The Geometry of Static Spacetimes in GR

Transcription

1 The Geometry of Static Spacetimes in GR or What we can learn from Newton Carla Cederbaum Tübingen University Séminaire Geometric C.O.R.P. June 13, 2013 Carla Cederbaum (Tübingen) Geometrostatics 1

2 Aim Gain better understanding of gravitating relativistic systems. In particular: their total mass their center of mass their relationship to gravitating Newtonian systems Carla Cederbaum (Tübingen) Geometrostatics 2

3 Setting We specialize to the following (rather general) setting of Physical Systems static isolated finite extension of source These model static stars or black holes. Carla Cederbaum (Tübingen) Geometrostatics 3

4 Mathematical Modelling Physical Systems static isolated finite extension of source Mathematical Models timelike Killing vector, hypersurface-orthogonal asymptotically flat matter density has compact support Carla Cederbaum (Tübingen) Geometrostatics 4

5 Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 5

6 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 6

7 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Reminder: Formal Structure of GR A relativistic system consists of a space-time 4-manifold M 4 a symmetric matter tensor field T µν a Lorentzian metric 4 g µν satisfying the Einstein equations 4 Ric µν R 4 g µν = 8πG c 4 T µν with gravitational constant G, speed of light c. Carla Cederbaum (Tübingen) Geometrostatics 7

8 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations 3+1 decomposition t t=1 t=0 Carla Cederbaum (Tübingen) Geometrostatics 8

9 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations 3+1 decomposition Theorem (Choquet-Bruhat et al.) The Einstein equations can be reformulated as a well-posed hyperbolic initial value problem (for suitable matter models). Remarks: involves (non-canonical) 3+1 decomposition, constraint equations involves choice of coordinates (lapse and shift) gives rise to phenomena like gravitational waves IVP approach is used in numerical simulations Carla Cederbaum (Tübingen) Geometrostatics 9

10 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Static General Relativity Definition A relativistic system (M 4, 4 g µν, T µν ) is called static if it possesses a global timelike Killing vector field X that is hypersurface-orthogonal: 4 (α X β) = 0 and X [α 4 β X γ] = 0 Frobenius: hypersurface-orthogonal integrability of distribution X Carla Cederbaum (Tübingen) Geometrostatics 10

11 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Lapse and 3-Metric Proposition Generic static spacetimes can be canonically decomposed into M 4 = R M 3, 4 g = N 2 c 2 dt g, with induced Riemannian metric 3 g and N := 4 g(x, X) > 0 the lapse function of the static spacetime. Carla Cederbaum (Tübingen) Geometrostatics 11

12 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Static Geometry: the Facts All time-slices (M 3, 3 g) are isometric. They are embedded into (M 4 = R M 3, 4 g) with vanishing second fundamental form. The lapse function N : M 3 R + is independent of time. The matter tensor induces time-independent matter variables ρ (matter density) and S (stress tensor). We think of a static spacetime as a tuple (M 3, 3 g, N, ρ, S). Carla Cederbaum (Tübingen) Geometrostatics 12

13 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations What Are the Main Equations? Static Metric Equations 3 N = 4πG c 2 ( N ρ + N 3 Ric = 3 2 N+ 4πG c 2 Vacuum Static Metric Equations 3 ) trs N c 2 3 N = 0 N 3 Ric = 3 2 N (ρ 3 g + 2c 2 ( S 3 trs 2 )) 3 g Carla Cederbaum (Tübingen) Geometrostatics 13

14 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations Wave-Harmonic Coordinates and Regularity The static metric equations form an elliptic system of PDEs in wave-harmonic coordinates: Definition Local coordinates x i on M 3 are called wave-harmonic if they satisfy 4 x i = 0 with respect to 4 g = N 2 c 2 dt g. Theorem (Müller zum Hagen) Any solution of the static metric equations is real analytic with respect to local wave-harmonic coordinates (for suitable matter models). Carla Cederbaum (Tübingen) Geometrostatics 14

