Σ) = inf E(Σ, X , T0)
We prove:
81 3,1 (2) Rigidity: m(Σ) = 0 if Σ is in R .
Quasilocal mass is defined to be the infimum of quasilocal energy E(Σ, X , T0) among all “admissible” (X , T0) (quasilocal observers).
m(Σ) = inf E(Σ, X , T0)
We prove:
(1) Positivity: m(Σ) ≥ 0 under dominant energy condition on spacetime and convexity assumptions on Σ.
82 Quasilocal mass is defined to be the infimum of quasilocal energy E(Σ, X , T0) among all “admissible” (X , T0) (quasilocal observers).
m(Σ) = inf E(Σ, X , T0)
We prove:
(1) Positivity: m(Σ) ≥ 0 under dominant energy condition on spacetime and convexity assumptions on Σ.
3,1 (2) Rigidity: m(Σ) = 0 if Σ is in R .
83 (4) The Euler-Lagrange equation for quasilocal energy can be solved for (X , T0) in both large and small sphere limit cases.
This produces a background configuration as a 2 + 1 hypersurface in the Minkowski space.
In fact, the quasilocal energy E(Sr , Xr , T0) gets linearized and acquires the Lorentzian symmetry at infinity.
µ lim E(Sr , Xr , T0) = T Pµ r→∞ 0
where Pµ = (P0, P1, P2, P3) is the AMD / Bondi-Sachs energy-momentum 4-vector, at spatial/null infinity.
(3) Quasilocal mass approaches the ADM mass and Bondi mass at spatial and null infinity, respectively and has the correct small sphere limits.
84 This produces a background configuration as a 2 + 1 hypersurface in the Minkowski space.
In fact, the quasilocal energy E(Sr , Xr , T0) gets linearized and acquires the Lorentzian symmetry at infinity.
µ lim E(Sr , Xr , T0) = T Pµ r→∞ 0
where Pµ = (P0, P1, P2, P3) is the AMD / Bondi-Sachs energy-momentum 4-vector, at spatial/null infinity.
(3) Quasilocal mass approaches the ADM mass and Bondi mass at spatial and null infinity, respectively and has the correct small sphere limits.
(4) The Euler-Lagrange equation for quasilocal energy can be solved for (X , T0) in both large and small sphere limit cases.
85 In fact, the quasilocal energy E(Sr , Xr , T0) gets linearized and acquires the Lorentzian symmetry at infinity.
µ lim E(Sr , Xr , T0) = T Pµ r→∞ 0
where Pµ = (P0, P1, P2, P3) is the AMD / Bondi-Sachs energy-momentum 4-vector, at spatial/null infinity.
(3) Quasilocal mass approaches the ADM mass and Bondi mass at spatial and null infinity, respectively and has the correct small sphere limits.
(4) The Euler-Lagrange equation for quasilocal energy can be solved for (X , T0) in both large and small sphere limit cases.
This produces a background configuration as a 2 + 1 hypersurface in the Minkowski space.
86 µ lim E(Sr , Xr , T0) = T Pµ r→∞ 0
where Pµ = (P0, P1, P2, P3) is the AMD / Bondi-Sachs energy-momentum 4-vector, at spatial/null infinity.
(3) Quasilocal mass approaches the ADM mass and Bondi mass at spatial and null infinity, respectively and has the correct small sphere limits.
(4) The Euler-Lagrange equation for quasilocal energy can be solved for (X , T0) in both large and small sphere limit cases.
This produces a background configuration as a 2 + 1 hypersurface in the Minkowski space.
In fact, the quasilocal energy E(Sr , Xr , T0) gets linearized and acquires the Lorentzian symmetry at infinity.
87 (3) Quasilocal mass approaches the ADM mass and Bondi mass at spatial and null infinity, respectively and has the correct small sphere limits.
(4) The Euler-Lagrange equation for quasilocal energy can be solved for (X , T0) in both large and small sphere limit cases.
This produces a background configuration as a 2 + 1 hypersurface in the Minkowski space.
In fact, the quasilocal energy E(Sr , Xr , T0) gets linearized and acquires the Lorentzian symmetry at infinity.
µ lim E(Sr , Xr , T0) = T Pµ r→∞ 0 where Pµ = (P0, P1, P2, P3) is the AMD / Bondi-Sachs energy-momentum 4-vector, at spatial/null infinity.
88 |H0| As long as |H| → 1, the limit of the quasilocal energy E(Σr , Xr , T0) is the same as the limit of
1 Z −hT0, I0i(|H0| − |H|) − (AH0 − AH )(∇τ). 8π Σr
where I0 is the unit timelike vector dual to H0
As τ = −hX , T0i, this is already linear in T0.
In general, suppose Σr is a family of surface in spacetime and a 3,1 family of isometric embedding Xr of Σr in R is given.
