Skip to main content
SpringerLink
Log in

Einstein’s Equations

  • Chapter
  • First Online:
Book cover Partial Differential Equations III

Part of the book series: Applied Mathematical Sciences ((AMS,volume 117))

  • 7433 Accesses

Abstract

In this chapter we discuss Einstein’s gravitational equations, which state that the presence of matter and energy creates curvature in spacetime, via

$${G}_{jk} = 8\pi \kappa {T}_{jk},$$
(0.1)

where Gjk = Ricjk − (1∕2)Sgjk is the Einstein tensor, Tjk 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R.Abraham and J.Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Mass., 1978.

    Google Scholar 

  2. R.Adler, M.Bazin, and M.Schiffer, Introduction to General Relativity, McGraw-Hill, New York, 1975.

    MATH  Google Scholar 

  3. V.Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.

    Book  Google Scholar 

  4. R.Arnowitt, S.Deser, and C.Misner, The dynamics of general relativity, pp.227–265 in L.Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, New York, 1962.

    Google Scholar 

  5. 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. A.Besse, Einstein Manifolds, Springer, New York, 1987.

    Book  Google Scholar 

  7. S.Chandrasekhar, An introduction to the theory of the Kerr metric and its perturbations, pp.370–453 in S.Hawking and W.Israel (eds.), General Relativity, an Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979.

    Google Scholar 

  8. S.Chandrasehkar, The Mathematical Theory of Black Holes, Oxford University Press, London, 1983.

    Google Scholar 

  9. Y.Choquet-Bruhat, Théorème d’existence pour certains systèmes d’équations aux derivées partielles non linéaires, Acta Math. 88(1952), 141–225.

    Google Scholar 

  10. Y.Choquet-Bruhat, Sur l’integration des équations d’Einstein, J. Rat. Mech. Anal. 5(1956), 951–966.

    Google Scholar 

  11. Y.Choquet-Bruhat, Théorème d’existence en mecanique des fluides relativistes, Bull. Soc. Math. France 86(1958), 155–175.

    Google Scholar 

  12. Y.Choquet-Bruhat, The Cauchy problem, pp.130–168 in L.Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, New York, 1962.

    Google Scholar 

  13. Y.Choquet-Bruhat, New elliptic systems and global solutions for the constraints equations in general relativity, Comm. Math. Phys. 21(1971), 211–218.

    Google Scholar 

  14. Y.Choquet-Bruhat, Global solutions of the constraints equations on open and closed manifolds, Gen. Relat. Grav. 5(1974), 49–60.

    Google Scholar 

  15. Y.Choquet-Bruhat, J.Isenberg, and V.Moncrief, Solutions of constraints for Einstein equations, CR Acad. Sci. Paris 315(1992), 349–355.

    Google Scholar 

  16. Y.Choquet-Bruhat and T.Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Comm. Math. Phys. 89(1983), 269–275.

    Google Scholar 

  17. Y.Choquet-Bruhat and J.York, The Cauchy Problem, pp.99–172 in A.Held (ed.), General Relativity and Gravitation, Vol.1, Plenum, New York, 1980.

    Google Scholar 

  18. Y.Choquet-Bruhat and J.York, Geometrical well posed systems for the Einstein equations, C R Acad. Sci. Paris Ser I Math. 321(1995), 1089–1095.

    Google Scholar 

  19. D.Christodoulou and S.Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton University Press, Princeton, N.J., 1993

    MATH  Google Scholar 

  20. D.Christodoulou and N.O’Murchadha, The boost problem in general relativity, Comm. Math. Phys. 80(1981), 271–300.

    Google Scholar 

  21. T.DeFelice and C.Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  22. D.DeTurck, The Cauchy problem for Lorentz metrics with prescribed Ricci curvature, Comp. Math. 48(1983), 327–349.

    Google Scholar 

  23. C.DeWitt and B.DeWitt (eds.), Black Holes, Gordon and Breach, New York, 1973.

    Google Scholar 

  24. J.Dimock, Scattering for the wave equation on the Schwarzschild metric, Gen. Relat. Grav. 17(1985), 353–369.

    Google Scholar 

  25. A.Eddington, The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, 1922.

    MATH  Google Scholar 

  26. T.Eguchi, P.Gilkey, and A.Hanson, Gravitation, gauge theories, and differential geometry, Phys. Rep., 66(1980) 6.

    Google Scholar 

  27. A.Einstein, Zur Allgemeinen Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1915), 778–786.

    Google Scholar 

  28. A.Einstein, Der Feldgleichungen der Gravitation, Preuss. Akad. Wiss. Berlin (1915), 844–847.

    Google Scholar 

  29. A.Einstein, Hamiltonschen Prinzip und allgemeine Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1916), 1111–1116.

