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Self-gravitating Elastic Bodies

  • Lars AnderssonEmail author
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 179)

Abstract

Extended objects in GR are often modelled using distributional solutions of the Einstein equations with point-like sources, or as the limit of infinitesimally small “test” objects. In this note, I will consider models of finite self-gravitating extended objects, which make it possible to give a rigorous treatment of the initial value problem for (finite) extended objects.

Keywords

Einstein Equation Elastic Body Free Boundary Condition Newtonian Gravity Reference Body 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Barack, TOPICAL REVIEW: Gravitational self-force in extreme mass-ratio inspirals. Class. Quantum Gravity 26(21), 213001 (2009)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    A.I. Harte, Motion in classical field theories and the foundations of the self-force problem (2014). (In this volume)Google Scholar
  3. 3.
    A. Pound, Motion of small bodies in curved spacetimes: an introduction to gravitational self-force (In this volume)Google Scholar
  4. 4.
    W.G. Dixon, The new mechanics of Myron Mathisson and its subsequent development (In this volume)Google Scholar
  5. 5.
    R.M. Wald, Introduction to gravitational self-force (2009)Google Scholar
  6. 6.
    L. Andersson, R. Beig, B.G. Schmidt, Elastic deformations of compact stars (2014)Google Scholar
  7. 7.
    D. Christodoulou, The Formation of Shocks in 3-dimensional Fluids. EMS Monographs in Mathematics. (European Mathematical Society, Zürich, 2007)Google Scholar
  8. 8.
    F. John, Formation of singularities in elastic waves. Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983) Lecture Notes in Physics, vol. 195, (Springer, Berlin, 1984), pp. 194–210Google Scholar
  9. 9.
    T.C. Sideris, The null condition and global existence of nonlinear elastic waves. Invent. Math. 123(2), 323–342 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2) 162(1), 109–194 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    D. Coutand, S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum (2010)Google Scholar
  12. 12.
    Y. Trakhinin, Local existence for the free boundary problem for the non-relativistic and relativistic compressible Euler equations with a vacuum boundary condition (2008)Google Scholar
  13. 13.
    H. Koch, Mixed problems for fully nonlinear hyperbolic equations. Math. Z. 214(1), 9–42 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    R. Beig, M. Wernig-Pichler, On the motion of a compact elastic body. Commun. Math. Phys. 271(2), 455–465 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    L. Andersson, T.A. Oliynyk, B.G. Schmidt, Dynamical elastic bodies in Newtonian gravity. Class. Quantum Gravity 28(23), 235006 (2011)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    H. van Elst, G.F.R. Ellis, B.G. Schmidt, Propagation of jump discontinuities in relativistic cosmology. Phys. Rev. D 62(10), 104023 (2000)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Lars Andersson, Todd A. Oliynyk, A transmission problem for quasi-linear wave equations. J. Differ. Equ. 256(6), 2023–2078 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    L. Andersson, T. Oliynyk, B. Schmidt, Dynamics of self-gravitating elastic bodies in general relativity (in preparation)Google Scholar
  19. 19.
    Yvonne Choquet-Bruhat, Helmut Friedrich, Motion of isolated bodies. Class. Quantum Gravity 23, 5941–5950 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. 20.
    S. Kind, I. Ehlers, Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid. Class. Quantum Gravity 10, 2123–2136 (1993)CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    A.D. Rendall, The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33, 1047–1053 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  22. 22.
    S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Dover, New York, 1987)Google Scholar
  23. 23.
    R. Meinel, M. Ansorg, A. Kleinwächter, G. Neugebauer, D. Petroff, Relativistic Figures of Equilibrium (Cambridge University Press, Cambridge, 2008)CrossRefzbMATHGoogle Scholar
  24. 24.
    Leon Lichtenstein, Gleichgewicthsfiguren rotirende flüssigkeiten (Springer, Berlin, 1933)CrossRefGoogle Scholar
  25. 25.
    Robert Beig, Bernd G. Schmidt, Celestial mechanics of elastic bodies. Math. Z. 258(2), 381–394 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Hand Lindblad, Karl Hakan Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary (2008)Google Scholar
