On foliations related to the center of mass in General Relativity Carla Cederbaum ICMP Montr´ eal, 24th of July 2018 Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 1 / 13
Isolated system in General Relativity Consider initial data ( M 3 , g, K, µ, J ) which are “optimally” asymptotically flat: M 3 ≈ R 3 \ ball ∋ � x g ij = δ ij + O 2 ( r − 1 2 − ε ) O 1 ( r − 3 2 − ε ) , K ij = O 0 ( r − 3 − ε ) µ, J = for some ε > 0 and r = | � x | → ∞ . Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 2 / 13
Expectations of a notion of center of mass Transforms like a point particle in Special Relativity under change of observer: ❀ equivariant transformation behavior under asymptotic boosts ˜ t t ˜ t = 1 t = 1 ˜ t = 0 t = 0 Equivariant transformation under spatial translations and rotations. Point particle-like evolution under Einstein evolution equations: d z ) = � dt ( E� P (ADM-energy E , ADM-momentum � P ) Newtonian limit c → ∞ of � z ( c ) recovers Newtonian center of mass of � z limiting Newtonian isolated system Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 3 / 13
Expectations of a notion of center of mass Transforms like a point particle in Special Relativity under change of observer: ❀ equivariant transformation behavior under asymptotic boosts ˜ t t ˜ t = 1 t = 1 ˜ t = 0 t = 0 Equivariant transformation under spatial translations and rotations. Point particle-like evolution under Einstein evolution equations: d z ) = � dt ( E� P (ADM-energy E , ADM-momentum � P ) Newtonian limit c → ∞ of � z ( c ) recovers Newtonian center of mass of � z limiting Newtonian isolated system Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 3 / 13
Expectations of a notion of center of mass Transforms like a point particle in Special Relativity under change of observer: ❀ equivariant transformation behavior under asymptotic boosts ˜ t t ˜ t = 1 t = 1 ˜ t = 0 t = 0 Equivariant transformation under spatial translations and rotations. Point particle-like evolution under Einstein evolution equations: d z ) = � dt ( E� P (ADM-energy E , ADM-momentum � P ) Newtonian limit c → ∞ of � z ( c ) recovers Newtonian center of mass of � z limiting Newtonian isolated system Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 3 / 13
Expectations of a notion of center of mass Transforms like a point particle in Special Relativity under change of observer: ❀ equivariant transformation behavior under asymptotic boosts ˜ t t ˜ t = 1 t = 1 ˜ t = 0 t = 0 Equivariant transformation under spatial translations and rotations. Point particle-like evolution under Einstein evolution equations: d z ) = � dt ( E� P (ADM-energy E , ADM-momentum � P ) Newtonian limit c → ∞ of � z ( c ) recovers Newtonian center of mass of � z limiting Newtonian isolated system Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 3 / 13
Status quo Different definitions of center of mass in the literature: Definition via Hamiltonian systems: Regge–Teitelboim ’74, Beig–´ O Murchadha ’87. ❀ does not transform equivariantly and does not converge in general Asymptotic foliation definition by Huisken–Yau ’96. ❀ see below Several others (Schoen, Corvino–Wu, Chen–Wang–Yau, . . . ). ❀ do not always converge and/or can not be computed in general Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 4 / 13
Status quo Different definitions of center of mass in the literature: Definition via Hamiltonian systems: Regge–Teitelboim ’74, Beig–´ O Murchadha ’87. ❀ does not transform equivariantly and does not converge in general Asymptotic foliation definition by Huisken–Yau ’96. ❀ see below Several others (Schoen, Corvino–Wu, Chen–Wang–Yau, . . . ). ❀ do not always converge and/or can not be computed in general Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 4 / 13
