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!1 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr : r!1 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. Ωncptset diffeomorphic to union of ends (complements of balls in 3 R ). 25 Z 1 i E = lim (gij;j − gjj;i )ν dSr r!1 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr : r!1 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. Ωncptset 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. Ωncptset 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!1 16π Sr Z 1 j Pi = lim (pij − δij pkk )ν dSr : r!1 8π Sr 27 (Ω; gij ; pij ) asymptotically flat spacelike hypersurface in spacetime.