    Google Scholar 

  30. C.Evans, Enforcing the momentum constraints during axisymmetric spacelike simulations, pp.194–205 in C.Evans, L.Finn, and D.Hobill (eds.), Frontiers in Numerical Relativity, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  31. C.Evans, L.Finn, and D.Hobill (eds.), Frontiers in Numerical Relativity, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  32. A.Fischer and J.Marsden, The Einstein evolution equation as a first-order symmetric hyperbolic quasilinear system, Comm. Math. Phys. 28(1972), 1–38.

    Google Scholar 

  33. A.Fischer and J.Marsden, The Einstein equations of evolution–a geometric approach, J. Math. Phys. 13(1972), 546–568.

    Google Scholar 

  34. A.Fischer and J.Marsden, The initial value problem and the dynamical formulation of general relativity, pp.138–211 in S.Hawking and W.Israel (eds.), General Relativity, an Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979.

    Google Scholar 

  35. M.Flato, R.Kerner, and A.Lichnerowicz, Physics on Manifolds, Kluwer, Boston, 1994.

    Book  Google Scholar 

  36. T.Frankel, Gravitational Curvature, W.H.Freeman, San Fransisco, 1979.

    MATH  Google Scholar 

  37. S.Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, Cambridge, 1989.

    Book  Google Scholar 

  38. S.Hawking and G.Ellis, The Large Scale Structure of Space-time, Cambridge University. Press, Cambridge, 1973.

    Book  Google Scholar 

  39. S.Hawking and W.Israel (eds.), General Relativity, an Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979.

    MATH  Google Scholar 

  40. S.Hawking and W.Israel (eds.), 300 Years of Gravitation, Cambridge University Press, Cambridge, 1987.

    MATH  Google Scholar 

  41. A.Held (ed.), General Relativity and Gravitation, Plenum, New York, 1980.

    Google Scholar 

  42. D.Hilbert, Die Grundlagen der Physik I, Nachr. Gesselsch. Wiss. zu Göttingen, (1915), 395–407.

    Google Scholar 

  43. 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.

    Google Scholar 

  44. L.Hughston and K.Tod, An Introduction to General Relativity, Student Texts #5, London Math. Soc., Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  45. J.Isenberg (ed.), Mathematics and General Relativity, AMS, Providence, R.I., 1988.

    MATH  Google Scholar 

  46. J.Isenberg and V.Moncrief, Some results on non constant mean curvature solutions of the Einstein constraint equations, pp.295–302 in M.Flato, R.Kerner, and A.Lichnerowicz, Physics on Manifolds, Kluwer, Boston, 1994.

    Google Scholar 

  47. C.Isham (ed.), Relativity, Groups, and Topology, North-Holland, Amsterdam, 1984.

    Google Scholar 

  48. B.Kostant, A Course in the Mathematics of General Relativity, ARK Publications, 1988.

    Google Scholar 

  49. D.Kramer, H.Stephani, M.MacCallum, and E.Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Univeristy Press, Cambridge, 1981.

    MATH  Google Scholar 

  50. M.Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. 119(1960), 1743–1745.

    Google Scholar 

  51. C.Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Gratitationsgleichungen, Phys. Z. 23(1922), 537–539.

    Google Scholar 

  52. A.Lichnerowicz, L’integration des équations de la gravitation relativiste et le problème des n corps, J. Math. Pures et Appl. 23(1944), 37–63.

    Google Scholar 

  53. A.Lichnerowicz, Théories Relativistes de la Gravitation et de L’Electromagnetisme, Masson et Cie, Paris, 1955.

    Book  Google Scholar 

  54. A.Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, Benjamin, New York, 1967.

    MATH  Google Scholar 

  55. A.Lichnerowicz, Shock waves in relativistic magnetohydrodynamics under general assumptions, J. Math. Phys. 17(1976), 2135–2142.

    Google Scholar 

  56. A.Lightman, W.Press, R.Price, and S.Teukolsky, Problem Book in Relativity and Gravitation, Princeton University Press, Princeton, N.J., 1975.

    MATH  Google Scholar 

  57. J.Miller and D.Sciama, Gravitational Collapse to the black hole state, pp. 359–391 in A.Held (ed.), General Relativity and Gravitation, Vol.2, Plenum, New York, 1980.

    Google Scholar 

  58. C.Misner, K.Thorne, and J.Wheeler, Gravitation, W.H.Freeman, New York, 1973.

    Google Scholar 

  59. 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.

