Einstein’s Equations

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 117)


In this chapter we discuss Einstein’s gravitational equations, which state that the presence of matter and energy creates curvature in spacetime, via
$${G_{jk}} = 8\pi \kappa {T_{jk}},$$
where G jk = Ricjk — (1/2)Sg jk is the Einstein tensor, T jk is the stress-energy tensor due to the presence of matter, and κ is a positive constant. In § 1 we introduce this equation and relate it to previous discussions of stress-energy tensors and their relation to equations of motion. We recall various stationary action principles that give rise to equations of motion and show that (0.1) itself results from adding a term proportional to the scalar curvature of spacetime to standard Lagrangians and considering variations of the metric tensor.


Vector Field Euler Equation Constraint Equation Scalar Curvature Fundamental Form 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AbM]
    R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Mass., 1978.zbMATHGoogle Scholar
  2. [ABS]
    R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity, McGraw-Hill, New York, 1975.Google Scholar
  3. [Ar]
    V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.zbMATHCrossRefGoogle Scholar
  4. [ADM]
    R. Arnowitt, S. Deser, and C. Misner, The dynamics of general relativity, pp. 227–265 in [Wit].Google Scholar
  5. [Bac]
    A. Bachelot, Scattering operator for Maxwell equations outside Schwarzschild black-hole, pp. 38–48 in Integral Equations and Inverse Problems, V. Petkov and L. Lazarov (eds.), Longman, New York, 1991.Google Scholar
  6. [Bes]
    A. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987.zbMATHCrossRefGoogle Scholar
  7. [Chan]
    S. Chandrasekhar, An introduction to the theory of the Kerr metric and its perturbations, pp. 370–453 in [HI1].Google Scholar
  8. [Chan2]
    S. Chandrasehkar, The Mathematical Theory of Black Holes, Oxford Univ. Press, London, 1983.Google Scholar
  9. [CBrl]
    Y. Choquet-Bruhat, Théorème d’existence pour certains systèmes d’équations aux derivées partielles non linéaires, Acta Math. 88(1952), 141–225.MathSciNetCrossRefGoogle Scholar
  10. [CBr2]
    Y. Choquet-Bruhat, Sur l’intégration des équations d’Einstein, J. Rat. Mech.Anal. 5(1956), 951–966.Google Scholar
  11. [CBr3]
    Y. Choquet-Bruhat, Théorème d’existence en mécanique des fluides relativistes, Bull. Soc. Math. France 86(1958), 155–175.MathSciNetzbMATHGoogle Scholar
  12. [CBr4]
    Y. Choquet-Bruhat, The Cauchy problem, pp. 130–168 in [Wit].Google Scholar
  13. [CBr5]
    Y. Choquet-Bruhat, New elliptic systems and global solutions for the constraints equations in general relativity, Comm. Math. Phys. 21(1971), 211–218.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [CBr6]
    Y Choquet-Bruhat, Global solutions of the constraints equations on open and closed manifolds, General Relativity and Gravitation 5(1974), 49–60.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [CBIM]
    Y Choquet-Bruhat, J. Isenberg, and V. Moncrief, Solutions of constraints for Einstein equations, CR Acad. Sci. Paris 315(1992), 349–355.MathSciNetzbMATHGoogle Scholar
  16. [CBR]
    Y Choquet-Bruhat and T. Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Comm. Math. Phys. 89(1983), 269–275.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [CBY1]
    Y Choquet-Bruhat and J. York, The Cauchy Problem, pp. 99–172 in [Hel], Vol. 1.Google Scholar
  18. [CBY2]
    Y Choquet-Bruhat and J. York, Hyperbolicity of the dynamical Einstein equations in the general case, Ann. Inst. Henri Poincaré, to appear.Google Scholar
  19. [CK]
