Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 4 (2014), 771-901.

The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates

Jared Speck

Full-text: Open access

Abstract

We study the coupling of the Einstein field equations of general relativity to a family of nonlinear electromagnetic field equations. The family comprises all covariant electromagnetic models that satisfy the following criteria: (i) they are derivable from a sufficiently regular Lagrangian; (ii) they reduce to the standard Maxwell model in the weak-field limit; (iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the (1+3)-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the standard Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave-coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations that have a special null structure, is based on an extension of a geometric energy-method framework developed by Christodoulou together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields and thus avoid the use of electromagnetic potentials.

Article information

Source
Anal. PDE, Volume 7, Number 4 (2014), 771-901.

Dates
Received: 29 November 2010
Revised: 18 September 2012
Accepted: 21 May 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731541

Digital Object Identifier
doi:10.2140/apde.2014.7.771

Mathematical Reviews number (MathSciNet)
MR3254347

Zentralblatt MATH identifier
1298.35225

Subjects
Primary: 35A01: Existence problems: global existence, local existence, non-existence 35Q76: Einstein equations
Secondary: 35L99: None of the above, but in this section 35Q60: PDEs in connection with optics and electromagnetic theory 35Q76: Einstein equations 78A25: Electromagnetic theory, general 83C22: Einstein-Maxwell equations 83C50: Electromagnetic fields

Keywords
Born–Infeld canonical stress energy currents global existence Hardy inequality Klainerman–Sobolev inequality Lagrangian field theory nonlinear electromagnetism null condition null decomposition quasilinear wave equation regularly hyperbolic vector field method weak null condition

Citation

Speck, Jared. The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates. Anal. PDE 7 (2014), no. 4, 771--901. doi:10.2140/apde.2014.7.771. https://projecteuclid.org/euclid.apde/1513731541