    Google Scholar 

  60. N.O’Murchadha and J.York, The initial-value problem of general relativity, Phys. Rev. D 10(1974), 428–446.

    Google Scholar 

  61. B.O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13(1966), 459–469.

    Google Scholar 

  62. B.O’Neill, Semi-Riemannian Geometry, Academic, New York, 1983.

    MATH  Google Scholar 

  63. J.Oppenheimer and J.Snyder, On continued gravitational contraction, Phys. Rev. 56(1939), 455–459.

    Google Scholar 

  64. J.Oppenheimer and G.Volkoff, On massive Neutron cores, Phys. Rev. 55(1939), 374–381.

    Google Scholar 

  65. R.Penrose, Gravitational collapse: The role of General Relativity, Revista del Nuovo Cimento 1(1969), 252–276.

    Google Scholar 

  66. R.Penrose, Techniques of Differential Topology in Relativity, Reg. Conf. Ser. in Appl. Math. #7, SIAM, Phila., 1972.

    Google Scholar 

  67. R.Penrose and W.Rindler, Spinors and Space-Time, Cambridge University Press, Cambridge, 1984.

    Book  Google Scholar 

  68. W.Rindler, Essential Relativity, Springer, New York, 1977.

    Book  Google Scholar 

  69. R.Sachs and H.Wu, General Relativity for Mathematicians, Springer, NewYork, 1977.

    Book  Google Scholar 

  70. J.Sanders and F.Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer, NewYork, 1985.

    Book  Google Scholar 

  71. R.Schoen and S.-T.Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math. Phys. 65(1979), 45–76.

    Google Scholar 

  72. B.Schultz, A First Course in General Relativity, Cambridge University Press, Cambridge, 1985.

    Google Scholar 

  73. 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. L.Smarr (ed.), Sources of Gravitational Radiation, Cambridge University Press, Cambridge, 1979.

    MATH  Google Scholar 

  75. L.Smarr and J.York, Kinematical conditions in the construction of spacetime, Phys. Rev. D 17(1978), 2529–2551.

    Google Scholar 

  76. J.Smoller and B.Temple, Global solutions of the relativistic Euler equation, Comm. Math. Phys. 156(1993), 67–99.

    Google Scholar 

  77. 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.

    Google Scholar 

  78. 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.

    Google Scholar 

  79. J.Stewart, Advanced General Relativity, Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  80. N.Strauman, General Relativity and Relativistic Astrophysics, Springer, NewYork, 1984.

    Book  Google Scholar 

  81. A.Taub (ed.), Studies in Applied Mathematics, MAA Studies in Math., Vol. 7, Printice Hall, Englewood Cliffs, N.J., 1971.

    Google Scholar 

  82. A.Taub, Relativistic hydrodynamics, pp.150–180 in A.Taub (ed.), Studies in Applied Mathematics, MAA Studies in Math., Vol. 7, Printice Hall, Englewood Cliffs, N.J., 1971.

    Google Scholar 

  83. A.Taub, High-frequency gravitational waves, two-timing, and averaged Lagrangians, pp.539–555 in A.Held (ed.), General Relativity and Gravitation, Vol.1, Plenum, New York, 1980.

    Google Scholar 

  84. M.Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.

    Book  Google Scholar 

  85. R.Wald, General Relativity, University of Chicago Press, Chicago, 1984.

    Book  Google Scholar 

  86. S.Weinberg, Gravitation and Cosmology, Wiley, New York, 1972.

    Google Scholar 

  87. G.Weinstein, On rotating black holes in equilibrium in general relativity, CPAM 43(1990), 903–948.

    Google Scholar 

  88. G.Weinstein, The stationary axisymmetric two-body problem in equilibrium in general relativity, CPAM 45(1992), 1183–1203.

    Google Scholar 

  89. H.Weyl, Space, Time, Matter, Dover, New York, 1952.

    Google Scholar 

  90. L.Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, NewYork, 1962.

    MATH  Google Scholar 

  91. 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.

    Google Scholar 

  92. J.York, Covariant decomposition of symmetric tensors in the theory of gravitation, Ann. Inst. H.Poincaré (Sec.A) 21(1974), 319–332.

    Google Scholar 

  93. J.York, Kinematics and dynamics of general relativity, pp.83–126 in L.Smarr (ed.), Sources of Gravitational Radiation, Cambridge University Press, Cambridge, 1979.

    Google Scholar 

  94. J.York, Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev. Lett. 28(1972), 1082–1085.

    Google Scholar 

  95. J.York, Boundary terms in the action principle of general relativity, Found. Phys. 16(1986), 249–258.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Michael E. Taylor

Authors
  1. Michael E. Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael E. Taylor .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Taylor, M.E. (2011). Einstein’s Equations. In: Partial Differential Equations III. Applied Mathematical Sciences, vol 117. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7049-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation