3+1 Formalism in General Relativity

Bases of Numerical Relativity

  • Eric Gourgoulhon

Part of the Lecture Notes in Physics book series (LNP, volume 846)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Éric Gourgoulhon
    Pages 1-3
  3. Éric Gourgoulhon
    Pages 5-28
  4. Éric Gourgoulhon
    Pages 29-54
  5. Éric Gourgoulhon
    Pages 55-71
  6. Éric Gourgoulhon
    Pages 73-99
  7. Éric Gourgoulhon
    Pages 101-132
  8. Éric Gourgoulhon
    Pages 133-156
  9. Éric Gourgoulhon
    Pages 157-183
  10. Éric Gourgoulhon
    Pages 185-219
  11. Éric Gourgoulhon
    Pages 221-251
  12. Éric Gourgoulhon
    Pages 253-270
  13. Back Matter
    Pages 273-294

About this book


This graduate-level, course-based text is devoted to the 3+1 formalism of general relativity, which also constitutes the theoretical foundations of numerical relativity. The book starts by establishing the mathematical background (differential geometry, hypersurfaces embedded in space-time, foliation of space-time by a family of space-like hypersurfaces), and then turns to the 3+1 decomposition of the Einstein equations, giving rise to the Cauchy problem with constraints, which constitutes the core of 3+1 formalism. The ADM Hamiltonian formulation of general relativity is also introduced at this stage. Finally, the decomposition of the matter and electromagnetic field equations is presented, focusing on the astrophysically relevant cases of a perfect fluid and a perfect conductor (ideal magnetohydrodynamics). The second part of the book introduces more advanced topics: the conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations, the Isenberg-Wilson-Mathews approximation to general relativity, global quantities associated with asymptotic flatness (ADM mass, linear and angular momentum) and with symmetries (Komar mass and angular momentum). In the last part, the initial data problem is studied, the choice of spacetime coordinates within the 3+1 framework is discussed and various schemes for the time integration of the 3+1 Einstein equations are reviewed. The prerequisites are those of a basic general relativity course with calculations and derivations presented in detail, making this text complete and self-contained. Numerical techniques are not covered in this book.


3+1 formalism and decomposition ADM Hamiltonian Cauchy problem with constraints Computational relativity and gravitation Foliation and slicing of spacetime Numerical relativity textbook

Authors and affiliations

  • Eric Gourgoulhon
    • 1
  1. 1.UMR 8102 du CNRS, Observatoire de ParisLab. Univers et Theories (LUTH)MeudonFrance

Bibliographic information