15 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 15

16 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Asymptotically Flat Ends An asymptotically flat end is a three-dimensional Riemannian manifold diffeomorphic to R 3 \ B 1 (0) satisfying fall-off conditions as r formulated as deviation from the Euclidean metric δ ij in weighted Sobolev spaces Carla Cederbaum (Tübingen) Geometrostatics 16

17 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Asymptotically Flat Manifolds An asymptotically flat manifold is Figure: example: black hole a geodesically complete three-dimensional Riemannian manifold that can be decomposed into a disjoint union of a compact set and a finite number of asymptotically flat ends. Carla Cederbaum (Tübingen) Geometrostatics 17

18 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Asymptotically Flat Manifolds An asymptotically flat manifold is compact set Figure: example: black hole a geodesically complete three-dimensional Riemannian manifold that can be decomposed into a disjoint union of a compact set and a finite number of asymptotically flat ends. Carla Cederbaum (Tübingen) Geometrostatics 17

19 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Asymptotically Flat Manifolds An asymptotically flat manifold is two ends Figure: example: black hole a geodesically complete three-dimensional Riemannian manifold that can be decomposed into a disjoint union of a compact set and a finite number of asymptotically flat ends. Carla Cederbaum (Tübingen) Geometrostatics 17

20 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions The Role of Asymptotically Flat Manifolds in GR In GR, such asymptotically flat manifolds appear as models of space-like time-slices of isolated systems (observer at infinity), e.g. stars or black holes. Figure: example: black hole Carla Cederbaum (Tübingen) Geometrostatics 18

21 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions The Role of Asymptotically Flat Manifolds in GR In GR, such asymptotically flat manifolds appear as models of t t=1 t=0 space-like time-slices of isolated systems (observer at infinity), e.g. stars or black holes. Figure: time-slices Carla Cederbaum (Tübingen) Geometrostatics 18

22 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions The Role of Asymptotically Flat Manifolds in GR In GR, such asymptotically flat manifolds appear as models of space-like time-slices of isolated systems (observer at infinity), e.g. stars or black holes. Figure: isolated system Carla Cederbaum (Tübingen) Geometrostatics 18

23 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions The Role of Asymptotically Flat Manifolds in GR In GR, such asymptotically flat manifolds appear as models of Figure: a star space-like time-slices of isolated systems (observer at infinity), e.g. stars or black holes. Carla Cederbaum (Tübingen) Geometrostatics 18

24 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions The Role of Asymptotically Flat Manifolds in GR In GR, such asymptotically flat manifolds appear as models of space-like time-slices of isolated systems (observer at infinity), e.g. stars or black holes. Figure: a black hole Carla Cederbaum (Tübingen) Geometrostatics 18

25 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Static Asymptotic Flatness Theorem (Kennefick & O Murchadha) Asymptotically Euclidean solutions of the static metric equations with compactly supported matter and N 1 as r are automatically asymptotically Schwarzschildean for some m R: N = 1 mg rc 2 + O 3( 1 r 2 ) and 3 g ij = (1 + 2mG rc 2 ) δ ij + O 2 ( 1 r 2 ) in asymptotically flat wave-harmonic coordinates as r. (Fall-off specified in weighted Sobolev spaces, m is ADM-mass of 3 g.) Carla Cederbaum (Tübingen) Geometrostatics 19

26 Static Isolated Relativistic ( Geometrostatic ) Systems The Boundary Conditions Recapitulation: Geometrostatics Definition (Geometrostatic Systems) A geodesically complete asymptotically flat solution (M 3, 3 g, N, ρ, S) of the static metric equations with compactly supported ρ and S and N 1 as r is called a geometrostatic system. Remarks: asymptotic flatness is defined in weighted Sobolev spaces definition can be extended to include black hole solutions name stresses the geometric viewpoint taken Carla Cederbaum (Tübingen) Geometrostatics 20

27 Static Isolated Relativistic ( Geometrostatic ) Systems Excursion: Static Isolated Newtonian Systems Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 21

28 Static Isolated Relativistic ( Geometrostatic ) Systems Excursion: Static Isolated Newtonian Systems Reminder: Formal Structure of NG Static Newtonian Gravity is governed by the equation Newton s Equation U = 4πGρ in R 3 where U is the Newtonian potential ρ is the matter density Carla Cederbaum (Tübingen) Geometrostatics 22