    D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of theMinkowski Space, Princeton Univ. Press, Princeton, N. J., 1993Google Scholar
  20. [CO]
    D. Christodoulou and N. O’Murchadha, The boost problem in general relativity, Comm. Math. Phys. 80(1981), 271–300.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [DC]
    T. DeFelice and C. Clarke, Relativity on Curved Manifolds, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
  22. [DeT]
    D. DeTurck, The Cauchy problem for Lorentz metrics with prescribed Ricci curvature, Comp. Math. 48(1983), 327–349.MathSciNetzbMATHGoogle Scholar
  23. [DD]
    C. DeWitt and B. DeWitt (eds.), Black Holes, Gordon and Breach, New York, 1973.Google Scholar
  24. [Dim]
    J. Dimock, Scattering for the wave equation on the Schwarzschild metric, Gen. Relativ. Grav. 17(1985), 353–369.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [Edd]
    A. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press, Cambridge, 1922.Google Scholar
  26. [EGH]
    T. Eguchi, P. Gilkey, and A. Hanson, Gravitation, gauge theories, and differential geometry. Physics Reports, Vol. 66, no. 6, Dec. 1980.Google Scholar
  27. [Einl]
    A. Einstein, Zur Allgemeinen Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1915), 778–786.Google Scholar
  28. [Ein2]
    A. Einstein, Der Feldgleichungen der Gravitation, Preuss. Akad. Wiss. Berlin (1915), 844–847.Google Scholar
  29. [Ein3]
    A. Einstein, Hamiltonschen Prinzip und allgemeine Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1916), 1111–1116.Google Scholar
  30. [Ev]
    C. Evans, Enforcing the momentum constraints during axisymmetric spacelike simulations, pp. 194–205 in [EFH].Google Scholar
  31. [EFH]
    C. Evans, L. Finn, and D. Hobill (cas.), Frontiers in Numerical Relativity, Cambridge Univ. Press, Cambridge, 1989.zbMATHGoogle Scholar
  32. [FMI]
    A. Fischer and J. Marsden, The Einstein evolution equation as a first-order symmetric hyperbolic quasilinear system, Comm. Math. Phys. 28(1972), 1–38.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [FM2]
    A. Fischer and J. Marsden, The Einstein equations of evolution-a geometric approach,/. Math. Phys. 13(1972), 546–568.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [FM3]
    A. Fischer and J. Marsden, The initial value problem and the dynamical formulation of general relativity, pp. 138–211 of [HI1].Google Scholar
  35. [FKL]
    M. Flato, R. Kerner, and A. Lichnerowicz, Physics on Manifolds, Kluwer, Boston, 1994.zbMATHCrossRefGoogle Scholar
  36. [Fran]
    T. Frankel, Gravitational Curvature, W. H. Freeman, San Fransisco, 1979.zbMATHGoogle Scholar
  37. [Fui]
    S. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge Univ. Press, Cambridge, 1989.zbMATHCrossRefGoogle Scholar
  38. [HE]
    S. Hawking and G. Ellis, The Large Scale Structure of Space-time, Cambridge Univ. Press, Cambridge, 1973.zbMATHCrossRefGoogle Scholar
  39. [HI1]
    S. Hawking and W. Israel (eds.), General Relativity, an Einstein Centenary Survey, Cambridge Univ. Press, Cambridge, 1979.zbMATHGoogle Scholar
  40. [HI2]
    S. Hawking and W. Israel (eds.), 300 Years of Gravitation, Cambridge Univ. Press, Cambridge, 1987.Google Scholar
  41. [Hel]
    A. Held (ed.), General Relativity and Gravitation, Plenum, New York, 1980.Google Scholar
  42. [Hilb]
    D. Hubert, Die Grundlagen der Physik I, Nachr. Gesselsch. Wiss. zu Göttingen, (1915), 395–407.Google Scholar
  43. [HKM]
    T. Hughes, T. Kato, and J. Marsden, Well-posed quasi-linear second order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. Anal. 63(1976), 273–294.MathSciNetGoogle Scholar
  44. [HT]
