Non-linear wave equations in General Relativity
This is an introductory course to non-linear wave equations arising in General Relativity from the point of view of analysis and partial differential equations. The course material builds on a previous course that I gave in the Spring 2024 on the Cauchy problem in mathematical General Relativity. The goal of this course is to show how non-linear hyperbolic wave equations can be related to the Einstein equations coupled to gauge field theories, such as the Yang-Mills equations. In particular, the Einstein-Yang-Mills equations will be recasted as a coupled system of non-linear wave equations with sources. We will explain how it is exactly the study of the non-linear structure of these source terms that would allow one to answer the question regarding the local existence and the global existence of solutions. We will show a method to obtain dispersive estimates for solutions of such non-linear hyperbolic equations, provided that one exploits the non-linear structure of the wave equations, and we will exhibit how this can be applied to perturbations of the Minkowski space-time governed by the Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates. We will not explain in this course the proof of the stability of Minkowski space-time in low dimensions, that is far much more complicated and involved, however we shall give elements that would allow one to think about the difficulty of the problem in the critical three space dimensions.
Keywords: non-linear wave equations, hyperbolic partial differential equations, Einstein equations, Yang-Mills fields, Einstein-Yang-Mills system, gauge transformations, Minkowski metric, wave coordinates, Lorenz gauge, rotations and Lorentz boosts, Klainerman-Sobolev inequality, weighted energy norms, bootstrap argument, decay, energy estimates, Hardy type inequality, commutator term, stability of Minkowski space-time, Grönwall type inequality.
Keywords: non-linear wave equations, hyperbolic partial differential equations, Einstein equations, Yang-Mills fields, Einstein-Yang-Mills system, gauge transformations, Minkowski metric, wave coordinates, Lorenz gauge, rotations and Lorentz boosts, Klainerman-Sobolev inequality, weighted energy norms, bootstrap argument, decay, energy estimates, Hardy type inequality, commutator term, stability of Minkowski space-time, Grönwall type inequality.
Lecturer
Date
19th September ~ 16th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Thursday | 13:30 - 15:05 | A3-2a-201 | ZOOM 13 | 637 734 0280 | BIMSA |
Prerequisite
Basic knowledge from my previous course on the Cauchy problem in mathematical General Relativity, graduate level knowledge in differential geometry and in Riemannian geometry, and basic knowledge in partial differential equations and analysis.
Syllabus
1. Reminders of pre-requisites:
— The Einstein equations, the Yang-Mills equations, the Einstein-Yang-Mills system, the Cauchy problem and the constraint equations, the gauges invariance of the equations.
2. The Einstein-Yang-Mills equations as a system on the perturbation of Minkowski space-time:
— Looking at the metric as a perturbation of the Minkowski space-time.
— The Einstein-Yang-Mills equations in a given system of coordinates.
3. The Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates as a non-linear hyperbolic system of partial differential equations.
4. Construction of the initial data and the gauges conditions constraints:
— The initial data for the Yang-Mills potential.
— The initial data for the metric.
— The propagation of the Lorenz gauge condition.
— The propagation of the wave coordinates condition.
— Construction of the initial data for the hyperbolic system given an initial data set that solves the Einstein-Yang-Mills constraints.
5. Set-up of analysis for proving decay for solutions of non-linear wave equations:
— The rotations and the Lorentz boosts.
— Weighted Klainerman-Sobolev inequality.
— Definition of the norms.
— The energy norm.
— The bootstrap argument.
— The bootstrap assumption.
— The big O notation.
6. A priori decay estimates:
— The spatial asymptotic behaviour of the fields on the initial hypersurface.
— Estimates on the time evolution of the fields.
7. Studying the structure of the source terms of the coupled non-linear wave equations.
8. Using the bootstrap assumption to exhibit the structure of the source terms of the Einstein-Yang-Mills system:
— Using the bootstrap assumption to exhibit the structure of the source terms for the Yang-Mills potential.
— Using the bootstrap assumption to exhibit the structure of the source terms for the metric.
— The source terms for n ≥ 5.
9. Energy estimates.
10. A Hardy type inequality.
11. The commutator term for n ≥ 4:
— Using the Hardy type inequality to estimate the commutator term.
12. The energy estimate for n ≥ 4:
— The main energy estimate for the fields for n ≥ 4.
13. The proof of global stability of Minkowski space-time for n ≥ 5:
— Using the Hardy type inequality for the space-time integrals of the source terms for n ≥ 5.
— Grönwall type inequality on the energy for n ≥ 5.
— The proof of the theorem for n ≥ 5.
— The Einstein equations, the Yang-Mills equations, the Einstein-Yang-Mills system, the Cauchy problem and the constraint equations, the gauges invariance of the equations.
2. The Einstein-Yang-Mills equations as a system on the perturbation of Minkowski space-time:
— Looking at the metric as a perturbation of the Minkowski space-time.
— The Einstein-Yang-Mills equations in a given system of coordinates.
3. The Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates as a non-linear hyperbolic system of partial differential equations.
