Self-gravitating elastic bodies

Transcription

1 Self-gravitating elastic bodies Lars Andersson PDEs, relativity and non-linear waves, Granada 2010 joint work with Robert Beig (Vienna), Bernd Schmidt (AEI) Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

2 Outline 1 Background 2 Classical Elasticity 3 Elastic bodies in GR 4 Open problems Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

3 Background Matter models: fluids collisionless matter (Vlasov) fields (YM, scalar field,... ) Newtonian case: Steady states of fluid configurations in Newtonian gravity may be complicated (Dedekind, Jacobi ellipsoids), cf. (Chandrasekhar, 1987), (Meinel, Ansorg, Kleinwächter, Neugebauer, & Petroff, 2008). (Lichtenstein, 1933) constructed steady rotating fluid configurations in Newtonian gravity. (Beig & Schmidt, 2008) constructed static and rotating elastic bodies in Newtonian gravity extending the work of Lichtenstein. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

4 Background GR case: Two spheres in steady rotation possible in Newtonian gravity, but must radiate in GR! Static fluids bodies are spherically symmetric (Masood-ul-Alam, 2007). (Lindblom, 1976) gave an (heuristic) argument showing that stationary fluids in GR are axi-symmetric (Heilig, 1995) constructed rotating fluid bodies in GR. Stars are usually modelled using fluid matter hence spherically symmetric or axi-symmetric Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

5 Background We see many non-symmetric steady states around us need a consistent model for non-symmetric extended bodies. Non-symmetric extended bodies in GR can be modelled using elastic matter. Relativistic elasticity has been studied since shortly after the introduction of relativity, cf. (Herglotz, 1911) (special relativity), (Rayner, 1963), (Carter & Quintana, 1972), (Kijowski & Magli, 1997), (Tahvildar-Zadeh, 1998). Neutron stars are rapidly spinning and modelled using an elastic crust containing a superfluid interior. Mountains on neutron stars are expected to be related to the glitches observed in the rotation speed of neutron stars giving enhanced gravitational wave emission. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

6 Results presented here Construction of static bodies in GR first construction of non-spherical static bodies in GR The bodies have a discontinuity in the density at the surface Construction of rotating bodies in GR Construction of static n-body configurations in GR. Initial value problem for self-gravitating elastic bodies in Newtonian gravity. ρ Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

7 Classical elasticity An elastic body is described in terms of configurations with respect to a reference body B, a domain in the extended body R 3 B φ µ B f 1 (B) f φ B = id R 3 B, x A M, x µ = (t, x i ), g µν f A Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

8 Classical elasticity Action for a hyperelastic body: 1 S = 2 ρv 2 nǫ where n = det f, and ǫ = ǫ(f, f) is the stored energy function ρ = nm is the mass density v 2 = v i v j δ ij, v i = φ i,af A,t = velocity Elastic stress tensor Euler-Lagrange equation τ j i = n ǫ f A f A,j,i ρ v i + j τ i j = 0, in f 1 (B), τ i j n j f 1 (B) = 0 Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

9 Classical elasticity The Euler-Lagrange equation corresponds to Newton s force law F = ma together with the zero traction boundary condition which allows for free motion of the body. The Cauchy problem for the elastic body is an initial-boundary value problem with Neumann type boundary conditions. Well-posedness (hyperbolicity) is determined by the positivity properties of the elasticity tensor L A i B j = 2 ǫ f A,i f B,j eg. pointwise stability L A i B j ξ A iξ B j Cξ A iξ B jδ AB δ ij Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

10 Classical elasticity Let γ AB = f A,if B,jδ ij and let λ i, i = 1, 2, 3 be the invariants of γ A B = γ AC δ B CB. The material is frame independent if ǫ = ǫ(f,γ AB ). isotropic if ǫ = ǫ(λ i ) Remarks: τ i j = τ i j (f, f) fully nonlinear PDE Polyconvexity (Ball, 1976/77): ǫ(f) = ˆǫ(F, CofF, det F), F = (φ i,a), with ˆǫ convex cancellations convergence of minimizing sequences Null condition small data global existence for nonlinear elastic waves in R 3 (Sideris, 1996) Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