29 Static Isolated Relativistic ( Geometrostatic ) Systems Excursion: Static Isolated Newtonian Systems Asymptotic Flatness and Regularity It is well-known that Theorem Any solution of the static Newtonian equations is real analytic with respect to the canonical (Euclidean) coordinates. Theorem Asymptotically flat solutions of Newton s equation satisfy U = mg r + O( 1 r 2 ) in canonical coordinates, where m is the Newtonian mass. Carla Cederbaum (Tübingen) Geometrostatics 23

30 Static Isolated Relativistic ( Geometrostatic ) Systems Excursion: Static Isolated Newtonian Systems Formal Analogy to Newton Newtonian δ U = 4πG matter δ Ric = 0 Geometrostatics 3 N = 4πG N/c 2 matter 3 Ric = 3 2 N/N + matter U = mg/r + O(r 2 ) δ ij = δ ij N = 1 mg/rc 2 + O(r 2 ) 3 g ij = ( 1 + 2mG/rc 2) δ ij +O(r 2 ) Question: How similar is U to N physically/geometrically? Carla Cederbaum (Tübingen) Geometrostatics 24

31 Geometric and Physical Properties Equipotential Surfaces Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 25

32 Geometric and Physical Properties Equipotential Surfaces Physical Interpretation (M 3, 3 g) is a time-slice, i. e. the status of a static system at any point of time in the eyes of the chosen observer X N describes how to measure time in order to see staticity. N is usually called the static potential. Is this name justified? Is N unique for given 3 g? Do the level sets of N relate to the dynamics of test bodies? Does N contain all information about the system? Carla Cederbaum (Tübingen) Geometrostatics 26

33 Geometric and Physical Properties Equipotential Surfaces Equipotential Surfaces in Newtonian Gravity A surface Σ R 3 is an equipotential surface in Newtonian gravity if any test body constrained to Σ and not subject to any exterior forces is not accelerated. the level sets of the Newtonian potential U are the only equipotential surfaces. force on test bodies satisfies test body restricted to Σ B no acceleration r Σ F = m test U. Carla Cederbaum (Tübingen) Geometrostatics 27

34 Geometric and Physical Properties Equipotential Surfaces Equipotential Surfaces in Geometrostatics Definition (Mimicking Newtonian Gravity, C.) A timelike curve in (R M 3, 4 g = N 2 c 2 dt g) is called a constrained test body if it is a critical point of the time functional T (µ) := b a µ(τ) dτ among all timelike curves of the form µ(τ) = (t(τ), x(τ)) with x(τ) Σ. A surface Σ M 3 is called an equipotential surface in (M 3, 3 g, N) if the spatial component x(τ) of every constrained test body is a geodesic in Σ with respect to the induced 2-metric. Carla Cederbaum (Tübingen) Geometrostatics 28

35 Geometric and Physical Properties Equipotential Surfaces The Role of the Level Sets of N Theorem (C.) A hypersurface Σ M 3 is an equipotential surface in ( 3 g, N) if and only if N const on Σ. Proof: Calculus of variations with Lagrange multipliers. Corollary The level sets of the Newtonian potential U and the lapse function N have the same physical interpretation! Carla Cederbaum (Tübingen) Geometrostatics 29

36 Geometric and Physical Properties Uniqueness Properties Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 30

37 Geometric and Physical Properties Uniqueness Properties Uniqueness of N in Geometrostatics Choquet-Bruhat s theorem on the initial value problem for the Einstein equations implies Corollary (C.) The Lorentzian metric of a geometrostatic system is independent of the lapse function N: If (R M 3, 4 g) is a static as. flat solution of Einstein s vacuum equations then it is uniquely characterized by its induced 3-metric 3 g. Thus, the equipotential surfaces of (R M 3, 4 g) are uniquely determined by 3 g. Proof: The constraint equations reduce to 3 R = 0 in our case (K 0). Surfaces of equilibrium only depend on 4 g. Carla Cederbaum (Tübingen) Geometrostatics 31

38 Geometric and Physical Properties Uniqueness Properties Uniqueness of N ctd. Theorem (C.) Let ( 3 g, N) and ( 3 g, Ñ) solve the static vacuum Einstein equations with N, Ñ 1 as r and suppose that 3 g is non-flat. Then N Ñ. Proof: (vacuum part here, remainder follows from elliptic theory) Levels of N, Ñ are each the only equipotential surfaces Ñ = f N for some function f : R R. 0 = 3 Ñ = 3 (f N) = f N 3 gradn 23 g + f N 3 N so that }{{} =0 f = 0 and thus Ñ = αn + β with α, β R. 3 2 Ñ = Ñ 3 Ric = (αn + β) 3 Ric = α 3 2 N + β 3 Ric = 3 2 Ñ + β 3 Ric and 3 Ric 0 gives β = 0. Finally N, Ñ 1 as r α = 1 N Ñ. Carla Cederbaum (Tübingen) Geometrostatics 32

39 Geometric and Physical Properties Uniqueness Properties Uniqueness of N ctd. Theorem (C.) Let ( 3 g, N) and ( 3 g, Ñ) solve the static vacuum Einstein equations with N, Ñ 1 as r and suppose that 3 g is non-flat. Then N Ñ. Proof: (vacuum part here, remainder follows from elliptic theory) Levels of N, Ñ are each the only equipotential surfaces Ñ = f N for some function f : R R. 0 = 3 Ñ = 3 (f N) = f N 3 gradn 23 g + f N 3 N so that }{{} =0 f = 0 and thus Ñ = αn + β with α, β R. 3 2 Ñ = Ñ 3 Ric = (αn + β) 3 Ric = α 3 2 N + β 3 Ric = 3 2 Ñ + β 3 Ric and 3 Ric 0 gives β = 0. Finally N, Ñ 1 as r α = 1 N Ñ. Carla Cederbaum (Tübingen) Geometrostatics 32

40 Geometric and Physical Properties Uniqueness Properties Uniqueness of N ctd. Theorem (C.) Let ( 3 g, N) and ( 3 g, Ñ) solve the static vacuum Einstein equations with N, Ñ 1 as r and suppose that 3 g is non-flat. Then N Ñ. Proof: (vacuum part here, remainder follows from elliptic theory) Levels of N, Ñ are each the only equipotential surfaces Ñ = f N for some function f : R R. 0 = 3 Ñ = 3 (f N) = f N 3 gradn 23 g + f N 3 N so that }{{} =0 f = 0 and thus Ñ = αn + β with α, β R. 3 2 Ñ = Ñ 3 Ric = (αn + β) 3 Ric = α 3 2 N + β 3 Ric = 3 2 Ñ + β 3 Ric and 3 Ric 0 gives β = 0. Finally N, Ñ 1 as r α = 1 N Ñ. Carla Cederbaum (Tübingen) Geometrostatics 32

41 Geometric and Physical Properties Uniqueness Properties Uniqueness of N ctd. Theorem (C.) Let ( 3 g, N) and ( 3 g, Ñ) solve the static vacuum Einstein equations with N, Ñ 1 as r and suppose that 3 g is non-flat. Then N Ñ. Proof: (vacuum part here, remainder follows from elliptic theory) Levels of N, Ñ are each the only equipotential surfaces Ñ = f N for some function f : R R. 0 = 3 Ñ = 3 (f N) = f N 3 gradn 23 g + f N 3 N so that }{{} =0 f = 0 and thus Ñ = αn + β with α, β R. 3 2 Ñ = Ñ 3 Ric = (αn + β) 3 Ric = α 3 2 N + β 3 Ric = 3 2 Ñ + β 3 Ric and 3 Ric 0 gives β = 0. Finally N, Ñ 1 as r α = 1 N Ñ. Carla Cederbaum (Tübingen) Geometrostatics 32