    L. Hughston and K. Tod, An Introduction to General Relativity, Student Texts #5, London Math. Soc, Cambridge Univ. Press, Cambridge, 1990.zbMATHGoogle Scholar
  45. [Is]
    J. Isenberg (ed.), Mathematics and General Relativity, AMS, Providence, R. I., 1988.zbMATHGoogle Scholar
  46. [IM]
    J. Isenberg and V. Moncrief, Some results on non constant mean curvature solutions of the Einstein constraint equations, pp. 295–302 in [FKL].Google Scholar
  47. [Ish]
    C. Isham (ed.), Relativity, Groups, and Topology, North-Holland, Amsterdam, 1984.Google Scholar
  48. [Kos]
    B. Kostant, A Course in the Mathematics of General Relativity, ARK Publications, 1988.Google Scholar
  49. [KSMH]
    D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Univ. Press, Cambridge, 1981.Google Scholar
  50. [Km]
    M. Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. 119(1960), 1743–1745.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [Lan]
    C. Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Grati-tationsgleichungen, Phys. Z. 23(1922), 537–539.zbMATHGoogle Scholar
  52. [Lich1]
    A. Lichnerowicz, L’intégration des équations de la gravitation relativiste et le problème des n corps, J. Math. Pures et Appl. 23(1944), 37–63.MathSciNetzbMATHGoogle Scholar
  53. [Lich2]
    A. Lichnerowicz, Théories Relativistes de la Gravitation et de LÎElectromag-netisme, Masson et Cie, Paris, 1955.Google Scholar
  54. [Lich3]
    A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, Benjamin, New York, 1967.zbMATHGoogle Scholar
  55. [Lich4]
    A. Lichnerowicz, Shock waves in relativistic magnetohydrodynamics under general assumptions, J. Math. Phys. 17(1976), 2135–2142.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [LPPT]
    A. Lightman, W. Press, R. Price, and S. Teukolsky, Problem Book in Relativity and Gravitation, Princeton Univ. Press, Princeton, N. J., 1975.zbMATHGoogle Scholar
  57. [MS]
    J. Miller and D. Sciama, Gravitational Collapse to the black hole state, pp. 359–391 in [Hel], Vol. 2.Google Scholar
  58. [MTW]
    C. Misner, K. Thome, and J. Wheeler, Gravitation, W. H. Freeman, New York, 1973.Google Scholar
  59. [OY1]
    N. O’Murchadha and J. York, Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys, 14(1973), 1551–1557.MathSciNetzbMATHCrossRefGoogle Scholar
  60. [OY2]
    N. O’Murchadha and J. York, The initial-value problem of general relativity, Phys. Rev. D 10(1974), 428–446.MathSciNetCrossRefGoogle Scholar
  61. [ON]
    B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13(1966), 459–469.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [ON2]
    B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.zbMATHGoogle Scholar
  63. [OS]
    J. Oppenheimer and J. Snyder, On continued gravitational contraction, Phys.Rev. 56(1939), 455–459.zbMATHCrossRefGoogle Scholar
  64. [OV]
    J. Oppenheimer and G. Volkoff, On massive Neutron cores, Phys. Rev. 55(1939), 374–381.zbMATHCrossRefGoogle Scholar
  65. [Peni]
    R. Penrose, Gravitational collapse: The role of General Relativity, Revista delNuovo Cimento 1(1969), 252–276.Google Scholar
  66. [Pen2]
    R. Penrose, Techniques of Differential Topology in Relativity, Reg. Conf. Ser. in Appl. Math. #7, SIAM, Phila., 1972.zbMATHCrossRefGoogle Scholar
  67. [PR]
    R. Penrose and W. Rindler, Spinors and Space-Time, Cambridge Univ. Press, Cambridge, 1984.zbMATHCrossRefGoogle Scholar
  68. [Rin]
    W. Rindler, Essential Relativity, Springer-Verlag, New York, 1977.zbMATHCrossRefGoogle Scholar
  69. [SW]
    R. Sachs and H. Wu, General Relativity for Mathematicians, Springer-Verlag, New York, 1977.zbMATHCrossRefGoogle Scholar
  70. [S V]
