Abstract
In this chapter we discuss Einstein’s gravitational equations, which state that the presence of matter and energy creates curvature in spacetime, via
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.
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References
R.Abraham and J.Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Mass., 1978.
R.Adler, M.Bazin, and M.Schiffer, Introduction to General Relativity, McGraw-Hill, New York, 1975.
V.Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
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.
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.
A.Besse, Einstein Manifolds, Springer, New York, 1987.
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.
S.Chandrasehkar, The Mathematical Theory of Black Holes, Oxford University Press, London, 1983.
Y.Choquet-Bruhat, Théorème d’existence pour certains systèmes d’équations aux derivées partielles non linéaires, Acta Math. 88(1952), 141–225.
Y.Choquet-Bruhat, Sur l’integration des équations d’Einstein, J. Rat. Mech. Anal. 5(1956), 951–966.
Y.Choquet-Bruhat, Théorème d’existence en mecanique des fluides relativistes, Bull. Soc. Math. France 86(1958), 155–175.
Y.Choquet-Bruhat, The Cauchy problem, pp.130–168 in L.Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, New York, 1962.
Y.Choquet-Bruhat, New elliptic systems and global solutions for the constraints equations in general relativity, Comm. Math. Phys. 21(1971), 211–218.
Y.Choquet-Bruhat, Global solutions of the constraints equations on open and closed manifolds, Gen. Relat. Grav. 5(1974), 49–60.
Y.Choquet-Bruhat, J.Isenberg, and V.Moncrief, Solutions of constraints for Einstein equations, CR Acad. Sci. Paris 315(1992), 349–355.
Y.Choquet-Bruhat and T.Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Comm. Math. Phys. 89(1983), 269–275.
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.
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.
D.Christodoulou and S.Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton University Press, Princeton, N.J., 1993
D.Christodoulou and N.O’Murchadha, The boost problem in general relativity, Comm. Math. Phys. 80(1981), 271–300.
T.DeFelice and C.Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, 1990.
D.DeTurck, The Cauchy problem for Lorentz metrics with prescribed Ricci curvature, Comp. Math. 48(1983), 327–349.
C.DeWitt and B.DeWitt (eds.), Black Holes, Gordon and Breach, New York, 1973.
J.Dimock, Scattering for the wave equation on the Schwarzschild metric, Gen. Relat. Grav. 17(1985), 353–369.
A.Eddington, The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, 1922.
T.Eguchi, P.Gilkey, and A.Hanson, Gravitation, gauge theories, and differential geometry, Phys. Rep., 66(1980) 6.
A.Einstein, Zur Allgemeinen Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1915), 778–786.
A.Einstein, Der Feldgleichungen der Gravitation, Preuss. Akad. Wiss. Berlin (1915), 844–847.
A.Einstein, Hamiltonschen Prinzip und allgemeine Relativitätstheorie, Preuss. Akad. Wiss. Berlin (1916), 1111–1116.
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.
C.Evans, L.Finn, and D.Hobill (eds.), Frontiers in Numerical Relativity, Cambridge University Press, Cambridge, 1989.
A.Fischer and J.Marsden, The Einstein evolution equation as a first-order symmetric hyperbolic quasilinear system, Comm. Math. Phys. 28(1972), 1–38.
A.Fischer and J.Marsden, The Einstein equations of evolution–a geometric approach, J. Math. Phys. 13(1972), 546–568.
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.
M.Flato, R.Kerner, and A.Lichnerowicz, Physics on Manifolds, Kluwer, Boston, 1994.
T.Frankel, Gravitational Curvature, W.H.Freeman, San Fransisco, 1979.
S.Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, Cambridge, 1989.
S.Hawking and G.Ellis, The Large Scale Structure of Space-time, Cambridge University. Press, Cambridge, 1973.
S.Hawking and W.Israel (eds.), General Relativity, an Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979.
S.Hawking and W.Israel (eds.), 300 Years of Gravitation, Cambridge University Press, Cambridge, 1987.
A.Held (ed.), General Relativity and Gravitation, Plenum, New York, 1980.
D.Hilbert, Die Grundlagen der Physik I, Nachr. Gesselsch. Wiss. zu Göttingen, (1915), 395–407.
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.
L.Hughston and K.Tod, An Introduction to General Relativity, Student Texts #5, London Math. Soc., Cambridge University Press, Cambridge, 1990.
J.Isenberg (ed.), Mathematics and General Relativity, AMS, Providence, R.I., 1988.