42 Geometric and Physical Properties Uniqueness Properties Newtonian Gravity Versus Geometrostatics Newtonian For given matter and Galilei coordinates (i.e. flat metric), U is unique. Geometrostatic For given matter and metric, N is unique. U, N have very similar uniqueness properties justifies name static potential used in the literature can also be proved with purely analytic methods or with a 3-geodesic method combined with method of continuity Carla Cederbaum (Tübingen) Geometrostatics 33

43 Geometric and Physical Properties Uniqueness Properties Even More: Uniqueness of 3 g Theorem (C.) 3 g is unique for given N and prescribed wave-harmonic asymptotically flat coordinates. Proof: Asymptotic analysis in weighted Sobolev spaces using the static metric equations. Properties of homogeneous harmonic polynomials. For given coordinates, N and 3 g are dual. Geometrostatic systems have 4 degrees of freedom. plausibility check for Bartnik s static metric extension conjecture Carla Cederbaum (Tübingen) Geometrostatics 34

44 Mass and Center of Mass Pseudo-Newtonian Gravity Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 35

45 Mass and Center of Mass Pseudo-Newtonian Gravity Pseudo-Newtonian Gravity Definition (Pseudo-Newtonian System) Let (M 3, 3 g, N, ρ, S) be a geometrostatic system. Let U := c 2 log N γ := N 2 3 g and call U the associated pseudo-newtonian potential and (M 3, γ, U, ρ, S) the associated pseudo-newtonian system. Carla Cederbaum (Tübingen) Geometrostatics 36

46 Mass and Center of Mass Pseudo-Newtonian Gravity Pseudo-Newtonian Equations Proposition (M 3, 3 g, N, ρ, S) satisfies the static metric equations iff (M 3, γ, U, ρ, S) satisfies the pseudo-newtonian equations ( ρ γ γ ) U = 4πG e + trs 2c 2 U c 2 In vacuum, these read γ Ric = 2 8πG du du + c4 c 4 (S γ trs γ). γ U = 0 γ Ric = 2 du du. c4 Carla Cederbaum (Tübingen) Geometrostatics 37

47 Mass and Center of Mass Pseudo-Newtonian Gravity Pseudo-Newtonian Fall-Off Proposition U and γ inherit the fall-off U = mg r + O 3 ( 1 r 2 ) and γ ij = δ ij + O 2 ( 1 r 2 ) in asymptotically flat harmonic coordinates; m R mass parameter. Coordinates are harmonic w.r.t. γ iff they are wave-harmonic. m = ADM-mass of 3 g Carla Cederbaum (Tübingen) Geometrostatics 38

48 Mass and Center of Mass Pseudo-Newtonian Gravity Formal Analogy to Newton Newtonian Pseudo-Newtonian δ U = 4πG ρ δ Ric = 0 γ U = 4πG matter γ Ric = c 4 (2 du du +matter) U = mg/r + O(r 2 ) δ ij = δ ij U = mg/r + O(r 2 ) γ ij = δ ij + O(r 2 c 4 ) Formally: c equations converge (if variables converge) Carla Cederbaum (Tübingen) Geometrostatics 39

49 Mass and Center of Mass Newtonian Limit Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 40

50 Mass and Center of Mass Newtonian Limit Newtonian Limit Ehlers constructed frame theory encompassing GR and NG Frame Theory allows rigorous definition of Newtonian limit Rendall, Olyinyk showed existence of converging families g,h,t,γ GR λ = 0: NG λ Carla Cederbaum (Tübingen) Geometrostatics 41

51 Mass and Center of Mass Newtonian Limit Newtonian Limit Ehlers constructed frame theory encompassing GR and NG Frame Theory allows rigorous definition of Newtonian limit Rendall, Olyinyk showed existence of converging families g,h,t,γ GR theory of interest λ = 0: NC λ Carla Cederbaum (Tübingen) Geometrostatics 41

52 Mass and Center of Mass Newtonian Limit Newtonian Limit Ehlers constructed frame theory encompassing GR and NG Frame Theory allows rigorous definition of Newtonian limit Rendall, Olyinyk showed existence of converging families g,h,t,γ GR theory of interest λ = 0: NC λ Carla Cederbaum (Tübingen) Geometrostatics 41

