The Cauchy problem in mathematical General Relativity

This is an introductory course to mathematical General Relativity from the point of view of differential geometry and partial differential equations. The goal of this course is to provide an understanding of the Cauchy problem of the Einstein equations coupled to gauge field theories, such as the Maxwell equations and the Yang-Mills equations. The Einstein-Maxwell and the Einstein-Yang-Mills constraint equations will be explained. We will also reach to a hyperbolic formulation of the evolution problem in General Relativity for perturbations of the Minkowski space-time governed by the Einstein-Yang-Mills equations in the Lorenz gauge and in wave coordinates.

Keywords: space-time, metric, Einstein equations, Maxwell fields, Yang-Mills fields, constraint equations, Gauss equations, Codazzi equations, Cauchy problem, gauge transformations, Minkowski metric, wave coordinates, Lorenz gauge.

Keywords: space-time, metric, Einstein equations, Maxwell fields, Yang-Mills fields, constraint equations, Gauss equations, Codazzi equations, Cauchy problem, gauge transformations, Minkowski metric, wave coordinates, Lorenz gauge.

Lecturer

Date

12th June ~ 12th July, 2024

Schedule

Weekday | Time | Venue | Online |
---|---|---|---|

Wednesday,Friday | 13:30 - 15:05 | A3-1a-204 | - |

Prerequisite

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:

- Elements of differential geometry: manifold, tensor, metric, Lorentzian manifold, covariant derivative, Lie derivative, connection of Levi-Civita, Christoffel symbols, Riemann tensor, Ricci tensor, hypersurface, second fundamental form, Minkowski metric.

2. The Einstein-Maxwell and the Einstein-Yang-Mills equations:

- General Relativity.

- The Yang-Mills equations: the Yang-Mills curvature, the Bianchi identities, the Yang-Mills stress-energy-momentum tensor, the Yang-Mills field equations.

- The Einstein-Yang-Mills system.

3. The Cauchy problem and the constraints for the Einstein-Yang-Mills system:

- The Cauchy problem.

- The constraint equations.

4. The gauges invariance of the equations:

- The invariance under gauge transformation.

- The diffeomorphism invariance.

5. 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.

6. The Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates as a non-linear hyperbolic system of partial differential equations.

7. 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.

- Elements of differential geometry: manifold, tensor, metric, Lorentzian manifold, covariant derivative, Lie derivative, connection of Levi-Civita, Christoffel symbols, Riemann tensor, Ricci tensor, hypersurface, second fundamental form, Minkowski metric.

2. The Einstein-Maxwell and the Einstein-Yang-Mills equations:

- General Relativity.

- The Yang-Mills equations: the Yang-Mills curvature, the Bianchi identities, the Yang-Mills stress-energy-momentum tensor, the Yang-Mills field equations.

- The Einstein-Yang-Mills system.

3. The Cauchy problem and the constraints for the Einstein-Yang-Mills system:

- The Cauchy problem.

- The constraint equations.

4. The gauges invariance of the equations:

- The invariance under gauge transformation.

- The diffeomorphism invariance.

5. 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.

6. The Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates as a non-linear hyperbolic system of partial differential equations.

7. 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.

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, 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, 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, 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, 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. Minkowski, “Raum und Zeit”, Physikalische Zeitschrift, 10. Jahrgang, 1909, 104–115.

[12] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998.

[13] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868).

[14] H. Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.

[15] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96:191–195, Oct 1954.

[16] 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, 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, 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, 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, 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. Minkowski, “Raum und Zeit”, Physikalische Zeitschrift, 10. Jahrgang, 1909, 104–115.

[12] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998.

[13] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868).

[14] H. Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.

[15] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96:191–195, Oct 1954.

[16] 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.