89 1 Z −hT0, I0i(|H0| − |H|) − (AH0 − AH )(∇τ). 8π Σr
where I0 is the unit timelike vector dual to H0
As τ = −hX , T0i, this is already linear in T0.
In general, suppose Σr is a family of surface in spacetime and a 3,1 family of isometric embedding Xr of Σr in R is given.
|H0| As long as |H| → 1, the limit of the quasilocal energy E(Σr , Xr , T0) is the same as the limit of
90 As τ = −hX , T0i, this is already linear in T0.
In general, suppose Σr is a family of surface in spacetime and a 3,1 family of isometric embedding Xr of Σr in R is given.
|H0| As long as |H| → 1, the limit of the quasilocal energy E(Σr , Xr , T0) is the same as the limit of
1 Z −hT0, I0i(|H0| − |H|) − (AH0 − AH )(∇τ). 8π Σr where I0 is the unit timelike vector dual to H0
91 In general, suppose Σr is a family of surface in spacetime and a 3,1 family of isometric embedding Xr of Σr in R is given.
|H0| As long as |H| → 1, the limit of the quasilocal energy E(Σr , Xr , T0) is the same as the limit of
1 Z −hT0, I0i(|H0| − |H|) − (AH0 − AH )(∇τ). 8π Σr where I0 is the unit timelike vector dual to H0
As τ = −hX , T0i, this is already linear in T0.
92 This gives a definition of total energy-momentum four-vector for an isolated system that is
(1) independent of asymptotically flat coordinates in physical spacetime.
(2) Lorentzian invariant.
(3) allowing more general asymptotics than ADM or Bondi.
93 (1) independent of asymptotically flat coordinates in physical spacetime.
(2) Lorentzian invariant.
(3) allowing more general asymptotics than ADM or Bondi.
This gives a definition of total energy-momentum four-vector for an isolated system that is
94 (2) Lorentzian invariant.
(3) allowing more general asymptotics than ADM or Bondi.
This gives a definition of total energy-momentum four-vector for an isolated system that is
(1) independent of asymptotically flat coordinates in physical spacetime.
95 (3) allowing more general asymptotics than ADM or Bondi.
This gives a definition of total energy-momentum four-vector for an isolated system that is
(1) independent of asymptotically flat coordinates in physical spacetime.
(2) Lorentzian invariant.
96 This gives a definition of total energy-momentum four-vector for an isolated system that is
(1) independent of asymptotically flat coordinates in physical spacetime.
(2) Lorentzian invariant.
(3) allowing more general asymptotics than ADM or Bondi.
97 Quasilocal mass by Hamilton-Jacobi method.
3,1 Isometric embedding of surfaces into R with convex shadows.
A canonical gauge choice so the physical surface and reference surface have the same expansion with respect to the observers.
A new definition of quasilocal mass that has both positivity and rigidity properties, and approaches expected limits.
Summary:
98 3,1 Isometric embedding of surfaces into R with convex shadows.
A canonical gauge choice so the physical surface and reference surface have the same expansion with respect to the observers.
A new definition of quasilocal mass that has both positivity and rigidity properties, and approaches expected limits.
Summary:
Quasilocal mass by Hamilton-Jacobi method.
99 A canonical gauge choice so the physical surface and reference surface have the same expansion with respect to the observers.
A new definition of quasilocal mass that has both positivity and rigidity properties, and approaches expected limits.
Summary:
Quasilocal mass by Hamilton-Jacobi method.
3,1 Isometric embedding of surfaces into R with convex shadows.
100 A new definition of quasilocal mass that has both positivity and rigidity properties, and approaches expected limits.
Summary:
Quasilocal mass by Hamilton-Jacobi method.
3,1 Isometric embedding of surfaces into R with convex shadows.
A canonical gauge choice so the physical surface and reference surface have the same expansion with respect to the observers.
101 Summary:
Quasilocal mass by Hamilton-Jacobi method.
3,1 Isometric embedding of surfaces into R with convex shadows.
A canonical gauge choice so the physical surface and reference surface have the same expansion with respect to the observers.
A new definition of quasilocal mass that has both positivity and rigidity properties, and approaches expected limits.
102 References
1. M.-T. Wang and S.-T. Yau, “Quasilocal mass in general relativity,” Phys. Rev. Lett. 102:021101 (2009) [arXiv:0804.1174]. 2. M.-T. Wang and S.-T. Yau, “Isometric embeddings into the Minkowski space and new quasi-local mass,” Comm. Math. Phys. 288, no. 3, 919 (2009) [arXiv:0805.1370]. 3. M.-T. Wang and S.-T. Yau, “Limit of quasilocal mass at spatial infinity,” Comm. Math. Phys. 296, no. 1, 271 (2010) [arXiv:0906.0200]. 4. P. Chen, M.-T. Wang, and S.-T. Yau, “Quasilocal energy-momentum at null infinity,” [arXiv:1002.0927].