53 Mass and Center of Mass Newtonian Limit Newtonian Limit of Geometrostatics Theorem (C.) Using Ehlers frame theory, one finds that in the Newtonian limit the pseudo-newtonian potential U(c) converges to the Newtonian potential U and the metric γ(c) converges to the flat metric δ. Remarks: in suitable asymptotically flat coordinates in suitably chosen uniform norms for families with a Newtonian limit Carla Cederbaum (Tübingen) Geometrostatics 42

54 Mass and Center of Mass Newtonian Limit Difficulties Definition of staticity in frame theory (Killing vector fields, hypersurface-orthogonality) Definitions of U, γ in frame theory Definition of a (locally C 1 ) Newtonian limit Handling coordinate dependence Carla Cederbaum (Tübingen) Geometrostatics 43

55 Mass and Center of Mass Newtonian Limit Physical Properties and the Newtonian Limit How do physical properties behave under the Newtonian limit along a family of pseudo-n. systems that has a Newtonian limit? Can we stretch the analogy between Newtonian and pseudo-newtonian theories further? Carla Cederbaum (Tübingen) Geometrostatics 44

56 Mass and Center of Mass Mass Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 45

57 Mass and Center of Mass Mass The Mass of a Newtonian System The mass of a system with density ρ is defined as m N := ρ dv. R 3 By U = 4πGρ and the divergence theorem rewrite m N = ρ dv = ρ dv = 1 U dv = 1 4πG 4πG R 3 C C C U ν dσ where C is any compact domain with C smooth and supp ρ C. Here: dv volume measure, dσ surface measure of C, ν outer unit normal to C Carla Cederbaum (Tübingen) Geometrostatics 46

58 Mass and Center of Mass Mass The Pseudo-Newtonian Mass In this spirit, we define Definition (Quasilocal Pseudo-Newtonian Mass) Let (γ, U) be as before. Let all geometric notions refer to γ. For any smooth surface Σ M 3 with outer unit normal ν define m PN (Σ) := 1 U 4πG ν dσ. 1 U The integral 4πG S ν dσ is well-known as the Komar-mass. Σ Carla Cederbaum (Tübingen) Geometrostatics 47

59 Mass and Center of Mass Mass The Newtonial Limit of Mass Just as in the Newtonian setting, we have Theorem (Pseudo-Newtonian Total Mass Theorem, C.) If Σ encloses the support of the matter, then m PN (Σ) = m ADM ( 3 g). In particular, m PN is independent of Σ and can be calculated "locally". Proof: Use divergence theorem, γ U = 0, U = mg/r + O(r 2 ) with m = m ADM ( 3 g), γ ij = δ ij + O(r 2 ). Theorem (Newtonian Limit of Mass Theorem, C.) For any sequence of space-times with a Newtonian limit, m ADM (c) = m PN (c) m N as c. Proof: U(c) U and γ(c) δ as c, see above. Carla Cederbaum (Tübingen) Geometrostatics 48

60 Mass and Center of Mass Center of Mass Contents 1 Static Isolated Relativistic ( Geometrostatic ) Systems The Equations The Boundary Conditions Excursion: Static Isolated Newtonian Systems 2 Geometric and Physical Properties Equipotential Surfaces Uniqueness Properties 3 Mass and Center of Mass Pseudo-Newtonian Gravity Newtonian Limit Mass Center of Mass Carla Cederbaum (Tübingen) Geometrostatics 49

61 Mass and Center of Mass Center of Mass The CoM of a Newtonian System The center of mass (CoM) of a system with density ρ is defined as z N := 1 ρ x dv m N R 3 w.r.t. Euclidean coordinates (x i ). For Σ bounding supp ρ, we can rewrite 4πG m N z N = U ν dσ + using Green s formula. Σ Σ U ν x dσ Carla Cederbaum (Tübingen) Geometrostatics 50