11 Elastic body in Newtonian gravity Action U 2 S = 8πG +ρu ρv 2 nǫ Euler-Lagrange equations ρ v i + i τ j i = ρ i U, in f 1 (B) τ i j n j f 1 (B) = 0 U = 4πGρχ f 1(B) Assuming suitable constitutive relations, the initial value problem for a Newtonian self-gravitating body is an elliptic-hyperbolic system with Neumann type boundary conditions. We prove well-posedness, this gives the first construction of self-gravitating dynamical bodies, cf. (Andersson, Oliynyk, & Schmidt, 2010). Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

12 Elastic body in Newtonian gravity An important step in the proof is to understand the regularity of the Newtonian potential U. Due to the jump in the matter density ρχ f 1(B) it must hold that 2 U has a jump discontinuity at f 1 (B). However, U has full regularity up to f 1 (B). Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

13 Equilibration Gauss law and the zero traction bundary condition gives for any Euclidean Killing field ξ i with ξ i,j = ξ [i,j] ξ j i i τ j = ξ j τ i j n i = 0, f 1 (B) f 1 (B) The body is static if the stress load balances the gravitational load i i τ j = b i := ρ j U In particular such a load must be equilibrated ξ i b i = 0, for any Killing field ξ i f 1 (B) Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

14 Self-gravitating elastic bodies in GR Action R g S = 16πG + Λ g. where Λ = Λ(f, f, g) is the energy density of the material in its own rest frame. Einstein equations G µν = 8πGT µν, where G µν = R µν 1 2 Rg µν, T µν = 2 Λ g µν Λg µν. Let γ AB = f A,µf B,νg µν. The relativistic number density is n = det(γ AB ) 1/2, and the stored energy function ǫ is given by Λ = nǫ. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

15 Self-gravitating elastic bodies in GR General covariance demands frame invariance, i.e. A material is isotropic if ǫ = ǫ(f,γ AB, g), γ AB = f A,µf B,νg µν ǫ = ǫ(λ 1,λ 2,λ 3 ) where λ i are the invariants of the matrix γ A B = γ AC δ B CB Fluids in GR correspond to a stored energy function of the form In this case ǫ = ǫ(detγ AB ) T µν = nǫu µ u ν + p(g µν + u µ u ν ) where f A,µu µ = 0, u µ u µ = 1. In particular, there are no transverse pressures. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

16 Static body in GR Assume (M, g µν ) is static. With M = R M, x µ = (t, x i ), a Kaluza-Klein reduction gives the spacetime metric g αβ dx α dx β = e 2U dt 2 + e 2U h ij dx i dx j where U, h ij depend only on x i. The action is S = M Then ξ µ µ = t, e 2U = ξ µ ξ µ. 1 h(rh 2 U 2 h 16πG )+ e U nǫ h M Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

17 Static body in GR The Euler-Lagrange equations are j (e U σ j i ) = e U (nǫ σ l l ) i U in f 1 (B), σ j i n j f 1 ( B) = 0 h U = 4πGe U (nǫ σ l l )χ f 1 (B) in R 3 S G ij = 8πG(Θ ij e U σ ij χ f 1 (B) ) in R3 S where Θ ij = 1 4πG [ iu j U 1 2 h ij U 2 ]. This system is equivalent to the 3+1 dimensional Einstein equations for the static elastic body. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

18 Static body in GR Let a relaxed reference body B be given. For small G, we construct a static self-gravitating body, i.e. a solution to the static Einstein-elastic equations, which is a deformation of B, cf. (Andersson, Beig, & Schmidt, 2006). The main steps in the construction are formulate the equations in the material frame work in harmonic coordinates This gives a system F(G, Z) = 0, where Z = (φ, Ū, h ij) are the fields in material frame. For suitable constitutive relations, the reduced system of Einstein-elastic equations is an elliptic boundary value problem. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