Export citation

References

  • I. Białynicki-Birula, “Nonlinear electrodynamics: Variations on a theme by Born and Infeld”, pp. 31–48 in Quantum theory of particles and fields, edited by B. Jancewicz and J. Lukierski, World Scientific, Singapore, 1983.
  • L. Bieri, An extension of the stability theorem of the Minkowski space in general relativity, PhD thesis, 17178, ETH Zürich, 2007.
  • G. Boillat, “Nonlinear electrodynamics: Lagrangians and equations of motion”, J. Math. Phys. 11:3 (1970), 941–951.
  • M. Born,\kern-0.5pt “Modified field equations with a finite radius of the electron”, Nature 132:3329 (1933), 282.\kern-0.5pt
  • M. Born and L. Infeld, “Foundations of the new field theory”, Proc. Roy. Soc. London A 144:852 (1934), 425–451.
  • Y. Choquet-Bruhat, “Un théorème d'instabilité pour certaines équations hyperboliques non linéaires”, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A281–A284.
  • Y. Choquet-Bruhat and D. Christodoulou, “Elliptic systems in $H\sb{s,\delta }$ spaces on manifolds which are Euclidean at infinity”, Acta Math. 146:1-2 (1981), 129–150.
  • Y. Choquet-Bruhat and R. Geroch, “Global aspects of the Cauchy problem in general relativity”, Comm. Math. Phys. 14 (1969), 329–335.
  • Y. Choquet-Bruhat and J. W. York, Jr., “The Cauchy problem”, pp. 99–172 in General relativity and gravitation, I, edited by A. Held, Plenum, New York, 1980.
  • Y. Choquet-Bruhat, P. T. Chruściel, and J. Loizelet, “Global solutions of the Einstein–Maxwell equations in higher dimensions”, Classical Quantum Gravity 23:24 (2006), 7383–7394.
  • D. Christodoulou, “Global solutions of nonlinear hyperbolic equations for small initial data”, Comm. Pure Appl. Math. 39:2 (1986), 267–282.
  • D. Christodoulou, The action principle and partial differential equations, Annals of Mathematics Studies 146, Princeton University Press, 2000.
  • D. Christodoulou, Mathematical problems of general relativity, I, European Mathematical Society, Zürich, 2008.
  • D. Christodoulou and S. Klainerman, “Asymptotic properties of linear field equations in Minkowski space”, Comm. Pure Appl. Math. 43:2 (1990), 137–199.
  • D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series 41, Princeton University Press, 1993.
  • P. T. Chruściel and E. Delay, “Erratum: “Existence of non-trivial, vacuum, asymptotically simple spacetimes””, Classical Quantum Gravity 19:12 (2002), 3389.
  • P. T. Chruściel and E. Delay, “Existence of non-trivial, vacuum, asymptotically simple spacetimes”, Classical Quantum Gravity 19:9 (2002), L71–L79.
  • J. Corvino, “Scalar curvature deformation and a gluing construction for the Einstein constraint equations”, Comm. Math. Phys. 214:1 (2000), 137–189.
  • T. de Donder, La gravifique einsteinienne, Gauthiers-Villars, Paris, 1921.
  • Y. Fourès-Bruhat, “Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires”, Acta Math. 88 (1952), 141–225.
  • H. Friedrich, “On the existence of $n$–geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Comm. Math. Phys. 107:4 (1986), 587–609.
  • G. W. Gibbons and C. A. R. Herdeiro, “Born–Infeld theory and stringy causality”, Phys. Rev. D $(3)$ 63:6 (2001), 064006–1–18.
  • L. H örmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications 26, Springer, Berlin, 1997.
  • F. John, “Blow-up for quasilinear wave equations in three space dimensions”, Comm. Pure Appl. Math. 34:1 (1981), 29–51.
  • S. Katayama, “Global existence for systems of wave equations with nonresonant nonlinearities and null forms”, J. Differential Equations 209:1 (2005), 140–171.
  • M. K.-H. Kiessling, “Electromagnetic field theory without divergence problems, I: The Born legacy”, J. Statist. Phys. 116:1-4 (2004), 1057–1122.
  • M. K.-H. Kiessling, “Electromagnetic field theory without divergence problems, II: A least invasively quantized theory”, J. Statist. Phys. 116:1-4 (2004), 1123–1159.
  • S. Klainerman, “The null condition and global existence to nonlinear wave equations”, pp. 293–326 in Nonlinear systems of partial differential equations in applied mathematics, I (Santa Fe, NM, 1984), edited by B. Nicolaenko et al., Lectures in Appl. Math. 23, Amer. Math. Soc., Providence, RI, 1986.
  • S. Klainerman and F. Nicolò, The evolution problem in general relativity, Progress in Mathematical Physics 25, Birkhäuser, Boston, 2003.
  • S. Klainerman and T. C. Sideris, “On almost global existence for nonrelativistic wave equations in $3$D”, Comm. Pure Appl. Math. 49:3 (1996), 307–321.
  • H. Lindblad, “A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time”, Proc. Amer. Math. Soc. 132:4 (2004), 1095–1102.
  • H. Lindblad, “Global solutions of quasilinear wave equations”, Amer. J. Math. 130:1 (2008), 115–157.
  • H. Lindblad and I. Rodnianski, “The weak null condition for Einstein's equations”, C. R. Math. Acad. Sci. Paris 336:11 (2003), 901–906.
  • H. Lindblad and I. Rodnianski, “Global existence for the Einstein vacuum equations in wave coordinates”, Comm. Math. Phys. 256:1 (2005), 43–110.
  • H. Lindblad and I. Rodnianski, “The global stability of Minkowski space-time in harmonic gauge”, Ann. of Math. $(2)$ 171:3 (2010), 1401–1477.
  • J. Loizelet, “Solutions globales des équations d'Einstein–Maxwell avec jauge harmonique et jauge de Lorentz”, C. R. Math. Acad. Sci. Paris 342:7 (2006), 479–482.
  • J. Loizelet, Problèms globaux en relativité generalé, Ph.D. thesis, Universitè Francois Rabelais, 2008, http://homepage.univie.ac.at/piotr.chrusciel/papers/TheseTitreComplet.pdf.
  • J. Loizelet, “Solutions globales des équations d'Einstein–Maxwell”, Ann. Fac. Sci. Toulouse Math. $(6)$ 18:3 (2009), 565–610.
  • A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences 53, Springer, New York, 1984.
  • J. Metcalfe and C. D. Sogge, “Global existence of null-form wave equations in exterior domains”, Math. Z. 256:3 (2007), 521–549.
  • J. Metcalfe, M. Nakamura, and C. D. Sogge, “Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition”, Japan. J. Math. $($N.S.$)$ 31:2 (2005), 391–472.
  • J. Pleba\`nski, Lectures on non-linear electrodynamics (Niels Bohr Institute and NORDITA, Copenhagen, 1968), NORDITA, Stockholm, 1970.
  • R. Schoen and S. T. Yau, “On the proof of the positive mass conjecture in general relativity”, Comm. Math. Phys. 65:1 (1979), 45–76.
  • R. Schoen and S. T. Yau, “Proof of the positive mass theorem, II”, Comm. Math. Phys. 79:2 (1981), 231–260.
  • J. Shatah and M. Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics 2, New York University Courant Institute of Mathematical Sciences, New York, 1998.
  • T. C. Sideris, “The null condition and global existence of nonlinear elastic waves”, Invent. Math. 123:2 (1996), 323–342.
  • C. D. Sogge, Lectures on non-linear wave equations, 2nd ed., International Press, Boston, MA, 2008. http://msp.org/idx/mr/2009i:35213MR 2009i:35213
  • J. Speck, “The non-relativistic limit of the Euler–Nordström system with cosmological constant”, Rev. Math. Phys. 21:7 (2009), 821–876.
  • J. Speck, “Well-posedness for the Euler–Nordström system with cosmological constant”, J. Hyperbolic Differ. Equ. 6:2 (2009), 313–358.
  • J. Speck, “The nonlinear stability of the trivial solution to the Maxwell–Born–Infeld system”, J. Math. Phys. 53:8 (2012), 083703–1–83.
  • M. E. Taylor, Partial differential equations, III: Nonlinear equations, Applied Mathematical Sciences 117, Springer, New York, 1996.
  • R. M. Wald, General relativity, University of Chicago Press, Chicago, 1984.
  • E. Witten, “A new proof of the positive energy theorem”, Comm. Math. Phys. 80:3 (1981), 381–402.
  • N. Zipser, The global nonlinear stability of the trivial solution of the Einstein–Maxwell equations, Ph.D. thesis, Harvard University, 2000, http://search.proquest.com/docview/304610196.