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.
C.Isham (ed.), Relativity, Groups, and Topology, North-Holland, Amsterdam, 1984.
B.Kostant, A Course in the Mathematics of General Relativity, ARK Publications, 1988.
D.Kramer, H.Stephani, M.MacCallum, and E.Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Univeristy Press, Cambridge, 1981.
M.Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. 119(1960), 1743–1745.
C.Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Gratitationsgleichungen, Phys. Z. 23(1922), 537–539.
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.
A.Lichnerowicz, Théories Relativistes de la Gravitation et de L’Electromagnetisme, Masson et Cie, Paris, 1955.
A.Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, Benjamin, New York, 1967.
A.Lichnerowicz, Shock waves in relativistic magnetohydrodynamics under general assumptions, J. Math. Phys. 17(1976), 2135–2142.
A.Lightman, W.Press, R.Price, and S.Teukolsky, Problem Book in Relativity and Gravitation, Princeton University Press, Princeton, N.J., 1975.
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.
C.Misner, K.Thorne, and J.Wheeler, Gravitation, W.H.Freeman, New York, 1973.
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.
N.O’Murchadha and J.York, The initial-value problem of general relativity, Phys. Rev. D 10(1974), 428–446.
B.O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13(1966), 459–469.
B.O’Neill, Semi-Riemannian Geometry, Academic, New York, 1983.
J.Oppenheimer and J.Snyder, On continued gravitational contraction, Phys. Rev. 56(1939), 455–459.
J.Oppenheimer and G.Volkoff, On massive Neutron cores, Phys. Rev. 55(1939), 374–381.
R.Penrose, Gravitational collapse: The role of General Relativity, Revista del Nuovo Cimento 1(1969), 252–276.
R.Penrose, Techniques of Differential Topology in Relativity, Reg. Conf. Ser. in Appl. Math. #7, SIAM, Phila., 1972.
R.Penrose and W.Rindler, Spinors and Space-Time, Cambridge University Press, Cambridge, 1984.
W.Rindler, Essential Relativity, Springer, New York, 1977.
R.Sachs and H.Wu, General Relativity for Mathematicians, Springer, NewYork, 1977.
J.Sanders and F.Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer, NewYork, 1985.
R.Schoen and S.-T.Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math. Phys. 65(1979), 45–76.
B.Schultz, A First Course in General Relativity, Cambridge University Press, Cambridge, 1985.
K.Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzber. Deut. Akad. Wiss. Berlin Kl. Math. Phys. Tech. (1916), 189–196.
L.Smarr (ed.), Sources of Gravitational Radiation, Cambridge University Press, Cambridge, 1979.
L.Smarr and J.York, Kinematical conditions in the construction of spacetime, Phys. Rev. D 17(1978), 2529–2551.
J.Smoller and B.Temple, Global solutions of the relativistic Euler equation, Comm. Math. Phys. 156(1993), 67–99.
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.
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.
J.Stewart, Advanced General Relativity, Cambridge University Press, Cambridge, 1990.
N.Strauman, General Relativity and Relativistic Astrophysics, Springer, NewYork, 1984.
A.Taub (ed.), Studies in Applied Mathematics, MAA Studies in Math., Vol. 7, Printice Hall, Englewood Cliffs, N.J., 1971.
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.
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.
M.Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.
R.Wald, General Relativity, University of Chicago Press, Chicago, 1984.
S.Weinberg, Gravitation and Cosmology, Wiley, New York, 1972.
G.Weinstein, On rotating black holes in equilibrium in general relativity, CPAM 43(1990), 903–948.
G.Weinstein, The stationary axisymmetric two-body problem in equilibrium in general relativity, CPAM 45(1992), 1183–1203.
H.Weyl, Space, Time, Matter, Dover, New York, 1952.
L.Witten (ed.), Gravitation: An Introduction to Current Research, Wiley, NewYork, 1962.
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.
J.York, Covariant decomposition of symmetric tensors in the theory of gravitation, Ann. Inst. H.Poincaré (Sec.A) 21(1974), 319–332.
J.York, Kinematics and dynamics of general relativity, pp.83–126 in L.Smarr (ed.), Sources of Gravitational Radiation, Cambridge University Press, Cambridge, 1979.
J.York, Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev. Lett. 28(1972), 1082–1085.
J.York, Boundary terms in the action principle of general relativity, Found. Phys. 16(1986), 249–258.
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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
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