  27. 27.
    V.A. Solonnikov, The problem on evolution of a self-gravitating isolated fluid mass that is not subject to the surface tension forces. J. Math. Sci. (N. Y.) 122(3), 3310–3330 (2004). Problems in mathematical analysisCrossRefMathSciNetGoogle Scholar
  28. 28.
    V.A. Solonnikov, On estimates for potentials related to the problem of stability of a rotating self-gravitating liquid. J. Math. Sci. (N. Y.) 154(1), 90–124 (2008). Problems in mathematical analysis. No. 37CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    A.K.M. Masood-ul-Alam, Proof that static stellar models are spherical. Gen. Relativ. Gravit. 39(1), 55–85 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. 30.
    Lee Lindblom, Stationary stars are axisymmetric. Astrophys. J. 208(3, part 1), 873–880 (1976)CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Uwe Heilig, On the existence of rotating stars in general relativity. Commun. Math. Phys. 166(3), 457–493 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    R. Beig, J.M. Heinzle, B.G. Schmidt, Helically symmetric N-Particle solutions in scalar gravity. Phys. Rev. Lett. 98(12), 121102\(-+\) (2007)Google Scholar
  33. 33.
    R. Beig, B.G. Schmidt, Helical solutions in scalar gravity. Gen. Relativity Gravit. 41, 2031–2043 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  34. 34.
    K. Uryū, F. Limousin, J.L. Friedman, E. Gourgoulhon, M. Shibata, Nonconformally flat initial data for binary compact objects. Phys. Rev. D 80(12), 124004\(-+\) (2009)Google Scholar
  35. 35.
    Gustav Herglotz, Über die mechanik des deformierbaren Körpers vom Standpunkte der Relativitätsteorie. Annalen der Physik 36, 493–533 (1911)CrossRefADSzbMATHGoogle Scholar
  36. 36.
    C.B. Rayner, Elasticity in general relativity. Proc. R. Soc. Ser. A 272, 44–53 (1963)CrossRefADSzbMATHMathSciNetGoogle Scholar
  37. 37.
    B. Carter, H. Quintana, Foundations of general relativistic high-pressure elasticity theory. Proc. R. Soc. Lond. Ser. A 331, 57–83 (1972)CrossRefADSzbMATHMathSciNetGoogle Scholar
  38. 38.
    Jerzy Kijowski, Giulio Magli, Unconstrained variational principle and canonical structure for relativistic elasticity. Rep. Math. Phys. 39(1), 99–112 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. 39.
    A. Shadi Tahvildar-Zadeh, Relativistic and nonrelativistic elastodynamics with small shear strains. Ann. Inst. H. Poincaré Phys. Théor. 69(3):275–307 (1998)Google Scholar
  40. 40.
    Jiseong Park, Spherically symmetric static solutions of the Einstein equations with elastic matter source. Gen. Relativity Gravit. 32(2), 235–252 (2000)CrossRefADSzbMATHGoogle Scholar
  41. 41.
    Lars Andersson, Robert Beig, Bernd G. Schmidt, Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61(7), 988–1023 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Lars Andersson, Robert Beig, Bernd G. Schmidt, Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63(5), 559–589 (2010)zbMATHMathSciNetGoogle Scholar
  43. 43.
    J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover Publications Inc., New York, 1994). Corrected reprint of the 1983 originalGoogle Scholar
  44. 44.
    C. Truesdell, W. Noll, The Non-linear Field Theories of Mechanics, 3rd edn. (Springer, Berlin, 2004). Edited and with a preface by Stuart S. AntmanCrossRefGoogle Scholar
  45. 45.
    J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976/77)Google Scholar
  46. 46.
    R. Agemi, Global existence of nonlinear elastic waves. Inventiones Mathematicae 142, 225–250 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  47. 47.
    S. Klainerman, I. Rodnianski, J. Szeftel, The bounded L2 curvature conjecture (2012)Google Scholar
  48. 48.
    L. Andersson, B.G. Schmidt, Static self-gravitating many-body systems in Einstein gravity. Class. Quantum Gravity 26(16), 165007\(-+\) (2009)Google Scholar
  49. 49.
    C. Cederbaum, Geometrostatics: the geometry of static space-times (2012)Google Scholar
  50. 50.
    R. Beig, R.M. Schoen, On static n-body configurations in relativity. Class. Quantum Gravity 26(7), 075014\(-+\) (2009)Google Scholar
  51. 51.
    R. Beig, G.W. Gibbons, R.M. Schoen, Gravitating opposites attract. Class. Quantum Gravity 26(22), 225013\(-+\) (2009)Google Scholar
  52. 52.
    Robert M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Albert Einstein InstitutePotsdamGermany

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