� � � � � ✁ � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � ✁ � � � � � � ✁ ✁ ✁ Excursion: Isolated systems in Newtonian Gravity z N ∈ R 3 of a mass density ρ and mass m N = ´ Center of mass � R 3 ρ dV � = 0 : 1 ˆ z N = � R 3 ρ � x dV. m N Can be reformulated: U Newtonian potential with U → 0 as r → ∞ : △ U = 4 πρ. If m N � = 0 : equipotential sets Σ U H=const foliate neighborhood of infinity. Recover � z N from z N = lim � � x dA. r 1 U → 0 Σ U Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 5 / 13
� � � � � ✁ � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � ✁ � � � � � � ✁ ✁ ✁ Excursion: Isolated systems in Newtonian Gravity z N ∈ R 3 of a mass density ρ and mass m N = ´ Center of mass � R 3 ρ dV � = 0 : 1 ˆ z N = � R 3 ρ � x dV. m N Can be reformulated: U Newtonian potential with U → 0 as r → ∞ : △ U = 4 πρ. If m N � = 0 : equipotential sets Σ U H=const foliate neighborhood of infinity. Recover � z N from z N = lim � � x dA. r 1 U → 0 Σ U Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 5 / 13
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � ✁ � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � ✁ Huisken–Yau definition of center of mass I Theorem (Huisken–Yau ’96; abstract CoM ) H=const Let ( M 3 , g ) be an asymptotically spherically symmetric Riemannian manifold of masse m > 0 . There exists an (almost) unique foliation of r 1 a neighborhood of infinity by stable spheres Σ H of constant mean curvature H (CMC). � 4 δ ij + O 4 ( 1 � 1 + m Asymptotic condition: g ij = r 2 ) . 2 r Generalizations: Ye, Metzger, Metzger–Eichmair, Huang, Nerz, . . . Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 6 / 13
Huisken–Yau definition of center of mass II Theorem (Huisken–Yau ’96; coordinate CoM ) Euclidean center � z H of Σ H and center of mass � z HY : � z H := x dA, � � z HY := lim H → 0 � z H . x (Σ H ) � R 3 x i (Σ H 3 ) x i (Σ H 2 ) � z H 1 � z H 3 0 � z H 2 x i (Σ H 1 ) inside Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 7 / 13
However: Theorem (C.–Nerz ’14) Der center of mass � z HY := lim H → 0 � z H does not always converge under the assumptions of Huisken–Yau. t Explicit counterexample: graphical timeslice in T ( � x ) Schwarzschild spacetime: x ) = � a · � x a � T ( � + sin(ln r ) , r { t = − 3 } x � a ∈ R 3 ,� � a � = 0 Figure: Logarithmic plot. ∈ L 1 in general, R scalar curvature of g . Reason: R � x / Same phenomenon in Newtonian setting by changing coordinates ∈ L 1 . asymptotically if ρ � x / Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 8 / 13
However: Theorem (C.–Nerz ’14) Der center of mass � z HY := lim H → 0 � z H does not always converge under the assumptions of Huisken–Yau. t Explicit counterexample: graphical timeslice in T ( � x ) Schwarzschild spacetime: x ) = � a · � x a � T ( � + sin(ln r ) , r { t = − 3 } x � a ∈ R 3 ,� � a � = 0 Figure: Logarithmic plot. ∈ L 1 in general, R scalar curvature of g . Reason: R � x / Same phenomenon in Newtonian setting by changing coordinates ∈ L 1 . asymptotically if ρ � x / Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 8 / 13
New development Theorem (C.–Sakovich ’18) Let ( M 3 , g, K, µ, J ) be initial data. Under optimal asymptotic flatness conditions and if the ADM-energy E � = 0 , there exists a unique foliation of a neighborhood of infinity by stable spheres Σ H of constant spacetime mean � g ( � H , � curvature H = H ) (STCMC). x ∈ L 1 , the euclidean center � Assuming µ � z H of Σ H and the center of mass z satisfies a : � � z H := � x dA, � z := lim H→ 0 � z H . x i (Σ H ) a Under a weak additional decay assumption on K which seem technical. Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 9 / 13
Coordinate STCMC-center of mass R 3 x i (Σ H 3 ) x i (Σ H 2 ) � z H 1 � z H 3 0 � z H 2 x i (Σ H 1 ) inside Carla Cederbaum Center of mass in General Relativity ICMP Montr´ eal 2018 10 / 13
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