    J. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.zbMATHCrossRefGoogle Scholar
  71. [S Y]
    R. Schoen and S .-T. Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math. Phys. 65(1979), 45–76.MathSciNetzbMATHCrossRefGoogle Scholar
  72. [Schu]
    B. Schultz, A First Course in General Relativity, Cambridge Univ. Press, Cambridge, 1985.Google Scholar
  73. [Schw]
    K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzber. Deut. Akad. Wiss. Berlin Kl. Math.-Phys. Tech. (1916), 189–196.Google Scholar
  74. [Sm]
    L. Smarr (ed.), Sources of Gravitational Radiation, Cambridge Univ. Press, Cambridge, 1979.zbMATHGoogle Scholar
  75. [SY]
    L. Smarr and J. York, Kinematical conditions in the construction of spacetime, Phys. Rev. D 17(1978), 2529–2551.MathSciNetCrossRefGoogle Scholar
  76. [ST]
    J. Smoller and B. Temple, Global solutions of the relativistic Euler equation, Comm. Math. Phys. 156(1993), 67–99.MathSciNetzbMATHCrossRefGoogle Scholar
  77. [ST2]
    J. Smoller and B. Temple, Shock-wave solutions of the Einstein equations: the Oppenheimer-Snyder model of gravitational collapse extended to the case of non-zero pressure, Arch. Rat. Mech. Anal. 128(1994), 249–297.MathSciNetzbMATHCrossRefGoogle Scholar
  78. [SWYM]
    J. Smoller, A. Wasserman, S.-T. Yau, and B. McLeod, Smooth static solutions of the Einstein/Yang-Mills equations, Comm. Math. Phys. 143(1991), 115–147.MathSciNetzbMATHCrossRefGoogle Scholar
  79. [Stew]
    J. Stewart, Advanced General Relativity, Cambridge Univ. Press, Cambridge, 1990.zbMATHGoogle Scholar
  80. [Str]
    N. Strauman, General Relativity and Relativistic Astrophysics, Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
  81. [Tau1]
    A. Taub (ed.), Studies in Applied Mathematics, MAA Studies in Math., Vol. 7, Printice Hall, Englewood Cliffs, N. J., 1971.Google Scholar
  82. [Tau2]
    A. Taub, Relativistic hydrodynamics, pp. 150–180 in [Taul].Google Scholar
  83. [Tau3]
    A. Taub, High-frequency gravitational waves, two-timing, and averaged La-grangians, pp. 539–555 in [Hel], Vol. 1.Google Scholar
  84. [Tay]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.zbMATHCrossRefGoogle Scholar
  85. [Wa]
    R. Wald, General Relativity, Univ. of Chicago Press, Chicago, 1984.zbMATHCrossRefGoogle Scholar
  86. [Wein]
    S. Weinberg, Gravitation and Cosmology, Wiley, New York, 1972.Google Scholar
  87. [We1]
    G. Weinstein, On rotating black holes in equilibrium in general relativity, CPAM 43(1990), 903–948.zbMATHGoogle Scholar
  88. [We2]
    G. Weinstein, The stationary axisymmetric two-body problem in equilibrium in general relativity, CPAM 45(1992), 1183–1203.zbMATHGoogle Scholar
  89. [Wey]
    H. Weyl, Space, Time, Matter, Dover, New York, 1952.Google Scholar
  90. [Wit]
    L. Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, New York, 1962.zbMATHGoogle Scholar
  91. [Yo1]
    J. York, Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, J. Math. Phys. 14(1973), 456–464.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [Yo2]
    J. York, Covariant decomposition of symmetric tensors in the theory of gravitation, Ann. Inst. H.Poincaré (Sec.A) 21(1974), 319–332.MathSciNetGoogle Scholar
  93. [Yo3]
    J. York, Kinematics and dynamics of general relativity, pp. 83–126 in [Sm].Google Scholar
  94. [Yo4]
    J. York, Role of conformai three-geometry in the dynamics of gravitation, Phys.Rev. Lett. 28(1972), 1082–1085.CrossRefGoogle Scholar
  95. [Yo5]
    J. York, Boundary terms in the action principle of general relativity, Foundations of Physics 16(1986), 249–258.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

Personalised recommendations