4. Construction of the initial data and the gauges conditions constraints:
— The initial data for the Yang-Mills potential.
— The initial data for the metric.
— The propagation of the Lorenz gauge condition.
— The propagation of the wave coordinates condition.
— Construction of the initial data for the hyperbolic system given an initial data set that solves the Einstein-Yang-Mills constraints.
5. Set-up of analysis for proving decay for solutions of non-linear wave equations:
— The rotations and the Lorentz boosts.
— Weighted Klainerman-Sobolev inequality.
— Definition of the norms.
— The energy norm.
— The bootstrap argument.
— The bootstrap assumption.
— The big O notation.
6. A priori decay estimates:
— The spatial asymptotic behaviour of the fields on the initial hypersurface.
— Estimates on the time evolution of the fields.
7. Studying the structure of the source terms of the coupled non-linear wave equations.
8. Using the bootstrap assumption to exhibit the structure of the source terms of the Einstein-Yang-Mills system:
— Using the bootstrap assumption to exhibit the structure of the source terms for the Yang-Mills potential.
— Using the bootstrap assumption to exhibit the structure of the source terms for the metric.
— The source terms for n ≥ 5.
9. Energy estimates.
10. A Hardy type inequality.
11. The commutator term for n ≥ 4:
— Using the Hardy type inequality to estimate the commutator term.
12. The energy estimate for n ≥ 4:
— The main energy estimate for the fields for n ≥ 4.
13. The proof of global stability of Minkowski space-time for n ≥ 5:
— Using the Hardy type inequality for the space-time integrals of the source terms for n ≥ 5.
— Grönwall type inequality on the energy for n ≥ 5.
— The proof of the theorem for n ≥ 5.
Reference
[1] L. Andersson and V. Moncrief, Future complete vacuum spacetimes, in The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel (2004).
[2] Y. C. Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non-linéaires., Acta Math. 88, 1952, 141–225.
[3] Y. C. Bruhat and R. Geroch, Global Aspects of the Cauchy Problem in General Relativity. Comm. Math. Phys. 14, 1969, 329–335.
[4] M. P. Do Carmo, Riemannian Geometry, Birkhäuser (1992).
[5] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171-191.
[6] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof, Comm. Math. Phys. 83 (1982), no. 2, 193-212.
[7] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik und Chemie 17, 1905, 891– 921.
[8] A. Einstein, Der Feldgleichungen des Gravitation, Preuss. Akad. Wiss. Berlin, Sitzber., 1915, 844– 847.
[9] S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time Cambridge University Press, 1973.
[10] J. Isenberg, The Initial Value Problem in General Relativity, in: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg (2014).
[11] H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Commun. Math. Phys. 256:43-110, 2005.
[12] H. Minkowski, “Raum und Zeit”, Physikalische Zeitschrift, 10. Jahrgang, 1909, 104–115.
[13] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998.
[14] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868).
[15] H. Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.
[16] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96:191–195, Oct 1954.
[17] R. Wald, General Relativity The University of Chicago Press, 1984.
[2] Y. C. Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non-linéaires., Acta Math. 88, 1952, 141–225.
[3] Y. C. Bruhat and R. Geroch, Global Aspects of the Cauchy Problem in General Relativity. Comm. Math. Phys. 14, 1969, 329–335.
[4] M. P. Do Carmo, Riemannian Geometry, Birkhäuser (1992).
[5] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171-191.
[6] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof, Comm. Math. Phys. 83 (1982), no. 2, 193-212.
[7] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik und Chemie 17, 1905, 891– 921.
[8] A. Einstein, Der Feldgleichungen des Gravitation, Preuss. Akad. Wiss. Berlin, Sitzber., 1915, 844– 847.
[9] S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time Cambridge University Press, 1973.
[10] J. Isenberg, The Initial Value Problem in General Relativity, in: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg (2014).
[11] H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Commun. Math. Phys. 256:43-110, 2005.
[12] H. Minkowski, “Raum und Zeit”, Physikalische Zeitschrift, 10. Jahrgang, 1909, 104–115.
[13] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998.
[14] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868).
[15] H. Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.
[16] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96:191–195, Oct 1954.
[17] R. Wald, General Relativity The University of Chicago Press, 1984.
Audience
Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
Sari Ghanem had his undergraduate education at Classe Préparatoire in Toulouse and completed his Master degree in Paris, and a second Master in the USA. Thereafter, he obtained his PhD in 2014 from University of Paris VII (Institut de Mathématiques de Jussieu) under the supervision of Frédéric Hélein (Jussieu) and Vincent Moncrief (Yale University). He then worked as a postdoctoral fellow at the Albert Einstein Institute (Max-Planck Institute for Gravitational Physics), and at the University of Grenoble in France. Thereafter, he founded the Al-Khwarizmi-Noether Institute for mathematics (https://akn-institute.org/the-institute/about/) of which he was the creator and chair on a voluntary basis, and worked at the Universities of Hamburg and of Lübeck in Germany. Now, he is a research Visiting Assistant Professor at BIMSA in China.