19 Static body in GR The linearized elasticity operator at the reference configuration is automatically equilibrated. This means we have a kernel and cokernel corresponding to the Killing fields of the Euclidean reference metric on M and on R 3 B. Apply a projection to get an isomorphims this allows to apply the implicit function theorem to construct a solution for small G to the projected system P B F(G, Z) = 0. Show that the solution to the projected system is automatically equilibrated, i.e. it is a solution to the full system. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

20 Rotating bodies in GR Neutron stars have a solid crust, modelled by elastic matter (Carter & Samuelsson, 2006; Frauendiener & Kabobel, 2007). Previous work considers only spherically symmetric bodies, allowing for linear axi-symmetric perturbations (Karlovini & Samuelsson, 2007). We construct rigidly rotating elastic bodies with minimal symmetry (Andersson, Beig, & Schmidt, 2008). Formulate Einstein equations for a rotating body using Kaluza-Klein reduction Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

21 Rotating bodies in GR Action S = M h ( ) R h 2 DU 2 h 16πG + e4u ω 2 h + ρ e U h, M Given a relaxed, axi-symmetric reference body, we construct for small rotation velocity Ω and small G a rigidly rotating deformation. The proof is an application of the implicit function theorem to a reduced projected system, together with an equilibration argument. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

22 Static n-body configurations in GR In Newtonian gravity there are many examples of self-gravitating many-body systems, consisting of rigid bodies. Examples of the form Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

23 Static n-body configurations in GR Given a static two-body configuration of self-gravitating rigid bodies in Newtonian gravity, satisfying certain non-degeneracy conditions, we construct for small G self-gravitating two-body configurations, cf. (Andersson & Schmidt, 2009). The proof makes use of the additional degree of freedom corresponding to the difference in the centers of mass and alignments of the bodies to achieve equilibration. The result generalizes immediately to n-body configurations. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

24 Open problems: Static configurations In Newtonian gravity, one proves easily that a two bodies separated by a plane cannot be in static equilibrium. This relates to Newton s principle actio est reactio which implies that each body must be equilibrated with respect to its own self-gravity. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

25 Open problems: Static configurations In GR, we lack the concept of force, and the problem of characterizing allowed n-body configurations is open. Partial results are given in (Beig & Schoen, 2009), (Beig, Gibbons, & Schoen, 2009) They show that two bodies separated by a totally geodesic surface cannot be in static equilibrium. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

26 Open problems: Extended bodies Well-posedness has been proved for non-gravitating fluid bodies with free boundary, cf. (Lindblad, 2005). Partial results exist for self-gravitating incompressible fluid bodies in Newtonian gravity (Lindblad & Nordgren, 2008). Dynamical self-gravitating bodies in Newtonian gravity have been constructed (Andersson et al., 2010). Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

27 Open problems: Extended bodies The corresponding problems in GR are open. The only results in this direction concern bounded fluid configurations under strong restrictions, (Choquet-Bruhat & Friedrich, 2006) (Einstein-dust, regular density), (Kind & Ehlers, 1993) (spherical symmetry, allows discontinuity at the boundary), (Rendall, 1992) (restricted equation of state, smooth density) Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

28 Open problems: Extended bodies The Einstein equations can be written as a hyperbolic system, information propagates along characteristics. Therefore the irregularity at the boundary of a finite body may radiate. There must be a geometric conspiracy at the boundary of the body. Well-posedness and long-time existence for self-gravitating elastic bodies provides a consistent approach to extended bodies and the n-body problem in GR Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