103 The areal radius for each SO(3) orbit form a smooth function on a b the quotient manifold Q with Lorentz (1, 1) metric gabdx dx . Each point p ∈ Q represents a standard 2-sphere Σ(p) with areal radius r(p).
The mean curvature vector of a sphere Σ(p) is
2 ∂ 2 H = − (∂ r)g ab = − ∇r, r a ∂xb r evaluating at p.
Spherically symmetric case
a b 2 2 gabdx dx + r dσ where dσ2 = dθ2 + sin2 θdφ2 is the standard metric on S2 in polar coordinates (θ, φ).
104 The mean curvature vector of a sphere Σ(p) is
2 ∂ 2 H = − (∂ r)g ab = − ∇r, r a ∂xb r evaluating at p.
Spherically symmetric case
a b 2 2 gabdx dx + r dσ where dσ2 = dθ2 + sin2 θdφ2 is the standard metric on S2 in polar coordinates (θ, φ).
The areal radius for each SO(3) orbit form a smooth function on a b the quotient manifold Q with Lorentz (1, 1) metric gabdx dx . Each point p ∈ Q represents a standard 2-sphere Σ(p) with areal radius r(p).
105 Spherically symmetric case
a b 2 2 gabdx dx + r dσ where dσ2 = dθ2 + sin2 θdφ2 is the standard metric on S2 in polar coordinates (θ, φ).
The areal radius for each SO(3) orbit form a smooth function on a b the quotient manifold Q with Lorentz (1, 1) metric gabdx dx . Each point p ∈ Q represents a standard 2-sphere Σ(p) with areal radius r(p).
The mean curvature vector of a sphere Σ(p) is
2 ∂ 2 H = − (∂ r)g ab = − ∇r, r a ∂xb r evaluating at p.
106 Two different definitions of quasilocal mass of Σ(p):
Misner-Sharpe, Hawking, Bartnik r m(p) = (1 − |∇r|2) 2 Kijowski, Epp, Liu-Yau, W-Yau
M(p) = r(1 − |∇r|).
Suppose the mean curvature vector of Σ(p) is spacelike
107 Misner-Sharpe, Hawking, Bartnik r m(p) = (1 − |∇r|2) 2 Kijowski, Epp, Liu-Yau, W-Yau
M(p) = r(1 − |∇r|).
Suppose the mean curvature vector of Σ(p) is spacelike
Two different definitions of quasilocal mass of Σ(p):
108 Kijowski, Epp, Liu-Yau, W-Yau
M(p) = r(1 − |∇r|).
Suppose the mean curvature vector of Σ(p) is spacelike
Two different definitions of quasilocal mass of Σ(p):
Misner-Sharpe, Hawking, Bartnik r m(p) = (1 − |∇r|2) 2
109 Suppose the mean curvature vector of Σ(p) is spacelike
Two different definitions of quasilocal mass of Σ(p):
Misner-Sharpe, Hawking, Bartnik r m(p) = (1 − |∇r|2) 2 Kijowski, Epp, Liu-Yau, W-Yau
M(p) = r(1 − |∇r|).
110 In particular, as long as M is bounded, M and m approach to the same limit as r goes to infinity.
r0 At horizon where ∇r = 0, we have M(r0) = r0 and m(r0) = 2 .
∂ On a Schwarzchild, along ∂r , M is monotone decreasing and 1 M(∞) = 2 M(r0) while m is a constant m(∞) = m(r0).
The relation between M and m is M2 m = M − 2r for any p ∈ Q.
111 r0 At horizon where ∇r = 0, we have M(r0) = r0 and m(r0) = 2 .
∂ On a Schwarzchild, along ∂r , M is monotone decreasing and 1 M(∞) = 2 M(r0) while m is a constant m(∞) = m(r0).
The relation between M and m is M2 m = M − 2r for any p ∈ Q.
In particular, as long as M is bounded, M and m approach to the same limit as r goes to infinity.
112 ∂ On a Schwarzchild, along ∂r , M is monotone decreasing and 1 M(∞) = 2 M(r0) while m is a constant m(∞) = m(r0).
The relation between M and m is M2 m = M − 2r for any p ∈ Q.
In particular, as long as M is bounded, M and m approach to the same limit as r goes to infinity.
r0 At horizon where ∇r = 0, we have M(r0) = r0 and m(r0) = 2 .
113 The relation between M and m is M2 m = M − 2r for any p ∈ Q.
In particular, as long as M is bounded, M and m approach to the same limit as r goes to infinity.
r0 At horizon where ∇r = 0, we have M(r0) = r0 and m(r0) = 2 .
∂ On a Schwarzchild, along ∂r , M is monotone decreasing and 1 M(∞) = 2 M(r0) while m is a constant m(∞) = m(r0).
114