Home
• Get access
• Cited by 5
• Print publication year: 2015
• Online publication date: June 2015

# 8 - Initial Data and the Einstein Constraint Equations

from Part Three - Gravity is Geometry, after all

## Summary

Introduction

Solutions of the Einstein equations evolve from initial data given on a three-dimensional manifold M. The initial position and velocity of the gravitational field are given by a Riemannian metric g and a symmetric (0, 2) tensor K. The metric g will be the metric induced on M as a spacelike hypersurface in the spacetime S which evolves from the data, and the tensor K will be the second fundamental form of M in S. Thus an initial data set is given by a triple (M, g, K). There is currently interest in higher-dimensional gravity in the physics community, so when convenient we will discuss initial data on an n-dimensional manifold Mn which will evolve to an (n + 1)-dimensional spacetime Sn + 1 (n ≥ 3).

A basic fact of life for the Einstein equations is that the initial data g and K cannot be freely specified, but must satisfy a system of n + 1 nonlinear partial differential equations. These are called the constraint equations, and Section 8.2 deals with recent progress on solving this set of equations. On the one hand the constraint equations present a complication in the study of the initial value problem since it is a difficult (and as yet unsolved) problem to fully analyze their solutions. On the other hand, it is because of the constraint equations that physical notions of energy and momentum can be defined. It is also because of them that geometric and topological restrictions hold in certain cases on the initial manifold M, and for black holes in Σ.

We do not have the space here to give a comprehensive survey of the initial value problem, so instead we have focused on several questions on which there has been recent progress and which are currently active areas of investigation. We have chosen to give brief outlines of the main ideas involved in the study of these specific questions rather than to attempt to touch on all aspects of the field.