62 Mass and Center of Mass Center of Mass The Pseudo-Newtonian CoM Definition (Quasilocal Pseudo-Newtonian CoM) Let (γ, U) be as before. Take all geometric notions w.r.t. γ and use γ-harmonic coordinates: γ x i = 0. Then we define ( ) 1 U x z PN (Σ) := x U dσ. 4πGm PN ν ν Here, ν and σ are the outer unit normal to and surface measure on Σ w.r.t. γ, respectively. Σ Remark: γ-harmonic coordinates are also wave-harmonic. Carla Cederbaum (Tübingen) Geometrostatics 51

63 Mass and Center of Mass Center of Mass Facts on the Pseudo-Newtonian CoM Again, we can prove theorems similar to the Newtonian one: 1st Pseudo-Newtonian CoM Theorem, C. ( ) 1 U x z PN := z PN (Σ) = x U dσ 4πGm PN ν ν is independent of the specific surface Σ enclosing the support of the matter and can be calculated "locally". Proof: Use divergence theorem/green s formula, γ U = 0, and γ x i = 0. Σ Carla Cederbaum (Tübingen) Geometrostatics 52

64 Mass and Center of Mass Center of Mass Facts on the Pseudo-Newtonian CoM ctd. 2nd Pseudo-Newtonian CoM Theorem, C. The (pseudo-newtonian) CoM z can be read off the asymptotics of U: U = mg r mg z x r 3 + O( 1 r 3 ). This expression transforms adequately under change of asymptotically flat γ-harmonic coordinates. A similar expansion holds for γ ij. Proof: theory of weighted Sobolev spaces, regularity arguments. Carla Cederbaum (Tübingen) Geometrostatics 53

65 Mass and Center of Mass Center of Mass Facts on the Pseudo-Newtonian CoM ctd. 3rd Pseudo-N. CoM Theorem, C. This CoM coincides with the CoM given by a foliation by spheres of constant mean curvature (Huisken & Yau, Metzger) and with the ADM center of mass: z PN = z CMC = z ADM. H=const 1 r Proof: Uses Huang s result z CMC = z ADM and asymptotics proved above. Carla Cederbaum (Tübingen) Geometrostatics 54

66 Mass and Center of Mass Center of Mass The Newtonian Limit of the Pseudo-Newtonian CoM Newtonian Limit of Pseudo-Newtonian CoM Theorem The Newtonian limit of the pseudo-newtonian CoM and therefore also of the CMC center of mass is the CoM of the Newtonian limit along any sequence of space-times with a Newtonian limit: z CMC (c) = z PN (c) z N as c. Carla Cederbaum (Tübingen) Geometrostatics 55

67 Mass and Center of Mass Center of Mass Summary U = 4πGρ static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

68 Mass and Center of Mass Center of Mass Summary 3 N = 4πG N matter c 2 N 3 Ric = 3 2 N + N matter γ U = 4πG matter γ Ric = c 4 (2dU du + matter) static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

69 Mass and Center of Mass Center of Mass Summary test body restricted to Σ B no acceleration r Σ static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

70 Mass and Center of Mass Center of Mass Summary m PN := 1 4πG Σ Theorem m PN = m ADM U ν dσ static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

71 Mass and Center of Mass Center of Mass Summary z PN := Theorem U = mg r Theorem 1 4πGm PN Σ z PN = z CMC = z ADM ( U x x U ν ν mg z x r 3 + O( 1 r 3 ) ) dσ static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

72 Mass and Center of Mass Center of Mass Summary Theorem U(c) U γ(c) δ m PN (c) m N z PN (c) z N static Newtonian Gravity with potential U Geometrostatics ( 3 g, N) or equivalently pseudo-newtonian Gravity (γ, U) equipotential surfaces mass as surface integral expressions for CoM Newtonian limit. Carla Cederbaum (Tübingen) Geometrostatics 56

73 Mass and Center of Mass Center of Mass Possible Generalizations Stationary asymptotically flat spacetimes Difficulty: Is there a "natural" pseudo-newtonian potential? Similar surface integral expressions? Surface integral expressions for linear/angular momentum? Carla Cederbaum (Tübingen) Geometrostatics 57

74 Mass and Center of Mass Center of Mass Thank you for your attention! U = const B U r Carla Cederbaum (Tübingen) Geometrostatics 58