29 References I Andersson, L., Beig, R., & Schmidt, B. (2006, November). Static self-gravitating elastic bodies in Einstein gravity. ArXiv General Relativity and Quantum Cosmology e-prints. Andersson, L., Beig, R., & Schmidt, B. (2008, November). Rotating elastic bodies in Einstein gravity. ArXiv e-prints. Andersson, L., Oliynyk, T., & Schmidt, B. (2010). Dynamical elastic bodies in newtonian gravity. Andersson, L., & Schmidt, B. G. (2009, August). Static self-gravitating many-body systems in Einstein gravity. Classical and Quantum Gravity, 26(16), Ball, J. M. (1976/77). Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4), Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

30 References II Beig, R., Gibbons, G. W., & Schoen, R. M. (2009, November). Gravitating opposites attract. Classical and Quantum Gravity, 26(22), Beig, R., & Schmidt, B. G. (2008). Celestial mechanics of elastic bodies. Math. Z., 258(2), Beig, R., & Schoen, R. M. (2009, April). On static n-body configurations in relativity. Classical and Quantum Gravity, 26(7), Carter, B., & Quintana, H. (1972). Foundations of general relativistic high-pressure elasticity theory. Proc. Roy. Soc. London Ser. A, 331, Carter, B., & Samuelsson, L. (2006). Relativistic mechanics of neutron superfluid in (magneto) elastic star crust. Class. Quant. Grav., 23, Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

31 References III Chandrasekhar, S. (1987). Ellipsoidal figures of equilibrium. New York : Dover, Choquet-Bruhat, Y., & Friedrich, H. (2006). Motion of Isolated bodies. Class. Quant. Grav., 23, Frauendiener, J., & Kabobel, A. (2007). The static spherically symmetric body in relativistic elasticity. Class. Quant. Grav., 24, Heilig, U. (1995). On the existence of rotating stars in general relativity. Comm. Math. Phys., 166(3), Herglotz, G. (1911). Über die mechanik des deformierbaren Körpers vom Standpunkte der Relativitätsteorie. Annalen der Physik, 36, Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

32 References IV Karlovini, M., & Samuelsson, L. (2007). Elastic stars in general relativity. IV: Axial perturbations. Class. Quant. Grav., 24, Kijowski, J., & Magli, G. (1997). Unconstrained variational principle and canonical structure for relativistic elasticity. Rep. Math. Phys., 39(1), Kind, S., & Ehlers, J. (1993). Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid. Classical Quantum Gravity, 10(10), Available from Lichtenstein, L. (1933). Gleichgewicthsfiguren rotirende flüssigkeiten. Berlin: Springer. Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

33 References V Lindblad, H. (2005). Well posedness for the motion of a compressible liquid with free surface boundary. Comm. Math. Phys., 260(2), Available from Lindblad, H., & Nordgren, K. H. (2008). A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary. (arxiv.org: ) Lindblom, L. (1976). Stationary stars are axisymmetric. Astrophys. J., 208(3, part 1), Masood-ul-Alam, A. K. M. (2007). Proof that static stellar models are spherical. Gen. Relativity Gravitation, 39(1), Available from Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

34 References VI Meinel, R., Ansorg, M., Kleinwächter, A., Neugebauer, G., & Petroff, D. (2008). Relativistic Figures of Equilibrium (Meinel, R., Ansorg, M., Kleinwächter, A., Neugebauer, G., & Petroff, D., Ed.). Rayner, C. B. (1963). Elasticity in general relativity. Proc. Roy. Soc. Ser. A, 272, Rendall, A. D. (1992, March). The initial value problem for a class of general relativistic fluid bodies. Journal of Mathematical Physics, 33, Sideris, T. C. (1996). The null condition and global existence of nonlinear elastic waves. Invent. Math., 123(2), Available from Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39

35 References VII Tahvildar-Zadeh, A. S. (1998). Relativistic and nonrelativistic elastodynamics with small shear strains. Ann. Inst. H. Poincaré Phys. Théor., 69(3), Lars Andersson (AEI) Self-gravitating elastic bodies Granada April 7, / 39