### Related content

[1] , , and , Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics, Class. Quant. Grav. 25 (2008), 075009, 15 pp.
[2] , , and , Jang's equation and its applications to marginally trapped surfaces, in: Complex analysis and dynamical systems IV: part 2. general relativity, geometry and PDE, Contemporary Mathematics, vol. 554, (AMS and Bar-Ilan), 2011.
[3] , , and , Local existence of dynamical and trapping horizons, Phys. Rev. Lett. 95 (2005), 111102.
[4] , , and , Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008), no. 4, 853–888.
[5] and , The area of horizons and the trapped region, Commun. Math. Phys. 290(2009), no. 3, 941–972.
[6] , , and , Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces, Commun. Math. Phys. 299(2010), no. 1, 89–127.
[7] , , and , Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122, (1961), 997–1006.
[8] , , and , The dynamics of general relativity, in Gravitation: an introduction to current research, 1962, pp. 227–265Wiley, New York. arXiv:gr-qc/0405109
[9] , Local existence and uniqueness for exterior static vacuum Einstein metrics, arXiv:1308.3642.
[10] , and , On the Bartnik extension problem for the static vacuum Einstein equations, Class. Quant. Grav. 30, (2013), 125005.
[11] , New definition of quasilocal mass, Phys. Rev. Lett., 62, (1989), 2346–2348.
[12] , Energy in general relativity, Lecture Notes at National Tsing Hua Univeristy, Hsinchu, Taiwan, July 1992.
[13] , Quasi-spherical metrics and prescribed scalar curvature, J. Diff. Geom. 37(1993), 31–71.
[14] , Phase space for the Einstein equations, Commun. Anal. Geom. 13, (2005), no. 5, 845–885.
[15] and , The constraint equations, in The Einstein equations and the large scale behavior of gravitational fields, 1–38, Birkhäuser, Basel, 2004.
[16] , and , Killing vectors in asymptotically flat space-times. I. Asymptot-ically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys. 37(1996), no. 4, 1939–1961.
[17] , Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Diff. Geom. 59(2001), no. 2, 177–267.
[18] , , , and , Generalized inverse mean curvature flows in space-time, Commun. Math. Phys. 272 (2007), no. 1, 119–138.
[19] , and , Time flat surfaces and the monotonicity of the spacetime Hawking mass, arXiv:1310.8638.
[20] , and , A Gibbons-Penrose inequality for surfaces in Schwarzschild spacetime, to appear in Commun. Math. Phys., arXiv:1303.1863.
[21] , , and , A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold, arXiv:1209.0669.
[22] , and , Jr., Quasilocal energy in general relativity, in Mathematical aspects of classical field theory (Seattle, WA, 1991), volume 132 of Contemporary Mathematics, pages 129–142. American Mathematical Society, Providence, RI, 1992.
[23] , and , Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D(3), 47(1993), no. 4, 1407–1419.
[24] , and , Moving observers, nonorthogonal boundaries, and quasilocal energy, Phys. Rev. D(3), 59(1999), no. 6, 064021.
[25] , The existence of non-trivial asymptotically flat initial data for vacuum spacetimes, Commun. Math. Phys. 57(1977), 83–96.
[26] , , , Rigidity of time-flat surfaces in the Minkowski spacetime, arXiv:1310.6081.
[27] , , and , Evaluating quasilocal energy and solving optimal embedding equation at null infinity, Commun. Math. Phys. 308(2011), no. 3, 845–863.
[28] , , and , Minimizing properties of critical points of quasi-local energy, to appear in Commun. Math. Phys., arXiv:1302.5321.
[29] , , and , Quasilocal angular momentum and center of mass in general relativity, arXiv:1312.0990.
[30] , , and , Conserved quantities in general relativity: from the quasi-local level to spatial infinity, arXiv:1312.0985.
[31] and , The Cauchy problem, General relativity and gravitation, Vol. 1, Plenum, New York, 1980, 99–172.
[32] , and , Construction of N-body initial data sets in general relativity, Commun. Math. Phys. 304(2011), 637–647.
[33] and , Existence of non-trivial, vacuum, asymptotically simple spacetimes, Class. Quant. Grav. 19(2002), L71–L79.
[34] and , On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (N. S.) No. 94 (2003), vi + 103 pp.
[35] , , and , Topological censorship for Kaluza–Klein space-times, Ann. Inst. Henri Poincaré 10(2009), no. 5, 893–912.
[36] , Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214(2000), 137–189.
[37] , On the existence and stability of the Penrose compactification, Ann. Inst. Henri Poincaré 8 (2007), 597–620.
[38] and , Scalar curvature and the Einstein constraint equations, in Surveys in geometric analysis and relativity, 145–188, Advanced Lectures in Mathathematics (ALM), 20, International Press, Somerville, MA, 2011.
[39] and , On the asymptotics for the vacuum Einstein constraint equations, J. Diff. Geom. 73(2006), 185–217.
[40] , , and , A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method, Duke Math. J. 161 (2012), 2669–2697.
[41] , The Plateau problem for marginally outer trapped surfaces, J. Diff. Geom. 83(2009), no. 3, 551–583.
[42], Existence, regularity, and properties of generalized apparent horizons, Commun. Math. Phys. 294(2010), no. 3, 745–760.
[43] , , and , Topological censorship from the initial data point of view, J. Diff. Geom. 95(2013), no. 3, 389–405.
[44] , , , and , The spacetime positive mass theorem in dimensions less than eight, 2011, arXiv:1110.2087v1.
[45] and , Jenkins–Serrin type results for the Jang equation, 2012, arXiv:1205.4301.
[46] and , A rotating black ring solution in five dimensions, Phys. Rev. Lett. 88(2002), no. 10, 101101, 4.
[47] , Angular momentum and an invariant quasilocal energy in general relativity, Phys. Rev. D 62 (2000), no. 12. 124108.
[48] , , and , Topological censorship, Phys. Rev. Lett. 71 (1993), no. 10, 1486–1489.
[49] , On the topology of the domain of outer communication, Class. Quant. Grav. 12 (1995), no. 10, L99–L101.
[50] , Rigidity of marginally trapped surfaces and the topology of black holes, Commun. Anal. Geom. 16 (2008), no. 1, 217–229.
[51] , , , and , Topological censorship and higher genus black holes, Phys. Rev. D(3) 60(1999), no. 10, 104039, 11.
[52] and , A generalization of Hawking's black hole topology theorem to higher dimensions, Commun. Math. Phys. 266(2006), no. 2, 571–576.
[53] , Energy extraction, Ann. N.Y. Acad. Sci. 224 (1973), 108–117.
[54] , Collapsing shells and the isoperimetric inequality for black holes, Class. Quant. Grav. 14 (1997), 2905–2915.
[55] and , Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. (1983), no. 58, 83–196 (1984).
[56] , Gravitational radiation in an expanding universe, J. Math. Phys. 9(1968), 598–604.
[57] and , The large scale structure of space-time, Cambridge University Press, London, 1973, Cambridge Monographs on Mathematical Physics, No. 1.
[58] , and , The gravitational Hamiltonian, action, entropy and surface terms, Class. Quant. Grav. 13(1996) (6), 1487–1498.
[59] , Residual finiteness for 3-manifolds, in Combinatorial group theory and topology (Alta, Utah, 1984), Ann. of Math. Stud., vol. 111, Princeton University Press, Princeton, NJ, 1987, pp. 379–396.
[60] , , and , Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions, Commun. Math. Phys. 288(2009), 547–613.
[61] (ed.), Black holes in higher dimensions, Cambridge University Press, London, 2012.
[62] , and , The inverse mean curvature flow and the Riemannian Penrose inequality, J. Diff. Geom. 59(2001), no. 3, 353–437.
[63] , Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Class. Quant. Grav. 12(1995), 2249–2274.
[64] and , Some results on nonconstant mean curvature solutions of the Einstein constraint equations, in Physics on manifolds (Paris 1992), Mathematical Physics Studies, vol. 15, Kluwer, Dordrecht, 1994, 295–302.
[65] , , and , On the topology of vacuum spacetimes, Ann. Henri Poincaré 4(2003), no. 2, 369–383.
[66] and , Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations, Class. Quant. Grav. 21(2004), S233–S241, A spacetime safari: essays in honor of Vincent Moncrief.
[67] , On the positivity of energy in general relativity, J. Math. Phys. 19(1978), no. 5, 1152–1155.
[68] , and , The positive energy conjecture and the cosmic censor hypothesis, J. Math. Phys. 18(1977), 41–44.
[69] , A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Rel. Grav. 29(1997), no. 3, 307–343.
[70] , L'intégration des equations de la gravitation relativiste et le problème des n corps, J. Math. Pures Appl. 23(1944), 37–63.
[71] , and , Positivity of quasilocal mass, Phys. Rev. Lett. 90(2003) no. 23, 231102
[72] , and , Positivity of quasilocal mass II, J. Amer. Math. Soc. 19(2006) 181–204.
[73] , , and , On the Penrose inequality for general horizons, Phys. Rev. Lett. 88(2002), no. 12, 121102.
[74] , Present status of the Penrose inequality, Class. Quant. Grav. 26(2009), 193001.
[75] , and , On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension, Class. Quant. Grav. 29(2012), 135005 (2012).
[76] , A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, Math. Res. Lett. 16(2009), 627–645.
[77] , A model problem for conformal parametrizations of the Einstein constraint equations, Commun. Math. Phys. 302(2011), 697–736.
[78] , On existence of static metric extensions in General Relativity, Commun. Math. Phys. 241(2003), 27–46.
[79] , , and , On geometric problems related to Brown-York and Liu-Yau quasilocal mass, Commun. Math. Phys. 298, (2010), no. 2, 437–459.
[80] , and , On second variation of Wang–Yau quasi-local energy, Ann. Inst. Henri Poincaré, onlinefirst, (2013), DOI: 10.1007/s00023-013-0279-z
[81] , , and , Critical points of Wang–Yau quasi-local energy, Ann. Inst. Henri Poincaré, 12(2011), no. 5, 987–1017.
[82] and , Black holes in higher-dimensional space-times, Annals of Physics 172(1986), no. 2, 304–347.
[83] , The Weyl and Minkowski problems in differential geometry in the large, Commun. Pure Appl. Math. 6(1953), 337–394.
[84] , , and , Comment on “Positivity of quasilocal mass”, Phys. Rev. Lett. 92(2004), 259001.
[85] , Semi-Riemannian geometry, Academic Press, New York, 1983.
[86] , Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14(1965), 57–59.
[87] , Naked singularities, Ann. N.Y. Acad. Sci. 224(1973), 125–134
[88] , Some unsolved problems in classical general relativity, Seminar on differential geometry, Ann. of Math. Stud., 102(1982), 631–668, Princeton University Press.
[89] , Regularity of a convex surface with given Gaussian curvature, Mat. Sbornik N.S., 31(73), (1952), 88–103.
[90] , Mean curvature in Riemannian geometry and general relativity, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 113–136.
[91] and , On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65(1979), no. 1, 45–76.
[92] and , On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28(1979), no. 1–3, 159–183.
[93] and , Positivity of the total mass of a general spacetime, Phys. Rev. Lett. 43(1979), 1457–1459.
[94] and , Proof of the positive mass theorem. II, Commun. Math. Phys. 79(1981), no. 2, 231–260.
[95] , and , Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Diff. Geom. 62(2002), 79–125.
[96] , and , Quasiconvex foliations and asymptotically flat metrics of non-negative scalar curvature, Commun. Anal. Geom. 12(2004), no. 3, 511–551.
[97] , Quasi-local energy-momentum and angular momentum in GR: a review article, Living Rev. Rel., 12(2009) no. 4, URL: relativity.livingreviews.org/Articles/lrr-2009-4
[98] , General relativity, University of Chicago Press, Chicago, IL, 1984.
[99] , Quasilocal mass and surface Hamiltonian in spacetime, arXiv:1211.1407.
[100] , and , Quasilocal mass in general relativity, Phys. Rev. Lett. 102 (2009), no. 2, no. 021101, 4 pp.
[101] , and , Isometric embeddings into the Minkowski space and new quasi-local mass, Com-mun. Math. Phys. 288(2009), no. 3, 919–942.
[102], and , Limit of quasilocal mass at spatial infinity, Commun. Math. Phys. 296(2010), no. 1, 271–283.
[103] , A new proof of the positive energy theorem, Commun. Math. Phys. 80(1981), 381–402.